"Discrete variational mechanics for solving optimal control problems"
Sina Ober-Blöbaum,Paderborn
Abstract:
The computation of optimal control policies to steer dynamical systems is of great interest in many different research areas, such as astrodynamics, robotics or biomotion.
For the numerical treatment, we present a local optimal control method, denoted by DMOC (Discrete Mechanics and Optimal Control) that is developed especially 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. The discretization of the Lagrange-d'Alembert principle leads to structure (symplectic-momentum) preserving time-stepping equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization
problem. For the solution of this problem standard nonlinear optimization techniques like sequential quadratic programming (SQP) can be used.
Besides the structure preserving properties that are handed down to the optimal control algorithm, we will derive approximation and convergence properties of the algorithm. For example, it can be shown, that the approximation order of the adjoint equations resulting from the necessary optimality conditions is the same as for the state equations due to the symplecticity of the discretization scheme.
For the treatment of holonomic constraints, as e.g. for multi-body dynamics, an extension of the approach to constrained mechanical systems is presented (DMOCC).
The numerical method is demonstrated by means of specific examples from astrodynamics and multi-body dynamics.
Further questions regarding mesh refinement strategies as well as the choice of good initial guesses using inherent dynamical properties such as symmetries and invariant manifolds will be addressed to.
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