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

Matrix Lie groups are an important class of manifolds commonly used in control and robotics, and the optimization of control policies on these manifolds is a fundamental problem. In this work, we propose a novel approach for trajectory optimization on matrix Lie groups using an augmented Lagrangian based constrained discrete Differential Dynamic Programming (DDP) algorithm. Our method involves lifting the optimization problem to the Lie algebra in the backward pass and retracting back to the manifold in the forward pass. In contrast to previous approaches which only addressed constraint handling for specific classes of matrix Lie groups, our method provides a general approach for nonlinear constraint handling for a generic matrix Lie groups. We also demonstrate the effectiveness of our method in handling external disturbances through its application as a Lie-algebraic feedback control policy on SE(3). The results show that our approach is able to effectively handle configuration, velocity and input constraints and maintain stability in the presence of external disturbances.