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2021 53 Pablo Pérez Muñoz Desarrollo de nuevas técnicas de verificación y compensación de sistemas de fabricación para metrología dimensional trazable en proceso Departamento Director/es Ingeniería de Diseño y Fabricación Santolaria Mazo, Jorge Albajez García, José Antonio © Universidad de Zaragoza Servicio de Publicaciones ISSN 2254-7606 Pablo Pérez Muñoz DESARROLLO DE NUEVAS TÉCNICAS DE VERIFICACIÓN Y COMPENSACIÓN DE SISTEMAS DE FABRICACIÓN PARA METROLOGÍA DIMENSIONAL TRAZABLE EN PROCESO Director/es Ingeniería de Diseño y Fabricación Santolaria Mazo, Jorge Albajez García, José Antonio Tesis Doctoral Autor 2019 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Zaragoza, 2019. Desarrollo de nuevas técnicas de verificación y compensación de sistemas de fabricación para metrología dimensional trazable en proceso Pablo Pérez Muñoz TESIS DOCTORAL Desarrollo de nuevas técnicas de verificación y compensación de sistemas de fabricación para metrología dimensional trazable en proceso Pablo Pérez Muñoz Dirigida por Dr. Jorge Santolaria Mazo Dr. José Antonio Albajez García Para la obtención del Título de Doctora por la Universidad de Zaragoza Zaragoza, Septiembre de 2019 D. Jorge Santolaria Mazo, Doctor por la Universidad de Zaragoza y Catedrático del departamento de Ingeniería de Diseño y Fabricación de la Universidad de Zaragoza. INFORMA: Que la tesis titulada “Desarrollo de nuevas técnicas de verificación y compensación de sistemas de fabricación para metrología dimensional trazable en proceso”, elaborada por D. Pablo Pérez Muñoz, ha sido realizada bajo mi dirección, se ajusta al proyecto de tesis inicialmente presentado y cumple los requisitos exigidos por la legislación vigente para optar al grado de Doctora por la Universidad de Zaragoza. Una vez finalizada, autorizo su presentación en la modalidad de compendio de publicaciones para ser evaluada por el tribunal correspondiente. Zaragoza, a 13 de Septiembre de 2019, Fdo. D. Jorge Santolaria Mazo 11 INDICE 1. INTRODUCCIÓN ....................................................................................................................... 19 1.1. Contexto ........................................................................................................................... 19 1.2. Motivación de la tesis....................................................................................................... 20 1.3. Alcance y objetivos ........................................................................................................... 20 1.4. Estructura ......................................................................................................................... 21 2. ESTADO DEL ARTE ................................................................................................................... 25 2.1. Introducción a la verificación de máquina herramienta .................................................. 25 2.2. Fuentes de error que afectan a la precisión de la máquina herramienta........................ 28 2.3. Errores geométricos ......................................................................................................... 28 2.4. Errores térmicos ............................................................................................................... 29 2.5. Error Máximo Permitido (MPE) ........................................................................................ 31 2.6. Compensación de errores ................................................................................................ 32 2.7. Modelo cinemático de la Máquina Herramienta ............................................................. 33 2.8. Conformidad, trazabilidad y medición de la incertidumbre ............................................ 36 2.9. Normativa Estandarizada ................................................................................................. 37 3. PRESENTACIÓN DE LAS PUBLICACIONES ................................................................................. 41 3.1. Justificación de unidad temática ...................................................................................... 41 3.2. Presentación de la publicación “Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements” ....................................................... 44 3.3. Presentación de la publicación “Analysis of the measurement capacity of a machine tool” ..................................................................................................................................... 48 3.4. Presentación de la publicación “Monte Carlo method to machine tool uncertainty evaluation” .......................................................................................................................... 50 3.5. Presentación de la publicación “Influence of laser tracker noise on the uncertainty of machine tool volumetric verification using the Monte Carlo method” .............................. 51 3.6. Presentación de la publicación “Study on Machine Tool Positioning Uncertainty Due to Volumetric Verification” ...................................................................................................... 54 3.7. Presentación de la publicación “Lateral error compensation for stitching-free measurement with focus variation microscopy” ................................................................ 56 4. PUBLICACIONES ....................................................................................................................... 63 4.1. Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements ..................................................................................................................... 63 4.2. Analysis of the measurement capacity of a machine tool ................................................ 73 4.3. Monte Carlo method to machine tool uncertainty evaluation ......................................... 81 4.4. Influence of laser tracker noise on the uncertainty of machine tool volumetric verification using the Monte Carlo method............................................................................................ 89 4.5. Study on machine tool positioning uncertainty due to volumetric verification ............... 99 4.6. Lateral error compensation for stitching-free measurement with focus variation microscopy......................................................................................................................... 117 5. METODOLOGÍA ...................................................................................................................... 129 5.1. Material y equipos utilizados ......................................................................................... 129 5.1.1. Máquina herramienta ............................................................................................. 129 5.1.2. Palpador .................................................................................................................. 130 5.1.3. Laser Tracker ........................................................................................................... 131 5.1.4. Alicona ..................................................................................................................... 133 5.2. Medición de los errores de estabilización del Laser Tracker ......................................... 135 5.3. Efecto de las turbulencias sobre las mediciones con Laser Tracker .............................. 141 5.4. Medición de la desviación de la punta del palpador ..................................................... 148 12 5.5. Medición de los errores mediante placa patrón ............................................................ 152 5.5.1. Proceso de medición ............................................................................................... 152 5.5.2. Cálculo de errores ................................................................................................... 154 5.5.3. Influencia de la temperatura ................................................................................... 158 5.6. Fuentes de incertidumbre .............................................................................................. 164 5.7. Medición de errores del ALICONA ................................................................................. 166 5.7.1. Medición de los errores con interferómetro .......................................................... 166 5.7.2. Análisis de las mediciones con y sin stitching ......................................................... 169 5.8. Verificación de máquina herramienta con el Laser Tracker fuera ................................. 180 6. ANEXOS ................................................................................................................................. 187 6.1. Índice de impacto de las publicaciones .......................................................................... 187 6.2. Justificación de la contribución en coautoría ................................................................. 188 BIBLIOGRAFÍA ............................................................................................................................ 189 13 INDICE DE FIGURAS Figura 2.1. Tendencia de la precisión alcanzable según Taniguchi [21]. ..................................... 25 Figura 2.2. Ejemplo de una pieza NAS979: círculo, diamante, cuadrado. .................................. 26 Figura 2.3. Verificación mediante barra de bolas telescópica. Fuente: Renishaw. .................... 27 Figura 2.4. Medición del error mediante un vector de posiciones, método diagonal por pasos. ..................................................................................................................................................... 27 Figura 2.5. Ejemplos de MPE según la norma ISO 10360-5 [109]. .............................................. 31 Figura 2.6. Compensación de errores mediante tabla de compensación. .................................. 32 Figura 2.7. Compensación de errores mediante a tiempo real mediante PC. ............................ 33 Figura 2.8. Compensación de errores mediante post-procesado. .............................................. 33 Figura 2.9. Esquema cinemáticos de máquinas herramienta: a) FXYZ, b) XFYZ, c) XYFZ, d) XYZF. ..................................................................................................................................................... 34 Figura 2.10. Movimiento de los ejes de la máquina herramienta y su esquema cinemático. .... 35 Figura 2.11. Representación de la zona de incertidumbre de un punto medido. ...................... 36 Figura 3.1. Comparación entre las mediciones y las ecuaciones modeladas. ............................ 45 Figura 3.2. Posición de los sensores (izda.) y temperatura de la carcasa (dcha.). ...................... 46 Figura 3.3. Configuración del ensayo de dilatación (izda.) y variación de la coordenada Z de un punto del LT2 medido con el LT1 (dcha.). ................................................................................... 46 Figura 3.4. Configuración de la medición de la placa. ................................................................. 48 Figura 3.5. Componentes de incertidumbre con y sin monitorización de temperatura ............ 49 Figura 3.6. Distribución del error final. ....................................................................................... 50 Figura 3.7. Procedimiento para calcular la incertidumbre. ......................................................... 51 Figura 3.8. Simulación de los puntos medidos con el error del ruido del Laser Tracker. ........... 52 Figura 3.9. Error Máximo Permisible. .......................................................................................... 53 Figura 3.11. Error residual de los puntos de verificación de 1000 tests. .................................... 55 Figura 3.12. Artefacto diseñado para medir los errores de los ejes X e Y del Alicona. ............... 57 Figura 3.13. Error inicial y residual de los puntos medidos. ....................................................... 58 Figura 5.1. Máquina herramienta ANAYAK VH1800. ................................................................ 129 Figura 5.2. Palpador de contacto. ............................................................................................. 130 Figura 5.3. Laser Tracker. .......................................................................................................... 131 Figura 5.4. Retrorreflector tipo SMR. ........................................................................................ 132 Figura 5.5. Nido centrante con base magnética. ...................................................................... 133 Figura 5.6. Alicona G5 Infinite Focus. ........................................................................................ 133 Figura 5.7. Variación máxima y mínima de la coordenada radial para ambos equipos. .......... 135 Figura 5.8. Variación de la coordenada radial del API y Leica a distancias distintas del nido. . 136 Figura 5.9. Configuración del ensayo y variación de la coordenada radial de cada punto. ...... 136 Figura 5.10. Variación de la coordenada radial después de compensar el error modelado. ... 137 Figura 5.11. Representación de como la expansión del cabezal modifica las distancias radiales. ................................................................................................................................................... 137 Figura 5.12. Variación de la coordenada radial después de compensar la dilatación térmica. 138 Figura 5.13. Variación de coordenada z del nido y cabezal a lo largo del tiempo. ................... 138 Figura 5.14. Variación de coordenada radial entre el cabezal y el nido a lo largo del tiempo. 139 14 Figura 5.15. Variación de la coordenada radial después de compensar la dilatación térmica entre nido y cabezal. ................................................................................................................. 139 Figura 5.16. Representación de la fuente del láser en cada modelo de Laser Tracker............. 140 Figura 5.17. Coordenada radial del ensayo a dos metros a lo largo del tiempo. ...................... 142 Figura 5.18. Dispersión de la medición con el retro-reflector a dos metros. ........................... 142 Figura 5.19. Ensayos a 3200 mm, 5000 mm y 6000 mm. .......................................................... 143 Figura 5.20. Relación entre la dispersión y la distancia medida cuando existe una perturbación. ................................................................................................................................................... 143 Figura 5.21. Coordenada radial del ensayo a varias temperaturas. ......................................... 144 Figura 5.22. Dispersión de la medición a varias temperaturas. ................................................ 144 Figura 5.23. Sensor de posición 0.25 segundos antes y después de la pérdida del láser. ........ 145 Figura 5.24. Valores obtenidos por el sensor de posición a distintas distancias de medición. 145 Figura 5.25. Desviación del rayo en un tramo de 2.5 m con aire turbulento. .......................... 146 Figura 5.26. Aumento de la distancia recorrida por el haz láser al desviarse en flujo turbulento. ................................................................................................................................................... 146 Figura 5.27. Desviación del rayo en un tramo de 2.5 m con aire no turbulento. ..................... 147 Figura 5.28. Aumento de la distancia recorrida por el haz láser al desviarse en flujo laminar. 147 Figura 5.29. Distintas puntas de palpador usadas. ................................................................... 148 Figura 5.30. Posición del centro de la esfera calibrada para distintas ángulos del cabezal. .... 148 Figura 5.31. Desviación euclídea (en milímetros) del centro de la esfera para cada posición angular del cabezal (en grados). ............................................................................................... 149 Figura 5.32. Desviación euclídea (en milímetros) del centro de la esfera para cada posición angular del cabezal (en grados). ............................................................................................... 150 Figura 5.33. Ejemplo de variación de desplazamiento de un palpador no isotrópico [145]. ... 151 Figura 5.34. Geometría de la placa patrón................................................................................ 152 Figura 5.35. Sentido de la medición (a) anti-horario y (b) horario. .......................................... 153 Figura 5.36. Sentido de la medición de la placa en los planos XZ o YZ. .................................... 154 Figura 5.37. Medición de la placa en el plano YZ. ..................................................................... 154 Figura 5.38. Ejemplo de perpendicularidad entre los movimientos lineales de los ejes X y Z. 155 Figura 5.39. Errores de la placa en los ejes XY. ......................................................................... 155 Figura 5.40. Error xPx para cada posición del eje X. ................................................................. 156 Figura 5.41. Error yPy para cada posición del eje Y. ................................................................. 156 Figura 5.42. Error xTy para cada posición del eje X. ................................................................. 157 Figura 5.43. Error yTx para cada posición del eje Y. .................................................................. 157 Figura 5.44. Error xRz para cada posición del eje X. ................................................................. 158 Figura 5.45. Desviación media de 6 mediciones de los 56 anillos antes y después de corregir los errores. ...................................................................................................................................... 158 Figura 5.46. Temperatura de la máquina herramienta, de la placa y ambiental. ..................... 159 Figura 5.47. Coordenada z del centro de un anillo esférico. ..................................................... 160 Figura 5.48. Temperatura de la columna durante la medición de 22 puntos cuatro veces consecutivas. ............................................................................................................................. 160 Figura 5.49. Variación de la coordenada z de los 22 puntos medidos cuatro veces consecutivas. ................................................................................................................................................... 161 Figura 5.50. Coordenada z tras aplicar la corrección de la expansión térmica. ........................ 161 Figura 5.51. Incremento de la coordenada Y respecto a la temperatura del cabezal. ............. 162 15 Figura 5.52. Incremento de la coordenada X respecto a la temperatura del cabezal. ............. 162 Figura 5.53. Efecto de la dilatación térmica de la máquina herramienta sobre la medición. .. 163 Figura 5.54. Fuentes de incertidumbre que afectan una verificación de máquina herramienta ................................................................................................................................................... 164 Figura 5.55. Piezas utilizadas para la verificación del error de posición del eje Z. ................... 168 Figura 5.56. Montaje para la verificación del error de posición del eje Z. ................................ 168 Figura 5.57. Error de posición eje Z en los dos sentidos de movimiento.................................. 169 Figura 5.58. Error de posición eje Z en los dos sentidos de movimiento.................................. 169 Figura 5.59. Dos posiciones del eje Z del Alicona: Z=0 mm (izda.) y Z=80 mm (dcha.). ............ 170 Figura 5.60. Desviación de las dos posiciones del eje Z. ........................................................... 170 Figura 5.61. Variación de la longitud del eje Z. ......................................................................... 171 Figura 5.62. Artefacto utilizado para las primeras mediciones con y sin stitching. .................. 172 Figura 5.63. Configuraciones del stitching estudiadas. ............................................................. 173 Figura 5.64. Artefacto diseñado para medir los errores de mediciones stitching free. ........... 174 Figura 5.65. Artefacto alineado con el eje X (izda.), con el eje Y (centro) y oblicuo (dcha.). .... 175 Figura 5.66. Parámetros de error calculados en el post-procesado. ........................................ 175 Figura 5.67. Desviación antes y después de corregir los errores (inicial y final respectivamente). ................................................................................................................................................... 176 Figura 5.68. Ensayos elegidos para el modelado de los errores. .............................................. 177 Figura 5.69. Error residual medio frente al número de iteraciones. ........................................ 178 Figura 5.70. Valores de los parámetros de error estimados para la lente 5×. .......................... 178 Figura 5.71. Errores en distancia de las mediciones antes y después de compensar los errores. ................................................................................................................................................... 179 Figura 5.72. Errores de posición (mm), antes y después de compensar los errores. ............... 179 Figura 5.73. Valores de los parámetros de error estimados para la lente 10×. ........................ 180 Figura 5.74. Medición del artefacto con stitching (izquierda) y sin stitching (derecha). .......... 180 Figura 5.75. Error en distancia antes y después de compensar los parámetros de error. ....... 181 Figura 5.76. Modelo cinemático de máquina herramienta XFYZ con Laser Tracker: a) verificación convencional y b) metodología propuesta. ........................................................... 182 Figura 5.77. Representación de las mediciones realizadas. ...................................................... 183 Figura 5.78. Mediciones en Spatial Analyzer. ........................................................................... 184 Figura 5.79. Montaje realizado para obtener mismo offset en las rectas y los planos. ........... 184 Figura 5.80. Error inicial. ........................................................................................................... 185 Figura 5.81. Error residual. ........................................................................................................ 185 16 INDICE DE TABLAS Tabla 2.1. Errores geométricos por tipo y eje de movimiento según la norma VDI 2617-3. ...... 29 Tabla 5.1. Características principales de los sistemas láser utilizados en la tesis ..................... 131 Tabla 5.2. Características de las distintas lentes del Alicona G5 Infinite Focus. ....................... 134 Tabla 5.3. Repetibilidad de los datos antes de corregir la punta .............................................. 149 Tabla 5.4. Repetibilidad de los datos después de corregir la punta ......................................... 150 Tabla 5.5. Distancia entre los agujeros de los extremos para cada estrategia en los dos ejes. 174 Tabla 5.6. Comparación de los diferentes modelados de los parámetros. ............................... 180 INTRODUCCIÓN 1. Introducción 19 1 INTRODUCCIÓN En este primer capítulo se define el contexto dentro del cual se ha desarrollado la investigación, la motivación de la misma, así como el alcance y los objetivos de la tesis. Finalmente se desglosa la estructura de la tesis mediante un breve resumen del contenido de los siguientes capítulos. 1.1 Contexto Una máquina herramienta es una máquina cuyo objetivo principal es la fabricación de piezas principalmente metálicas o de otros materiales rígidos generalmente cortando, perforando, rectificando, o aplicando otras formas de deformación [1]. Las máquinas herramienta pueden operarse manualmente o mediante control automático. Las primeras máquinas utilizaban volantes para estabilizar su movimiento y poseían sistemas complejos de engranajes y palancas para controlar la máquina y las piezas en que trabajaba. A principios de la década de 1950 se desarrollaron los sistemas de control numérico [2]. Fue en el Instituto de Tecnología de Massachusetts (MIT) donde se automatizó por primera vez una fresadora. Las máquinas de control numérico utilizaban tarjetas perforadas para controlar su movimiento. En la década de 1960 se añadieron computadoras para aumentar la flexibilidad del proceso. Tales máquinas se denominaron máquinas de Control Numérico por Computadora (CNC). Estas máquinas han adquirido una gran flexibilidad pudiendo realizar un amplio rango de tareas, repetir secuencias una y otra vez con precisión así como producir piezas mucho más complejas que las que podría hacer el operario más experimentado. Las máquinas herramienta están constituidas por múltiples elementos que permiten los movimientos de traslación o rotación ya sea de la pieza o de la propia herramienta. Como consecuencia de un mal montaje entre los elementos propios o debido a diversos factores pueden aparecer errores que reducen la precisión de la máquina herramienta [3-11]. En 1980 R. J. Hocken [12] definió “error”, en términos de desviaciones de una máquina herramienta, como la diferencia entre la respuesta real de la máquina a una instrucción realizada de acuerdo al protocolo aceptado por dicha máquina y la respuesta esperada por el protocolo de la misma instrucción. La fabricación de piezas mediante arranque de viruta debe ir acompañada de un proceso de verificación para comprobar que se cumplen las tolerancias requeridas [13]. Existen tres técnicas principalmente para validar la verificación periódica de la máquina herramienta: 1. Realizando algún tipo de ensayo de mecanizado, por ejemplo, NAS979: prueba de corte compuesto. 2. Inspección in situ con galgas o artefactos mediante una sonda de contacto. 3. Procedimientos específicos utilizando equipos de medición metrológicos. Actualmente los procesos de fabricación y de verificación de pieza se realizan en máquinas y entornos distintos. La fabricación se realiza en talleres con máquinas herramienta mientras que la verificación se realiza en laboratorios metrológicos con máquinas de medición por coordenadas. Sin embargo, hoy en día, las máquinas herramienta tienen la posibilidad de instalar en su cabezal porta-herramientas un palpador para medición por contacto, lo cual permitiría realizar la verificación de pieza inmediatamente después e incluso durante para 2. Estado del arte 26 Los errores geométricos pueden ser medidos principalmente usando dos métodos: mediante métodos directos o mediante métodos indirectos. Los métodos directos requieren múltiples mediciones con sus respectivos montajes diferentes para calcular cada error, lo cual hace lento a este método en comparación con los métodos indirectos que con la medición de una única nube de puntos que englobe el espacio de trabajo ya permite la estimación de los errores. Sin embargo, son los métodos directos los que proporcionan datos reales del comportamiento de cada error, los métodos indirectos proporcionan el valor óptimo del conjunto de errores. En 1966 la Asociación de Industrias Aeroespaciales de América definió una serie de pruebas de corte estandarizadas para máquinas de fresado por control numérico en el Estándar Nacional Aeroespacial NAS979 [23]. La máquina herramienta se utiliza para fabricar una pieza de prueba en condiciones controladas. El rendimiento de la máquina se evalúa luego verificando la precisión dimensional de la pieza de prueba en una máquina de medir por coordenadas. La sección 4.3.3.5.1 de esta norma define una prueba de corte compuesto que involucra el mecanizado de perfiles circulares, de diamante y de forma cuadrada bajo control CNC. Figura 2.2. Ejemplo de una pieza NAS979: círculo, diamante, cuadrado. NAS979 ahora se considera obsoleta y sin reemplazo directo. Sin embargo, pruebas de círculo, diamante y cuadrado similares se definen ahora en la Norma Internacional ISO 10791-7 2014 [24] y en la Norma Americana ASME B5.54 2005 [25]. En 1991 Knapp et al [26] y Trapet et al [27] estudiaron distintos artefactos para la verificación de máquinas de medir por coordenadas y máquinas herramienta con el fin de que el usuario pudiera comprobar la precisión especificada y garantizada por el proveedor después de la instalación de la máquina y durante los años de operación. En 1982 Bryan et al [28,29] propuso la utilización de una barra de bolas telescópica como patrón para el cálculo de errores de una máquina herramienta mediante la realización de trayectorias circulares. Este método resulta económico y fácil de usar pero abarca un limitado volumen de trabajo por lo que resulta útil para maquinas herramienta pequeñas y medianas, pero queda descartado para las máquinas herramientas de grandes dimensiones. 2. Estado del arte 27 Figura 2.3. Verificación mediante barra de bolas telescópica. Fuente: Renishaw. La comparación de la trayectoria realizada con respecto a la esperada determina el error resultante, que proporciona un resultado numérico simple que indica la capacidad de la máquina herramienta. Tal es el valor de la prueba de barra de bolas telescópica, que se ha incluido en numerosas Normas nacionales e internacionales [25, 30-32]. En 1988 Zhan et al [33] desarrollaron un método para realizar la verificación volumétrica de una máquina de tres ejes lineales mediante la medición de 22 líneas distribuidas en el volumen de trabajo. En el año 2001 Chen et al [34] optimiza este método reduciendo el número de líneas de 22 a 15. Por otra parte, son varios los investigadores que optaron por estudiar el error volumétrico de una máquina herramienta a partir de la medición láser de un vector de posiciones [35-38]. En el año 2003 Chapman [39] presentó las restricciones de este método. En el año 2005 J.A. Soons [40] y O. Svoboda en 2006 [41] confirmaron que un mal alineamiento del espejo o del láser generaban importantes errores en este método. Para solucionar estas limitaciones, entre 2009 y 2010 Ibaraki et al mejoraron el método diagonal por pasos para 2D y 3D realizando mediciones lineales adicionales paralelas a los ejes de la máquina [42,43]. Figura 2.4. Medición del error mediante un vector de posiciones, método diagonal por pasos. 2. Estado del arte 28 Desde la década de 1970, los sistemas de interferómetro láser se han empleado para procedimientos de calibración de precisión. Pero la incorporación del interferómetro láser a sistemas de medición con mayor grado de libertad como el Laser Tracker o el Laser Tracer permitió la verificación de máquina herramienta mediante sistemas de medición sin contacto [44-53]. Independientemente del sistema de medida empleado, la aproximación volumétrica consiste en medir una nube de puntos, ya sea ordenada o aleatoria, distribuidos por todo el volumen de trabajo, lo cual permite obtener información representativa del error global de la máquina herramienta. Por otra parte, la verificación geométrica busca la determinación de los errores geométricos de la máquina mediante la medición de las coordenadas espaciales de ciertos puntos. 2.2 Fuentes de error que afectan a la precisión de la máquina herramienta Las distintas fuentes de error que afectan a la precisión de una máquina herramienta se dividen principalmente en dos grupos: errores cuasi-estáticos y errores dinámicos [11]. Los errores cuasi-estáticos son los que se producen entre la herramienta y la pieza. Estos errores están relacionados con la estructura de la máquina herramienta y pueden variar lentamente con el tiempo y la temperatura y son causados por fuentes geométricas, cinemáticas, térmicas, etc. [54-59]. Por otra parte, los errores dinámicos son causados por el movimiento del husillo, fuerzas de corte, vibraciones y errores del control [60-63]. A lo largo de los años, se han estudiado las diversas fuentes de error y se estima que aproximadamente un 70% del error total de una máquina herramienta lo componen las fuentes geométricas y térmicas [3,64-68], por lo que estas dos fuentes de error se van a considerar y estudiar con mayor detalle. Como el objetivo último es que la máquina herramienta actúe como una máquina de medir por coordenadas, se hace especial hincapié en los errores cuasi-estáticos. 2.3 Errores geométricos Los errores geométricos son una causa de desviación que aparece durante el funcionamiento de la máquina. Estas desviaciones pueden ser debidas a diversas causas, como por ejemplo una fabricación inexacta de las guías de movimiento de la máquina, errores de montaje, flexión de componentes de la máquina debido a cargas estáticas, deformación de componentes a lo largo del tiempo, desgastes, causando errores en las trayectorias de movimiento de los ejes, etc. Los movimientos de los ejes de una máquina herramienta pueden describirse mediante seis grados de libertad, tres de rotación y tres de traslación. Esto hace que los errores de una máquina herramienta de 3 ejes lineales se puedan definir mediante 21 errores geométricos, estos son los correspondientes a sus seis grados de libertad por cada eje además de otros tres errores de perpendicularidad, uno por cada par de ejes. La notación de los errores geométricos esta estandarizada por las normas ISO 841 [69] y por la VDI 2617-3 [70]. 2. Estado del arte 29 Tabla 2.1. Errores geométricos por tipo y eje de movimiento según la norma VDI 2617-3. Tipo de error Eje de movimiento X Y Z Traslación Posicionamiento xTx yTy zTz Rectitud horizontal xTy yTx zTx Rectitud vertical xTz yTz zTy Rotación Vuelco xRx yRy zRz Cabeceo xRy yRx zRx Rumbo xRz yRz zRy Perpendicularidad xWy yWz xWz Los métodos para calcular los errores geométricos se clasifican principalmente en dos grupos: métodos directos [32,71-73] y métodos indirectos [74-80]. UNE-ISO 230-1:2014 [81] y la norma ASME B5.54-2005 [20] son dos normas internacionales donde se especifican métodos para medir y calcular los errores geométricos de una máquina herramienta usando métodos directos en condiciones cuasi-estáticas o sin carga. Con estos métodos se determina el valor de un error en cada posición del espacio de trabajo de la máquina. Sin embargo, con los métodos indirectos se determina la influencia conjunta de los errores en base al movimiento de los ejes y el modelo cinemático, de manera que no tienen un significado físico real, pero determinan una solución óptima que puede ser extrapolada a todo el volumen de trabajo. Por esta razón los sistemas de verificación volumétrica que usan laser tracer [74-76], laser tracker [77-79] o barras de bolas [80] como sistemas de medición son cada vez más usados en lugar de los métodos directos que usan interferómetros láser, niveles, etc., especialmente para la evaluación de máquinas herramienta de largo recorrido. 2.4 Errores térmicos En los años 60 los errores térmicos fueron formalmente reconocidos como la mayor fuente de error sobre las piezas fabricadas [82,83]. Los tres principales mecanismos que producen un cambio de temperatura estructural son: calor generado internamente por la máquina, cambio de temperatura medio ambiental y fuentes de radiación externa [84]. Los efectos térmicos dentro de un entorno de mecanizado tienen una gran influencia en la estructura de una máquina herramienta y, por lo tanto, afectan la precisión de la pieza mecanizada. Cualquier forma de expansión y contracción creará una distorsión en la máquina herramienta y afectará a su precisión de posicionamiento [10,84,85]. A lo largo de los años, los investigadores han demostrado que esta distorsión térmica de la máquina herramienta puede generar entre el 40 y el 70% de todos los errores dimensionales en piezas mecanizadas con precisión. Por lo tanto, cualquier cambio mínimo de temperatura puede hacer una diferencia significativa en la calidad mecanizada de un componente [4]. El problema fundamental relacionado con la medición de posición en ejes lineales cuando están formados por un husillo de bolas con sistema de recirculación, es la expansión térmica que se crea en el husillo en servicio. La expansión térmica de este husillo puede dar como resultado un error de posicionamiento de una magnitud de 100 μm, dependiendo de la naturaleza y la longitud del programa de pieza del CNC real [82,87,88]. Es de destacar que 2. Estado del arte 30 después de cada operación de mecanizado de un nuevo programa CNC de piezas de cualquier nivel de complejidad y longitud razonables, los husillos de bolas requerirán aproximadamente una hora para alcanzar una condición térmicamente estable. Esta condición de incremento térmico también es aplicable a las interrupciones en el ciclo operativo de mecanizado. Como norma general, la expansión térmica de un husillo de bolas, se puede decir que "por cada metro de longitud de un husillo de bolas en frío, se espera que crezca entre 0.5 y 1 μm, después de cada doble carrera del recorrido” [89]. Esta expansión térmica se acumulará, dentro de ciertas restricciones de tiempo. En 1995 Sato et al [90] introdujeron el diseño de un mecanismo de tornillo de avance activo que puede controlar o eliminar el juego de estos sistemas de movimiento lineal. Esta fue una importante mejora de los husillos de cara al posicionamiento ultra-preciso. La norma ISO 3408 [91] clasifica los husillos a bolas según su precisión, siendo la clase 1 los más precisos y la clase 10 los menos precisos. Esta clasificación de la precisión del mecanizado de los husillos a bolas, se define por la desviación longitudinal en función de la longitud de la rosca. Considerando que, la clase 7 es un husillo a bolas de baja precisión, aún puede ser utilizado en aproximadamente el 80% de las aplicaciones industriales. La clase 5 reduce estos errores a la mitad, siendo la desviación de 51 µm para un husillo de entre 1600 mm y 2000 mm de recorrido útil. La clase 3 a su vez reduce este error una vez más a la mitad, lo que brinda una precisión extremadamente alta, por ello se utiliza para la mayoría de las máquinas herramienta CNC de alta calidad de hoy en día. Estas, junto a otras fuentes de calor internas, provocan un incremento de la temperatura de la máquina significativamente superior a los cambios en la temperatura ambiental. Diversos autores han tratado de modelar las máquinas herramienta mediante análisis con elementos finitos (FEA) [92-98]. Este método de análisis permite simular el comportamiento de los elementos estructurales de la máquina ante gradientes de temperatura, pero supone un reto importante tanto el modelado de los elementos estructurales como el definir las condiciones reales que se va a encontrar la máquina durante el mecanizado para que los resultados sean realistas. Una simplificación de los métodos por elementos finitos es el método de descomposición [99], que consiste en descomponer la máquina en sus principales elementos estructurales y analizar su comportamiento térmico por separado para obtener la respuesta conjunta al sumar los comportamientos de los distintos componentes por separado. Para estudiar el comportamiento térmico de la máquina herramienta algunos autores realizaron simplificaciones para estimar estadísticamente estos errores mediante el uso de unos pocos sensores de temperatura [100,101]. Lee et al estudiaron cual era el número óptimo de sensores necesarios para modelar adecuadamente la temperatura de la máquina herramienta, concluyendo que dicho número era entre tres y cuatro [102]. Por otra parte, otros investigadores optaron por el uso de una cámara termográfica para obtener una visión aún más global de la temperatura de los elementos estructurales de la máquina [103,104]. Algunos autores han usado las mediciones de los sensores de temperatura para modelar los errores térmicos mediante filtros Kalman [105,106]. El filtro de Kalman es un algoritmo que sirve para poder identificar el estado oculto (no medible) de un sistema dinámico lineal. Este tipo de filtros son capaces de elegir la ganancia K de realimentación del error de forma óptima 2. Estado del arte 31 cuando se conocen las varianzas de los ruidos que afectan al sistema. Dado que el filtro de Kalman es un algoritmo recursivo, puede aplicarse en tiempo real usando únicamente las mediciones de entrada. Otra solución propuesta consiste en modelar los errores térmicos mediante redes neuronales [107,108]. Estas son un modelo computacional consistente en un conjunto de unidades conectadas entre sí para transmitirse señales. La información de entrada atraviesa la red neuronal, donde se somete a diversas operaciones, produciendo unos valores de salida. Debido al gran impacto que las influencias de la temperatura tienen en la incertidumbre de la medición para cualquier característica del componente medido en el taller, deben corregirse a 20 °C. Sin embargo, corregir la temperatura dentro del taller de producción puede dar lugar a problemas prácticos imprevistos. En el taller, la temperatura puede variar durante la jornada laboral, a medida que se realizan mediciones dimensionales adicionales. 2.5 Error Máximo Permitido (MPE) El error máximo permitido de una máquina de medir por coordenadas se define como el valor extremo del error de la indicación de dicha máquina para mediciones longitudinales, permitido por las especificaciones y regulaciones para la máquina. El error máximo permitido en una medición de longitud se puede expresar de una de las tres formas siguientes [109]: 1. 𝑀𝑃𝐸=± 𝑚í𝑛𝑖𝑚𝑜 𝑒𝑛𝑡𝑟𝑒 (𝐴+𝐿 𝐾 ⁄ ) 𝑦 𝐵; 2. 𝑀𝑃𝐸=± (𝐴 + 𝐿 𝐾 ⁄); 3. 𝑀𝑃𝐸=± 𝐵; Donde “𝐴” es una constante positiva expresada en µm y proporcionada por el fabricante; “𝐾” es una constante positiva adimensional dada por el fabricante; “𝐵” es el error máximo permitido en µm, determinado por el fabricante. Estas expresiones pueden aplicarse en cualquier posición y/o dirección dentro del volumen de la máquina de medir por coordenadas. Figura 2.5. Ejemplos de MPE según la norma ISO 10360-5 [109]. La prueba se debe realizar con 5 componentes certificados (bloques patrones) de diversas longitudes, orientándolos en 7 direcciones distintas en el volumen de medición y medir cada uno 3 veces para obtener un total de 105 mediciones. Todos los resultados de las 105 mediciones tienen que estar dentro de los límites especificados por el fabricante (MPE). No se admiten las mediciones fuera de los límites. La ventaja de este ensayo con respecto a otros es que aunque esté limitada a la medición de longitudes, el resultado de la medición tiene una trazabilidad directa a la unidad de longitud, el metro. 2. Estado del arte 32 Esta norma está pensada para ser aplicada a máquinas de medición por coordenadas, pero dado que el objetivo de esta tesis es que la máquina herramienta funcione como una máquina de medir por coordenadas, realizar una estimación del error máximo permisible de la máquina herramienta nos acercaría al objetivo de obtener mediciones trazables. 2.6 Compensación de errores La compensación de errores, ya sean geométricos o cinemáticos, ha sido ampliamente estudiada tanto para máquina herramienta como para máquina de medir por coordenadas [49-51,110-112]. La capacidad de compensación de los controles limita en la actualidad la aplicación de las técnicas de compensación cinemática adecuadas para el uso preciso y trazable de los sistemas de fabricación como sistemas de inspección directa. La capacidad de compensación de los controles numéricos es variable y se basa en su versión más básica en tablas de interpolación de error que van desde los errores de posición únicamente hasta tablas de compensación por interpolación de 6 errores por eje y errores de perpendicularidad de los ejes. Figura 2.6. Compensación de errores mediante tabla de compensación. La principal limitación de la compensación de errores mediante tablas es que la información debe ser cargada en el equipo, lo cual consume memoria. Además, la información introducida es discreta y los errores en puntos intermedios son obtenidos mediante interpolación lineal. La utilización de controles de arquitectura abierta basada en ordenadores ofrece mayor flexibilidad. Este sistema permite introducir modelo cinemático y los parámetros de los errores [113-120]. Durante el mecanizado, el ordenador recibe señales de la posición real de los ejes, estas se introducen en el modelo matemático y este proporciona una señal correctiva con la que retroalimenta los servomotores para un posicionamiento correcto de los ejes mientras el programa es ejecutado. Este método es especialmente complicado y costoso. 2. Estado del arte 33 Figura 2.7. Compensación de errores mediante a tiempo real mediante PC. Por otra parte, el modelo cinemático de la máquina puede ser usado como post-procesador de un programa CNC e introducir los errores geométricos en él, obteniendo un CNC modificado para que acuda a las posiciones correctas una vez corregidas [121]. Figura 2.8. Compensación de errores mediante post-procesado. Los sistemas de fabricación asistidos por ordenador (CAM) han evolucionado permitiendo la corrección de errores en post-proceso. Las trayectorias para la fabricación de una pieza son introducidas en el programa de diseño asistido por ordenador, el cual las analiza e introduce los errores, finalmente este devuelve un nuevo programa CNC con las nuevas trayectorias que debe seguir la máquina herramienta para fabricar la pieza con los errores compensados. Este procedimiento es muy flexible para su aplicación. Durante un tiempo, las máquinas herramienta sólo tenían una capacidad de corrección limitada frente a las máquinas de medición por coordenadas. Esto es debido a las limitaciones de potencia de cálculo, ya que en las máquinas herramientas la compensación tiene que ser en tiempo real. Mientras que las máquinas de medir por coordenadas la compensación puede realizarse a posteriori. Actualmente existen diversos sistemas de control numérico comerciales, como son por ejemplo Siemens, Fagor o Heidenheim. 2.7 Modelo cinemático de la Máquina Herramienta El modelo cinemático es el modelo matemático o la ecuación que describe el movimiento del sistema mecánico. En ingeniería mecánica es lo que define el conjunto de cuerpos rígidos conectados entre sí que describen el movimiento deseado del sistema. Esto incluye sus posiciones, velocidades y aceleraciones, dadas sobre un sistema coordenado de referencia, pero no tiene en cuenta las fuerzas que actúan sobre él. 2. Estado del arte 34 La estructura de una máquina se determina por la combinación de sus diferentes elementos estructurales. Estos elementos pueden modelarse mediante una cadena cinemática, que simboliza el flujo de movimientos de estructuras cinemáticas en serie o paralelas [122,123]. A lo largo de los años ha ido evolucionando la manera de determinar el modelo cinemático. A principios de la década de los sesenta, el modelo cinemático se obtenía mediante relaciones trigonométricas entre los diferentes ejes de movimiento de la máquina [124]. En la década de los setenta se implantó la utilización de la cadena cinemática basado en matrices de transformación homogéneas [125,126]. En 1986 Ferreira et al [127] introdujeron dos simplificaciones que aún hoy se siguen utilizando: la consideración de que los componentes de la máquina herramienta actúan como un sólido rígido y la aproximación para ángulos pequeños en los errores geométricos de rotación. El modelo matemático de la cadena cinemática de una máquina herramienta o máquina de medir por coordenadas se utiliza para la compensación de sus errores geométricos [11]. Asumiendo el movimiento de la máquina como un sólido rígido, el movimiento puede modelarse mediante matrices de transformación y de rotación [3,128,129]. Las máquinas herramientas se pueden clasificar de acuerdo con el movimiento de la pieza y de la herramienta. En el caso de las máquinas herramientas de 3 ejes, estas pueden clasificarse como FXYZ, XFYZ, XYFZ y XYZF. Donde F determina la parte fija de la máquina. Los ejes que se encuentran a la izquierda de la F son aquellos que se mueven con la pieza, mientras que los que se encuentran a la derecha representan los ejes que se mueven con la herramienta. Figura 2.9. Esquema cinemáticos de máquinas herramienta: a) FXYZ, b) XFYZ, c) XYFZ, d) XYZF. El esquema cinemático representa los dos caminos que tienen la pieza y la herramienta para encontrarse. La máquina herramienta analizada en este trabajo tiene una configuración XFYZ, lo que significa que la pieza de trabajo se anclará o apoyará en la mesa cuyo movimiento corresponde al eje X, mientras que la herramienta está instalada de solidariamente con los ejes de movimiento Y y Z. Sin embargo, la máquina de medición por variación focal Alicona tiene un modelo cinemático del tipo YXFZ, lo que significa, por una parte que el eje X “reposa” sobre el eje Y, y ambos ejes se encargan de desplazar la pieza, y por otra parte que la herramienta, o en este caso, las lentes de medición son solidarias al eje Z. 2. Estado del arte 35 Figura 2.10. Movimiento de los ejes de la máquina herramienta y su esquema cinemático. Con la cadena cinemática es posible determinar las ecuaciones que rigen el movimiento de la máquina herramienta con respecto a la pieza [130,131]. La siguiente ecuación muestra el modelo matemático para la configuración XFYZ: 𝑊 =𝑅(𝑋)∙(−𝑋+𝑌)+𝑅(𝑋)∙𝑅−1(𝑌)∙𝑍+[𝑅(𝑋)∙𝑅−1(𝑌)∙𝑅−1(𝑍)]∙𝑇 (2.1) Donde: 𝑋=(𝑥+𝑥𝑇𝑥 𝑥𝑇𝑦 𝑥𝑇𝑧 ), (2.2) 𝑌=(𝑦𝑇𝑥+𝑥𝑊𝑧∙𝑦 𝑦+𝑦𝑇𝑦 𝑦𝑇𝑧 ), (2.3) 𝑍=(𝑧𝑇𝑥+𝑥𝑊𝑧∙𝑧 𝑧𝑇𝑦+𝑦𝑊𝑧∙𝑧 𝑧+𝑧𝑇𝑧 ), (2.4) 𝑅(𝑋)=( 1 𝑥𝑅𝑧 −𝑥𝑅𝑦 −𝑥𝑅𝑧 1 𝑥𝑅𝑥 𝑥𝑅𝑦 −𝑥𝑅𝑥 1 ), (2.5) 𝑅(𝑌)=( 1 𝑦𝑅𝑧 −𝑦𝑅𝑦 −𝑦𝑅𝑧 1 𝑦𝑅𝑥 𝑦𝑅𝑦 −𝑦𝑅𝑥 1 ), (2.6) 𝑅(𝑍)=( 1 𝑧𝑅𝑧 −𝑧𝑅𝑦 −𝑧𝑅𝑧 1 𝑧𝑅𝑥 𝑧𝑅𝑦 −𝑧𝑅𝑥 1 ), (2.7) Cuyas componentes de error están definidas en la Tabla 2.1 de este capítulo. El desarrollo de estas ecuaciones determina el movimiento de la máquina herramienta y conduce al cálculo de los 21 errores geométricos de la misma. 3. Presentación de las publicaciones 42 en el modelo cinemático y en el modelado del ruido del Laser Tracker para luego usar el método Monte Carlo para el cálculo de la incertidumbre. El artículo “Influence of laser tracker noise on the uncertainty of machine tool volumetric verification using the Monte Carlo method” profundiza más en el tema ofreciendo más información sobre las simulaciones. Puede verse cómo la incertidumbre de medida adquiere la forma característica de un elipsoide y finalmente se presenta un valor del máximo error permitido (MPE) basándose en los datos obtenidos de las simulaciones. Por último, “Study on Machine Tool Positioning Uncertainty Due to Volumetric Verification” presenta un análisis detallado de las distintas fuentes de incertidumbre y se introducen estas en el modelo cinemático de la máquina herramienta, obteniendo así un modelo cinemático más realista, que tiene en cuenta las incertidumbre presentes en la verificación de la máquina herramienta. Además, en este artículo se estudia la localización óptima del Laser Tracker en el espacio de trabajo para minimizar las incertidumbres durante el proceso de verificación. La última publicación que se va a presentar lleva por título “Lateral error compensation for stitching-free measurement with focus variation microscopy”. Este artículo se centra en la auto-calibración de un sistema de medición distinto, en este caso se trata de una máquina de medición que utiliza la tecnología de microscopía por variación focal. Sin embargo, la idea principal que se persigue es la misma, la auto-calibración de un sistema de medición mediante un artefacto y a través del modelo cinemático de la propia máquina. Aunque actualmente la microscopía por variación focal no es una opción disponible para máquina herramienta, en el futuro podría plantearse la instalación de un sistema similar en el cabezal porta-herramientas de la máquina herramienta para posibilitar una verificación rápida y precisa del volumen de trabajo. Las publicaciones “Lateral error compensation for stitching-free measurement with focus variation microscopy”, “Analysis of the measurement capacity of a machine tool”, “Monte Carlo method to machine tool uncertainty evaluation”, “Influence of laser tracker noise on the uncertainty of machine tool volumetric verification using the Monte Carlo method” y “Study on Machine Tool Positioning Uncertainty Due to Volumetric Verification” se apoyan principalmente en el uso del modelo cinemático de la máquina para la estimación de errores. En el caso de la primera publicación se estudia el modelo cinemático de un sistema de medición por variación focal. Las otras cuatro publicaciones, analizan el modelo cinemático del sistema máquina herramienta con Laser Tracker. Todas las publicaciones incluidas en la tesis presentan una unidad temática sólida dentro del ámbito de la verificación y la estimación de errores geométricos de sistemas de medición. En cuanto a los objetivos específicos de la tesis, previamente explicados en la sección 1.3, han sido cubiertos por cada una de las publicaciones del siguiente modo: En el primer artículo, “Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements”, se afrontan los objetivos 1, 2, 6 y 7. Mientras que en el artículo “Lateral error compensation for stitching-free measurement with focus variation microscopy” se abordan los objetivos 1, 4, 5 y 6. El trabajo presentado en los artículos “Analysis of the measurement capacity of a machine tool”, “Monte Carlo method to machine tool uncertainty evaluation”, 3. Presentación de las publicaciones 43 “Influence of laser tracker noise on the uncertainty of machine tool volumetric verification using the Monte Carlo method” y “Study on Machine Tool Positioning Uncertainty Due to Volumetric Verification”se han llevado a cabo los objetivos 1, 2, 3, 4, 5, 6, 7 y 8. Las publicaciones presentadas se ven completadas por una serie de trabajos realizados que se presentan en el capítulo 5 de la tesis, el de metodología. Las secciones 5.2 y 5.3 completan el artículo “Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements”. La primera de estas dos secciones contiene detalles que se comentan en el artículo aunque no se muestren los datos, como por ejemplo que el fenómeno se observa a diferentes distancias y no sólo en el nido. Además se profundiza en el modelo de corrección al observar un error que se había sobre-compensado y en esta sección se corrige. Por otra parte, la sección 5.3 contiene más información sobre el estudio de las turbulencias en el láser, presentando más ensayos a distintas distancias y distintas temperaturas, además de un simulador de la curvatura del rayo. La sección 5.4 muestra los datos de desalineamiento de la punta del palpador al ser instalada y como se ha logrado corregir este error que es fundamental corregir antes de realizar las mediciones de los artículos “Analysis of the measurement capacity of a machine tool”, “Monte Carlo method to machine tool uncertainty evaluation” y “Influence of laser tracker noise on the uncertainty of machine tool volumetric verification using the Monte Carlo method”. Estos artículos también se ven apoyados por la sección 5.5 donde se calculas los errores de la máquina herramienta mediante diversas mediciones de la placa. También se estudian los efectos de la temperatura en las mediciones con máquina herramienta. En la sección 5.6 se profundiza en las fuentes de incertidumbre estudiadas en el artículo “Study on Machine Tool Positioning Uncertainty Due to Volumetric Verification”. La sección 5.7 aporta más información que completa el artículo “Lateral error compensation for stitching-free measurement with focus variation microscopy”. Finalmente, la sección 5.8 presenta la línea de trabajo por la que se pretende seguir, la cual busca una verificación volumétrica rápida y precisa, dicho procedimiento consiste en la medición de una recta y un plano de forma independiente, pero que se interrelacionan en un punto y con el modelo cinemático de la máquina herramienta que se ha dividido en dos subcadenas cinemáticas. 3. Presentación de las publicaciones 44 3.2 Presentación de la publicación “Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements” En este artículo se analizan diversos aspectos del funcionamiento de un Laser Tracker, una herramienta que en las últimas décadas se ha vuelto indispensable para realizar la verificación volumétrica de la máquina herramienta por métodos indirectos. Por lo que las conclusiones obtenidas de este primer artículo serán tenidas en cuenta cuando dicho sistema de medición sea utilizado en futuros trabajos. El primer aspecto analizado trata sobre la estabilización térmica inicial del Laser Tracker. Esto es, que al encender el Laser Tracker el fabricante recomienda un tiempo de espera de entre 15 y 20 minutos. De hecho, durante ese periodo, el software no permite realizar medición alguna. Sin embargo, se ha comprobado que durante los siguientes minutos el Laser Tracker aún no ha alcanzado una estabilización apropiada, pudiendo generarse errores de entre 20 y 80 µm. El segundo aspecto estudiado tiene en consideración que las máquinas herramientas normalmente se encuentran en talleres de mecanizado donde las condiciones ambientales fluctúan y pueden afectar de manera considerable los valores de una medición. Por esta razón es importante analizar el efecto de las corrientes de aire turbulentas en los valores de una medición con Laser Tracker. En la introducción se mencionan las tres principales normas estandarizadas que tratan acerca del funcionamiento de los Laser Tracker: la ASME B89.4.19-2005, la norma VDI/VDE 2617-10 y la norma ISO 10360-10. A pesar de que estas tres normas tienen en cuenta ciertos errores relacionados con las variaciones de temperatura, ninguna de ellas tiene en cuenta posibles errores generados por la deformación térmica del Laser Tracker ni se estudia el tiempo de calentamiento óptimo. Los Laser Trackers están listos para capturar datos después del periodo de estabilización de la fuente del láser que dura entre 15 y 20 minutos. Sin embargo, se ha observado que el usuario debe esperar al menos 2 horas o más para conseguir unos valores de medición estables. Durante este proceso de estabilización la fuente del láser actúa como un foco de calor, esto hace que aumente la temperatura en el interior del equipo provocando que las lentes y algunas partes mecánicas se dilaten. Este error genera un offset en la coordenada radial de la medición similar al death path error. Para estimar este error se ha medido un punto fijo a lo largo del tiempo. Las mediciones se han realizado con una frecuencia de muestreo de 2 puntos por minuto, iniciando la toma de datos inmediatamente después de finalizar la estabilización térmica teórica. Ambos equipos han sufrido variaciones en la coordenada radial con una tendencia exponencial negativa: 𝜌(𝑡)=𝐴+𝐵∙(1−𝑒−𝑡 𝜏 ⁄) (3.1) Donde 𝐴 es el valor inicial de la coordenada 𝜌 en milímetros, 𝐵 es el valor máximo de variación que la coordenada 𝜌 experimenta, también en milímetros, 𝑡 es el instante de tiempo en minutos y 𝜏 es la constante de tiempo en minutos. 3. Presentación de las publicaciones 45 De este modo, se han modelado las ecuaciones que determinan el error en la coordenada radial de los dos Laser Trackers, siendo la ecuación del API Tracker3: 𝜌(𝑡)=159.798+0.019∙(1−𝑒−𝑡 30 ⁄) (3.2) Y la del Leica Geosystems: 𝜌(𝑡)=154.999+0.075∙(1−𝑒−𝑡 90 ⁄) (3.3) La Figura 3.1 (Figura 2b en el artículo), compara los valores de variación de coordenada radial de ambos equipos con los valores que se obtendrían con las ecuaciones que se han modelado. Figura 3.1. Comparación entre las mediciones y las ecuaciones modeladas. En este tipo de ecuaciones exponenciales, el 95% del valor final se alcanza en el instante de tiempo 3τ. Esto significa que para obtener mediciones estables es necesario esperar 90 minutos en el caso del API Tracker3 y 270 minutos en el caso del Leica Geosystems. En caso contrario, las mediciones se verán afectadas por un offset no despreciable, que alcanza un valor máximo de 19 µm para el API Tracker3 y 75 µm para el Leica Geosystems. Además de la dilatación de componentes internos del equipo, dependiendo de la configuración del Laser Tracker, el incremento de la temperatura en el interior del equipo puede producir también un error debido a la expansión de la carcasa exterior. En el caso del API la fuente del láser se encuentra en la caja externa y el láser llega hasta el cabezal a través de un cable de fibra óptica. Sin embargo, el Leica Geosystems tiene la fuente del láser situada en el interior de la carcasa, debajo del cabezal del Laser Tracker. Esto provoca que al encender la fuente láser se caliente la estructura provocando que la dilatación desplace hacia arriba el cabezal con el espejo que dirige el rayo y el origen de coordenadas. Para comprobar esto, primero se monitorizaron las temperaturas de varios puntos de la carcasa del Leica Geosystems durante el periodo de estabilización. Los datos de la Figura 3.2 se comenzaron a tomar en el mismo instante en que se encendió la fuente del láser. Durante el periodo en el cual el software no permite realizar mediciones la temperatura apenas aumenta 0 10 20 30 40 50 60 70 80 030 60 90 120 150 180 210 240 270 ∆ρ / µm Tiempo / minutos LT2 LT1 Eq.3 Eq.2 3. Presentación de las publicaciones 46 0.5 ᵒC. Sin embargo, algunos puntos del cabezal aumentan a lo largo del periodo estimado de estabilización térmica entre 3 ᵒC y 4 ᵒC. Figura 3.2. Posición de los sensores (izda.) y temperatura de la carcasa (dcha.). Para verificar que el aumento de temperatura provoca expansión térmica del cabezal del Laser Tracker en el eje Z, se realizó un ensayo con la configuración mostrada en la siguiente Figura. Figura 3.3. Configuración del ensayo de dilatación (izda.) y variación de la coordenada Z de un punto del LT2 medido con el LT1 (dcha.). Con el API previamente estabilizado, se midió la variación de coordenada z del retrorreflector situado sobre el cabezal del Leica Geosystems durante su periodo de estabilización. El retrorreflector fue fijado a un punto del cabezal del Leica Geosystems y en el instante en que se encendió, se inició la toma de datos con el API Tracker3. La Figura 3.3 (dcha.) muestra la variación en el tiempo de la coordenada Z de dicho punto. Este desplazamiento del cabezal del Laser Tracker se traduce en un desplazamiento del origen de coordenadas. La sección 2.3 del artículo presenta un modelo de corrección en dos pasos para corregir los errores existentes en el caso de que el usuario decidiera no esperar el periodo de estabilización térmica que se ha visto. El primer paso consiste en corregir el efecto de la dilatación térmica de la carcasa, en el caso de que se produjera. El segundo paso consiste en la corrección del error interno, modelado como una exponencial negativa. En el caso del API, el 21,5 22 22,5 23 23,5 24 24,5 25 25,5 26 060 120 180 240 300 360 420 Temperatura / ᵒC Tiempo / minutos m0 m1 m2 m3 -10 0 10 20 30 40 50 60 70 80 060 120 180 240 300 360 420 ∆z / µm Tiempo / minutos 3. Presentación de las publicaciones 47 primer paso no habría que realizarlo, ya que no se produce dilatación de la carcasa al tener la fuente del láser fuera. Sin embargo, para el Leica Geosystems, es necesario aplicar los dos pasos. El experimento presentado para mostrar el modelo de corrección se llevó a cabo midiendo la posición de cuatro puntos fijos cada 5 minutos, además se ha querido mostrar una peculiaridad: Tras 100 minutos de medición se decidió resetear el interferómetro mediante la función “Measure the Home Position” y tras esto seguir con la medición durante 15 minutos más. Esto se realizó con el propósito de comprobar que tras aplicar el reseteo se corrige el error producido por las dilataciones internas, pero no se corrige el efecto de la expansión térmica de la carcasa. Además se comprobó cómo a pesar de haber reseteado el interferómetro, la tendencia que seguían los datos seguía siendo la misma que antes de resetear, ya que esta tendencia no cesará hasta que se alcance una temperatura interior realmente estable. Finalmente se muestran los valores de la variación de la coordenada radial antes y después de realizar la corrección en dos pasos. En la siguiente sección se explica cómo el índice de refracción del aire puede modificar la longitud de onda del láser y por lo tanto afectar al valor de la medición. El ensayo mostrado en la Figura 9 del artículo consistió en la medición de un punto fijo cuya coordenada radial inicial era aproximadamente 2 metros. Durante los primeros 120 segundos la medición se realizó en condiciones normales de un laboratorio de metrología, pero pasado ese periodo se encendió una fuente de calor que afectaba directamente a un pequeño tramo que atravesaba el rayo. Inmediatamente la coordenada radial se redujo hasta 8 µm, y la desviación estándar de los datos pasó de valores que rondaban los 5 µm a valores que alcanzaban los 70 µm. Una vez observado el efecto de la turbulencia en el rayo y como aumenta considerablemente la desviación estándar de la medición, se presenta una estimación de la incertidumbre que se esperaría obtener con el Laser Tracker de acuerdo con la GUM. La incertidumbre estimada tiene un valor en torno a los observados en la medición antes de que la perturbación disparase el valor de la desviación estándar. Finalmente se quiere demostrar que estas perturbaciones en el aire que recorre el rayo generan fluctuaciones que afectan a la medición. Para ello se han realizado varios ensayos a diferentes distancias en los que se observaba el sensor de posición PSD antes y después de introducir la perturbación. También se ha medido que valores alcanzaba a cada distancia la desviación estándar antes de perderse el rayo para la misma perturbación. La conclusión obtenida es que cuanto mayor es la distancia que recorre el láser, para una misma perturbación, mayor es la desviación estándar y más fácil resulta perder la señal debido a un flujo turbulento de aire a diferente temperatura. 3. Presentación de las publicaciones 48 3.3 Presentación de la publicación “Analysis of the measurement capacity of a machine tool” En este artículo se aborda por primera vez la idea de verificación de piezas en proceso. Para lo cual se realiza la medición de un patrón, en este caso una placa calibrada, mediante una sonda palpador instalada en el porta-herramientas de la máquina herramienta. Simultáneamente, un Laser Tracker, en este caso se ha elegido el Leica Geosystems, se ha situado solidariamente a la mesa (eje X de la máquina herramienta), para que mida los movimientos del cabezal mientras este toma los datos de la placa. Las conclusiones obtenidas en el artículo “Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements” se han tenido en cuenta al realizar los ensayos presentados en este artículo. De hecho, se ha respetado el periodo de calentamiento observado en el Laser Tracker utilizado y antes de iniciar las mediciones, se esperó más de 4 horas, para que los componentes internos del equipo se estabilizasen térmicamente. Pasado dicho periodo, se reinicializó el sistema mediante la orden “Measure Home Position” que recalibra el equipo ya que mide una posición conocida cuyo valor ha sido aportado por el fabricante. La placa ha sido medida como se indica en la Figura 1 del artículo (Figura 3.4), midiendo en espiral cada dos agujeros. En el texto se explica que esta ha sido la estrategia seguida porque se ha comprobado mediante simulaciones que de este modo el efecto del juego se ve reducido. Figura 3.4. Configuración de la medición de la placa. Antes de analizar los datos medidos y los datos obtenidos con el Laser Tracker, se han realizados dos sencillas correcciones para eliminar errores que no sean inherentes a la máquina herramienta. El primero de los dos errores es el de desalineamiento de la placa. Este error ha sido corregido con una matriz de rotación, de modo que el eje X de la placa coincida con el eje X de la máquina herramienta. El segundo error que se ha corregido antes de analizar los datos, es la posible expansión térmica que se haya producido en la propia placa. Para ello durante el proceso de medición se ha monitorizado la temperatura de la placa. Y teniendo en cuenta la posición de cada punto respecto al punto de anclaje, el coeficiente de dilatación de la placa y la temperatura de la placa en el instante de la medición de cada punto, se ha realizado una compensación para que los errores de la placa, en este caso por dilatación térmica no interfieran con los errores propios de la máquina herramienta. 3. Presentación de las publicaciones 49 A continuación, en el apartado de resultados se muestra el valor de unas simulaciones en las cuales a los resultados obtenidos con el Laser Tracker se les añade el error estimado del ruido en una medición. Se han simulado 1000 ensayos, por lo que se han obtenido errores en 28000 puntos, ya que se han medido 28 agujeros de la placa calibrada. Estos errores iniciales se han optimizado tras calcular las funciones de aproximación de los errores de la máquina herramienta. La desviación estándar de este error residual se considera parte de la incertidumbre del proceso de medición. En el siguiente apartado, se ha analizado la incertidumbre del proceso de medición. Para ello se han tomado las ecuaciones que ofrece la ISO/TS 15530-3:2011 Titulada: “Especificación geométrica de productos (GPS). Máquinas de medición de coordenadas (CMM): Técnicas para la determinación de la incertidumbre de medida. Parte 3: Utilización de piezas calibradas o de patrones de medida”. Una vez calculados todos los componentes de incertidumbre asociados a la incertidumbre expandida, se han presentado tanto el error sistemático obtenido en el espacio de trabajo utilizado en la medición, como el valor de la incertidumbre expandida de dicho espacio de trabajo. Figura 3.5. Componentes de incertidumbre con y sin monitorización de temperatura Finalmente se ha realizado una observación acerca de la componente de incertidumbre asociada a variaciones del material (uw), concretamente se ha analizado cómo aumentaría dicha componente de la incertidumbre si no se hubiera monitorizado la temperatura de la placa durante la medición. Resulta interesante que mientras esta componente representa un 6.5% de la incertidumbre total si la temperatura se monitoriza y compensan sus efectos, esta componente puede llegar a representar la mitad de la incertidumbre si no se monitoriza. 0 10 20 30 40 50 60 70 80 90 100 Monitoring Temperature without Monitoring Temperature % of influence ucal uw up 3. Presentación de las publicaciones 50 3.4 Presentación de la publicación “Monte Carlo method to machine tool uncertainty evaluation” Este artículo sigue la misma línea que “Analysis of the measurement capacity of a machine tool” ya que se basa en los datos del mismo ensayo. En dicho artículo se explicaba el proceso de medición las correcciones iniciales y termina mostrando gráficamente los valores de error sistemático e incertidumbre de la máquina herramienta. Sin embargo, “Monte Carlo method to machine tool uncertainty evaluation” se centra principalmente en la estimación de una de las principales componentes de la incertidumbre que afectan a una verificación volumétrica, la cual es determinada usando el método Monte Carlo. El método Monte Carlo es una herramienta que emplea una gran cantidad de recursos computacionales para simular una gran cantidad de datos pseudo-aleatorios. De este modo se simulan complejos sistemas desde el punto de vista probabilístico. El apartado 2 del artículo enumera las principales limitaciones del cálculo de incertidumbres mediante la GUM. Algunas de estas limitaciones son los modelos matemáticos no lineales o la validación del teorema central del límite. El suplemento 1 de la GUM recomienda el uso del método Monte Carlo cuando las incertidumbres no puedan ser evaluadas como Tipo A, Tipo B o incertidumbres combinadas. El siguiente apartado muestra las ecuaciones que modelan la cadena cinemática que incluye el conjunto máquina herramienta con el Laser Tracker solidario al eje X. En el modelo cinemático se incluyen los 21 errores geométricos presentes en una máquina herramienta de 3 ejes. A los datos medidos se les añade el ruido del Laser Tracker mediante 1000 simulaciones de la toma de datos de los 28 agujeros medidos de la placa, obteniendo así 28000 datos, los cuales después de ser optimizados proporcionan el dato de incertidumbre de calibración. Figura 3.6. Distribución del error final. Finalmente el artículo presenta como conclusión la necesidad de usar el método Monte Carlo para la estimación de la incertidumbre del proceso de calibración de una máquina herramienta debido al comportamiento no lineal de su cadena cinemática. 3. Presentación de las publicaciones 51 3.5 Presentación de la publicación “Influence of laser tracker noise on the uncertainty of machine tool volumetric verification using the Monte Carlo method” En este artículo se profundiza más en el ensayo con el cual se han realizado los dos artículos anteriores. En la introducción se explica la normativa existente, tanto la relacionada con los errores geométricos (ISO 841, VDI 2617-3 y UNE-ISO 230) como la que trata sobre estimación de incertidumbres (UNE-ISO/TR 230-9, ISO/TS 14253-2). También se presenta la bibliografía de diversos autores que han tratado previamente acerca de ambos aspectos. El siguiente apartado explica los métodos para estimar la incertidumbre de medida. Para ello se explican las diferencias entre la GUM y el método Monte Carlo y sus ámbitos de aplicación. Además, en este artículo, a diferencia de los dos anteriores, se desarrolla cada uno de los pasos a seguir cuando el método Monte Carlo va a ser usado. En el de metodología, se explica la fórmula empleada para la estimación de la incertidumbre: 𝑈=𝑘·√𝑢𝑐𝑎𝑙 2+𝑢𝑝 2+𝑢𝑏 2+𝑢𝑤 2, (3.4) Y también se explica cómo se estima cada una de las componentes que se han tenido en cuenta: incertidumbre de calibración, incertidumbre asociada al proceso de medición, incertidumbre asociada al error sistemático y por último la incertidumbre asociada al material, la cual incluye incertidumbres asociadas al coeficiente de dilatación, errores de forma, rugosidad y elasticidad. La Figura 3.7 muestra un diagrama de flujo que explica claramente el procedimiento seguido para calcular la incertidumbre a partir de los puntos medidos, sus puntos nominales y el ruido del Laser Tracker introducido. En este apartado además, se presentan las ecuaciones con las que se calcula la incertidumbre. Figura 3.7. Procedimiento para calcular la incertidumbre. 3. Presentación de las publicaciones 58 desplazamiento recoge información cada pocos micrómetros, esto le proporciona al eje una precisión, según el fabricante, de décimas de micrómetro. El modelo cinemático se ha realizado en Matlab. En el programa se aportan los datos de diversas mediciones del patrón, colocado en diferentes posiciones y diferentes direcciones, para que abarque tanto el eje X como el Y, así como varias mediciones oblicuas a ambos ejes. El algoritmo utilizado para la optimización es Levenberg-Marquardt, ya que proporciona una solución mediante mínimos cuadrados a este tipo de problemas no lineales. La lente utilizada para la medición de este artefacto es la 5×. De todos los ensayos medidos, se han seleccionado tres de ellos para calcular los parámetros de los errores geométricos y el resto de los ensayos se usarán para validar dichos parámetros utilizando los errores estimados para realizar una corrección. De los tres ensayos seleccionados para calcular los parámetros de error, se ha seleccionado uno con el patrón alineado con el eje X, otro con el patrón alineado con el eje Y, y finalmente un tercer ensayo con el patrón oblicuo a ambos ejes. Esta primera modelización presenta problemas de convergencia, como se puede observar en la Figura 5. Esto es debido a que algunos parámetros interfieren con otros por la forma en que han sido modelados. Para solucionar este problema se han desarrollado las ecuaciones del modelo y aquellos parámetros que multiplican a otros parámetros se han despreciado, ya que el orden de magnitud con el que afectarían a la coordenada es muy pequeño. De esta forma, las ecuaciones 10, 11 y 12 representan las tres coordenadas del modelo y la ecuación 13 representa el modelo en formato matricial. La ecuación 14 presenta una reordenación de los parámetros de la ecuación 13, esta última tiene por un lado los términos lineales y por otro se le suman los términos angulares que van multiplicados por una coordenada. Finalmente, la ecuación 15 muestra el modelo con los parámetros tal y como han sido modelados. Una vez simplificado el modelo y modelado en Matlab, se ha realizado la optimización y se han obtenido los valores de los parámetros. Con estos errores, se ha realizado una corrección de todas las distancias (Figura 8). La Figura 3.5 muestra los errores de distancias antes de realizar la corrección y después de realizar la corrección. Figura 3.12. Error inicial y residual de los puntos medidos. -20 0 20 40 60 80 100 120 140 160 180 200 020 40 60 80 100 120 140 160 180 Error /µm Distance measured /mm Before correction After correction 3. Presentación de las publicaciones 59 Que los errores del eje X sean mayores que los errores del eje Y probablemente sea debido al propio montaje de la máquina de medición, cuyo eje X está montado sobre el eje Y lo que provocaría que los errores de Y se propagaran al eje X. Lo siguiente que se ha presentado es un cálculo similar, pero en este caso utilizando como objeto medido el artefacto calibrado de 20 mm × 20 mm utilizado en el estudio inicial del comportamiento del stitching. Para estos ensayos se han realizado las mediciones con la lente 10× de este modo se han calculado los errores para mediciones con esta lente. La Figura 11 del artículo contiene los valores estimados, mientras que la Figura 12 muestra los valores en distancia antes y después de realizar la corrección con los valores estimados. Al haber logrado reducir considerablemente el error de las mediciones stitching-free, se considera que la metodología propuesta es válida y se puede aplicar en cualquier máquina de medición sin contacto como la presentada en este trabajo usando un artefacto que no necesita estar calibrado. PUBLICACIONES 62 UNCORRECTED PROOF ARTICLE INFO Article history: Received 7 July 2016 Received in revised form 19 September 2016 Accepted 10 October 2016 Available online xxx Keywords: Laser tracker Metrology Thermal stabilization Warm-up period Turbulence ABSTRACT During the several last years, Laser Trackers have become more common as a measurement tool in the manufacture and assembly of large components such as aircraft wings and ship hulls, as well as for error mapping in coordinate measuring machines and machine tools. Most of these processes cannot be developed in a controlled metrological laboratory but must be implemented directly on a shop floor. Therefore, the process of stabilization of the Laser Tracker has been studied in several experimental tests, and it has been observed that the warm-up time suggested by the manufacturer is not enough. During the first hours of the measurement process two types of thermal errors significantly affect the measurements, causing inaccuracies of between 20 and 80 μm, depending on the equipment used and the positions of the measured points. These thermal errors are systematic and repeatable; therefore they can be estimated and compensated for each measurement system. Because environmental conditions on a shop floor cannot be controlled, once the Laser Tracker is stabilized, the effects of ambient air in measurements have also been studied, focusing on the effect of turbulent flows on the beam path. It has been observed that this turbulence may cause radial distance drifts on the order of micrometers, deflection of the beam trajectory and signal loss. © 2016 Published by Elsevier Ltd. Journal of Manufacturing Systems xxx (2016) xxx-xxx Contents lists available at ScienceDirect Journal of Manufacturing Systems journal homepage: www.elsevier.com Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements Pablo Pérez Muñoz ⁎, JoséAntonio Albajez García, Jorge Santolaria Mazo Department of Design and Manufacturing Engineering, University of Zaragoza, Calle Mar í a de Luna 3, 50018 Zaragoza, Spain 1. Introduction Laser Tracker systems, as portable coordinate measuring machines, have become a notably important metrological tool for industries that must manage large scale components, such as aerospace, wind power, automotive or even machine tool manufacture [1–6]. One common feature of all these sectors is the need to make precise measurements on the shop floor. This is due to the difficulty of moving the large parts involved to an environmentally controlled metrological laboratory. A tracking interferometer system basically consists of a laser interferometer with two rotary axes that provide the tracking capability, and a target retroreflector (corner-cube or cat’s eye type). As a result, this system measures the position of the target on a spherical coordinate system ( ρ ,Ɵ, φ ). Therefore, the measurement uncertainty of these systems is dependent on the laser system itself, which measures the distance to target ( ρ ), and the rotary encoders, which provide the angular position (Ɵ,φ). The angular encoders are widely recognized as one of the main sources of uncertainty [7,8] and the reason of the development of the multilateration techniques [6]. Nevertheless, in the continual effort to attain higher accuracies, some improvements related to distance-measuring uncertainty are still possible. In a factory workshop without a temperature-controlled environment, the temperature can significantly fluctuate along the day. In [2] is reported an example of an aircraft assembly facility with temperature variations of 8 °over four hours and vertical gradients of 2.2°. ⁎⁎ Corresponding author. Email address: [email protected] (P.P. Muñoz) During the aircraft assembly process, if the beginning and the ending temperatures of the measurement survey vary by more than 2.2°, then the survey is considered void and has to be repeated. Environmental conditions have a variety of effects on the measuring process of large components. On one hand, the measured part may present significant dimensional distortions due to temperature variations. These are very difficult to compensate for because they are usually non-uniformly distributed. On the other hand, in non-controlled environments, fluctuations in atmospheric conditions affect the wavelength of the laser beam and, therefore, the stability of the measurement. These effects can be a dominant performance-limiting factor [9]. And lastly, temperature also affects the components of the laser tracker itself: mechanical parts, lenses, the laser source, etc. There are three major manufacturers of Laser Trackers: Leica Geosystems, Faro Inc. and API. Each manufacturer has its own design and system characteristics [6]. The University of Zaragoza has an API Tracker3 LTS-3000 and a Leica Geosystems LTD600. These two Laser Trackers were used in this work and designated as LT1 and LT2, respectively. The initial thermal stabilization of both Laser Trackers was studied. In order to analyse the effect of air turbulences on the Laser Tracker measurements, only LT1 was used because it allows access to tracking system data. Fig. 1 shows both Laser Trackers and their coordinate systems. These two Laser Trackers were chosen for their different structural configurations. LT1 has the laser source inside the head, while LT2 has the laser source fixed inside the housing, with a gimballed mirror to direct the laser beam to the retroreflector. The technical specifications of both Laser Trackers are described in Table 1. There are three main standards concerning performance evaluation of Laser Trackers. ASME B.89.4.19-2005 [10] is the American http://dx.doi.org/10.1016/j.jmsy.2016.10.002 0278-6125/© 2016 Published by Elsevier Ltd. UNCORRECTED PROOF 2 Journal of Manufacturing Systems xxx (2016) xxx-xxx Fig. 1. Laser Trackers used on the experiments: (a) LT1 with his Cartesian and Spherical coordinate system; (b) LT2 with his Cartesian and Spherical coordinate system. Table 1 Laser Trackers specifications. LT1 LT2 Distance Range 0–30 m 0–40 m Horizontal ±320° ±235° Vertical 77°/−60° ±45° Distance Resolution (IFM) 1 μm 1.26 μm Distance Accuracy ±5μm/m 10 μm±0.5 μm/m Angular Resolution 0.07 arc sec 0.14 arc sec National Standard, focused specifically on the use of Laser Trackers as industrial measurement tools. It prescribes methods for the performance evaluation of laser-based spherical coordinate measurement systems and provides a basis for performance comparisons among such systems. These methods and procedures require that the Laser Tracker specifications be accompanied by environmental conditions, including minimum temperature, maximum temperature, and temperature gradients (spatial gradients in °C/m and temporal gradients in °C/h.). However, the standards methods and procedures do not account for other types of temperature-related errors such as those that might arise from the thermal deformation of the Laser Tracker or the warm-up time. The other two standards, VDI/VDE 2617-10 [11] and ISO 10360-10 [12], do not even consider the effect of air temperature on the laser beam (it is assumed the refractive index is properly estimated). Nevertheless, the concern of manufacturers with these thermal issues can be seen in different patents [13–15] trying to address these problems. Laser tracker users are also aware of this situation. It can be observed in [16,17] some examples analysing the warm-up time of two different Laser Trackers from an experimental point of view that clearly show the need for longer stabilization times. However, in [16] and [17] no explanation is given for this phenomenon. Therefore, the first part of work here presented focuses on the warm-up behaviour of two different Laser Trackers. It has been evaluated how the increase of temperature when the laser is switched on affects the measurements in two ways: with an ‘internal error’and with an ‘external error’. The internal error is a consequence of the increase of the temperature inside the Laser Tracker and behaves similarly to the ‘death path error’on linear laser interferometers [18], whereas the external error is a consequence of the thermal deformation of the Laser Tracker. If these two errors are not taken into account, the user of the Laser Tracker may be assuming an important error on his measurements. On the second part of this paper another relevant thermal effect has been addressed. The influence of air and its turbulences on measurements has been studied in order to expose the importance of avoiding heat sources and air flows on the shop floor while the measurement process is being performed. The environmental conditions influence a laser beam through changes in the refraction index. This results not only in a different value for the measured distance but also in a bending of the ray (a procedure to calculate these effects can be seen in [10]). Currently, the dependency of the refractive index on light wavelength and air temperature, pressure and humidity (and even CO2content) is well known thanks to such models as the ones proposed by various authors: Edlén [19], Birch and Downs [20,21], or Ciddor [22] which provide theoretical uncertainties of approximately 10−7 or 10−8. Therefore, laser systems usually have one weather station to provide the necessary values for these environmental variables (the accuracy of each attribute measured by the weather station can be observed on Table 2). Thus, these equations are especially useful under controlled laboratory conditions where environmental parameters may be considered uniform along the beam path and a single measuring point is representative of the whole environment. Nevertheless, Laser Trackers are used in large workshops where spatial and temporal gradients, which a single weather station cannot detect, are likely to be found. This problem may be compounded by the presence of air turbulence. Moreover, the signal quality can be greatly impaired in the presence of turbulent air [1]. Thus, from the point of view of Laser Trackers users, the only available solutions consist on the use of fans or air homogenizers (laminar flow) and data averaging over time [1,23]. Nevertheless, this problem has been analysed more deeply in other research areas, such as in geodesy and long range interferometry, where the laser beams have to travel distances in the range of kilometres [24]. Thus, there are several studies about fluctuations in laser beams due to thermal turbulences that are going to be described. “Turbulence can be described as the random mixing of air particles in the atmosphere due to either rapid or small-scale spatial and temporal temperature-related refractive index fluctuations”[25]. These refractive index fluctuations cause random phase perturbations of the laser beam that can lead to beam distortion. In addition, laser propagation through turbulent media can result in scintillation, beam wander and beam spreading [26]. In 2008, Wang et al. [27] found that in the case of a strong atmospheric turbulence within the light fields, the laser beam loses his Gaussian-shaped distribution as optical vortices are formed. And even after propagation over a sufficiently long-distance for the beam to recover its Gaussian-shaped distribution, the output beam’s light is totally incoherent due to the turbulent atmosphere. In 2013 Ji et al. [28] studied changes in the centroid position of laser beams propagating through a turbulent atmosphere and noticed that a cross wind causes a decentred field phase distortion. In 2014 SC. Ndlovu et al. [29] found changes in the spatial Table 2 Weather Station accuracy. LT1 LT2 Temperature ±0.3 °C±0.3 °C Air pressure ±0.1 KPa ±0.1 KPa Relative humidity ±5%HR ±2.5%HR UNCORRECTED PROOF Journal of Manufacturing Systems xxx (2016) xxx-xxx 3 intensity profile of a propagating laser beam due to thermal turbulence, even over small distances. Nevertheless, Laser trackers have the very distinctive capability of tracking the laser beam compared to conventional laser systems. This means that turbulence effects can also be felt not only in the distance range information but also in the angular data provided by the encoders. Their tracking capability is possible due to a 2D Position Sensitive Detector (PSD) that provides feedback signal to the rotatory motors control [30]. This keeps the returned signal of the retro-reflector in the centre of the PSD and, as a consequence, the light beam also ends centred into the retro-reflector. Optical Position Sensitive Detectors are a kind of photodiodes (1D and 2D) which outputs are proportional to the centroid position of a light spot projected on their surface. The position information is calculated from the relative magnitudes of a few photocurrent signals provided by the PSD. PSDs are widely used in commercial and industrial applications where low-cost or high-speed position sensing is needed [31]. The information provided by the Laser Tracker PSD in our experiments has been used to see how turbulences cause a deflection of the laser beam. Turbulence causes the beam to blur. The more divergent the beam becomes, the less light falls onto the reflector, and less light is therefore returned to the Position Sensitive Detector, which in the end can no longer detect the reflector signal [30]. This loss of the beam signal may occur because the beam’s fluctuations are faster than the Laser Tracker system can follow. The magnitude of these fluctuations can be seen in the magnitude of the root-mean-square error (σ) of the measurements. Since both sources of error (thermal stabilization and air turbulences) have a non-negligible effect on measurements and their uncertainty, especially when the measurement process is conducted on a shop floor where environmental conditions are not controlled, both of these sources of error should be calculated and taken into account. 2. Warm-up behaviour 2.1. Internal error Laser Trackers become ready to work after a warm-up process of approximately 15 to 20 min which allows the laser source to stabilize. Nevertheless, we found that users should wait for at least two hours or even longer to obtain stable measuring data [16,17]. In order to determine the real warm-up time of each instrument, the retrore flector was placed on the birdbath nest (a location where the interferometer distance is set based on previous calibration) as shown in Fig. 2(a). Then, output data were sampled just after the equipment was prepared with the settings in Table 3 and a frequency of one point each 30 s. For 270 min data were collected in a metrological laboratory with a temperature of 20 °C±1°C. The Variation in radial distance is represented in Fig. 2(b). After several tests, it was observed that this phenomenon is repeatable and a model is proposed for each system. The radial distance can be fitted according to this general equation: Where is the initial value of ρ in the birdbath nest (defined by the manufacturer) in millimeters, is the maximum value of the radial distance variation in millimeters, is the instant of time of the measurement in minutes and is the time constant in minutes. For LT1 the equation modeled is For LT2 the equation is modeled as Fig. 2(b) shows mean variation on time of the ρ coordinate (radial distance) of the birdbath nest for both Laser Trackers and the variation of warm-up curves calculated with Eqs. (2) and (3). In exponential equations such as this, 95% of the final value is reached at time 3 . This finding means that to obtain stable results, it is necessary to wait 90 min for LT1 and 270 min for LT2. Otherwise, data collected at different times will have some variation. This variation in the radial coordinate should not be neglected, as it has values of 0.019 millimeters for LT1 and 0.075 millimeters for LT2 when stabilized. This measurement error in the radial component, which occurs during the first hours after the laser source of the Laser Tracker is turned on, is due to the temperature increase inside the housing, which causes expansion of the distance between some components, such the lenses or beam splitter. However, this ‘internal error’, caused by a thermal drift of the components inside the housing, is an Fig. 2. Warm-up test: (a) Configuration of the warm-up test; (b) Variation on time of radial distance from the origin of coordinates to the birdbath nest of each Laser Tracker. (1) (2) (3) UNCORRECTED PROOF 4 Journal of Manufacturing Systems xxx (2016) xxx-xxx Table 3 Experiment settings. Retroreflector SMR 1.5” Samples per point 200 samples/point Sampling frequency 100 Hz offset that increases along time and only can be corrected by rebooting the Laser Tracker using the function ‘Measure Home Position’when the system has thermally stabilized. 2.2. External error The internal heat does not only affect the components inside the tracker, but it also affects the external housing, increasing its temperature and causing thermal dilatation in its structure and, depending on the configuration of the Laser Tracker, a displacement of its origin of coordinates. This effect could be considered as an ‘external error’ that depends on the position of the heat source. In the case of LT1 the heat source is inside the Laser Tracker head and coincides with the theoretical intersection of the axes of rotation. Therefore the gimbal point remains steady. Nevertheless, since LT2 has its heat source in the housing (under the Laser Tracker head) when the Laser Tracker is switched on, the temperature inside the housing increases and thermal expansion causes the structure to displaces the head and the gimballed mirror up, altering the origin of coordinates of the Laser Tracker. Temperature sensors were placed on the housing at the positions denoted as m0, m1, m2 and m3 (see Fig. 3(a)). During the thermal study of the housing, the air temperature was 21,7 °C, but as seen in Fig. 3(b), during the warm-up period defined by the manufacturer, the housing temperature rises 0.4°, while during the calculated warm-up period the housing temperature rises, depending on the point, between 3 and 4°. To verify that this temperature increase causes expansion of the structure of LT2, an experiment was performed with the settings from Table 3 and the configuration shown in Fig. 4(a). The first step is to turn on LT1 and wait almost 90 min until the unit is fully warmed up in order to minimize errors. In one set of experiments the retroreflector was fixed on the birdbath nest of LT2 and in other set of experiments the retroreflector was fixed above the LT2 head as shown in Fig. 4(a). LT2 is turned on and during its warm up period LT1 measures the variation of these fixed once per minute. The variation of the height of those points (the Z coordinate in the Cartesian system of coordinates) is represented in Fig. 4(b). It can be observed that during the first 120 min of the test the birdbath nest and the upper surface of the LT2 rose 60 μm. After another 150 min both points were stabilized, with the birdbath nest Fig. 3. LT2 thermal study during warm-up tests: (a) Position of the temperature sensors; (b) Temperature of the housing. Fig. 4. Structure dilatation test: (a) Configuration of the test; (b) Variation on time of the Z-coordinate in a Cartesian coordinate system of two points of LT2 measured with LT1. UNCORRECTED PROOF Journal of Manufacturing Systems xxx (2016) xxx-xxx 5 reaching a displacement of 70 μm and the upper surface of LT2 a displacement of 77 μm. The 7 μm of difference between the upper surface and the birdbath nest is due to the direction of the thermal expansion on the + Z-axis. This displacement difference would generate an error in the internal offset that the Laser Tracker is unable to detect or correct. It should be noticed in Fig. 4(b), that after the warm-up period (20 min after the laser source is turned on), at the moment when the measurement process can be started, both points on LT2 have risen only 5 μm due to thermal expansion (which represents around 7% of the final value) and from then until the moment when LT2 stabilizes, the upper surface of LT2 rises 72 μm. 2.3. Correction model of the warm-up behaviour 2.3.1. Configuration and data collection In order to observe how the errors caused by internal heat affects the Laser Tracker’s measurements, a series of fixed bases were placed at different distances and heights over a Coordinate Measuring Machine in a metrological laboratory. The four points measured are designated A, B, C and D, they are on the same straight line to minimize angular errors, but at different radial distances and at different heights. These different distances (A, B and C) and heights (D) are shown in Fig. 5(a). The initial radial distances ( ρ coordinate) of each point are 1352.205 mm for point A, 1799.363 mm for point B, 2402.274 mm for point C and 1658.840 mm for point D. It should be noted that bases A, B and C have a negative Z-coordinate on the Cartesian coordinate system because they are below the origin of LT2′s coordinate system, whereas the base D has a positive Z-value because it is above the LT2′s origin. The test was performed with the settings shown in Table 3 in a metrological laboratory with an air temperature of 20 °C±1°C. One radial distance value for each point A, B, C and D are collected every 5 min. After the first 95 min the retroreflector is brought to the nest and the function ‘Measure Home Position’was launched to reset the interferometer, and, after that, distance values for A, B, C and D were taken for another 15 min, resetting the interferometer again before collecting new data. That means that during the first 95 min the Laser Tracker is producing both types of errors discussed above, internal errors and external errors. However, during the last 15 min of the experiment, since the Laser Tracker is reset before each measurement, the internal error is corrected and the Laser Tracker is producing only external errors. The data show that all measurements increase over time, and at the moment when the Laser Tracker is reset, bringing the retroreflector to its birdbath nest (at minute 95 of this experiment), all of the measurements decrease by 45 μm, which is exactly the value of the internal error of the ρ coordinate in the birdbath nest in minute 95 as seen in Fig. 2(b) for LT2. However, even after applying the correction for internal error, there is a gap of 35 μm between A and D. That is because the Laser Tracker is suffering thermal dilatation and external error is not being corrected. 2.3.2. Correction model In this experiment, correction is applied in two steps. First, the external error induced by the thermal expansion of the LT2 housing will be removed for all the data. The second step consists of a correction of the internal error for the first 95 min of the experiment because in the last 15 min this error is corrected when the interferometer is been rebooted. The ρ coordinate can be calculated as Where x0, y0and z0are the Cartesian coordinates of the origin. The Laser Tracker assumes that the origin is fixed and each coordinate is equal to zero. Thus for the Laser Tracker the radial distance is always However, with the experiment presented in section 2.2 (Fig. 4), it can be observed that the Laser Tracker expands on the + Z-axis and thus the origin of coordinates shifts upwards. The first step in the correction involves calculating the new value of the ρ coordinate for each point ( ρ’ ) at each moment considering that the origin of coordinates of LT2 is moving on the + Z-axis due to dilatation of the housing, as seen in Fig. 4(b). To correct for this thermal expansion we fix the origin of coordinates subtracting the value of the height of expansion. Next, the new value of the ρ coordinate is calculated as Fig. 5. Measuring process test: (a) Configuration of the test; (b) Variation on time of radial distance ( ρ coordinate) of the points A, B, C and D along time. (4) (5) P. Pérez et al. / Procedia Manufacturing 13 (2017) 434–441 435 Available online at www.sciencedirect.com ScienceDirect Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Manufacturing Engineering Society International Conference 2017, MESIC 2017, 28-30 June 2017, Vigo (Pontevedra), Spain Analysis of the measurement capacity of a machine tool P. Péreza,*, S. Aguadob, J.A. Albajeza,J. Velazqueza, J. Santolariaa, J.J. Aguilara aDepartment of Design and Manufacturing Engineering, University of Zaragoza, C/ María de Luna s/n, Zaragoza 50018, Spain bCentro Universitario de la Defensa. University of Zaragoza, Academia General Militar, Ctra. De Huesca s/n, Zaragoza 50090, Spain Abstract Industrial sectors that demand manufacturing of high quality components within specified tolerances are looking for cost reductions without affecting the quality of the product. The verification of workpieces is normally carried out in post-process with coordinate measuring machines which increase the manufacturing cycle time. Machine tools can carry out contact measuring operations with a probe, and since there is a growing need to inspect the workpieces in process, using the machine tool itself for verification while the workpiece remains clamped to the machine can lead to an improvement in manufacturing times, reduction of costs and energy saving. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Keywords: Machine Tool; Measurement; Uncertainty. Nomenclature MT Machine Tool CMM Coordinate Measuring Machine LT Laser Tracker * Corresponding author. Tel.: +34 365141396 E-mail address: [email protected] Available online at www.sciencedirect.com ScienceDirect Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Manufacturing Engineering Society International Conference 2017, MESIC 2017, 28-30 June 2017, Vigo (Pontevedra), Spain Analysis of the measurement capacity of a machine tool P. Péreza,*, S. Aguadob, J.A. Albajeza,J. Velazqueza, J. Santolariaa, J.J. Aguilara aDepartment of Design and Manufacturing Engineering, University of Zaragoza, C/ María de Luna s/n, Zaragoza 50018, Spain bCentro Universitario de la Defensa. University of Zaragoza, Academia General Militar, Ctra. De Huesca s/n, Zaragoza 50090, Spain Abstract Industrial sectors that demand manufacturing of high quality components within specified tolerances are looking for cost reductions without affecting the quality of the product. The verification of workpieces is normally carried out in post-process with coordinate measuring machines which increase the manufacturing cycle time. Machine tools can carry out contact measuring operations with a probe, and since there is a growing need to inspect the workpieces in process, using the machine tool itself for verification while the workpiece remains clamped to the machine can lead to an improvement in manufacturing times, reduction of costs and energy saving. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Keywords: Machine Tool; Measurement; Uncertainty. Nomenclature MT Machine Tool CMM Coordinate Measuring Machine LT Laser Tracker * Corresponding author. Tel.: +34 365141396 E-mail address: [email protected] 2 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 1. Introduction Industrial sectors such as aeronautics, automotive, renewable energies, etc. demand the manufacturing of largescale components with high precision. The transportation of these large-scale components to an environmentally controlled metrological laboratory is a difficult task which leads to increases in manufacturing times and increase in costs, and sometimes the workpieces are too large to fit in a Coordinate Measuring Machine (CMM). They have to be measured in process. To reach this goal, traceable dimensional metrology techniques must be incorporated in the Machine Tool (MT) in order for the resultant manufacturing program to produce the desired output within the specified tolerance [1]. The integration of the workpiece verification process into the MT can reduce the manufacturing time as there is no need of transportation of the workpiece to a measurement laboratory. While the workpiece remains clamped in the MT, the same coordinate system used during the manufacturing process can be used in measurements and in reworks, reducing manufacturing times, machining waste materials and therefore reducing costs without losing product quality. MT errors are the difference between the actual tool path and the desired path. Errors can be classified in two categories namely quasi-static errors and dynamic errors. Quasi-static errors are those between the tool and the workpiece that are slowly varying with time and related to the structure of the machine tool itself. These sources include the geometric errors, kinematic errors, thermally induced errors, etc., on the other hand, dynamic errors are caused by sources such as spindle error motion, vibrations, controller errors, etc. [2]. For 3-axis MT, there are 21 components of geometric and kinematic errors: each axis has 6 errors of movement, 3 of rotation and 3 of translation over each axis, and between each pair of axis there is a squareness error. Geometrical errors can be measured individually with direct measurement techniques or all together with indirect measurements. UNE-ISO 230-1:2014 [3] is an international standard that specifies methods for testing the accuracy of machine tools with direct measurements, operating either under no-load or under quasi-static conditions. Indirect measurement produces a global correction of the MT workspace based on multi-axis movement and its kinematic model [4]. In previous work [4,5] measuring a mesh of points of the MT workspace allowed obtaining the approximation functions of each geometric error. A volumetric verification based on non-linear optimization was applied improving MT accuracy. With the approximation functions of the geometric errors obtained in [5] we have developed a program to simulate 1000 volumetric verifications with slight variations on the input parameters in order to estimate the MT uncertainty. We determinate an uncertainty area (U) for each point of the MT workspace, for the case of a threedimensional measurement, the obtained shape is the ellipsoid with axes ux(P), uy(P), uz(P). The ellipsoid represents the volume in which it is more likely to find the true value of the measured point. Once the uncertainty for each point of the workspace is determined, it is necessary to check that measurements carried out with the MT are within the uncertainties calculated to assure the traceability on measurements. With this purpose, an object must be measured with the MT and check if the measures are within the uncertainty area. 2. Experimental procedure The methods and procedures are going to be applied over a 3-axis MT with a XYFZ configuration but can be extrapolated to other cases; in our particular case the MT it is an ANAYAK VH-1800 with computer numerical control (CNC) Fagor 8025. The MT has integrated in its software a matrix of error compensation that can compensate position errors and therefore improve the accuracy. However, during the test presented in this paper, the matrix of compensation is disabled in order to identify the MT geometrical errors. The object that is going to be measured is a calibrated hole plate of outside dimensions 460 x 460 mm made from aluminum. The nominal distance between rings centers is 50 mm. It is important to know the coefficient of thermal expansion of the plate corpus (αplate=24·10-6 K-1) and monitor the plate temperature during the measurements in order to compensate for possible errors due to thermal expansion. Twenty eight holes of the calibrated plate are measured. For each hole, four points are measured to determine the best fitted circle center. Each point is measured at the same time by the MT, using a probe, and with a Laser Tracker (LT), with the retro-reflector magnetically attached to the probe. When the probe makes contact with the hole plate on the point that is going to be measured, the MT pauses so the LT can also measure the same point. 436 P. Pérez et al. / Procedia Manufacturing 13 (2017) 434–441 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 3 The plate hole was clamped into the MT the day before the measurement to be thermally stabilized. The MT is switched on around one hour before starting measurements for an appropriate warm-up. The origin of the MT coordinate system is set on the center of the first hole, as shown in Fig. 1. The measurement strategy consist on first locate the probe near to the center of the hole to be measured. Next, the probe descends to a determined Z-coordinate (all the holes will be measured at the same Z-coordinate) the order of measurement of the four points are: first the point situated on +X with respect to the center, after that the point situated on –X, after that the probes goes to the center and measures the point on located +Y and finally the point on –Y. Once the four points of the hole have been measured the probe goes to the center of the hole, ascends on the coordinate +Z and goes to the next hole to be measured. The holes are measured in spiral as shown in Fig. 1 because it has been simulated that with this strategy the effect of clearance is reduced. Fig. 1. Measurement configuration and strategy. With the four points taken the best fitted circle center is calculated in each hole. As the plate was misaligned, it is necessary to rotate the coordinates by multiplying with a rotation matrix:            100 0cossin 0sincos   R (1) Where θ is the angle of misalignment between the measured coordinates and to the nominal coordinates. However, as the measurements are going to be compared with its nominal coordinates, which are referred to the plate at 20 ºC, it is necessary to correct the thermal expansion of the plate. For simplicity, a linear behavior is considered starting from the clamping point (which is located on the position X=200 mm, Y=250 mm), according to the equation:   201 0  plateplatef Tdd  (2) Where d0 and df are the distance between the clamping point and the center of the hole before and after applying the correction, αplate is the coefficient of thermal expansion of the plate and Tplate is the temperature of the plate in ºC at the moment when the hole is being measured. 4 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 Fig. 2. Correction of the thermal expansion. As the misalignment and the thermal expansion of the hole plate are not MT errors, they have been compensated to make possible the comparison between the nominal coordinates and the coordinates measured with the MT. Since the measurements are being carried out under quasi-static conditions, the error obtained will be due to the MT geometrical, kinematic and thermally induced errors. 3. Results The methods and procedures are going to be applied over a 3-axis MT with a XYFZ configuration but can be extrapolated to other cases; 3.1. Simulations With the nominal coordinates of the calibrated plate, the measured coordinates obtained measuring 28 holes and the equations of the kinematic model; it is possible to develop a volumetric verification to determine the approximation function of error of the system MT+LT. To obtain the uncertainty, we use the Monte Carlo Method performing simulations varying slightly the input parameters such as LT noise [6]. One thousand test are generated, therefore we have 28000 simulated hole centres with which we have calculated the new 1000 approximation functions of errors. Fig. 3 shows the error distribution of these 28000 points. Fig. 3. Error distribution of the simulated mesh before the optimization. With the 1000 approximation functions of error simulated, non-linear optimization can be used to reduce the MT errors. After applying the optimization compensating the errors, the result obtained are the best fitted coordinates of the holes centres. The error distribution of the 28000 points after optimization can be seen in Fig. 4. P. Pérez et al. / Procedia Manufacturing 13 (2017) 434–441 437 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 3 The plate hole was clamped into the MT the day before the measurement to be thermally stabilized. The MT is switched on around one hour before starting measurements for an appropriate warm-up. The origin of the MT coordinate system is set on the center of the first hole, as shown in Fig. 1. The measurement strategy consist on first locate the probe near to the center of the hole to be measured. Next, the probe descends to a determined Z-coordinate (all the holes will be measured at the same Z-coordinate) the order of measurement of the four points are: first the point situated on +X with respect to the center, after that the point situated on –X, after that the probes goes to the center and measures the point on located +Y and finally the point on –Y. Once the four points of the hole have been measured the probe goes to the center of the hole, ascends on the coordinate +Z and goes to the next hole to be measured. The holes are measured in spiral as shown in Fig. 1 because it has been simulated that with this strategy the effect of clearance is reduced. Fig. 1. Measurement configuration and strategy. With the four points taken the best fitted circle center is calculated in each hole. As the plate was misaligned, it is necessary to rotate the coordinates by multiplying with a rotation matrix:            100 0cossin 0sincos   R (1) Where θ is the angle of misalignment between the measured coordinates and to the nominal coordinates. However, as the measurements are going to be compared with its nominal coordinates, which are referred to the plate at 20 ºC, it is necessary to correct the thermal expansion of the plate. For simplicity, a linear behavior is considered starting from the clamping point (which is located on the position X=200 mm, Y=250 mm), according to the equation:   201 0  plateplatef Tdd  (2) Where d0 and df are the distance between the clamping point and the center of the hole before and after applying the correction, αplate is the coefficient of thermal expansion of the plate and Tplate is the temperature of the plate in ºC at the moment when the hole is being measured. 4 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 Fig. 2. Correction of the thermal expansion. As the misalignment and the thermal expansion of the hole plate are not MT errors, they have been compensated to make possible the comparison between the nominal coordinates and the coordinates measured with the MT. Since the measurements are being carried out under quasi-static conditions, the error obtained will be due to the MT geometrical, kinematic and thermally induced errors. 3. Results The methods and procedures are going to be applied over a 3-axis MT with a XYFZ configuration but can be extrapolated to other cases; 3.1. Simulations With the nominal coordinates of the calibrated plate, the measured coordinates obtained measuring 28 holes and the equations of the kinematic model; it is possible to develop a volumetric verification to determine the approximation function of error of the system MT+LT. To obtain the uncertainty, we use the Monte Carlo Method performing simulations varying slightly the input parameters such as LT noise [6]. One thousand test are generated, therefore we have 28000 simulated hole centres with which we have calculated the new 1000 approximation functions of errors. Fig. 3 shows the error distribution of these 28000 points. Fig. 3. Error distribution of the simulated mesh before the optimization. With the 1000 approximation functions of error simulated, non-linear optimization can be used to reduce the MT errors. After applying the optimization compensating the errors, the result obtained are the best fitted coordinates of the holes centres. The error distribution of the 28000 points after optimization can be seen in Fig. 4. 438 P. Pérez et al. / Procedia Manufacturing 13 (2017) 434–441 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 5 Fig. 4. Error distribution of the simulated mesh after the optimization. The average error of each hole is considered as a systematic error (b) in that position of the MT workspace, while the standard deviation is part of the uncertainty of the measurement procedure (up). 3.2. Uncertainty evaluation The standard ISO/TS 15530-3:2011 [7] provides an experimental technique for the uncertainty evaluation of CMM measurements. The discussed standard describes the uncertainty evaluation procedure for both parts: experiment and calculation. The expanded uncertainty is calculated as: 2222 wbpcal uuuukU  (3) Where ucal is the standard uncertainty associated with the uncertainty of the calibration, up is the standard uncertainty resulting from the measurement procedure, ub is the standard uncertainty associated with the systematic error b, uw is the standard uncertainty resulting from material and manufacturing variations (due to variations of the expansion coefficient, form errors, roughness and elasticity) and k=2 for a coverage probability of 95%. If the manufacturer do not provide the value of ucal, it can be estimated with the maximum permissible error of the CMM used in the calibration of the plate. In our case, the manufacturer provided that value: 500 45.1 L u cal  (4) The value of up can be calculated in each workspace position as two times the standard deviation of the simulated mesh of points at that position. In our case this value takes values between 4.2 and 15.95 µm, depending on the workspace position. According to the Guide to the expression of Uncertainty in Measurement (GUM) [8], ub is calculated as a type A uncertainty, therefore: n u b   (5) Where σ is the standard deviation of the systematic error b and n is the number of simulated values. The value of ub is negligible as σ has a small value and n=1000. The value of uw has to be estimated. As we have compensated the thermal expansion, our uncertainty due to thermal expansion is related with the possible error that the sensor is committing measuring the temperature of the plate. Assuming that the accuracy of our sensor is ± 0.2 ºC with a rectangular distribution, uw can be calculated as: 6 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 mmm LTL u w   11.100111.0 3 4002.01024 33 6         (6) If the MT user is not measuring the temperature of the workpiece and compensating the effect of thermal expansion, this lack of information should be added in this term of uncertainty. The result of a measurement should be expressed as: UbyY  (7) Where Y is the expression of the measurement, y is measured value, b is the systematic error and U is the expanded uncertainty. Fig. 5 shows the systematic error (b) in every position of the MT workspace. Fig. 5. Systematic error of the workspace. This systematic error is a residual error that remains on the MT after applying the compensation according to the kinematic model and the approximation functions. Therefore, should be added on the expression of a measurement result with the expanded uncertainty, which value in the workspace is shown in Fig. 6. P. Pérez et al. / Procedia Manufacturing 13 (2017) 434–441 439 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 5 Fig. 4. Error distribution of the simulated mesh after the optimization. The average error of each hole is considered as a systematic error (b) in that position of the MT workspace, while the standard deviation is part of the uncertainty of the measurement procedure (up). 3.2. Uncertainty evaluation The standard ISO/TS 15530-3:2011 [7] provides an experimental technique for the uncertainty evaluation of CMM measurements. The discussed standard describes the uncertainty evaluation procedure for both parts: experiment and calculation. The expanded uncertainty is calculated as: 2222 wbpcal uuuukU  (3) Where ucal is the standard uncertainty associated with the uncertainty of the calibration, up is the standard uncertainty resulting from the measurement procedure, ub is the standard uncertainty associated with the systematic error b, uw is the standard uncertainty resulting from material and manufacturing variations (due to variations of the expansion coefficient, form errors, roughness and elasticity) and k=2 for a coverage probability of 95%. If the manufacturer do not provide the value of ucal, it can be estimated with the maximum permissible error of the CMM used in the calibration of the plate. In our case, the manufacturer provided that value: 500 45.1 L u cal  (4) The value of up can be calculated in each workspace position as two times the standard deviation of the simulated mesh of points at that position. In our case this value takes values between 4.2 and 15.95 µm, depending on the workspace position. According to the Guide to the expression of Uncertainty in Measurement (GUM) [8], ub is calculated as a type A uncertainty, therefore: n u b   (5) Where σ is the standard deviation of the systematic error b and n is the number of simulated values. The value of ub is negligible as σ has a small value and n=1000. The value of uw has to be estimated. As we have compensated the thermal expansion, our uncertainty due to thermal expansion is related with the possible error that the sensor is committing measuring the temperature of the plate. Assuming that the accuracy of our sensor is ± 0.2 ºC with a rectangular distribution, uw can be calculated as: 6 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 mmm LTL u w   11.100111.0 3 4002.01024 33 6         (6) If the MT user is not measuring the temperature of the workpiece and compensating the effect of thermal expansion, this lack of information should be added in this term of uncertainty. The result of a measurement should be expressed as: UbyY  (7) Where Y is the expression of the measurement, y is measured value, b is the systematic error and U is the expanded uncertainty. Fig. 5 shows the systematic error (b) in every position of the MT workspace. Fig. 5. Systematic error of the workspace. This systematic error is a residual error that remains on the MT after applying the compensation according to the kinematic model and the approximation functions. Therefore, should be added on the expression of a measurement result with the expanded uncertainty, which value in the workspace is shown in Fig. 6. 440 P. Pérez et al. / Procedia Manufacturing 13 (2017) 434–441 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 7 Fig. 6. Expanded uncertainty of the workspace. In our simulations, the component up is the one that has the most influence (around 76.2% of the total uncertainty comes from this component while uw represents around the 6.5% and ucal the 17.3%). But that is because the temperature was monitored and therefore the uncertainty is reduced, if the user do no monitor the temperature and do not correct the effect of thermal expansion, assuming that temperature of the plate is inside the interval of 20 ± 2 ºC, the uw component would be greater and would even have the same importance as the up component of uncertainty. Fig. 7. Percentage of influence of each component of the expanded uncertainty. The standard uncertainty resulting from material and manufacturing variations can also be reduced by using a calibrated workpiece made of materials with lower coefficient of thermal expansion such as Invar or Zerodur. 0 10 20 30 40 50 60 70 80 90 100 Monitoring Temperature without Monitoring Temperature %ofinfluence ucal uw up 8 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 4. Conclusions The capability of a MT as a measuring machine depends on a proper estimation of a measurement uncertainty. With the standard ISO/TS 15530-3:2011 [7] and the indications of the “Guide to the expression of uncertainty in measurements” (GUM) [8], we have estimated the uncertainty area in a delimited area of the MT workspace. We have determinate a strategy of measurement and we have measured a calibrated object. After performing a volumetric verification the approximation functions of the geometric errors can be calculated and using Monte Carlo Method we have estimated the uncertainty. Applying non-linear optimization the coordinates measured can be corrected; to assure the traceability of the MT the corrected coordinates must be inside the uncertainty volume estimated. In this paper the different components that affect uncertainty have been discussed and calculated for a real milling machine with XFYZ configuration. Acknowledgements This work was supported by the Spain Government with the project: DPI2013-46979-C2-1-P: METRAP, and the Funds of the scholarship BES-2014-070480. References [1] A.P. Longstaff, S. Fletcher, S. Parkinson, A. Myers, Int. J. Metrol. Qual. Eng. 4 (2013) 177-184. [2] R. Ramesh, M.A. Mannan, A.N. Poo, Int. J. Mach. Tools Manuf. 40 (2000) 1235-1256. [3] UNE-ISO 230-1:2014, Geometric accuracy of machines operating under no-load or quasi-static conditions. [4] S. Aguado, J. Santolaria, J. Aguilar, D. Samper, J. Velazquez, Proceedings of 6th Manufacturing Society International Conference (MESIC 2015), Barcelona, 2015. [5] S. Aguado , J. Santolaria, D. Samper, J. Aguilar, J. Velazquez, J. Manuf. Systems. 40 (2016) 26-36. [6] S. Aguado, P. Perez, J.A. Albajez, J. Velazquez, J. Santolaria. Proceedings of 7th Manufacturing Society International Conference (MESIC 2017). Vigo, 2017. [7] ISO/TR 15530-3:2011 Coordinate measuring machines (CMM): Technique for determining the uncertainty of measurement – Part 3: Use of calibrated workpieces or measurement standards. [8] Evaluation of measurement data: Guide to the expression of Uncertainty in Measurement (GUM). JCGM 100:2008. P. Pérez et al. / Procedia Manufacturing 13 (2017) 434–441 441 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 7 Fig. 6. Expanded uncertainty of the workspace. In our simulations, the component up is the one that has the most influence (around 76.2% of the total uncertainty comes from this component while uw represents around the 6.5% and ucal the 17.3%). But that is because the temperature was monitored and therefore the uncertainty is reduced, if the user do no monitor the temperature and do not correct the effect of thermal expansion, assuming that temperature of the plate is inside the interval of 20 ± 2 ºC, the uw component would be greater and would even have the same importance as the up component of uncertainty. Fig. 7. Percentage of influence of each component of the expanded uncertainty. The standard uncertainty resulting from material and manufacturing variations can also be reduced by using a calibrated workpiece made of materials with lower coefficient of thermal expansion such as Invar or Zerodur. 0 10 20 30 40 50 60 70 80 90 100 Monitoring Temperature without Monitoring Temperature %ofinfluence ucal uw up 8 P.Pérez et al. / Procedia Manufacturing 00 (2017) 000–000 4. Conclusions The capability of a MT as a measuring machine depends on a proper estimation of a measurement uncertainty. With the standard ISO/TS 15530-3:2011 [7] and the indications of the “Guide to the expression of uncertainty in measurements” (GUM) [8], we have estimated the uncertainty area in a delimited area of the MT workspace. We have determinate a strategy of measurement and we have measured a calibrated object. After performing a volumetric verification the approximation functions of the geometric errors can be calculated and using Monte Carlo Method we have estimated the uncertainty. Applying non-linear optimization the coordinates measured can be corrected; to assure the traceability of the MT the corrected coordinates must be inside the uncertainty volume estimated. In this paper the different components that affect uncertainty have been discussed and calculated for a real milling machine with XFYZ configuration. Acknowledgements This work was supported by the Spain Government with the project: DPI2013-46979-C2-1-P: METRAP, and the Funds of the scholarship BES-2014-070480. References [1] A.P. Longstaff, S. Fletcher, S. Parkinson, A. Myers, Int. J. Metrol. Qual. Eng. 4 (2013) 177-184. [2] R. Ramesh, M.A. Mannan, A.N. Poo, Int. J. Mach. Tools Manuf. 40 (2000) 1235-1256. [3] UNE-ISO 230-1:2014, Geometric accuracy of machines operating under no-load or quasi-static conditions. [4] S. Aguado, J. Santolaria, J. Aguilar, D. Samper, J. Velazquez, Proceedings of 6th Manufacturing Society International Conference (MESIC 2015), Barcelona, 2015. [5] S. Aguado , J. Santolaria, D. Samper, J. Aguilar, J. Velazquez, J. Manuf. Systems. 40 (2016) 26-36. [6] S. Aguado, P. Perez, J.A. Albajez, J. Velazquez, J. Santolaria. Proceedings of 7th Manufacturing Society International Conference (MESIC 2017). Vigo, 2017. [7] ISO/TR 15530-3:2011 Coordinate measuring machines (CMM): Technique for determining the uncertainty of measurement – Part 3: Use of calibrated workpieces or measurement standards. [8] Evaluation of measurement data: Guide to the expression of Uncertainty in Measurement (GUM). JCGM 100:2008. ScienceDirect Available online at www.sciencedirect.com Available online at www.sciencedirect.com ScienceDirect Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia * Paulo Afonso. Tel.: +351 253 510 761; fax: +351 253 604 741 E-mail address: [email protected] 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Manufacturing Engineering Society International Conference 2017, MESIC 2017, 28-30 June 2017, Vigo (Pontevedra), Spain Costing models for capacity optimization in Industry 4.0: Trade-off between used capacity and operational efficiency A. Santanaa, P. Afonsoa,*, A. Zaninb, R. Wernkeb a University of Minho, 4800-058 Guimarães, Portugal bUnochapecó, 89809-000 Chapecó, SC, Brazil Abstract Under the concept of "Industry 4.0", production processes will be pushed to be increasingly interconnected, information based on a real time basis and, necessarily, much more efficient. In this context, capacity optimization goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability and value. Indeed, lean management and continuous improvement approaches suggest capacity optimization instead of maximization. The study of capacity optimization and costing models is an important research topic that deserves contributions from both the practical and theoretical perspectives. This paper presents and discusses a mathematical model for capacity management based on different costing models (ABC and TDABC). A generic model has been developed and it was used to analyze idle capacity and to design strategies towards the maximization of organization’s value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity optimization might hide operational inefficiency. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Keywords: Cost Models; ABC; TDABC; Capacity Management; Idle Capacity; Operational Efficiency 1. Introduction The cost of idle capacity is a fundamental information for companies and their management of extreme importance in modern production systems. In general, it is defined as unused capacity or production potential and can be measured in several ways: tons of production, available hours of manufacturing, etc. The management of the idle capacity Procedia Manufacturing 13 (2017) 585–592 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 10.1016/j.promfg.2017.09.105 10.1016/j.promfg.2017.09.105 2351-9789 Available online at www.sciencedirect.com ScienceDirect Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Manufacturing Engineering Society International Conference 2017, MESIC 2017, 28-30 June 2017, Vigo (Pontevedra), Spain Monte Carlo method to machine tool uncertainty evaluation S. Aguadoa, P. Pérezb, J.A. Albajezb, J. Velázquezb, J. Santolariab aCentro Universitario de la Defensa. Universidad de Zaragoza. Academia General Militar. Ctra. Huesca s/n, Zaragoza 50090, Spain bDepartment of Design and Manufacturing Engineering. Universidad de Zaragoza. Marìa de Luna 3, Zaragoza 50018, Spain Abstract Currently machine tools are not only a way to make different parts based on material removal processes. These ones can be used as a measurement system too. In this way, overall inspection time is reduced and equipment productivity is increased. Nevertheless, the use of machine tool probes as measurement tool in manufacturing parts required previous works. Firstly, the machine tool accuracy should be improved, in order to reduce the influence of its geometric errors. This way, volumetric verification based on laser tracker measurement has increased strongly in the last few years, especially in long range machine tools. Secondly, calibration uncertainty should be calculated to provide measurement uncertainty. This way, the paper presents a new tool able to analyze the effect of different influence verification parameters in calibration uncertainty based on Monte Carlo method. Using real tests carried out on a milling machine and its geometric errors, the influence or laser tracker measurement noise in calibration uncertainty is studied using Monte Carlo method. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Keywords: Volumetric verification; laser tracker; machine tool; uncertainty; Monte Carlo. 1. Introduction Machine tool calibration (MT) is defined as the process from which the influence MT geometric errors is obtained. This way, the MT accuracy is increased reducing the influence of these systematic behavior through software compensation. * S. Aguado. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: saguad[email protected] Available online at www.sciencedirect.com ScienceDirect Procedia Manufacturing 00 (2017) 000–000 www.elsevier.com/locate/procedia 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Manufacturing Engineering Society International Conference 2017, MESIC 2017, 28-30 June 2017, Vigo (Pontevedra), Spain Monte Carlo method to machine tool uncertainty evaluation S. Aguadoa, P. Pérezb, J.A. Albajezb, J. Velázquezb, J. Santolariab aCentro Universitario de la Defensa. Universidad de Zaragoza. Academia General Militar. Ctra. Huesca s/n, Zaragoza 50090, Spain bDepartment of Design and Manufacturing Engineering. Universidad de Zaragoza. Marìa de Luna 3, Zaragoza 50018, Spain Abstract Currently machine tools are not only a way to make different parts based on material removal processes. These ones can be used as a measurement system too. In this way, overall inspection time is reduced and equipment productivity is increased. Nevertheless, the use of machine tool probes as measurement tool in manufacturing parts required previous works. Firstly, the machine tool accuracy should be improved, in order to reduce the influence of its geometric errors. This way, volumetric verification based on laser tracker measurement has increased strongly in the last few years, especially in long range machine tools. Secondly, calibration uncertainty should be calculated to provide measurement uncertainty. This way, the paper presents a new tool able to analyze the effect of different influence verification parameters in calibration uncertainty based on Monte Carlo method. Using real tests carried out on a milling machine and its geometric errors, the influence or laser tracker measurement noise in calibration uncertainty is studied using Monte Carlo method. © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. Keywords: Volumetric verification; laser tracker; machine tool; uncertainty; Monte Carlo. 1. Introduction Machine tool calibration (MT) is defined as the process from which the influence MT geometric errors is obtained. This way, the MT accuracy is increased reducing the influence of these systematic behavior through software compensation. * S. Aguado. Tel.: +0-000-000-0000 ; fax: +0-000-000-0000 . E-mail address: [email protected] © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 586 S. Aguado et al. / Procedia Manufacturing 13 (2017) 585–592 2 S. Aguado / Procedia Manufacturing 00 (2017) 000–000 Currently there are two different ways to obtain MT geometric errors. First one, determines the influence of each error from each axis in a particular position of the workspace of the MT [1]. Second one, indirect measurement method, obtain the joint influence of MT geometric errors based on multi-axis movement and MT kinematic model [2]. Meanwhile direct measurement provides the real physical behaviour of each error, indirect one provides a join optimum values. However, the relationship between geometric errors obtained using direct measurement is not studied and approximation functions obtained are directly extrapolated to all MT workspace. Similarly, each error needs an own assembly measurement procedure and data treatment; increasing substantially verification time. These are the principal reasons why volumetric verification (VV) based on indirect measurement errors using laser tracer, laser tracker or ball bar as measurement systems, are daily more popular than geometric verification, based on indirect measurement using laser interferometer, levels, etc. Calibration process result is associated with calibration uncertainty value. It characterizes results dispersion in relation with geometric errors obtained and sources of errors that affect it. This one is considered especially relevant in different manufacturing and quality assurance processes. It is required when the MT is used as measurement system; providing metrological characteristic required to obtain a traceable measurement system. The International Organization for Standardization (ISO) has developed and published different guidelines for the representation of measurement uncertainty (GUM), such as the UNE-ISO / TR 230-9 [3] standard for measurement uncertainty estimation for machine tool test, or ISO / TS 14253-2 [4], widely accepted. It combines the estimation of the different sources of error and their associated typical uncertainties, to determine the typical uncertainty associated with the overall process. This way, accuracy and metrological characteristic of a MT as measurement system are related to measurement system used, machine tool and calibration conditions. The GUM provides the basic framework for evaluating uncertainty in measurement, but it does not work properly in non-lineal process such as MT calibration based on VV. As errors that affect to VV have a random and probabilistic behavior, Monte Carlo method is recommended to obtain its uncertainty. This paper presents a new simulation software developed to study how different factors with influence in volumetric verification affect to calibration uncertainty. The software allows the use of different probabilistic error functions (PDFs) to characterize the behaviour of each error source. Within different sources of uncertainty, this paper is focused on the study of laser tracker measurement noise influence. So, using a real milling machine with XFYZ configuration, a LT Leica LT 600 and a probe as measurement system and our own developed software, real tests have been carried out. 2. Comparison of the GUM and Monte Carlo Method to determine the uncertainty of a machine tool volumetric verification process 2.1. Volumetric verification and influence factors Volumetric verification is based on an intensive process of parameters identification through the kinematic model of the MT. Minimizing the difference between theoretical and real pair of points, through the MT kinematic model, the joint influence of MT geometric errors are obtained. Their behavior are modeled minimizing the mean square volumetric error of the machine (Ev) using non-lineal optimization techniques [2]. As shows Fig. 1, principals’ uncertainty sources with influence on machine tool verification are divided in three groups: machine tool, measurement and verification, and measurement system uncertainties. S. Aguado / Procedia Manufacturing 00 (2017) 000–000 3 Fig 1. Volumetric verification scheme taking into consideration uncertainty sources. 2.2. Main differences between GUM and Monte Carlo The Monte Carlo Method (MCM) is a tool that uses the computational capacity of currents computers to simulate a high account of pseudo random numbers. This way, it allows to simulate complex system from a probabilistic point of view [5]. Meanwhile the GUM is focused on evaluate Type A, Type B and combined uncertainties, the MCM uses a large number of samples, with different probabilistic functions, to obtain the final uncertainty distribution (Fig. 2). Fig 2. Propagation of uncertainties based on GUM – left – propagation of distribution based on MCM– right. However, the estimation of uncertainties using GUM is based on assumptions that are not always fulfilled. These adequacy limitations of the GUM are given by: The non-linearity of the mathematical model that describes the process. When the model presents strong elements of non-linearity, the approximation made by the GUM approach may not be enough to correctly estimate the uncertainty output. S. Aguado et al. / Procedia Manufacturing 13 (2017) 585–592 587 2 S. Aguado / Procedia Manufacturing 00 (2017) 000–000 Currently there are two different ways to obtain MT geometric errors. First one, determines the influence of each error from each axis in a particular position of the workspace of the MT [1]. Second one, indirect measurement method, obtain the joint influence of MT geometric errors based on multi-axis movement and MT kinematic model [2]. Meanwhile direct measurement provides the real physical behaviour of each error, indirect one provides a join optimum values. However, the relationship between geometric errors obtained using direct measurement is not studied and approximation functions obtained are directly extrapolated to all MT workspace. Similarly, each error needs an own assembly measurement procedure and data treatment; increasing substantially verification time. These are the principal reasons why volumetric verification (VV) based on indirect measurement errors using laser tracer, laser tracker or ball bar as measurement systems, are daily more popular than geometric verification, based on indirect measurement using laser interferometer, levels, etc. Calibration process result is associated with calibration uncertainty value. It characterizes results dispersion in relation with geometric errors obtained and sources of errors that affect it. This one is considered especially relevant in different manufacturing and quality assurance processes. It is required when the MT is used as measurement system; providing metrological characteristic required to obtain a traceable measurement system. The International Organization for Standardization (ISO) has developed and published different guidelines for the representation of measurement uncertainty (GUM), such as the UNE-ISO / TR 230-9 [3] standard for measurement uncertainty estimation for machine tool test, or ISO / TS 14253-2 [4], widely accepted. It combines the estimation of the different sources of error and their associated typical uncertainties, to determine the typical uncertainty associated with the overall process. This way, accuracy and metrological characteristic of a MT as measurement system are related to measurement system used, machine tool and calibration conditions. The GUM provides the basic framework for evaluating uncertainty in measurement, but it does not work properly in non-lineal process such as MT calibration based on VV. As errors that affect to VV have a random and probabilistic behavior, Monte Carlo method is recommended to obtain its uncertainty. This paper presents a new simulation software developed to study how different factors with influence in volumetric verification affect to calibration uncertainty. The software allows the use of different probabilistic error functions (PDFs) to characterize the behaviour of each error source. Within different sources of uncertainty, this paper is focused on the study of laser tracker measurement noise influence. So, using a real milling machine with XFYZ configuration, a LT Leica LT 600 and a probe as measurement system and our own developed software, real tests have been carried out. 2. Comparison of the GUM and Monte Carlo Method to determine the uncertainty of a machine tool volumetric verification process 2.1. Volumetric verification and influence factors Volumetric verification is based on an intensive process of parameters identification through the kinematic model of the MT. Minimizing the difference between theoretical and real pair of points, through the MT kinematic model, the joint influence of MT geometric errors are obtained. Their behavior are modeled minimizing the mean square volumetric error of the machine (Ev) using non-lineal optimization techniques [2]. As shows Fig. 1, principals’ uncertainty sources with influence on machine tool verification are divided in three groups: machine tool, measurement and verification, and measurement system uncertainties. S. Aguado / Procedia Manufacturing 00 (2017) 000–000 3 Fig 1. Volumetric verification scheme taking into consideration uncertainty sources. 2.2. Main differences between GUM and Monte Carlo The Monte Carlo Method (MCM) is a tool that uses the computational capacity of currents computers to simulate a high account of pseudo random numbers. This way, it allows to simulate complex system from a probabilistic point of view [5]. Meanwhile the GUM is focused on evaluate Type A, Type B and combined uncertainties, the MCM uses a large number of samples, with different probabilistic functions, to obtain the final uncertainty distribution (Fig. 2). Fig 2. Propagation of uncertainties based on GUM – left – propagation of distribution based on MCM– right. However, the estimation of uncertainties using GUM is based on assumptions that are not always fulfilled. These adequacy limitations of the GUM are given by: The non-linearity of the mathematical model that describes the process. When the model presents strong elements of non-linearity, the approximation made by the GUM approach may not be enough to correctly estimate the uncertainty output.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              sensors Article Study on Machine Tool Positioning Uncertainty Due to Volumetric Verification Sergio Aguado 1,* , Pablo Pérez 2, JoséAntonio Albajez 2, Jorge Santolaria 2and Jesús Velazquez 2 1Centro Universitario de la Defensa, Academia General Militar, Ctra. Huesca S/N, 50090 Zaragoza, Spain 2Design and Manufacturing Engineering Department, University of Zaragoza, 50018 Zaragoza, Spain *Correspondence: [email protected] Received: 15 April 2019; Accepted: 21 June 2019; Published: 26 June 2019   Abstract: Volumetric verification is based on the machine tool (MT) kinematic model, along with its geometric errors. Although users often ignore the uncertainty of verification, the use of the MT as a traceable measurement system in the manufacturing process has increased the need for professionals to be aware of it. This paper presents an improvement in the MT kinematic model, introducing in it the influence of verification uncertainty sources. These sources have been classified into four groups: the MT, the measurement system itself, the measurement strategy, and the optimization strategy. As the developed model exhibits non-linear behavior, the Monte Carlo method was used to determine the influence of the measurement system on verification uncertainty using synthetic tests. In this manner, an improved estimation of the MT uncertainty can be obtained. Therefore, if the MT is used as a traceable measurement system, its accuracy should not be higher than the laser tracker (LT) verification influence. It hence shows the importance of LT influence. Keywords: Monte Carlo method; machine tool uncertainty; calibration; measurement in process; traceability 1. Introduction The accuracy of the machined parts and the feedback information of the manufacturing process are two of the most critical considerations for any manufacturer. These considerations are the result of a number of influences and sources of error, such as the cutting conditions, the part type, and the machine tool (MT) characteristics. These errors can be classified as either random or systematic errors, including quasi-static and dynamic ones. Quasi-static errors are divided into geometric, kinematic, and thermal errors. Geometric errors are the result of structural elements and they reduce the positioning accuracy of the MT. The direction of motion generated by joints, couplings, gears, and stiffness errors cause deformations and generate kinematic errors. Thermal errors are the result of temperature gradients in the structure of a machine or part, which generate dimensional changes that affect accuracy. Dynamic errors are caused by sources such as the vibration of the MT structure, the spindle movement errors, or the software. Therefore, the accuracy of the MT is provided by random and systematic errors, which are not compensated after verification processes. Uncertainty should be calculated to evaluate whether additional manufacturing tolerances can be achieved. When an MT is used as a measurement system with a probe to determine part dimensions, the MT measurement uncertainty determines the conformance zone. Figure 1, whose upper bar shows specifications and lower bar the influence of uncertainty, shows that false acceptance could occur if the uncertainty interval is outside the tolerance zone, but the measurement remains within the tolerance limits. However, false rejection could occur when the measured value is within the tolerance zone, but Sensors 2019,19, 2847; doi:10.3390/s19132847 www.mdpi.com/journal/sensors Sensors 2019,19, 2847 8 of 17 Sensors 2019, 19, x FOR PEER REVIEW 7 of 16 errors sources as environmental errors, data captures, and simplifications. If all these errors sources are analyzed together, two principal groups can be done. The first one is related to systematic errors, such as environmental conditions. Pressure, temperature, and humidity produce a variation of the refraction index of the air, which provides an error in the laser wavelength estimation, which affects the measurement distance. However, these errors present a systematic behavior that can be analytically compensated due to a meteorological LT station and control. Other systematic errors that cannot be compensated by LT control, such as warmup time, can be characterized and its influence reduced [18]. The second one consists of random error sources, such as uncertainty from radical and angular encoders. The influence of these errors sources is defined as measurement noise. In this way, manufacturers provide specifications for their LTs, and their accuracy is combined based on the ISO DIS 10360-10 and ASME B 89.4.19-2005 standards [22,23]. It is the joint effect of the uncertainties of the LT components and depends on the employed encoders and sensors. As shown in the equation of movement of the MT (Equation (2)), the influence of the relation between the LT and the MT on the verification (Figure 4) is derived from a rotation matrix 𝑅(𝑙𝑡) defined by the Euler angles, α, β, and δ, and a translational vector, 𝑇     . This vector and matrix are calculated as shown in Equations (21) and (22), respectively: 𝑇     =󰇟𝐷,𝐿,𝐻󰇠, (21) 𝑅       =󰇭 cos (𝛽)cos (𝛿) −cos(𝛽)sin (𝛿) sin (𝛽) cos(𝛼)sin(𝛿)+sin(𝛼)sin(𝛽)cos (𝛿) cos(𝛼)cos(𝛿)−sin(𝛼)sin(𝛽)sin (𝛿) −sin(𝛼)cos (𝛽) sin(𝛼)sin(𝛿)−cos(𝛼)sin(𝛽)cos (𝛿) sin(𝛼)cos(𝛿)+cos(𝛼)sin(𝛽)sin (𝛿) cos(𝛼)cos (𝛽)󰇮. (22) Figure 4. LT measurement uncertainty and location parameters. Influence of systematic errors can be reduced, but measured points are affected by LT measurement uncertainties originating from angular encoders and the radial distance, as per Equations (23), random errors (Figure 4). This equation links the data from the encoders and the radial distance with their uncertainty; the uncertainty is provided by means of a point measured in Cartesian coordinates: 󰇯𝑢 𝑢 𝑢󰇰=󰇯sin𝜃∙cos𝜑𝑟 ∙cos𝜃∙cos𝜑𝑟 ∙sin𝜃∙sin𝜑 sin𝜃∙sin𝜑𝑟 ∙cos𝜃∙sin𝜑𝑟 ∙sin𝜃∙cos𝜑 cos𝜃𝑟 ∙sin𝜃0 󰇰∙󰇯𝑢 𝑢 𝑢 󰇰. (23) In this case, r represents the radial measured distance, 𝑢 the radial uncertainty, θ the azimuth angle, 𝑢 the azimuth angle uncertainty, φ the polar angle, and 𝑢 the polar angle uncertainty. The uncertainty of a measured point is presented in Equation (24). Figure 4. LT measurement uncertainty and location parameters. Influence of systematic errors can be reduced, but measured points are affected by LT measurement uncertainties originating from angular encoders and the radial distance, as per Equations (23), random errors (Figure 4). This equation links the data from the encoders and the radial distance with their uncertainty; the uncertainty is provided by means of a point measured in Cartesian coordinates:           u2 x u2 y u2 z           =          sin2θ·cos2ϕr2·cos2θ·cos2ϕr2·sin2θ·sin2ϕ sin2θ·sin2ϕr2·cos2θ·sin2ϕr2·sin2θ·cos2ϕ cos2θr2·sin2θ0          ·          u2 r u2 θ u2 ϕ           . (23) In this case, rrepresents the radial measured distance, ur the radial uncertainty, θ the azimuth angle, uθ the azimuth angle uncertainty, ϕ the polar angle, and uϕ the polar angle uncertainty. The uncertainty of a measured point is presented in Equation (24). u2 p=u2 x+u2 y+u2 z. (24) Moreover, a retro-reflector as an element of a measurement system is a source of uncertainty [ 19 ]. The LT laser beam should strike at the center of a reflector. If not, the influence of the retro-reflector uncertainty (uSMR) must be modelled as a normal distribution and included in the equations. The employed work procedure is an additional uncertainty source; however, it cannot be modelled. Moreover, metrology software allows one to define the measurements that modify the frequency and samples per point; the software provides an uncertainty for each measured point, i.e., uMS,x,uMS,y,uMS,z. 2.2.3. Uncertainty Related to the Measurement Strategy As has been described in the previous section, the LT should be located to improve data accuracy. However, if actual verification is conducted, the LT position is obtained using a least-squares adjustment between the MT nominal coordinates and the actual points measured using LT. As the LT-measured points are affected by different sources of error (such as the measured noise), the rotation matrix R(lt) and the translational vector TLT are affected as well. Moreover, the least-squares adjustment is a mathematical tool that minimizes the difference between points, producing its own uncertainty. Thus, random elements are added to the R(lt)and TLT components, following a PDF (Equations (25)–(26)). R(lt)=R(lt)nominal +URlt (25) Sensors 2019,19, 2847 9 of 17 TLT =TLT,nominal +UTlt (26) The other source of error related to the measurement strategy is the use of multilateration to improve data accuracy. Studies have shown different results depending on the LT and the employed measurement technique, the measurement mode (IFM—Interferometer or ADM—Absolute Distance Measurement), and the spatial angle between the LT locations. Multilateration prevents angular measurement noise and increases the radial one [ 24 ]; however, its uncertainty has not yet been modelled as a PDF related to spatial angles between LTs beams. Due to MT characteristic, tests presented in Section 3were not carried out based on the multilateration technique, they were done on a singular LT location. 2.2.4. Uncertainty Related to the Optimization Strategy The optimization strategy, convergence criteria, and phase optimization (Figure 5) cannot be modelled as previous uncertainty sources; nonetheless, they affect the MT accuracy depending on its configuration and verification design [ 24 ]. Therefore, if these are changed, the MT capacity will be modified. Sensors 2019, 19, x FOR PEER REVIEW 8 of 16 𝑢=𝑢+𝑢+𝑢. (24) Moreover, a retro-reflector as an element of a measurement system is a source of uncertainty [19]. The LT laser beam should strike at the center of a reflector. If not, the influence of the retro-reflector uncertainty (𝑢) must be modelled as a normal distribution and included in the equations. The employed work procedure is an additional uncertainty source; however, it cannot be modelled. Moreover, metrology software allows one to define the measurements that modify the frequency and samples per point; the software provides an uncertainty for each measured point, i.e., 𝑢, ,𝑢, ,𝑢, . 2.2.3. Uncertainty Related to the Measurement Strategy As has been described in the previous section, the LT should be located to improve data accuracy. However, if actual verification is conducted, the LT position is obtained using a leastsquares adjustment between the MT nominal coordinates and the actual points measured using LT. As the LT-measured points are affected by different sources of error (such as the measured noise), the rotation matrix 𝑅(𝑙𝑡) and the translational vector 𝑇     are affected as well. Moreover, the leastsquares adjustment is a mathematical tool that minimizes the difference between points, producing its own uncertainty. Thus, random elements are added to the 𝑅(𝑙𝑡) and 𝑇     components, following a PDF (Equations (25)–(26)). 𝑅(𝑙𝑡)=𝑅 (𝑙𝑡)+𝑈      (25) 𝑇     =𝑇,               +𝑈      (26) The other source of error related to the measurement strategy is the use of multilateration to improve data accuracy. Studies have shown different results depending on the LT and the employed measurement technique, the measurement mode (IFM—Interferometer or ADM—Absolute Distance Measurement), and the spatial angle between the LT locations. Multilateration prevents angular measurement noise and increases the radial one [24]; however, its uncertainty has not yet been modelled as a PDF related to spatial angles between LTs beams. Due to MT characteristic, tests presented in Section 3 were not carried out based on the multilateration technique, they were done on a singular LT location. 2.2.4. Uncertainty Related to the Optimization Strategy The optimization strategy, convergence criteria, and phase optimization (Figure 5) cannot be modelled as previous uncertainty sources; nonetheless, they affect the MT accuracy depending on its configuration and verification design [24]. Therefore, if these are changed, the MT capacity will be modified. Figure 5. Characterization Procedure Scheme. Approximation functions that characterize MT geometric errors may be obtained, thus minimizing the difference between the real and the nominal points. The difference becomes minimized using an iterative process of parameter identification through the MT equation of movements. Therefore, the objective function is the difference between pairs of points (Equation (27)). Figure 5. Characterization Procedure Scheme. Approximation functions that characterize MT geometric errors may be obtained, thus minimizing the difference between the real and the nominal points. The difference becomes minimized using an iterative process of parameter identification through the MT equation of movements. Therefore, the objective function is the difference between pairs of points (Equation (27)). ve,LT =Pn i=1q(xi,n−xi,r)2+(yi,n−yi,r)2+(zi,n−zi,r)2 n(27) The residual error after verification will depend on the verification design, i.e., the identification method, the approximation functions used, the optimization strategy, the defined initial parameters, and the employed convergence criteria (Figure 6). The adequacy of the optimization strategy in the verification of the MT configuration has been studied in past research works [ 25 ]. Considering this, the best optimization strategy for the XFYZ MT configuration was implemented and was not modified in the verification process of the tests presented in this paper (Section 4) as a prior step for the estimation of the verification uncertainty. Sensors 2019,19, 2847 10 of 17 Sensors 2019, 19, x FOR PEER REVIEW 9 of 16 𝑣,= ∑(𝑥,−𝑥,)+(𝑦,−𝑦,)+(𝑧,−𝑧,)  𝑛 (27) The residual error after verification will depend on the verification design, i.e., the identification method, the approximation functions used, the optimization strategy, the defined initial parameters, and the employed convergence criteria (Figure 6). Figure 6. Optimization strategy scheme. The adequacy of the optimization strategy in the verification of the MT configuration has been studied in past research works [25]. Considering this, the best optimization strategy for the XFYZ MT configuration was implemented and was not modified in the verification process of the tests presented in this paper (Section 4) as a prior step for the estimation of the verification uncertainty. 2.3. Monte Carlo Method for Uncertainty Evaluation Every measurement should be provided with an estimation of its measurement uncertainty. Currently, a standard guide exists for the estimation of the measurement uncertainty, namely the “Guide to the expression of uncertainty in measurement” (GUM) [26]. However, it should not be used for non-linear models, such as the MT studied in this paper. Supplement 1 of GUM recommends the use of the Monte Carlo method (MCM) in such cases. The Monte Carlo method employs a large number of samples generated according to different propagation probabilistic functions (Figure 7) to obtain the final uncertainty distribution through the measurement equation, i.e., Equation (2). Figure 7. Propagation of uncertainties based on GUM (Guide to the expression of uncertainty in measurement) (left); propagation of distributions based on the Monte Carlo method (MCM) (right). To estimate the volumetric verification uncertainty using MCM, the first step is the definition of statistical distributions of measured-input quantities. The principal ones are divided into MT errors, Figure 6. Optimization strategy scheme. 2.3. Monte Carlo Method for Uncertainty Evaluation Every measurement should be provided with an estimation of its measurement uncertainty. Currently, a standard guide exists for the estimation of the measurement uncertainty, namely the “Guide to the expression of uncertainty in measurement” (GUM) [ 26 ]. However, it should not be used for non-linear models, such as the MT studied in this paper. Supplement 1 of GUM recommends the use of the Monte Carlo method (MCM) in such cases. The Monte Carlo method employs a large number of samples generated according to different propagation probabilistic functions (Figure 7) to obtain the final uncertainty distribution through the measurement equation, i.e., Equation (2). Sensors 2019, 19, x FOR PEER REVIEW 9 of 16 𝑣,= ∑(𝑥,−𝑥,)+(𝑦,−𝑦,)+(𝑧,−𝑧,)  𝑛 (27) The residual error after verification will depend on the verification design, i.e., the identification method, the approximation functions used, the optimization strategy, the defined initial parameters, and the employed convergence criteria (Figure 6). Figure 6. Optimization strategy scheme. The adequacy of the optimization strategy in the verification of the MT configuration has been studied in past research works [25]. Considering this, the best optimization strategy for the XFYZ MT configuration was implemented and was not modified in the verification process of the tests presented in this paper (Section 4) as a prior step for the estimation of the verification uncertainty. 2.3. Monte Carlo Method for Uncertainty Evaluation Every measurement should be provided with an estimation of its measurement uncertainty. Currently, a standard guide exists for the estimation of the measurement uncertainty, namely the “Guide to the expression of uncertainty in measurement” (GUM) [26]. However, it should not be used for non-linear models, such as the MT studied in this paper. Supplement 1 of GUM recommends the use of the Monte Carlo method (MCM) in such cases. The Monte Carlo method employs a large number of samples generated according to different propagation probabilistic functions (Figure 7) to obtain the final uncertainty distribution through the measurement equation, i.e., Equation (2). Figure 7. Propagation of uncertainties based on GUM (Guide to the expression of uncertainty in measurement) (left); propagation of distributions based on the Monte Carlo method (MCM) (right). To estimate the volumetric verification uncertainty using MCM, the first step is the definition of statistical distributions of measured-input quantities. The principal ones are divided into MT errors, Figure 7. Propagation of uncertainties based on GUM (Guide to the expression of uncertainty in measurement) (left); propagation of distributions based on the Monte Carlo method (MCM) (right). To estimate the volumetric verification uncertainty using MCM, the first step is the definition of statistical distributions of measured-input quantities. The principal ones are divided into MT errors, measurement strategy errors, measurement system errors, and optimization strategy, as shown in Figure 3. All of them have been presented and analyzed in depth in Section 2.2. Secondly, the mathematical model that describes the process must be defined; in this case, the movement equation provided by the MT kinematic chain affected by the input uncertainties has been presented in Section 2.2. Next, using verification conditions, tests were conducted for the measurement data; moreover, the PDFs for the input quantities were estimated. At this stage, the verification tests were created and the MCM simulation was set up and run. When all tests had been completed, the last step was to summarize and present the results. Several researchers have merely taken the first step towards studying MTs, considering different distributions functions, thermal variations, or measurement noise [ 2 – 4 , 23 ] Figure 8shows the working Sensors 2019,19, 2847 11 of 17 principle of the development of the software used for the estimation of the verification uncertainty of an MT using LT as measurement system through volumetric verification. Sensors 2019, 19, x FOR PEER REVIEW 10 of 16 measurement strategy errors, measurement system errors, and optimization strategy, as shown in Figure 3. All of them have been presented and analyzed in depth in Section 2.2. Secondly, the mathematical model that describes the process must be defined; in this case, the movement equation provided by the MT kinematic chain affected by the input uncertainties has been presented in Section 2.2. Next, using verification conditions, tests were conducted for the measurement data; moreover, the PDFs for the input quantities were estimated. At this stage, the verification tests were created and the MCM simulation was set up and run. When all tests had been completed, the last step was to summarize and present the results. Figure 8. Uncertainty evaluation and confidence intervals of the parameters. Several researchers have merely taken the first step towards studying MTs, considering different distributions functions, thermal variations, or measurement noise [2–4,23] Figure 8 shows the working principle of the development of the software used for the estimation of the verification uncertainty of an MT using LT as measurement system through volumetric verification. 3. Simulations Results To use an MT as a measurement system with traceability, it needs to improve its positioning accuracy. It will depend on the PDF of its final verification volumetric error, which should be obtained to estimate its uncertainty. To study the influence of LT measurement uncertainty in LT positioning and verification uncertainty, MCM was used. The best LT position related to measurement uncertainty was obtained using Equation (24). The values of 𝑢 and 𝑢 were derived from the normal probability distribution, i.e., µ = 20 µrad and σ = 3 µrad, and 𝑢 was defined as 4 µm ± 0.8 µm/m for radial coordinate. Therefore, each point to measure will yield different uncertainty values in each test. To achieve this, simulation tests were conducted in a three-axis MT with an XFYZ configuration. The Monte Carlo method tests were defined as follows: • MT workspace to verify: 800 mm ≤ X ≤ 1200 mm, 100 mm ≤ Y ≤ 500 mm, and 200 mm ≤ Z ≤ 400 mm. • The verification mesh contained 75 verification points with intervals of 100 mm in all axes. • Available space to locate LT: –2000 mm ≤ D ≤ –500 mm, 350 mm ≤ H ≤ 2000 mm, –2000 mm ≤ L ≤ 2000 mm. • LT measurement range characteristic: 0.5 m ≤ r ≤ 15 m, –45° ≤ θ ≤ 45°; –235° ≤ φ ≤ 235°. • LT could not be located inside the MT workspace or body for verification. Figure 8. Uncertainty evaluation and confidence intervals of the parameters. 3. Simulations Results To use an MT as a measurement system with traceability, it needs to improve its positioning accuracy. It will depend on the PDF of its final verification volumetric error, which should be obtained to estimate its uncertainty. To study the influence of LT measurement uncertainty in LT positioning and verification uncertainty, MCM was used. The best LT position related to measurement uncertainty was obtained using Equation (24). The values of uθ and uϕ were derived from the normal probability distribution, i.e., µ =20 µ rad and σ =3 µ rad, and ud was defined as 4 µ m ± 0.8 µ m/m for radial coordinate. Therefore, each point to measure will yield different uncertainty values in each test. To achieve this, simulation tests were conducted in a three-axis MT with an XFYZ configuration. The Monte Carlo method tests were defined as follows: • MT workspace to verify: 800 mm ≤ X ≤ 1200 mm, 100 mm ≤ Y ≤ 500 mm, and 200 mm ≤ Z ≤400 mm. •The verification mesh contained 75 verification points with intervals of 100 mm in all axes. • Available space to locate LT: − 2000 mm ≤ D ≤ − 500 mm, 350 mm ≤ H ≤ 2000 mm, − 2000 mm ≤ L ≤2000 mm. •LT measurement range characteristic: 0.5 m ≤r≤15 m, −45◦≤θ≤45◦;−235◦≤ϕ≤235◦. •LT could not be located inside the MT workspace or body for verification. •The number of Monte Carlo tests was 100.000 in the X-axis and Y-axis directions. The main objective of these tests was not only to obtain the best LT location, but also to be able to associate the LT position with the influence of LT measurement uncertainty in the verification mesh. In this manner, although measurement uncertainty would be smaller than the maximum error introduced in most of the verification points, an approximate value below which the MT measurement/position accuracy is within the uncertainty zone (Figure 1) would be obtained. Figures 9and 10 show the LT location and the maximum error introduced to the verification mesh for each Monte Carlo test where LTs are located along the Y-axis and X-axis of the MT, respectively. Sensors 2019,19, 2847 12 of 17 Each point represents the best location to a definite test, and its color represents the maximum error introduced on it by LT. This value is obtained as the maximum absolute value of Equation (1). Sensors 2019, 19, x FOR PEER REVIEW 11 of 16 • The number of Monte Carlo tests was 100.000 in the X-axis and Y-axis directions. The main objective of these tests was not only to obtain the best LT location, but also to be able to associate the LT position with the influence of LT measurement uncertainty in the verification mesh. In this manner, although measurement uncertainty would be smaller than the maximum error introduced in most of the verification points, an approximate value below which the MT measurement/position accuracy is within the uncertainty zone (Figure 1) would be obtained. Figures 9 and 10 show the LT location and the maximum error introduced to the verification mesh for each Monte Carlo test where LTs are located along the Y-axis and X-axis of the MT, respectively. Each point represents the best location to a definite test, and its color represents the maximum error introduced on it by LT. This value is obtained as the maximum absolute value of Equation (1). If both figures are compared, there is a central zone with Z values lower than 700 mm in Figure 9 and 1100 mm in Figure 10, where there is a higher concentration of lower maximum errors. In the same manner, there are five columns that rise in the Z-axis, with a higher concentration of upper values. In both cases, these columns are located in the same planes (Y and X) as that of the verification points. Figure 9. Best LT position and maximum error obtained on each Monte Carlo test: Y-axis direction. Figure 10. Best LT position and maximum error obtained on each Monte Carlo test: X-axis direction. However, certain differences may be observed between both figures. Lower values in Figure 9 lie between 60 and 80 µm; the lowest values of Figure 10 lie between 40 and 60 µm. This means that the influence of noise has been reduced by approximately 30%. In the same manner, the maximum values in Figure 9 are approximately 30 µm higher than the maximum values in Figure 10. The color maps presented in Figure 9 and Figure 10 provide an efficient manner to observe the spatial areas where LT should be placed. However, to identify which frequency obtained the highest values and which PDFs best characterized the behavior, a histogram of both cases is presented in Figure 11. Figure 9. Best LT position and maximum error obtained on each Monte Carlo test: Y-axis direction. Sensors 2019, 19, x FOR PEER REVIEW 11 of 16 • The number of Monte Carlo tests was 100.000 in the X-axis and Y-axis directions. The main objective of these tests was not only to obtain the best LT location, but also to be able to associate the LT position with the influence of LT measurement uncertainty in the verification mesh. In this manner, although measurement uncertainty would be smaller than the maximum error introduced in most of the verification points, an approximate value below which the MT measurement/position accuracy is within the uncertainty zone (Figure 1) would be obtained. Figures 9 and 10 show the LT location and the maximum error introduced to the verification mesh for each Monte Carlo test where LTs are located along the Y-axis and X-axis of the MT, respectively. Each point represents the best location to a definite test, and its color represents the maximum error introduced on it by LT. This value is obtained as the maximum absolute value of Equation (1). If both figures are compared, there is a central zone with Z values lower than 700 mm in Figure 9 and 1100 mm in Figure 10, where there is a higher concentration of lower maximum errors. In the same manner, there are five columns that rise in the Z-axis, with a higher concentration of upper values. In both cases, these columns are located in the same planes (Y and X) as that of the verification points. Figure 9. Best LT position and maximum error obtained on each Monte Carlo test: Y-axis direction. Figure 10. Best LT position and maximum error obtained on each Monte Carlo test: X-axis direction. However, certain differences may be observed between both figures. Lower values in Figure 9 lie between 60 and 80 µm; the lowest values of Figure 10 lie between 40 and 60 µm. This means that the influence of noise has been reduced by approximately 30%. In the same manner, the maximum values in Figure 9 are approximately 30 µm higher than the maximum values in Figure 10. The color maps presented in Figure 9 and Figure 10 provide an efficient manner to observe the spatial areas where LT should be placed. However, to identify which frequency obtained the highest values and which PDFs best characterized the behavior, a histogram of both cases is presented in Figure 11. Figure 10. Best LT position and maximum error obtained on each Monte Carlo test: X-axis direction. If both figures are compared, there is a central zone with Z values lower than 700 mm in Figure 9 and 1100 mm in Figure 10, where there is a higher concentration of lower maximum errors. In the same manner, there are five columns that rise in the Z-axis, with a higher concentration of upper values. In both cases, these columns are located in the same planes (Y and X) as that of the verification points. However, certain differences may be observed between both figures. Lower values in Figure 9lie between 60 and 80 µ m; the lowest values of Figure 10 lie between 40 and 60 µ m. This means that the influence of noise has been reduced by approximately 30%. In the same manner, the maximum values in Figure 9are approximately 30 µm higher than the maximum values in Figure 10. The color maps presented in Figures 9and 10 provide an efficient manner to observe the spatial areas where LT should be placed. However, to identify which frequency obtained the highest values and which PDFs best characterized the behavior, a histogram of both cases is presented in Figure 11. Sensors 2019, 19, x FOR PEER REVIEW 12 of 16 Figure 11. Maximum error for each best LT position. Figure 11 presents the 𝑢values obtained from Equation (24); hence, no negative values exist, in addition to the LT position and the maximum error introduced for each one. Although measurement uncertainty from radial and angular encoders has been defined as a normal distribution, none of the presented PDFs describes the real distribution of maximum errors properly. To study the influence of LT measurement uncertainty on the verification uncertainty, the verification of the same mesh was simulated 1000 times. The working principle was presented in Section 2, and the parameters required were: • Equation of movement of the MT and its kinematic model (Equation (2)). • Verification mesh of points (in this case, the same used to determine the LT position). • Laser tracker position: d = –563.52 mm, l = 194.37 mm, h = 677.78 mm, α = 0.0339°, β = 0.0343°, and δ = 59.4652°, obtained in LT position from previous tests. • Generation functions that characterize geometrics of the MT were obtained from real verification. • Optimization strategy. To identify the geometric errors of the MT, a one phase optimization procedure is used as is shown in Figure 6. From information of a single LT, the influence of squareness, translation, and rotation error are considered together. Figure 12 shows verification point errors in the X, Y, and Z axes in the LT coordinate system. These errors are the joint influence of its errors and LT noise. Although the probability distribution of LT components is normal, the joint influence of geometrical and measurement error cannot fit a normal distribution. Figure 12. Initial error of MT verification points of 1000 tests. Figure 13 analyses the initial error distribution along the verification mesh (Equation (1)). For each verification point, there are 1000 samples affected by the same geometric error; however, with a different LT noise, there is a cloud of measured points around nominal positions. It can be observed that geometric errors have increased from the reference point, in this case the corner of the cube, following a systematic behavior. So, the main aim of volumetric verification is to compensate for this Figure 11. Maximum error for each best LT position. Sensors 2019,19, 2847 13 of 17 Figure 11 presents the upvalues obtained from Equation (24); hence, no negative values exist, in addition to the LT position and the maximum error introduced for each one. Although measurement uncertainty from radial and angular encoders has been defined as a normal distribution, none of the presented PDFs describes the real distribution of maximum errors properly. To study the influence of LT measurement uncertainty on the verification uncertainty, the verification of the same mesh was simulated 1000 times. The working principle was presented in Section 2, and the parameters required were: •Equation of movement of the MT and its kinematic model (Equation (2)). •Verification mesh of points (in this case, the same used to determine the LT position). • Laser tracker position: d =–563.52 mm, l =194.37 mm, h =677.78 mm, α =0.0339 ◦ , β =0.0343 ◦ , and δ=59.4652◦, obtained in LT position from previous tests. • Generation functions that characterize geometrics of the MT were obtained from real verification. • Optimization strategy. To identify the geometric errors of the MT, a one phase optimization procedure is used as is shown in Figure 6. From information of a single LT, the influence of squareness, translation, and rotation error are considered together. Figure 12 shows verification point errors in the X, Y, and Z axes in the LT coordinate system. These errors are the joint influence of its errors and LT noise. Although the probability distribution of LT components is normal, the joint influence of geometrical and measurement error cannot fit a normal distribution. Sensors 2019, 19, x FOR PEER REVIEW 12 of 16 Figure 11. Maximum error for each best LT position. Figure 11 presents the 𝑢values obtained from Equation (24); hence, no negative values exist, in addition to the LT position and the maximum error introduced for each one. Although measurement uncertainty from radial and angular encoders has been defined as a normal distribution, none of the presented PDFs describes the real distribution of maximum errors properly. To study the influence of LT measurement uncertainty on the verification uncertainty, the verification of the same mesh was simulated 1000 times. The working principle was presented in Section 2, and the parameters required were: • Equation of movement of the MT and its kinematic model (Equation (2)). • Verification mesh of points (in this case, the same used to determine the LT position). • Laser tracker position: d = –563.52 mm, l = 194.37 mm, h = 677.78 mm, α = 0.0339°, β = 0.0343°, and δ = 59.4652°, obtained in LT position from previous tests. • Generation functions that characterize geometrics of the MT were obtained from real verification. • Optimization strategy. To identify the geometric errors of the MT, a one phase optimization procedure is used as is shown in Figure 6. From information of a single LT, the influence of squareness, translation, and rotation error are considered together. Figure 12 shows verification point errors in the X, Y, and Z axes in the LT coordinate system. These errors are the joint influence of its errors and LT noise. Although the probability distribution of LT components is normal, the joint influence of geometrical and measurement error cannot fit a normal distribution. Figure 12. Initial error of MT verification points of 1000 tests. Figure 13 analyses the initial error distribution along the verification mesh (Equation (1)). For each verification point, there are 1000 samples affected by the same geometric error; however, with a different LT noise, there is a cloud of measured points around nominal positions. It can be observed that geometric errors have increased from the reference point, in this case the corner of the cube, following a systematic behavior. So, the main aim of volumetric verification is to compensate for this Figure 12. Initial error of MT verification points of 1000 tests. Figure 13 analyses the initial error distribution along the verification mesh (Equation (1)). For each verification point, there are 1000 samples affected by the same geometric error; however, with a different LT noise, there is a cloud of measured points around nominal positions. It can be observed that geometric errors have increased from the reference point, in this case the corner of the cube, following a systematic behavior. So, the main aim of volumetric verification is to compensate for this error influence on MT accuracy. Moreover, there are no strong variations in the color of each point cloud; hence, the influence of geometric errors is greater than the measurement noise. Sensors 2019,19, 2847 14 of 17 Sensors 2019, 19, x FOR PEER REVIEW 13 of 16 error influence on MT accuracy. Moreover, there are no strong variations in the color of each point cloud; hence, the influence of geometric errors is greater than the measurement noise. Figure 13. Initial color map of verification points of 1000 tests on laser tracker coordinate system. Figure 14 shows the final errors in the verification points of all simulated tests. If we compare it with Figure 12, it can be seen that the final error distribution presents a normal distribution, particularly the final error in the distance and the error in the Y coordinate. Figure 14. Final error of MT verification points of 1000 tests in total. Figure 15 shows the final residual error in the verification points of all simulated tests, when the influence of geometric errors is compensated. A comparison of Figures 13 and 15 shows that the errors have been greatly minimized. Moreover, the color of the cloud of each verification point is not homogeneous. Although their real geometric errors are the same, each test provides a different geometric error approximation function for each error. This is because of the effects of the LT measurement noise on the approximation functions and therefore on the final positioning of MT accuracy. Figure 13. Initial color map of verification points of 1000 tests on laser tracker coordinate system. Figure 14 shows the final errors in the verification points of all simulated tests. If we compare it with Figure 12, it can be seen that the final error distribution presents a normal distribution, particularly the final error in the distance and the error in the Y coordinate. Sensors 2019, 19, x FOR PEER REVIEW 13 of 16 error influence on MT accuracy. Moreover, there are no strong variations in the color of each point cloud; hence, the influence of geometric errors is greater than the measurement noise. Figure 13. Initial color map of verification points of 1000 tests on laser tracker coordinate system. Figure 14 shows the final errors in the verification points of all simulated tests. If we compare it with Figure 12, it can be seen that the final error distribution presents a normal distribution, particularly the final error in the distance and the error in the Y coordinate. Figure 14. Final error of MT verification points of 1000 tests in total. Figure 15 shows the final residual error in the verification points of all simulated tests, when the influence of geometric errors is compensated. A comparison of Figures 13 and 15 shows that the errors have been greatly minimized. Moreover, the color of the cloud of each verification point is not homogeneous. Although their real geometric errors are the same, each test provides a different geometric error approximation function for each error. This is because of the effects of the LT measurement noise on the approximation functions and therefore on the final positioning of MT accuracy. Figure 14. Final error of MT verification points of 1000 tests in total. Figure 15 shows the final residual error in the verification points of all simulated tests, when the influence of geometric errors is compensated. A comparison of Figures 13 and 15 shows that the errors have been greatly minimized. Moreover, the color of the cloud of each verification point is not homogeneous. Although their real geometric errors are the same, each test provides a different geometric error approximation function for each error. This is because of the effects of the LT measurement noise on the approximation functions and therefore on the final positioning of MT accuracy. The maximum final error after verification is approximately 50 µ m, as shown in Figure 15. If compared with the error introduced by the LT measurement noise at its location (Figure 10), the error has a similar value. Hence, the verification uncertainty of the MT is 50 µ m, although the final mean volumetric error of the verification tests is a normal distribution with X13.77 µm and σ=5.02 µm. Sensors 2019,19, 2847 15 of 17 Sensors 2019, 19, x FOR PEER REVIEW 14 of 16 Figure 15. Final color map of verification points of 1000 tests on laser tracker coordinate system. The maximum final error after verification is approximately 50 µm, as shown in Figure 15. If compared with the error introduced by the LT measurement noise at its location (Figure 10), the error has a similar value. Hence, the verification uncertainty of the MT is 50 µm, although the final mean volumetric error of the verification tests is a normal distribution with 𝑋 13.77 µm and σ = 5.02 µm. 4. Conclusions To use an MT as a measurement system in a process with traceability requires the improvement of its positioning accuracy and the determination of its measurement uncertainty. The measurement uncertainty of the MT will be equal or higher than the verification uncertainty. Main uncertainty sources related to MT verification can be divided into four groups: the MT, measurement system, measurement strategy, and optimization strategy. Uncertainty from the MT and the measurement systems affected nominal and actual measured points. The remaining uncertainty sources did not cause physical consequences; however, they affected the resolution of mathematical errors. We have developed a new kinematic model where the influences of these sources of uncertainty were considered via the modelling of their behavior. As the developed kinematic model had a non-linear behavior, GUM could not be implemented. Instead, the Monte Carlo method was used to estimate verification uncertainty. The measurement noise from LT was modelled as a normal probability distribution; however, simulated tests demonstrated that its influence on the verification points may not be properly described by any of the presented PDFs used to model the real distribution of maximum errors. A suitable LT location along the MT workspace would provide an average maximum error reduction of approximately 30%. With the LT at this suitable area, tests showed that the initial-error behavior cannot be modelled as a normal distribution, neither in distance nor in any of the directions of the axes of the machine. Nevertheless, there is a normal cloud of points around nominal positions, i.e., the noise measurements. For each test, different geometric approximation functions were employed with different residual errors. These had a normal behavior, which represented the uncertainty verification; it could be modelled as a normal distribution with 𝑋 = 13.77 µm and σ = 5.02 µm. This means that with a confidence level of 95%, the uncertainty value would be lower than 23.81 µm. In this manner, falsely accepted and false-rejection zones would be identified. However, in a few tests, certain verification points presented a maximum position error of approximately 50 µm, which was similar to the maximum error introduced in the LT position tests. Although the error was smaller than 50 µm in most tests, accuracy greater than this value cannot be ensured. Author Contributions: All authors contributed to the development of theoretical model, S.A. and J.S conceived and designed the experiments; S.A performed the experiments; S.A, P.P. J.A.A. analyzed the data; J.V contributed reagents/materials/analysis tools; S.A and P.P wrote the paper. All authors contributed to the editing of the manuscript. Figure 15. Final color map of verification points of 1000 tests on laser tracker coordinate system. 4. Conclusions To use an MT as a measurement system in a process with traceability requires the improvement of its positioning accuracy and the determination of its measurement uncertainty. The measurement uncertainty of the MT will be equal or higher than the verification uncertainty. Main uncertainty sources related to MT verification can be divided into four groups: the MT, measurement system, measurement strategy, and optimization strategy. Uncertainty from the MT and the measurement systems affected nominal and actual measured points. The remaining uncertainty sources did not cause physical consequences; however, they affected the resolution of mathematical errors. We have developed a new kinematic model where the influences of these sources of uncertainty were considered via the modelling of their behavior. As the developed kinematic model had a non-linear behavior, GUM could not be implemented. Instead, the Monte Carlo method was used to estimate verification uncertainty. The measurement noise from LT was modelled as a normal probability distribution; however, simulated tests demonstrated that its influence on the verification points may not be properly described by any of the presented PDFs used to model the real distribution of maximum errors. A suitable LT location along the MT workspace would provide an average maximum error reduction of approximately 30%. With the LT at this suitable area, tests showed that the initial-error behavior cannot be modelled as a normal distribution, neither in distance nor in any of the directions of the axes of the machine. Nevertheless, there is a normal cloud of points around nominal positions, i.e., the noise measurements. For each test, different geometric approximation functions were employed with different residual errors. These had a normal behavior, which represented the uncertainty verification; it could be modelled as a normal distribution with X =13.77 µ m and σ =5.02 µ m. This means that with a confidence level of 95%, the uncertainty value would be lower than 23.81 µ m. In this manner, falsely accepted and false-rejection zones would be identified. However, in a few tests, certain verification points presented a maximum position error of approximately 50 µ m, which was similar to the maximum error introduced in the LT position tests. Although the error was smaller than 50 µ m in most tests, accuracy greater than this value cannot be ensured. Author Contributions: All authors contributed to the development of theoretical model, S.A. and J.S. conceived and designed the experiments; S.A. performed the experiments; S.A., P.P. J.A.A. analyzed the data; J.V. contributed reagents/materials/analysis tools; S.A. and P.P. wrote the paper. All authors contributed to the editing of the manuscript. Funding: This work was supported by the Spanish Government (Ministry of Economy and Competitiveness) through the research project SIMULTI—Development of a high precision telescopic instrument based on SIMultaneous laser MULTIlateration for production systems verification, Scientific and Technical Research of Excellence Development National Program, and by the Aragon Government (Department of Industry and Sensors 2019,19, 2847 16 of 17 Innovation) through the Research Activity Grant for research groups recognized by the Aragon Government (T56_17R Manufacturing Engineering and Advanced Metrology Group). Conflicts of Interest: The authors declare no conflict of interest. References 1. Forbes, A.B. Measurement uncertainty and optimized conformance assessment. Measurement 2006 , 39, 808–814. [CrossRef] 2. Ahn, K.G.; Cho, D.W. An analysis of the volumetric error uncertainty of a three-axis machine tool by beta distribution. Int. J. Mach. Tools Manuf. 2000,40, 2235–2248. [CrossRef] 3. 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P é rez Muñoz, P.; Albajez Garc í a, J.A.; Santolaria Mazo, J. Analysis of the initial thermal stabilization and air turbulences effects on Laser Tracker measurements. J. Manuf. Syst. 2016,41, 277–286. [CrossRef] 19. Ouyang, J.; Liu, W.; Qu, X.; Yan, Y. The effect of beam incident angles on cube corner retro-reflector measuring accuracy. In Optical Design and Testing III; SPIE: Bellingham, WA, USA, 2007; Volume 6834, pp. 1–9. 20. Gallagher, B.B. Optical Shop Applications for Laser Tracker Metrology Systems. Available online: www.loft. optics.arizona.edu/documents/journal_articles/2003_Ben_Gallagher.pdf (accessed on 26 June 2019). 21. Ouyang, J.; Liu, W.; Qu, X.; Yan, Y. The effect of beam incident angles on cube corner retro-reflector measuring accuracy. In Proceedings of the Photonics Asia 2007, Optical Design and Testing III, Beijing, China, 11–15 November 2007; pp. 1–9. 22. Muralikrishnan, B.; Sawyer, D.; Blackburn, C.; Phillips, S.; Borchardt, B.; Estler, W.T. ASME B.89.4.19 Performance evaluation tests and geometric misalignments in laser trackers. J. Res. Natl. Inst. Stand. Technol. 2009,114, 21–35. [CrossRef] [PubMed] Sensors 2019,19, 2847 17 of 17 23. ISO 10360-10:2016—Geometrical Product Specifications (GPS)—Acceptance and Reverification Test for Coordinate Measuring Systems (CMS)—Part 10: Laser Trackers for Measuring Point-to-Point Distances; ISO: Geneva, Switzerland, 2016. 24. ISO/IEC Guide 98-3:2008—Uncertainty of Measurement—Part 3: Guide to the Expression of Uncertainty in Measurement (GUM:1995); ISO: Geneva, Switzerland, 2008. 25. Aguado, S.; Santolaria, J.; Samper, D.; Velazquez, J.; Aguilar, J.J. Empirical analysis of the efficient use of geometric error identification in a machine tool by tracking measurement techniques. Meas. Sci. Technol. 2016,27, 035002. [CrossRef] 26. Knapp, W. Measurement Uncertainty and Machine Tool Testing. CIRP Ann. 2002,51, 459–462. [CrossRef] © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). P Pérez etal 6 grows, another decreases to compensate its effect), this way the optimal solution cannot be found and the solution is determined by the maximum number of iterations allowed by the optimization stopping criteria (figure 5). Setting this value in one iteration, the 27 parameters of the geometric errors can be calculated and a correction can be performed. Figure 6 shows the initial error when measuring distances between centres, and the residual error after Figure 6. Distance measurement errors before and after the correction with the parameters obtained with one iteration. Figure 7. Geometrical errors estimated for 5× magnification objective lens. Figure 8. Distance measurement errors before and after the correction with the ten parameters of kinematic model. Meas. Sci. Technol. 30 (2019) 065002 P Pérez etal 7 applying the correction. The error components have been obtained after optimising the non-linear function with a total number of 45 equations. To verify that the correction is giving optimum results in the whole work space, the correction has been applied to the measurements over 162 calottes in different positions. It can be seen that the correction obtained is optimum for most of the points (with residual errors between +5 µm and −8 µm), although there is a case which has some points only partially compensated and its residual error reaches the value of −25 µm. This is due to the interference among some parameters. To find which parameters are interfering, we develop the equationsof the kinematic model (equations (7)–(9)): X P= x−xTx −yTx + xRy · [ xTz + yTz −z + yRx· ( y + yTy ) −x·xWz −y·yWz +yRy ·yTx]+yRz ·(y+yTy) +xRz ·[y−xTy +yTy +x·xWy +yRz ·yTx. +yRx · (z − yTz +y · yWz)] − yRy · (z − yTz +y · yWz) , (7) Y P =y−xTy +yTy +xRz ·[xTx −x+yTx −yRz ·(y+yTy)+yRy ·(z−yTz +y·yWz)] −xRx ·[xTz +yTz −z+yRx ·(y+yTy)−x· xWz −y·yWz +yRy ·yTx]+x·xWy +yRz ·yTx +yRx · (z − yTz +y · yWz), (8) Z P =z−yTz −xTz −xRy ·[xTx −x+yTx −yRz ·(y+yTy) +yRy ·(z−yTz +y·yWz)] −yRx ·(y+yTy) +x·xWz +y·yWz −yRy ∗yTx −xRx ·[y−xTy +yTy +x · xWy +yRz · yTx +yRx · (z − yTz +y · yWz)] . (9) Neglecting the coupled parameters, we obtain the following: XP=x−xTx −yTx −xRy ·z+yRz ·y+xRz ·y−yRy ·z, (10) YP=y−xTy +yTy −xRz ·x+xRx ·z+x·xWy +yRx ·z, (11) Z P =z−yTz −xTz +xRy ·x−yRx ·y +x·xWz +y·yWz −xRx ·y. (12) The simplified model becomes Figure 9. Histogram of the error in distance measurements: (a) nine parameters, (b) 27 parameters.   X P YP Z P   =   x y z   −   xTx +yTx xTy −yTy yTz +xTz   +   0yRz +xRz −xRy −yRy −xRz +xWy 0xRx +yRx xRy +xWz − yRx − xRx +yWz 0   ·   x y z  . (13) As we are measuring with a z-coordinate as approximately constant, we cannot obtain accurate information about the parameters that are coupled to the z-coordinate: xRy, yRy, xRx and yRx. Moreover, some of those parameters interfere with other parameters in the model, for example, yRx and xRx interfere with yWz, or xRy interferes with xWz. Therefore, xRy, yRy, xRx and yRx have been neglected. The parameter xRz interferes with the perpendicularity xWy and the rotational error yRz. This information can be used to simplify some polynomials of the parameter used in the model in equation(13), thus   X P YP ZP   =   x y z   −   xTx +yTx −yTy +xTy yTz +xTz   +   0yRz 0 xWy 00 xWz yWz 0   ·   x y z  . (14) The parameters xTz0 and yTz0 may be treated as one combined parameter (xTz0 + yTz0). The parameters yWz, xWz, xWy, yRz, xTy, yTx, xTz and yTz have been modelled as zeroorder polynomials (constant terms); xTx and yTy have been modelled as linear polynomials and their constant terms xTx0 and yTy0 have been combined with the parameters yTx0 and xTy0 respectively:   X P YP ZP   =   xTx 0+ yTx 0 yTy0+xTy0 yTz0+xTz0   +   ( 1 − xTx 1) yRz 0 0 xWy (1+yTy1)0 xWz yWz 1   ·   x y z  . (15) These simplifications make the model more robust, reaching an optimum solution with only three iterations. Figure7 shows the geometrical errors of the lateral stage of the FVM obtained from the proposed methodology. A total of nine rotational and translational errors of the lateral stage have been estimated. With the estimation of the errors, it is possible to perform an error compensation of any lateral measurements, where the measurements are carried out without stitching. The parameters estimated are used in the original kinematic model (equation (1)) to calculate the corrected position of each centre. With the corrected positions, the distance measurement error can be calculated (figure 8). A significant error reduction of the lateral measurements without stitching can be obtained with the proposed error compensation methodology and with the uncalibrated artefact. Figure9(a) shows that the residual error obtained in distance measurement with the nine-parameter kinematic model is improved compared to the residual error in distance measurement obtained with the 27-parameter kinematic model (figure 9(b)). Though from a physical point of view all the errors are completely independent variables, from a mathematical point of view a problem with parameter redundancy arises: the higher the order of the polynomials used to model Meas. Sci. Technol. 30 (2019) 065002 P Pérez etal 8 the errors, the greater this redundancy, therefore the simplification is justified. These results have been obtained for measurements with the 5× magnification objective lens. As these geometrical errors are due to manufacturing errors and misalignments, they may change when other magnification objective lenses are used. The same procedure has been used to estimate the errors for the 10× magnification objective lens. In this case, the artefact used in section2 has been measured twice (figure 2), with and without stitching, as shown in figure10. Figure 11 shows the new estimation of the geometrical errors obtained with the 10× magnification objective lens. The difference between the coordinates measured with and without stitching is considered as the initial error. The coordinates of the workpiece obtained on the measurement without stitching (figure 10(b)) are corrected by applying the kinematic model (equation (1)) and the estimated errors are obtained (figure 11). Comparing the geometric errors between the 5× magnification objective and 10× magnification objective lenses (figures 7 and 11), it can be seen that xTx and yTy have similar values; these two components are the ones more related to the overlapping of image tiles. The translational components have smaller values for the 10× magnification objective than Figure 12. Errors before and after the correction on the calibrated stainless steel artefact. Figure 10. Measured areas: (a) measuring the calottes and the area in between the calottes, (b) measuring only the calottes. Figure 11. Geometrical errors estimated for 10× magnification objective lens. Meas. Sci. Technol. 30 (2019) 065002 P Pérez etal 9 those obtained for the 5× magnification objective, but the signs and relationships between them are similar: xTy0 and yTx0 have negative values and magnitudes three times higher than xTz0 + yTz0, which has a positive value. The squareness errors yWz and xWz are different for both lenses but xWy has a similar value. This error may be because yWz and xWz are optical configuration related errors, so different lenses will have different perpendicularity errors and xWy will depend on the xy-stage so it does not change with different lenses. But it should also be taken into account that volumetric solutions do not have a real physical equivalence as they only provide optimum values for the joint set of all parameters. In figure12, the errors in distance measurements before and after applying the correction of the calibrated stainless steel artefact (figure 2) are represented. The initial error is defined as the difference between the coordinates measured with stitching (figure 10(a)) and the coordinates measured without stitching (figure 10(b)). The residual error is defined as the difference between the coordinates measured with stitching (figure 10(a)) and coordinates obtained after correcting the measurement without stitching (figure 10(b)), with the geometrical errors estimated (figure 11). The lateral error of a measurement without stitching is reduced from an amplitude of 18 µm over measurements of 20 mm, to an amplitude of 2.5 µm over the whole space measured. Moreover, a considerable amount of computational time and data storage saving are gained with the non-stitching measurements. In the first case (measurement with stitching), the number of image tiles measured is 225 and in the second case it is 49. 4. Conclusions and future work Generally, it is worth noting that measurement without image stitching is not the normal operating mode of commonly available FVM instruments. Moreover, typical measurements with the instrument are usually not over a large area of (180 × 180) mm. Nevertheless, we have shown that it is possible to characterise the xy-stage of a FVM by performing two types of measurement (with stitching and without stitching) with an uncalibrated artefact. The methodology allows compensation of lateral errors of non-stitching measurements of different features on a surface. The corrections obtained can significantly reduce the error of lateral measurements. The lateral error of the uncompensated non-stitching measurements can reach values of 200 µm over a measurement length of 200 mm. Typically measurements are not performed over such large ranges ((200 × 200) mm), and thus, the maximum errors are not representative for common FVM measurements. Nevertheless, after the compensation, the residual error is less than 15 µm. The correction allows measurement of relevant features independently of features in between them so that a considerable saving on computation and data storage can be obtained. Future work includes the manufacture of an artefact with features at different z heights to extend the analysis to the z-axis so that 3D dimensional error compensation can be applied. Acknowledgments This work was supported by the funds of the Ministerio de Economia, Industria y Competitividad, EEBB-I-17-12430 scholarship that allowed the realisation of a research stay at the University of Nottingham and the Engineering and Physical Sciences Research Council (grant number EP/M008983/1). ORCID iDs Pablo Pérez https://orcid.org/0000-0002-1093-8233 References [1] AlburaytA, SyamWP and LeachRK 2018 Lateral scale calibration for focus variation microscopy Meas. Sci. 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Sci. Technol. 25085008 Meas. Sci. Technol. 30 (2019) 065002 METODOLOGÍA 5. Metodología 129 5 METODOLOGÍA En esta sección se van a presentar los equipos utilizados en el trascurso de la investigación y los trabajos en los que se apoyan los artículos. Primero se ha estudiado el Laser Tracker ya que es una herramienta que se utiliza para verificar la máquina herramienta y va a ser usada para comprobar las posiciones del cabezal respecto a la mesa durante una medición con palpador. El palpador es el siguiente elemento analizado en esta sección, siendo la desviación de la punta el principal problema que nos encontramos y que corregimos antes de realizar las mediciones. En esta sección también se presentan los ensayos de medición de la placa y como se pueden calcular los errores de la máquina herramienta con estos datos. Finalmente se realiza un estudio del Alicona, una máquina de medición por variación focal con potencial para realizar una verificación de máquina herramienta si se logra instalar un sistema similar en el cabezal porta-herramientas. 5.1 Material y equipos utilizados 5.1.1 Máquina herramienta Los ensayos que aparecen en esta tesis han sido realizados con una fresadora ANAYAK modelo VH1800 con un control numérico FAGOR 8025-MS perteneciente al servicio de mecánica de precisión de la Universidad de Zaragoza. El control numérico de esta máquina herramienta tiene la ventaja de que permite el acceso a los parámetros de compensación de errores. La cadena cinemática de la máquina herramienta es XFYZ, lo que significa que la pieza se mueve solidariamente con el eje X mientras que la herramienta lo hace con los ejes Y y Z. El fresado, es por lo tanto un proceso de mecanizado en el que el husillo principal gira la herramienta mientras la pieza describe movimientos de traslación. Figura 5.1. Máquina herramienta ANAYAK VH1800. En el husillo, además de las herramientas para el arranque de viruta, puede colocarse un palpador para medición por contacto proporcionando así a la máquina herramienta la capacidad de medición de una máquina de medir por coordenadas. 5. Metodología 130 5.1.2 Palpador Las mediciones realizadas en esta tesis se han realizado con un palpador discreto touchtrigger. Estos palpadores se componen principalmente de tres elementos: el cuerpo del palpador, la aguja del palpador y la punta. El cuerpo del palpador es la parte que contiene los órganos móviles y los sistemas de transducción. Está conectado de forma mecánica y electrónica a la máquina de medición. La aguja del palpador es un vástago rígido y ligero hecho de acero, fibra de carbono o cerámica cuyo extremo finaliza en la punta del palpador, que normalmente tiene forma de esfera y es la parte del palpador que tiene contacto con la pieza a medir. La punta puede estar hecha de varios materiales como rubí sintético, nitruro de silicio o circonita sintética. El error de esfericidad de estas puntas será una componente de la incertidumbre de medida de la máquina herramienta. Las bolas de rubí se fabrican con varios niveles de precisión, definidos por su “grado” (cuanto más bajo es el grado, mejor es la bola). Las dos especificaciones de bola que más se utilizan son el grado 5 y el grado 10 cuyos valores de esfericidad son 0.13 µm y 0.25 µm respectivamente. Figura 5.2. Palpador de contacto. Es conveniente que el palpador sea tan corto y rígido como sea posible. La razón de que un palpador largo no se utilice en todas las aplicaciones es que hay una disminución de la precisión que aumenta cuanto mayor es la flexión del palpador. Esta flexión permite que la sonda se mueva una pequeña distancia con respecto a la pieza después de que se produzca el contacto físico y antes de que se produzca el disparo. Es difícil calcular el efecto de esto sobre la incertidumbre de medida sin llevar a cabo pruebas empíricas. Además la fuerza necesaria para disparar la sonda es variable en las diferentes direcciones. La mayoría de las sondas no se disparan en el momento en que se produce el contacto entre el palpador y la pieza, sino que necesitan que se produzca una fuerza que venza la carga del muelle que está dentro del mecanismo del sensor. Cuando el palpador de la sonda toca una superficie, una vez superada la fuerza de disparo, se transmite una señal al control CNC de la máquina, quedando automáticamente registrada la posición de palpado de los ejes. La orden de control numérico que se emplea para el palpado es G75, con la cual la máquina se moverá hasta la posición indicada o hasta recibir la señal de contacto con la pieza. Durante el movimiento de palpación no es posible variar la velocidad de avance con el conmutador, por lo 5. Metodología 131 que una medida de seguridad importante previa al palpado es establecer una velocidad de avance segura para el palpado. 5.1.3 Laser Tracker El Laser Tracker es un sistema portátil de medición que reporta coordenadas 3D en un sistema de coordenadas esférico. Los Laser Tracker poseen una gran precisión y tienen un alto rango de medición, por lo que se convierten en una herramienta de gran utilidad para la verificación de una máquina herramienta. Figura 5.3. Laser Tracker. En 1987 el científico del NIST Kam Lau desarrolló el primer sistema de interferómetro láser combinado con un sistema con servo motor que permitía al láser realizar el seguimiento de un objetivo móvil y medir tanto la distancia como el ángulo del objetivo. Este sistema de medición tridimensional se ha vuelto esencial para las mediciones de precisión a gran escala en especial en los procesos de fabricación del sector aeroespacial, donde prácticamente han reemplazado las técnicas tradicionales de medición y ensamblaje. Los Laser Tracker también son ampliamente utilizados en los sectores de automoción, naval, robótica, industria pesada, en el sector energético y con las máquinas herramienta. En la Universidad de Zaragoza se disponen de dos modelos de Laser Tracker, un API Tracker3 LTS-3000 y un Leica Geosystems LTD600. Las características principales de ambos modelos se muestran en la siguiente tabla: Tabla 5.1. Características principales de los sistemas láser utilizados en la tesis API Tracker3 Leica LTD600 Rango de medición 30 m 40 m Rango Horizontal ±320ᵒ ±235ᵒ Rango Vertical 77ᵒ / -60ᵒ ±45ᵒ IFM Sí Sí ADM Sí No Precisión angular 0.07 arc sec 0.14 arc sec Precisión absoluta ± 5 µm/m 10 µm ± 0.5 µm/m 5. Metodología 138 𝜌=√𝑥2+𝑦2+(𝑧−𝑧𝐿𝑒𝑖𝑐𝑎)2, (5.4) donde 𝑧𝐿𝑒𝑖𝑐𝑎 toma los valores de la Figura 4 del artículo teniendo en cuenta que el primer punto está en 𝑡=15 𝑚𝑖𝑛, que sería el primer instante en que sería posible medir. La siguiente Figura muestra cómo quedaría la variación de la coordenada radial después de compensar este segundo error presente en la medición: Figura 5.12. Variación de la coordenada radial después de compensar la dilatación térmica. Estas dos correcciones son términos lineales y el orden de aplicación es independiente. De hecho, en el artículo primero se corrige la dilatación térmica y después la variación de la coordenada radial debida al calentamiento interno. En dicho artículo se corrigió la coordenada radial hasta este punto ya que lo primero que se pensó del error restante podía ser debido a variaciones de las condiciones ambientales. Sin embargo, después se vio que aún faltaba un pequeño error por compensar, o más bien que se había sobre-compensado. La siguiente Figura muestra que al modelar el error, la dilatación de la carcasa provocó que la distancia radial aumentase realmente con el tiempo: Figura 5.13. Variación de coordenada z del nido y cabezal a lo largo del tiempo. Esto se comprobó con un ensayo similar al mostrado en la Figura 4 del artículo, pero en esta ocasión el API en lugar de medir la variación de la coordenada z con el retrorreflector situado en el cabezal del Leica, midió esta variación con el retrorreflector situado en el nido del Leica. -30 -20 -10 0 10 20 30 40 50 60 015 30 45 60 75 90 105 120 ∆ρ / µm Tiempo / minutos A B C D 5. Metodología 139 Al comparar las dos curvas, puede hallarse la variación de la distancia radial a lo largo del tiempo entre nido y origen de coordenadas del Leica, como muestra la siguiente Figura: Figura 5.14. Variación de coordenada radial entre el cabezal y el nido a lo largo del tiempo. Esta variación de la coordenada z entre nido y cabezal antes se ha considerado como un error debido al calentamiento, sin embargo, no es así, sino que se trata de una distancia que realmente ha aumentado, por lo que si no se tiene en cuenta, se habría sobredimensionado ligeramente el error compensado. Si se compensa este error, la variación de las coordenadas radiales de los cuatro puntos medidos en el ensayo queda finalmente como muestra la siguiente Figura: Figura 5.15. Variación de la coordenada radial después de compensar la dilatación térmica entre nido y cabezal. Ahora los cuatro puntos sí que parecen tener una coordenada radial constante a lo largo del tiempo. Por lo que habría que tener en cuenta las siguientes tres componentes de error para compensar correctamente los efectos térmicos durante las primeras horas de uso del Laser Tracker:  Variación de la coordenada radial debido al calentamiento interno del láser.  Variación de la coordenada radial debido a la dilatación térmica del equipo.  Variación de la distancia entre el nido y el cabezal del equipo. 0 2 4 6 8 10 12 14 060 120 180 240 300 360 ∆ρ entre cabezal y nido / µm Tiempo / minutos -30 -20 -10 0 10 20 30 40 50 60 015 30 45 60 75 90 105 120 ∆ρ / µm Tiempo / minutos A B C D 5. Metodología 140 La siguiente Figura muestra una representación de donde están localizadas físicamente las fuentes del láser en cada equipo. Figura 5.16. Representación de la fuente del láser en cada modelo de Laser Tracker. La caja del Leica es simplemente el controlador del Laser Tracker, mientras que la caja del API, a parte del controlador contiene la fuente del láser, el cual llega al cabezal a través de un cable de fibra óptica. Esta diferencia en la configuración estructural hace al Leica más sensible a sufrir errores derivados de los gradientes de temperatura causados por la propia fuente del láser que, como se ha podido comprobar, dilata la estructura. 5. Metodología 141 5.3 Efecto de las turbulencias sobre las mediciones con Laser Tracker En los talleres de trabajo donde suelen estar situadas las máquinas herramienta, las condiciones no son las típicas de un laboratorio metrológico, si no que hay importantes cambios de temperatura, corrientes de aire, vibraciones, o incluso puede entrar luz solar, que también provocaría cambios en el índice de refracción del aire de la zona de trabajo. Al realizar una verificación con un Laser Tracker en un taller, el haz láser atravesará diversas zonas a distintas temperaturas y probablemente con corrientes de aire turbulentas si hay motores en marcha cerca, como los propios motores que proporcionan movimiento a la máquina herramienta. Por lo que se ha decidido estudiar la influencia sobre las mediciones realizadas con Laser Tracker cuando el haz láser atraviesa una zona de aire turbulento con un gradiente de temperatura. Para aislar el efecto de estas fluctuaciones, pese a que lo que se quiere hallar es posibles errores generados al realizar mediciones en talleres, los ensayos se han realizado en un laboratorio metrológico con control de temperatura ambiental a 20 ± 1 ᵒC. De este modo, se va a poder estudiar el efecto de las perturbaciones introducidas de forma aislada. Para aumentar la temperatura del aire en una zona puntual del espacio se han utilizado en algunos ensayos una estufa y en otros ensayos un secador normal de pelo, pero nunca apuntando directamente con el flujo de aire al espacio que atraviesa el láser, ya que al ser demasiado turbulento el flujo provoca la pérdida del haz de forma inmediata. Los tramos de aire atravesados por el láser, al tener distintas temperaturas, tendrán también distinto índice de refracción. Edlén demostró en 1966, que el valor de índice de refracción del aire depende de la temperatura, la presión, la humedad y otros factores [142]: 𝑛=1+7.86∙10−4∙𝑝 273+𝑡 −1.5∙10−11∙ℎ∙(𝑡2+160), (5.5) Donde 𝑛 es el índice de refracción del aire, 𝑝 la presión ambiental en kilo pascales, 𝑡 la temperatura del aire en grados Celsius y ℎ la humedad relativa del aire en porcentaje. A su vez, el índice de refracción va a modificar la longitud de onda del láser de acuerdo con la siguiente ecuación: 𝑛∙𝜆𝑎𝑖𝑟=𝜆𝑣𝑎𝑐, (5.6) Por lo tanto, el índice de refracción va a afectar a la longitud de onda del láser, lo cual va a afectar a la medición independientemente de si se mide en modo IFM (Interferometría) o si se mide en modo ADM (Absolute Distance Meter). En el primer caso porque el valor de la medición se obtiene mediante interferencia de onda y la longitud de onda del láser va a variar durante el recorrido del láser. En el segundo porque el tiempo de vuelo del haz depende de la velocidad del láser y de su longitud de onda, al variar esta última, variará el tiempo de vuelo y por lo tanto el valor de la medición obtenido. Por otra parte, los Laser Tracker poseen una estación meteorológica que estima que el índice de refracción de la zona de trabajo es constante, sin embargo si el rayo atraviesa una zona con otro índice de refracción, el valor medido debería calcularse integrando el espacio de acuerdo a la siguiente ecuación: 5. Metodología 142 𝜌𝑚𝑒𝑑𝑖𝑑𝑜=∫𝑛𝑖(𝑥) 𝑛𝐿𝑇 ∙𝜌𝑖∙𝑑𝑥, (5.7) Donde 𝜌𝑖 es la distancia real de un tramo 𝑖 con un índice de refracción 𝑛𝑖 y 𝑛𝐿𝑇 es el índice de refracción estimado por el Laser Tracker para todo el espacio medido. Las porciones de aire a mayor temperatura que atraviesa el láser tienen un índice de refracción menor. Esto hace que la longitud de onda del láser sea mayor en ese tramo. De modo que el Laser Tracker, al asumir que la longitud de onda es constante en todo el espacio, obtenga un valor de la distancia menor que el real. En el artículo se presentó el siguiente ensayo donde en un determinado momento (a los 120 segundos del inicio de la medición) se encendió una fuente de calor. Al aumentar la temperatura, el Laser Tracker introdujo un error en la medición de 8 µm y finalmente debido a las turbulencias del aire el rayo se perdió. Figura 5.17. Coordenada radial del ensayo a dos metros a lo largo del tiempo. La Figura 5.17 muestra el valor de la coordenada radial a lo largo del tiempo del punto fijo situado a dos metros del Laser Tracker. El Laser Tracker obtuvo cada valor de la medición como la media de 200 muestras. La siguiente Figura muestra los valores de la dispersión: Figura 5.18. Dispersión de la medición con el retro-reflector a dos metros. 2000,007 2000,009 2000,011 2000,013 2000,015 2000,017 2000,019 030 60 90 120 150 ρ / mm Tiempo / segundos 0,000 0,010 0,020 0,030 0,040 0,050 0,060 0,070 030 60 90 120 150 σ / mm Tiempo / segundos 5. Metodología 143 En el momento en que entra en funcionamiento la fuente de calor, la distancia medida por el Laser Tracker se reduce 8 µm y la dispersión de los valores aumenta considerablemente. En condiciones normales la dispersión tiene valores en torno a los 4 µm. En el momento en que entra la perturbación, esta aumenta hasta alcanzar 70 µm, momento en que se perdió la señal. Las siguiente Figura muestra ensayos similares, pero con el retro-reflector a diferentes distancias iniciales. Figura 5.19. Ensayos a 3200 mm, 5000 mm y 6000 mm. En todos estos ensayos la fuente de calor aplicada es la misma, por lo que el tramo de aire que se calienta es similar, esto hace que la variación de la coordenada radial de todos los ensayos sea aproximadamente la misma (≈8 µm). Sin embargo, cuanto mayor es la distancia radial, mayor es la dispersión de los valores cuando el rayo atraviesa la perturbación. La siguiente Figura muestra la relación entre la dispersión y la longitud medida. Figura 5.20. Relación entre la dispersión y la distancia medida cuando existe una perturbación. 0 50 100 150 200 250 300 350 0 1 2 3 4 5 6 σ / µm distancia / m 5. Metodología 144 Al realizar una verificación volumétrica con un Laser Tracker en un taller propenso a tener perturbaciones y gradientes de temperatura, cuanto mayor sea la distancia que recorre el láser, mayor será la dispersión de los datos. Esto tiene como consecuencia directa un aumento de la probabilidad de que el Laser Tracker pierda la señal durante el proceso de medición. Los ensayos mostrados hasta ahora, como ya se ha comentado, han sido realizados introduciendo una perturbación similar en cada uno de ellos. El siguiente ensayo pretende mostrar cómo afectaría sobre una misma medición dos perturbaciones diferentes en cuanto a temperatura. Para ello se ha utilizado la estufa, que se puede regular a dos intensidades. La siguiente Figura muestra el valor de la coordenada radial a lo largo del ensayo. T0 representa los datos antes de introducir la perturbación. T1 representa los datos una vez encendida la estufa y T2 representa los datos con la estufa a máxima potencia. Este ensayo finalizó a los 83 segundos con la pérdida del láser. Figura 5.21. Coordenada radial del ensayo a varias temperaturas. Figura 5.22. Dispersión de la medición a varias temperaturas. De este ensayo se concluye que la temperatura de la perturbación introducida va a afectar a la medición de dos formas. Por una parte, cuanto mayor sea la temperatura menor será la distancia medida, como ya se dedujo de la ecuación 5.7. Y por otra parte, la temperatura también afecta al valor de la dispersión de los datos, a mayor temperatura mayor dispersión. 6000,650 6000,652 6000,654 6000,656 6000,658 6000,660 6000,662 6000,664 010 20 30 40 50 60 70 80 90 ρ / mm Tiempo / segundos T2 T1 T0 0,000 0,050 0,100 0,150 0,200 0,250 0,300 010 20 30 40 50 60 70 80 90 σ / mm Tiempo / segundos T2 T1 T0 5. Metodología 145 Además de estas variaciones en la distancia medida y en la dispersión de los datos, las turbulencias provocan que el rayo se desvíe de acuerdo a la ley de Snell simplemente por el hecho de atravesar varios medios a distintos índices de refracción. Este desvío puede observarse en el sensor de posición (PSD) en un ensayo similar al realizado en las Figuras 5.17 y 5.18, con una velocidad de lectura de 1 dato cada 0.01 segundos. Figura 5.23. Sensor de posición 0.25 segundos antes y después de la pérdida del láser. La Figura 5.23 muestra el movimiento del haz láser sobre el PSD en el momento en que el haz se pierde, concretamente muestra en azul los 0.25 segundos antes de perder la señal y en rojo los 0.25 segundos después de perderse el rayo, por lo que los dos tipos de datos mostrados están afectados por la perturbación. De hecho, antes de introducir la perturbación los datos del PSD tenían una oscilación máxima entre ± 1 µm. Este fenómeno al igual que los vistos anteriormente se verá incrementado cuanto mayor sea la distancia a la que se encuentra el retrorreflector, como muestra la siguiente Figura: Figura 5.24. Valores obtenidos por el sensor de posición a distintas distancias de medición. -200 -100 0 100 200 -200 -100 0 100 200 PSD Y / µm PSD X / µm NOT measuring measuring -500 -400 -300 -200 -100 0 100 200 300 400 500 -500-400-300-200-100 0 100 200 300 400 500 PSD Y / µm PSD X / µm 3 metros 2 metros 1 metro 5. Metodología 146 Algunos autores recomiendan usar ventiladores para homogeneizar el aire en la zona de trabajo [143], sin embargo, con lo visto en estos ensayos, mover flujos de aire de forma turbulenta puede tener consecuencias negativas en una medición con Laser Tracker, especialmente si el aire movido está a diferente temperatura y por lo tanto a distinto índice de refracción. Se ha desarrollado un programa de Matlab que simula la desviación del haz láser al atravesar diversos tramos de aire con distintos índices de refracción. Aquí se muestran algunos de los resultados obtenidos: Figura 5.25. Desviación del rayo en un tramo de 2.5 m con aire turbulento. Figura 5.26. Aumento de la distancia recorrida por el haz láser al desviarse en flujo turbulento. El histograma de la Figura 5.26 muestra el recorrido extra que realiza el rayo debido a la desviación del rayo. De media recorre 0.5 µm extra en una distancia de 2.5 metros con un tramo de aire con flujo turbulento. La siguiente Figura muestra los datos de la simulación 5. Metodología 147 cuando la temperatura del recorrido del láser es la misma que en la Figura 5.25, pero sin embargo el flujo de aire no es turbulento: Figura 5.27. Desviación del rayo en un tramo de 2.5 m con aire no turbulento. Figura 5.28. Aumento de la distancia recorrida por el haz láser al desviarse en flujo laminar. El histograma de la Figura 5.28 muestra el recorrido extra que realiza el rayo debido a la desviación del rayo. De media recorre 0.0475 µm extra en una distancia de 2.5 metros con un tramo de aire con flujo laminar.