We study semidefinite relaxations for collision-free motion planning. We focus on a point robot moving from start to goal through spherical obstacles in $\mathbb{R}^n$, subject to path continuity constraints and squared derivative costs; a setting that is conceptually simple yet captures the hardness of collision-free motion planning. We formulate this problem exactly as a nonconvex problem over polynomial curves, and present a natural semidefinite relaxation. We contribute two key theoretical insights; to our knowledge this is the first theoretical analysis of semidefinite relaxations for collision-free motion planning. First, we show that solving the convex relaxation is equivalent to solving, to global optimality, a related motion planning problem in a potentially higher-dimensional space. This geometric interpretation yields necessary and sufficient conditions for tightness, and a clear intuition for when the relaxation is loose. Second, we show that the relaxation admits a symmetry reduction that makes it significantly smaller than one might expect, with positive semidefinite cone sizes that scale linearly with the polynomial degree and are independent of the ambient dimension. The resulting relaxation is 10 to 100 times faster than direct nonlinear programming transcriptions solved with SNOPT and IPOPT, exhibits significantly lower variance in solve times, and reliably finds a locally optimal path for the original problem. We demonstrate its effectiveness as a convex steering function in an RRT planner for minimum-snap quadrotor planning with $C^4$ continuous trajectories.

翻译：我们研究了面向无碰撞运动规划的半定松弛方法。研究对象是空间中一个点状机器人从起点运动至终点的情形，其间需避开$\mathbb{R}^n$中的球状障碍物，同时满足路径连续性约束与平方导数代价——这一场景虽概念简单，却体现了无碰撞运动规划的复杂性。我们将该问题精确建模为多项式曲线上的非凸优化问题，并提出其自然的半定松弛形式。本文贡献两项关键理论洞见：据我们所知，这是首次针对无碰撞运动规划的半定松弛方法进行理论分析。第一，证明求解该凸松弛等价于在更高维空间中求解一个相关运动规划问题的全局最优解。这一几何解释给出了松弛紧致性的充要条件，并清晰揭示了松弛非紧时的直观机理。第二，证明该松弛具有对称性约简特性，使其规模远小于预期——正半定锥的尺寸随多项式次数线性增长，且与空间维度无关。相比直接使用SNOPT和IPOPT求解非线性规划转录，所得松弛方法速度快10至100倍，求解时间方差显著降低，并能稳定获得原问题的局部最优路径。我们将其作为凸导向函数嵌入RRT规划器中，实现了$C^4$连续轨迹的最小加加速度四旋翼飞行器规划，验证了方法的有效性。