Discrete mechanics and optimal control for constrained systems

Abstract

SUMMARYThe equations of motion of a controlled mechanical system subject to holonomic constraints may be formulated in terms of the states and controls by applying a constrained version of the Lagrange-d'Alembert principle. This paper derives a structure-preserving scheme for the optimal control of such systems using, as one of the key ingredients, a discrete analogue of that principle. This property is inherited when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced discrete equations serve as nonlinear equality constraints for the minimization of a given objective functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. In particular, for the optimal control of multibody systems, a force formulation consistent with the joint constraints is introduced. This enables one to prove the consistency of the evolution of momentum maps. Using a two-link pendulum, the method is compared with existing methods. Further, it is applied to a satellite reorientation maneuver and a biomotion problem.

Optimal Control for a Pitcher’s Motion Modeled as Constrained Mechanical System

“…The simplified model we investigate was already introduced in [1] and [2]. The kinematic chain representing the pitcher's arm consists of three parts: the collarbone, the upper and the forearm.…”

“…Secondly, we present the different joint connections and the resulting constraints on the system. Besides the use of joints that have been introduced in [1,12] within the framework of the discrete nullspace method, a new hinge joint is presented. Thirdly, the muscle model used for one of the multi-body models is described in detail.…”

“…(1) and Eqn. (3) in combination with desired boundary constraints is now transformed into a finite dimensional constrained optimization problem by using a global discretization of the states and controls.…”

“…As a first step in that direction, in this work we use a recently developed method, namely Discrete Mechanics and Optimal Control for Constrained Systems (DMOCC [1)) to determine the optimal motion of a pitcher's arm that allows him to max-enforced using Lagrange multipliers within the discrete variational principle. However, the presence of the Lagrange multipliers in the set of unknowns enlarges the number of equations.…”

Planned contacts and collision avoidance in optimal control problems

“…This work will show how the non-smooth alternative can be included in the context of optimal control problems using DMOCC, see [2]. The optimal control problem formulation gives more freedom than the forward dynamics problem, since leaving the exact placement of time nodes (within certain bounds) to the optimiser leaves the freedom to assume that contact takes place at a certain time node without loss of generality, i.e.…”