Ergodic control for active sensing of 2D Euclidean surfaces

Ergodic control for active sensing of 2D Euclidean surfaces Veronica Campana1,3, Andrea Del Prete1,3, Luigi Palopoli2,3, Daniele Fontanelli1,3 Abstract—This work proposes an active sensing strategy for efficient surface defects detection in industrial settings, using robotic manipulators. The approach leverages prior knowledge of the underlying manufacturing process to focus inspection on the most relevant surface regions. The sensing trajectories are planned through ergodic control, balancing exploration and exploitation toward areas with higher defects likelihood. Index Terms—Active sensing, ergodic control, motion control I. INTRODUCTION In manufacturing applications, efficiently detecting potential defects on product surfaces is crucial to ensure quality control and reduce losses. In many industrial settings, products processed in sequence often exhibit defects that are not independent but show spatial or temporal correlations. Such correlations may arise from different factors, like storage conditions of a stack of material, machine malfunctions, or misalignment of raw materials within the production line. Accurately modeling defects correlations and leveraging additional data from the manufacturing process are essential for developing detection procedures that may yield to more effective results than uniform inspection methods, as they exploit the structure of the underlying process. Robotic manipulators can perform defect-searching inspection tasks in industrial scenarios, and designing control strategies that guide the robot to efficiently sense product surfaces improves detection performance. Active sensing is a feedback control strategy, which consists in planning the control action to maximize the information acquired, by focusing on the most informative surface areas [1]. In this study, we propose a framework that exploits the information about the correlations of defects in consecutive surfaces to implement an efficient active sensing algorithm with a robotic manipulator. Specifically, in the proposed algorithm, visual measurements from previously inspected surfaces are used to build a prior distribution that guides sensing of the current surface. Control actions in this framework are planned using ergodic control. Ergodic control aligns the timeaveraged behavior of a system to its space-averaged behavior [2], ensuring that the trajectory of the system covers the surface proportionally to a target probability distribution. We compute the sensing ergodic trajectory using a target PDF, This research has been funded by the European Union project MAGICIAN (GA no. 101120731). 1Department of Industrial Engineering, Universit` a di Trento, Trento, Italy. [email protected]. 2Department of Information Engineering and Computer Science, Universit` a di Trento, Trento, Italy. 3IDRA Labs @ University of Trento. derived from the state belief of the surface, with higher values in the areas in which it is more likely to detect defects. II. FRAMEWORK A. State Belief Update using Kalman filter The state belief associated with each surface is obtained by updating with Kalman filter the belief from the previously sensed surface with the new measurements acquired there. We consider each surface area to be subdivided into cells of fixed dimension. We propose to model the state belief as a multidimensional Gaussian PDF with ndimensions, where nis the number of cells. The state associated to each surface is denoted as x∈Rnand i-th element of the state xiidentifies the percentage of the i-th cell that is covered by defects. For the first surface of the sequence, if we assume no a-priori information, we consider a state estimate vector containing 0 in the all elements, meaning absence of information. In the prediction step of the Kalman filter, the estimate covariance evolves according to the system dynamics and the process noise covariance is added to it, to reflect uncertainty of the model. We propose to design the process noise covariance matrix to have non-zeros entries corresponding to the unvisited cells, so that the uncertainty in those elements of the belief state increases accordingly. The measurement matrix considered in the Kalman filter update formula is a selection matrix, which selects the elements of the state corresponding to the visited cells. Therefore, the state estimate and the state estimate covariance are updated in correspondence of the visited cells. Specifically, when a measurement is collected from the i-th cell, the i-th diagonal element in the covariance matrix is updated narrowed, meaning greater confidence in its value. B. Ergodic trajectory The ergodic trajectory used to sense each surface is obtained considering a target PDF obtained from the state belief. The ergodic metric minimized to obtain the sensing trajectory is the one presented in [3], computed as the sum of the weighted squared difference of the Fourier coefficients of the target PDF and of the distribution representing the trajectory of the system. We propose to model the target PDF as a Gaussianmixture model (GMM) probability distribution. From now on, the target PDF for the ergodic control is denoted as Φ(x).Φ(x) must capture the information that drives the ergodic trajectory to perform both exploitation and exploration of the area of interest. Specifically, the ergodic trajectory should focus on regions where, based on the belief state, defects are most likely 2025 I-RIM Conference October 17-19, Rome, Italy ISBN: 9788894580570 10.5281/zenodo.17629730 105 to be found, while also spending significant time in areas that were less frequently explored in the past. Therefore, we want Φ(x)to be characterized by high values in correspondence of the cells in which a defect was found in the previous surfaces, namely where the belief state shows a high expected value of the state estimate and a low estimate variance. On the other hand, to promote exploration, Φ(x)must also have high values for cells that were not previously visited and consequently show a high variance, which means less confidence for that cell. The Gaussian components of Φ(x)have fixed means coinciding with the centers of the cells and identical variances. The information on the state estimate and estimate covariance for each surface can be encoded in the weights associated with each Gaussian component. Specifically, to balance the information given by the state estimate and the covariance of the estimate, the formula to assign the value to the vector of all the weights wis given as follows: w=βP−1ˆx+ (1 −β)P1.(1) The first term encourages exploitation, assigning higher values to the cells in which the variance is lower and the state estimate is higher. In contrast, the second term encourages exploration, giving more importance to the cells with higher variance. The parameter β∈[0,1] allows to tune the function and obtain a smooth trade-off between exploitation and exploration. The weights are further normalized such that the sum of the weights of all the components is equal to 1. III. PRELIMINARY RESULTS For preliminary experiments, we tested our active sensing with ergodic control algorithm using a point-mass system, moving in a 2D rectangular Euclidean space. We applied the proposed approach considering subsequent surfaces with correlated defects. In the preliminary results presented in Figure 1, defects are generated by stationary sources, that produce random defects while maintaining the same overall distribution over time. The black scattered points in the background represent the simulated defects, while the plotted ergodic trajectory is optimized with respect to the underlying target PDF. We assume that as the ergodic trajectory passes over a cell, it is possible to measure the proportion of the cell covered by defects. The target PDF is iteratively updated: for each surface, it is obtained updating the PDF associated with the previous surface using the measurements acquired in the visited cells. On the first surface, the target PDF approximates a uniform distribution. As exploration progresses, the update strategy balances exploration and exploitation. Indeed, initially it favors mainly unvisited areas, while in subsequent surfaces higher values of the target PDF are associated with areas where defects were previously found, while also in this case ensuring inspection of areas that were less recently visited. The proposed active sensing strategy will be used to guide the robot in sensing 2D surfaces modeled as previously described. Therefore, the ongoing work focuses on incorporating the kinematics of the robotic manipulator into the ergodic 0.0 0.5 1.0 X (m) 0.0 0.5 1.0 Y (m) Surface: 1 Optimized trajectory Initial state 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 0.0 0.5 1.0 X (m) 0.0 0.5 1.0 Y (m) Surface: 2 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.0 0.5 1.0 X (m) 0.0 0.5 1.0 Y (m) Surface: 3 0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 0.0 0.5 1.0 X (m) 0.0 0.5 1.0 Y (m) Surface: 4 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 0.0 0.5 1.0 X (m) 0.0 0.5 1.0 Y (m) Surface: 5 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 0.0 0.5 1.0 X (m) 0.0 0.5 1.0 Y (m) Surface: 6 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 Fig. 1: Preliminary results for the proposed active sensing strategy using optimized ergodic trajectories to sense subsequent surfaces with correlated defects. trajectory optimization, acquiring surface measurements with a camera mounted on the robot end-effector. IV. CONCLUSION AND FUTURE WORK This work proposes an active sensing framework for efficient robotic inspection of surfaces with correlated defects. Such scenarios are often observed in industrial applications, where the proposed strategy can lead to effective defect detection. Ergodic control was shown to be a well-suited technique to plan the control action and efficiently guide the robot through the most relevant areas. Future work will first focus on experimental validation on the actual robotic manipulator. Moreover, further extensions will involve the application of the proposed strategy to the inspection of curved surfaces and the deployment of the framework with multi-agent systems. REFERENCES [1] Chris Kreucher, Keith Kastella, Alfred O. Hero III, ”Sensor management using an active sensing approach”, Signal Processing, Volume 85, Issue 3, 2005, Pages 607-624, ISSN 0165-1684, https://doi.org/10.1016/j.sigpro.2004.11.004. [2] L. M. Miller, Y. Silverman, M. A. MacIver and T. D. 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