An Analytical Method For Sensitivity Analysis Of Rigid Multibody System Dynamics Using Projective Geometric Algebra

The analytical sensitivity analysis, i.e., the analytical first-order partial derivatives of dynamical equations, is one key to improving descent-based optimization methods for motion planning and control of robots. This paper proposes an efficient algorithm that recursively evaluates the analytic gradient of the dynamical equations of a multibody system. The theory of projective geometric algebra (PGA) is used to generate the algorithm. It provides a systemic and geometrically intuitive interpretation for the multibody system dynamics, and the resulting algorithm is highly efficient, with concise formula. The algorithm is first applied to the open-chain system and extended for the cases when kinematic loops are contained. The runtime varying with respect to the degree of freedom (DOF) of the system is analyzed. The results are compared with that obtained from the algorithm based on spatial vector algebra (SVA) using open-source MATLAB codes. A 2-DOF serial robot, a 3-DOF robot with a kinematic loop and the PUMA560 robot are used for the validation of the minimum-effort motion planning, and it is verified that the proposed algorithm improves the efficiency.