CDS/CIMMS Seminar: From efficient mechanical integration of multibody dynamics towards its optimal control

Dr. Sigrid Leyendeker, Aeronautics, Caltech

The regard of rigid bodies and multibody systems as constrained mechanical systems, where the constraints can be distinguished into internal constraints, representing the rigidity of each body and external constraints related to their coupling by joints, has proved convenient since it circumvents the use of rotational parameters throughout the time discretisation. The formulation of the dynamics as finite dimensional Hamiltonian system subject to holonomic constraints  provides the possibility to use different methods for the  constraint enforcement, which differ significantly in the following categories:  dimension of the system of nonlinear equations, condition number of the  iteration matrix during the iterative solution procedure, exactness of the constraint fulfillment and computational costs. The discrete null space  method provides an integration scheme for the constrained equations of motion,  which has proven to perform excellently in the mentioned categories. It relies on the derivation of a discrete form of the differential algebraic equations emanating from the use of the Lagrange multiplier method for the constraint enforcement. Thereby conservation properties of the real motion (energy, momentum maps or symplecticity) are inherited by the solution of the time-stepping scheme. After the discretisation has been completed, a size reduction of the time-stepping scheme is performed using the discrete null space method, whereby the constraint forces are eliminated. The resulting discrete system has the minimal possible dimension which equals exactly the number of degrees of freedom of the mechanical system.
                                                                                      
Thus the reduced time-stepping equations are exceptionally well-suited to be used as equality constraints in a nonlinear optimisation problem which determines the actuation forces. Together with initial and final conditions on the configuration and conjugate momentum, they serve as nonlinear equality constraints for the minimization of a given cost 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.