Lie Algebraic Cost Function Design for Control on Lie Groups

Abstract

This paper presents a control framework on Lie groups by designing the control objective in its Lie algebra. Control on Lie groups is challenging due to its nonlinear nature and difficulties in system parameterization. Existing methods to design the control objective on a Lie group and then derive the gradient for controller design are non-trivial and can result in slow convergence in tracking control. We show that with a proper left-invariant metric, setting the gradient of the cost function as the tracking error in the Lie algebra leads to a quadratic Lyapunov function that enables globally exponential convergence. In the PD control case, we show that our controller can maintain an exponential convergence rate even when the initial error is approaching π in SO(3). We also show the merit of this proposed framework in trajectory optimization. The proposed cost function enables the iterative Linear Quadratic Regulator (iLQR) to converge much faster than the Differential Dynamic Programming (DDP) with a well-adopted cost function when the initial trajectory is poorly initialized on SO(3).

Progress in symmetry preserving robot perception and control through geometry and learning

“…The detailed proof of the theorems are presented in the work of Teng et al (2022b). For the proposed MPC, we can follow the same steps and estimate the region of attraction.…”

“…The experimental results confirm that the proposed approach provides faster convergence when rotation and position are controlled simultaneously. Future work will implement the trajectory optimization using this geometric control framework proposed by Teng et al (2022b) for robot control. Another interesting research direction is to incorporate learning into this framework (Shi et al, 2019;Li et al, 2022;Ma et al, 2022;O'Connell et al, 2022;Power and Berenson, 2022;Rodriguez et al, 2022).…”

“…Section 4 provides an overview of the error-state Model Predictive Control (MPC) on Lie groups and the stability analysis by a Lyapunov function expressed in the Lie algebra (Teng et al, 2022a;Teng et al, 2022b). We derive the linearized configuration error dynamics and equations of motion in the Lie algebra (tangent space at the identity) that, given an initial condition, are globally valid and independent of the system trajectory.…”

Trajectory Optimization on Matrix Lie Groups with Differential Dynamic Programming and Nonlinear Constraints

“…They demonstrated that DDP has significantly better convergence rates compared to sequential quadratic programming (SQP) methods. Teng et al [18] further improved the convergence performance of DDP for matrix groups by designing the control objective in its Lie algebra. Both of these approaches [13], [18] formulate the trajectory optimization on matrix Lie groups in an unconstrained framework.…”

“…Teng et al [18] further improved the convergence performance of DDP for matrix groups by designing the control objective in its Lie algebra. Both of these approaches [13], [18] formulate the trajectory optimization on matrix Lie groups in an unconstrained framework. In order to address this limitation, Liu et al [14] extended the work [13] by imposing SO(3) pose constraints.…”