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.

翻译：母体 Lism 母体是控制和机器人中常用的重要方块, 优化这些方块的控制政策是一个根本问题。 在这项工作中, 我们提出一种新颖的方法, 利用基于拉格朗加的、 强化的、 分散的、 分散的、 不同动态的编程(DDP) 算法, 优化矩阵 Lie 群的轨迹。 我们的方法是将优化问题提升到后转通道的Lie 代数, 并收回到前转通道的方块 。 与以往的方法相比, 我们的方法只处理特定方块的制约性处理, 我们的方法为非线性约束性矩阵 Lie 群的处理提供了一种总体方法。 我们还展示了我们处理外部扰动的方法的有效性, 其应用方式是用Li- algebraic 反馈控制 SE(3) 政策。 结果显示, 我们的方法能够有效地处理配置、 速度和输入制约, 并在外部扰动时保持稳定性 。