Optimal dynamics of actuated kinematic chains. Part 1: Dynamic modelling and symbolic differentiations

The problem of modelling dynamics of open-loop multibody systems is addressed in the present Part-1 of the paper with optimization of motion in mind. The optimization technique implemented in the complementary Part-2 is the Pontryagin Maximum Principle (PMP) which requires deriving equations of motion in state space form, and makes necessary to carry out higher order differentiation in order to formulate some optimality conditions. So as to fulfil these requirements, an algebraic differentiation technique is developed in the present paper, which results in formulating, in the same global computational scheme, Lagrangian and Hamiltonian equations of motion together with the Jacobian matrix of phase-velocities involved in the conditions for optimality stated by the PMP. All formulations required are formally exact and essentially non-redundant which will ensure safer and faster numerical processing. The structure of the final algorithm was used to develop a computerized symbolic formulation of the entire optimization problem. The file which results can be used directly by the numerical solver. Such a symbolic computation code has proved to be an essential tool to cope with the huge complexity of formulations involved in the statement of the optimization problem dealt with in part 2 of the paper.