Recursive inverse dynamics sensitivity analysis of open-tree-type multibody systems

We present a first-order recursive approach to sensitivity analysis based on the application of the direct differentiation method to the inverse Lagrangian dynamics of rigid multibody systems. Our method is simple and efficient and is characterized by the following features. Firstly, it describes the kinematics of multibody systems using branch connectivity graphs and joint-branch connectivity matrices. For most mechanical systems with an open-tree kinematic structure, this method turns out to be more efficient compared to other kinematic descriptions employing joint or link connectivity graphs. Secondly, a recursive sensitivity analysis is presented for a dynamic system with an open-tree kinematic structure and inverse dynamic equations described in terms of the Lagrangian formalism. Thirdly, known approaches to recursive inverse dynamic and sensitivity analyses are modified to include dynamic systems with external forces and torques acting simultaneously at all joints. Finally, the proposed method for sensitivity analysis is easy to implement and computationally efficient. It can be utilized to evaluate the derivatives of the dynamic equations of multibody systems in gradient-based optimization algorithms. It also allows less experienced users to perform sensitivity analyses using the power of high-level programming languages such as MATLAB. To illustrate the method, simulation results for a human body model are discussed. The shortcomings of the method and possible directions for future work are outlined.