An analytical approach to the sensitivity analysis of semi-recursive ODE formulations for multibody dynamics

Sensitivity analysis is an extremely powerful tool in many applications such as in the optimization of the dynamics of multibody systems with gradient-based methods. Sensitivity calculations are computationally burdensome and, depending on the method chosen for differentiation and the set of dynamic equations, they could result highly inefficient. Semi-recursive dynamic methods are seldom studied analytically in terms of sensitivity analysis due to their complexity, even though their dynamic performance is usually among the most efficient. This work explores the sensitivity analysis of a particular multibody-dynamics formulation, the semi-recursive Matrix R formulation, which is based on the nullspace of constraint equations and leads to a system of ordinary differential equations. As a result, two sets of sensitivity equations are proposed, one based on the direct differentiation method (DDM) and other on the Adjoint Variable Method (AVM), being these sensitivity formulations the main novelty of this work. The main derivatives required in the sensitivity equations are listed in this document, paying special attention to conciseness, correctness and completeness. The methods proposed have been implemented in the general purpose multibody library MBSLIM ( Multibody Systems in Laboratorio de Ingenier & iacute;a Mec & aacute;nica), and their performance has been tested in two numerical experiments, a five-bar benchmark problem and a four-wheeled buggy vehicle. A review and generalization of constrained and unconstrained kinematic problems in relative coordinates is provided as an introduction to the generation of the semi-recursive Matrix R equations of motion. Due to the importance of the selection of the set of independent coordinates, a more general description of the Matrix R method is presented as a novel contribution as well.