Variation-Based Linearization of Nonlinear Systems Evolving on $SO(3)$ and $\mathbb S^{2}$

Abstract

In this paper, we propose a variation-based method to linearize the nonlinear dynamics of robotic systems, whose configuration spaces contain the manifolds S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> 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: 1) a 3-D pendulum: 2) a spherical pendulum; and 3) a quadrotor with a suspended load, whose dynamics evolve on SO(3), S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , and SE(3) × S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> , 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-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.