Riemannian Direct Trajectory Optimization of Rigid Bodies on Matrix Lie Groups

Constrained Riemannian direct trajectory optimization of rigid bodies formulated on matrix Lie groups.The rigid body dynamics can be formulated in generalized coordinates (q R n ) or on maximal coordinates (product space of SO(3) R 3 ).We compare the landscape of a quadratic cost Rz -z 2 defined on M = SO(3) projected on S 2 = SO(3)/SO(2) with different coordinates.With generalized coordinates represented by Euler angles, the landscape is highly complicated with saddle points.With the matrix Lie group coordinates, the decision variable is in a symmetric homogeneous space.In this work, we, for the first time, introduce constrained Riemannian optimization to solve motion planning of rigid bodies modeled on matrix Lie groups.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.