Natural Coordinates in the Optimal Control of Multibody Systems

Abstract

The formulation of multibody dynamics in terms of natural coordinates (NCs) leads to equations of motion in the form of differential-algebraic equations (DAEs). A characteristic feature of the natural coordinates approach is a constant mass matrix. The DAEs make possible (i) the systematic assembly of open-loop and closed-loop multibody systems, (ii) the design of state-of-the-art structure-preserving integrators such as energy-momentum or symplectic-momentum schemes, and (iii) the direct link to nonlinear finite element methods. However, the use of NCs in the optimal control of multibody systems presents two major challenges. First, the consistent application of actuating joint-forces becomes an issue since conjugate joint-coordinates are not directly available. Second, numerical methods for optimal control with index-3 DAEs are still in their infancy. The talk will address the two aforementioned issues. In particular, a new energy-momentum consistent method for the optimal control of multibody systems in terms of NCs will be presented.

Conservation of generalized momentum maps in mechanical optimal control problems with symmetry

“…where use has been made of (15). Inserting from the adjoint equations (10) and the control equations (11) into the last equation yields…”

“…Structure-preserving numerical integrators can be directly used in optimal control problems by applying the direct transcription approach. For example, EM integrators have been used in the solution of optimal control problems by Bottasso and Croce [14], Betsch et al [15], and Koch and Leyendecker [16]. Similarly, symplectic-momentum variational integrators have been applied in the framework of the direct approach to optimal control problems by Leyendecker et al [17] and Ober-Blöbaum et al [18].…”

Continuum Mechanical Considerations for Rigid Bodies and Fluid-Structure Interaction Problems

“…Note that the consistent application of external torque (30) has originally been proposed in Betsch et al [2]. At last, we derive the constraint forces based on (20), written as follows…”

Analysis of the influence of coordinate and dynamic formulations on solving biomechanical optimal control problems

“…Different numerical methods have been proposed to solve these problems [27] . Direct collocation is one of the most used methods in predictive dynamics [19,[28][29][30] . As the previous cited studies show, there are various ways to formulate the optimal control problems (different coordinates, dynamic formulations or degrees of complexity).…”