Matrix Lie groups are an important class of manifolds commonly used in control and robotics, and optimizing control policies on these manifolds is a fundamental problem. In this work, we propose a novel and computationally efficient approach for trajectory optimization on matrix Lie groups using an augmented Lagrangian-based constrained discrete Differential Dynamic Programming.  The method involves lifting the optimization problem to the Lie algebra during the backward pass and retracting back to the manifold during the forward pass. Unlike previous approaches that addressed constraint handling only for specific classes of matrix Lie groups, the proposed method provides a general solution for nonlinear constraint handling across generic matrix Lie groups. 

We evaluate the proposed DDP algorithm through extensive experiments, demonstrating its efficacy in managing constraints within a rigid-body mechanical system on SE(3), its computational superiority compared to existing optimization solvers, robustness under external disturbances as a Lie-algebraic feedback controller, and effectiveness in trajectory optimization tasks including realistic quadrotor scenarios as underactuated systems and deformable objects whose deformation dynamics are represented in SL(2). The experimental results validate the generality, stability, and computational efficiency of our proposed method.

Citation

To cite this work, please use the following BibTex entry:

@article{alcan2024cddplie,

  title={Constrained Trajectory Optimization on Matrix Lie Groups via Lie-Algebraic Differential Dynamic Programming}, 

  author={Alcan, Gokhan and Abu-Dakka, Fares J and Kyrki, Ville},

  journal={arXiv preprint arXiv:2301.02018},

  year={2024}

}