2022/03/03 by Singh, Shubham, Russell, Ryan P., Wensing, Patrick M.
#FOS: Computer and information sciences #Robotics (cs.RO)
paper · doi:10.48550/arxiv.2203.01497
Optimization-based robot control strategies often rely on first-order dynamics approximation methods, as in iLQR. Using second-order approximations of the dynamics is expensive due to the costly second-order partial derivatives of the dynamics with respect to the state and control. Current approaches for calculating these derivatives typically use automatic differentiation (AD) and chain-rule accumulation or finite-difference. In this paper, for the first time, we present analytical expressions for the second-order partial derivatives of inverse dynamics for open-chain rigid-body systems with floating base and multi-DoF joints. A new extension of spatial vector algebra is proposed that enables the analysis. A recursive algorithm with complexity of O(Nd2) is also provided where N is the number of bodies and d is the depth of the kinematic tree. A comparison with AD in CasADi shows speedups of 1.5-3× for serial kinematic trees with N> 5, and a C++ implementation shows runtimes of ≈51μs for a quadruped.