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arXiv 2205.04422cs.RO
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基于凸优化的绕障碍物运动规划

Motion Planning around Obstacles with Convex Optimization

Massachusetts Institute of Technology(麻省理工学院);Harvard University(哈佛大学)

Tobia Marcucci, Mark Petersen, David von Wrangel, Russ Tedrake

摘要:

轨迹优化为动态约束下的高维空间运动规划提供了成熟的工具。然而,当面对复杂的配置空间,混乱的障碍,机器人专家通常回落到基于采样的规划,斗争在非常高的维度和连续的微分约束。事实上,障碍是许多教科书中的例子有问题的非凸性的概率优化问题的来源。在这里,我们表明,凸优化,事实上,可以用来可靠地规划周围的障碍物的轨迹。具体来说,我们考虑规划问题的碰撞避免的约束,以及成本惩罚和硬约束的形状,持续时间和速度的轨迹。结合Bézier曲线的性质和最近提出的在凸集图(GCS)中寻找最短路径的框架,我们将规划问题表示为紧凑的混合整数优化。与现有的混合整数规划器形成鲜明对比,我们的程序的凸松弛是非常紧的,并且其解决方案的廉价舍入通常足以设计全局最优轨迹。这将混合整数规划还原为简单的凸优化,并自动为规划的轨迹提供最优性边界。我们建议的计划GCS,其基本的优化框架后。我们演示了GCS在模拟各种机器人平台,包括四旋翼飞行通过建筑物和双臂机械手(14个自由度)在有限的空间移动。使用数值实验的七度的自由度机械手,我们表明,GCS可以通过在更短的时间内找到更高质量的轨迹优于广泛使用的基于采样的规划。

英文摘要:

Trajectory optimization offers mature tools for motion planning in high-dimensional spaces under dynamic constraints. However, when facing complex configuration spaces, cluttered with obstacles, roboticists typically fall back to sampling-based planners that struggle in very high dimensions and with continuous differential constraints. Indeed, obstacles are the source of many textbook examples of problematic nonconvexities in the trajectory-optimization problem. Here we show that convex optimization can, in fact, be used to reliably plan trajectories around obstacles. Specifically, we consider planning problems with collision-avoidance constraints, as well as cost penalties and hard constraints on the shape, the duration, and the velocity of the trajectory. Combining the properties of Bézier curves with a recently-proposed framework for finding shortest paths in Graphs of Convex Sets (GCS), we formulate the planning problem as a compact mixed-integer optimization. In stark contrast with existing mixed-integer planners, the convex relaxation of our programs is very tight, and a cheap rounding of its solution is typically sufficient to design globally-optimal trajectories. This reduces the mixed-integer program back to a simple convex optimization, and automatically provides optimality bounds for the planned trajectories. We name the proposed planner GCS, after its underlying optimization framework. We demonstrate GCS in simulation on a variety of robotic platforms, including a quadrotor flying through buildings and a dual-arm manipulator (with fourteen degrees of freedom) moving in a confined space. Using numerical experiments on a seven-degree-of-freedom manipulator, we show that GCS can outperform widely-used sampling-based planners by finding higher-quality trajectories in less time.

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