Computing optimal, collision-free trajectories for high-dimensional systems is a challenging problem. Sampling-based planners struggle with the dimensionality, whereas trajectory optimizers may get stuck in local minima due to inherent nonconvexities in the optimization landscape. The use of mixed-integer programming to encapsulate these nonconvexities and find globally optimal trajectories has recently shown great promise, thanks in part to tight convex relaxations and efficient approximation strategies that greatly reduce runtimes. These approaches were previously limited to Euclidean configuration spaces, precluding their use with mobile bases or continuous revolute joints. In this paper, we handle such scenarios by modeling configuration spaces as Riemannian manifolds, and we describe a reduction procedure for the zero-curvature case to a mixed-integer convex optimization problem. We demonstrate our results on various robot platforms, including producing efficient collision-free trajectories for a PR2 bimanual mobile manipulator.

翻译：计算高维系统的最优无碰撞轨迹是一个具有挑战性的问题。基于采样的规划方法在高维空间中效果不佳，而轨迹优化器则可能因优化景观中固有的非凸性陷入局部极小值。近年来，利用混合整数规划封装这些非凸性并寻找全局最优轨迹的方法展现出巨大潜力，这得益于紧凸松弛和高效近似策略大幅缩短了运行时间。然而，这类方法此前仅限于欧几里得配置空间，无法用于移动基座或连续旋转关节等场景。本文通过将配置空间建模为黎曼流形来处理此类问题，并针对零曲率情形描述了将其归约为混合整数凸优化问题的方法。我们在多种机器人平台上验证了结果，包括为PR2双手机器人生成高效的无碰撞轨迹。