Optimal control of multibody systems using an energy preserving direct transcription method

We consider the solution of path planning problems for multibody systems, using (a) a differential-algebraic formulation, (b) a direct transcription process, and (c) an energy-coherent temporal discretization. The differential-algebraic nature of the equations ensures that problems with arbitrary topologies can be solved with the proposed methodology. The use of a direct transcription process allows for an easy implementation of the method in existing standard multibody software with minimal modifications, since the equations of optimal control are not needed. Furthermore, we show that the governing optimal control equations are exactly recovered as the temporal mesh size tends to zero. Finally, the use of an energy preserving scheme ensures nonlinear unconditional stability in the solution of the resulting multipoint boundary value problems, which ensures superior numerical robustness to the numerical procedures. We test the proposed methodology with the help of representative examples, and we verify the second order accuracy of all computed solution fields, including positions, velocities, controls and final time.