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：针对高维系统计算最优无碰撞轨迹是一个具有挑战性的问题。基于采样的规划器难以应对高维性，而轨迹优化器则可能因优化景观中固有的非凸性而陷入局部极小值。近期，利用混合整数规划来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，这在一定程度上得益于紧凸松弛策略及能大幅缩减运行时的高效近似方法。然而，这些方法此前局限于欧几里得构型空间，无法应用于移动基座或连续旋转关节等场景。本文通过将构型空间建模为黎曼流形来处理此类场景，并针对零曲率情况提出了一种归约为混合整数凸优化问题的约简方法。我们在多种机器人平台上验证了所提方法的效果，包括为PR2双臂移动机械手生成高效的无碰撞轨迹。