Discrete mechanics and optimal control: An analysis

Abstract

Abstract. The optimal control of a mechanical system is of crucial importance in many application areas. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion sequences in robotics and biomechanics. In most cases, some sort of discretization of the original, infinite-dimensional optimization problem has to be performed in order to make the problem amenable to computations. The approach proposed in this paper is to directly discretize the variational description of the system's motion. The resulting optimization algorithm lets the discrete solution directly inherit characteristic structural properties from the continuous one like symmetries and integrals of the motion. We show that the DMOC (Discrete Mechanics and Optimal Control) approach is equivalent to a finite difference discretization of Hamilton's equations by a symplectic partitioned Runge-Kutta scheme and employ this fact in order to give a proof of convergence. The numerical performance of DMOC and its relationship to other existing optimal control methods are investigated.Mathematics Subject Classification. 49M25, 49N99, 65K10.

Discrete mechanics and optimal control for constrained systems

“…Then, the order of the discrete Lagrangian and the discrete virtual work is given by the order of the quadrature rule in use. In [25], (Ober-Blöbaum et al, submitted) it is shown for DMOC (unconstrained dynamics) that a discrete Lagrangian and discrete virtual work of order lead to an optimal control scheme of order . ‡ That means, the state and control trajectories as well as the Lagrange multipliers resulting from the Pontryagin maximum principle are approximated with an accuracy of O(h ).…”

“…Then the body's Euler tensor with respect to the center of mass can be related to the inertia tensor J via E = 1 2 (tr J)I−J, where I denotes the 3×3 identity matrix. The principal values of the Euler tensor, E i , together with the body's total mass M are ingredients in the rigid-body's constant symmetric positive-definite mass matrix The angular momentum of the rigid body can be computed as L = u×p +d I ×p I (25) where summation convention is used to sum over the repeated index I . Null space matrix: An account of rigid-body dynamics is given in [36,42] where also the null space matrix…”

“…Computation of p + n+1 and via p − n the discrete Legendre transform (14) and insertion into the definition of angular momentum (25) yield…”

“…Thereby, the state and control variables are discretized directly in order to transform the optimal control problem into a finite-dimensional nonlinear constrained optimization problem that can be solved by standard nonlinear optimization techniques such as sequential quadratic programming (see [14,15]). In contrast to other methods such as, for example, shooting [16][17][18], multiple shooting [19,20] or collocation methods [21,22], relying on a direct integration of the associated ordinary differential equations parameterized by states and controls or the controls only (see also [23] and [24] for an overview of the current state of the art), a recently developed method DMOC (see [25,26]) is used here. It is based on the discretization of the variational structure of the mechanical system directly.…”

Optimal Control on Stable Manifolds for a Double Pendulum

“…In most applications, optimal control problems have to be solved by numerical methods. In this contribution, we will use DMOC (Discrete Mechanics and Optimal Control, [2]), a direct optimal control method for mechanical systems. (Un)stable Manifolds of the Natural Dynamics The natural, i.e.…”