Inference of contractility evolution on planar worm locomotion

ECCOMAS Thematic Conference on Multibody Dynamics July 24 - 28, 2023, Lisbon, Portugal Inference of contractility evolution on planar worm locomotion Jose J. Muñoz 1, Albert Jiménez-Blanco2, Ashutosh Bijalwan3 1Dept. Mathematics Universitat Politècnica de Catalunya Jordi Girona 31. 08034 Barcelona, Spain Lab. Càlcul Numèric (LaCàn) Centre Int. Mèt. Num. Eng. (CIMNE) Inst. de Mat. UPC-BarcelonaTech (IMTech) [email protected] 2Faculty of Mathematics and Statistics Universitat Politècnica de Catalunya Pau Gargallo, 14. 08028 Barcelona, Spain. [email protected] 3Centre Int. Mèt. Num. Eng. (CIMNE) Gran Capità s/n, 08034 Barcelona, Spain. ashutosh.bijalw[email protected] EXTENDED ABSTRACT 1 Introduction Small slender organisms have the ability to propel autonomously in a very efficient manner. By using their lateral muscles, they bend their longitudinal body, and due to their distinct tangential and lateral friction, the body is subjected to propulsive forces [1]. The set of bending modes (eigenworms) [2] and friction coefficients has been measured computationally and experimentally [3, 4]. However, a full mechanical explanation and simulation reproducing the worm locomotion is still missing. We here present a worm model composed of straight segments, which have associated longitudinal and bending elastic potential. By adding the worm active bending through a set of self-equilibrated forces, and the non-isotropic friction, we are able to reproduce worm locomotion. From a set of simulations and experimental images, and resorting to machine learning techniques, we also aim to infer the set of bending moments that best match the observed the worm dynamics. We comment on the different strategies employed and their suitability for solving the inverse dynamics problem at hand. 2 Worm Mechanics The worm backbone is modelled as an elastic slender body made of Nsegments with longitudinal and bending elastic energy given by, Wel l=1 2 n ∑ i=1 kl(li−l0)2 ,Wel θ=1 2 n−1 ∑ i=1 kθθ2 i with θithe relative angle between segment i+1 and i,li=||x x xi+1−x x xi||, and (kl,kθ)the stretching and bending stiffness, respectively. The body is subjected to a non-isotropic friction which is given by longitudinal and normal friction coefficients (µτ,µn, and a frictional tensor ν ν ν=µττ τ τ⊗τ τ τ+µnn n n⊗n n n. After neglecting inertial forces, the equations of motion can be written as ∇x x xi(Wel l+Wel θ)+ µ µ µ˙ x x xi=m m mi,i=1,...,N(1) The external forces m m miare a set of self-equilibrated forces that mimic the effect of the bending moment Miat the interior nodes i=2,...,N−1. See Figure 1a and [5] for further details. Figure 1b shows an illustrative set of deformations that result in the motion of the centre of mass ¯ x x x. These are obtained by solving the non-linear ordinary differential equations in 1 with an implicit algorithm. 0 0.5 1 -0.2 -0.1 0 0.1 0.2 (a) (b) Figure 1: (a) Scheme of mechanical model. (b) Simulation of worm locomotion from mechanical model and using a wave of bending moments. Extracted from [5]. 3 Moment inference Through a set of simulations, we aim to infer the nodal moments Mi(tn)at each time-step tnthat best match a sequence of worm motions. This inverse problem is solved resorting to machine learning techniques and the training of a neural network with a sufficiently large set of simulations. In order to reduce the training data, we pull back the worm configuration x x x={x x x1,...,x x xN} onto the position ¯ x x x=0 0 0, and rotate the worm such that the vector x x xn−x x x1is aligned with the x−axis. We also test different training and fitting strategies, depending on the variables employed (total or incremental displacements u u uand moment M M M), which are summarised in Table 1. Name Input Output NNDisp ∆uexp ,u u uexp n,Mn∆M NN1 u u uexp Mn+1 NN2 u u uexp ,u u uexp n+1Mn+1 NN3 ∆u u uexp ,u u uexp n,MnMn+1 Table 1: Different strategies employed in the training of the neural network. u u uexp ndenotes “experimental” displacements at time tn. Operator ∆(•) = (•)n+1−(•)ndenotes incremental quantities. The performance of each strategy is given in Figure 2, where we have applied the inference techniques to a synthetically generated set of worm motions. The Figure shows the reproduced configuration (a), the evolution of the moments (b) and the error with respect to the synthetically generated deformations (denoted as “experimental” in Table 1). (a) (b) (c) Figure 2: (a) Snapshot of worm deformation. (b) Extracted moments along worm backbone. (c) Evolution of error Ebetween inferred worm positions and synthetically generated positions. 4 Conclusions In this work we show that ML techniques can be employed to derive the worm activity. So far the methodology has been applied to a simulated worm locomotion. In future work we will apply the techniques to experimentally measured worm configurations. Acknowledgments This work has been financially supported by the Spanish Ministry of Science and Innovation unde grants CEX2018-000797-S and PID2020-116141GB-I00. References [1] Bilbao, A., Patel, A.K. Roll maneuvers are essential for active reorientation of Caenorhabditis elegans in 3D media, Proc. Nat. Acad. Sc. (PNAS), 2018. [2] Stephens, G., Johnson-Kerner, B., Bialek, W., Ryu, W.: Dimensionality and dynamics in the behavior of c. elegans. PLOS Comp. Biol. 4, e1000028, 2008. [3] Kandhari, A., Huang, Y., Quinn, R.D. Body stiffness in orthogonal directions oppositely affects worm-like robot turning and straight-line locomotion, Bioinspiration and biomimetics, 13: 026003, 2018. [4] Rabers, Y., Backholm, M., Dalnoki-Veress, K., Ruy, W.S. Direct Measurements of Drag Forces in C. elegans Crawling Locomotion, Bioph. J. 107(8):1980–1987, 2014. [5] Muñoz, J.J., Condamin, L., Doste, D. On the net displacement of contact surface centroid in contractile bodies, Mech. Res. Commun., 119:103809 2022. [6] A. Bijalwan, A., Muñoz, J.J. A control Hamiltonian preserving discretisation for optimal control. Multibody System Dynamics, 2023. Under review.