Riemannian Direct Trajectory Optimization of Rigid Bodies on Matrix Lie Groups

TL;DR

This paper presents a Riemannian optimization framework for rigid body trajectory planning on matrix Lie groups, ensuring topological correctness and improving computational efficiency in robotics applications.

Contribution

It introduces a novel Riemannian trajectory optimization method using Lie Group Variational Integrators and RIPM, addressing topological issues and enhancing speed.

Findings

Faster convergence than traditional methods by an order of magnitude.

Ensures topological correctness and avoids singularities in rotation representations.

Linear complexity in planning horizon and system degrees of freedom.

Abstract

Designing dynamically feasible trajectories for rigid bodies is a fundamental problem in robotics. Although direct trajectory optimization is widely applied to solve this problem, inappropriate parameterizations of rigid body dynamics often result in slow convergence and violations of the intrinsic topological structure of the rotation group. This paper introduces a Riemannian optimization framework for direct trajectory optimization of rigid bodies. We first use the Lie Group Variational Integrator to formulate the discrete rigid body dynamics on matrix Lie groups. We then derive the closed-form first- and second-order Riemannian derivatives of the dynamics. Finally, this work applies a line-search Riemannian Interior Point Method (RIPM) to perform trajectory optimization with general nonlinear constraints. As the optimization is performed on matrix Lie groups, it is correct-by-construction to respect the topological structure of the rotation group and be free of singularities. The paper demonstrates that both the derivative evaluations and Newton steps required to solve the RIPM exhibit linear complexity with respect to the planning horizon and system degrees of freedom. Simulation results illustrate that the proposed method is faster than conventional methods by an order of magnitude in challenging robotics tasks.