A monolithic optimal control method for displacement tracking of Cosserat rod with application to reconstruction of C. elegans locomotion

Abstract

This article considers an inverse problem for a Cosserat rod where we are given only the position of the centreline of the rod and must solve for external forces and torques as well as the orientation of the cross sections of the centreline. We formulate the inverse problem as an optimal control problem using the position of the centreline as an objective function with the external force and torque as control variables, with meaningful regularisation of the orientations. A monolithic, implicit numerical scheme is proposed in the sense that primal and adjoint equations are solved in a fully-coupled manner and all the nonlinear coefficients of the governing partial differential equations are updated to the current state variables. The forward formulation, determining rod configuration from external forces and torques, is first validated by a numerical benchmark; the solvability and stability of the inverse problem are then tested using data from forward simulations. The proposed optimal control method is motivated by reconstruction of the orientations of a rod’s cross sections, with its centreline being captured through imaging protocols. As a case study, we take the locomotion of the nematode, Caenorhabditis elegans. In this study we take laboratory data for its centreline and infer its cross-section orientation (muscle locations) with the control force and torque being interpreted as the reaction force, activated by C. elegans’ muscles, from the surrounding fluids. This method thus combines the mathematical modelling and laboratory data to study the locomotion of C. elegans, which gives us insights into the potential anatomical orientation of the worm beyond what can be observed through the laboratory data. The paper is completed with several additional remarks explaining the theoretical and technical details of the model.