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

Sina Ober-Blobaum* Control and Dynamical Systems California Institute of Technology Pasadena, CA 91125 [email protected] In this contribution, a recently developed optimal control method for constrained mechanical systems is applied to determine optimal motions and muscle force evolutions for a pitcher's arm. The method is based on a discrete constrained version of the Lagrange-d'Alembert principle leading to structure preserving time-stepping equations. A reduction technique is used to derive the nonlinear equality constraints for the minimization of a given objective function, Different multi-body models for the pitcher's arm are investigated and compared with respect to the motion itself, the control effort, the pitch velocity, and the pitch duration time. In particular. the use of a muscle model allows for an identification of limits on the maximal forces that ensure more realistic optimal pitch motions. INTRODUCTION The computation of optimal motion sequences of biomechanical multi-body systems is of great interest in many different research areas, especially the optimal control of the motion of the human body itself. For several reasons it is important to understand the muscle activation and coordination, e.g. to construct prothesis and implants in modern medical surgery. 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• Address all correspondence to this author. Julia Timmermann Heinz Nixdorf Institute, Control Engineering and Mechatronics University of Paderborn 597 D-33098 Paderborn, Germany [email protected] imize the velocity of his pitch, In extension to previous work in [1,21 we also take the interaction of the muscles into account. This leads on the one hand to more insight into the optimal time evolution of the muscle forces acting in the joints, on the other hand the muscle model automatically provides bounds on the maximal producible forces leading to a more realistic model. Optimal Control for Constrained Mechanical Systems DMOC (Discrete Mechanics and Optimal Control, (2,3]) is a local optimal control method developed for mechanical systems. It is based on the discretization of the variational structure of the mechanical system directly in contrast to other methods like, e.g, shooting, multiple shooting, or collocation methods. These methods rely on a direct integration of the associated ordinary differential equations (see e.g. [4-7]). In the context of variational integrators (see [8]), the discretization of the Lagranged' Alembert principle leads to structure (symplectic-momentum) preserving time-stepping equations. These serve as equality constraints for the resulting finite dimensional nonlinear optimization problem that can be solved by standard nonlinear optimization techniques like sequential quadratic programming (SQP, see e.g. [9, 10]). The extension to constrained mechancial systems (DMOCC) was developed in [I, Ill. Within the constrained formulation, the system is described in terms of redundant coordinates subject to holonomic constraints. For multi-body systems, the couplings between the bodies are characterized by holonomic constraints describing the kinematic conditions arising from the specific joint connections. These configuration constraints are Copyright © 2009 by ASME 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. To reduce the number of unknowns (configurations and torques at the time nodes) and thereby the dimension of the discrete system, the discrete null space method in conjunction with a nodal reparametrization in generalized coordinates is used. This method was introduced in r 12] for the simulation of multi-body systems without control. This procedure on the one hand leads to lower computational cost for the optimization algorithm and on the other hand inherits the conservation properties from the constrained scheme. The benefit of exact constraint fulfillment, correct computation of the change in momentum maps and good energy behavior is guaranteed by the resulting optimization algorithm as shown in [I]. In particular, these are important benefits for the optimal control of high dimensional rigid body systems with joint constraints.