Variation-Based Linearization of Nonlinear Systems Evolving on SO(3) and S²

In this paper, we propose a variation-based method to linearize the nonlinear dynamics of robotic systems, whose configuration spaces contain the manifolds S and SO(3), along dynamically-feasible reference trajectories. The proposed variation-based linearization results in an implicitly time-varying linear system, representing the error dynamics, that is globally valid. We illustrate this method through three different systems, a 3D pendulum, a spherical pendulum, and a quadrotor with a suspended load, whose dynamics evolve on SO(3), S, and SE(3) × S respectively. We show that for these systems, the resulting time-varying linear system obtained as the linearization about a reference trajectory is controllable for all possible reference trajectories. Finally, a Linear Quadratic Regulator (LQR)-based controller is designed to attenuate the error so as to locally exponentially stabilize tracking of a reference trajectory for the nonlinear system. Several simulations results are provided to validate the effectiveness of this method.