An emerging class of trajectory optimization methods enforces collision avoidance by jointly optimizing the robot's configuration and a separating hyperplane. However, as linear separators only apply to convex sets, these methods require convex approximations of both the robot and obstacles, which becomes an overly conservative assumption in cluttered and narrow environments. In this work, we unequivocally remove this limitation by introducing nonlinear separating hypersurfaces parameterized by polynomial functions. We first generalize the classical separating hyperplane theorem and prove that any two disjoint bounded closed sets in Euclidean space can be separated by a polynomial hypersurface, serving as the theoretical foundation for nonlinear separation of arbitrary geometries. Building on this result, we formulate a nonlinear programming (NLP) problem that jointly optimizes the robot's trajectory and the coefficients of the separating polynomials, enabling geometry-aware collision avoidance without conservative convex simplifications. The optimization remains efficiently solvable using standard NLP solvers. Simulation and real-world experiments with nonconvex robots demonstrate that our method achieves smooth, collision-free, and agile maneuvers in environments where convex-approximation baselines fail.

翻译：一类新兴的轨迹优化方法通过联合优化机器人构型与分离超平面来保证避碰。然而，由于线性分离器仅适用于凸集，这类方法需要对机器人和障碍物进行凸近似，这在杂乱狭窄的环境中会成为一种过于保守的假设。本研究通过引入以多项式函数参数化的非线性分离超曲面，明确地消除了这一限制。我们首先推广了经典的分离超平面定理，证明欧几里得空间中任意两个不相交的有界闭集均可被多项式超曲面分离，这为任意几何形状的非线性分离奠定了理论基础。基于此结果，我们构建了一个非线性规划问题，该问题联合优化机器人轨迹与分离多项式的系数，从而在无需保守凸简化的情况下实现几何感知的避碰。该优化问题仍可通过标准NLP求解器高效求解。针对非凸机器人进行的仿真与实物实验表明，在凸近似基线方法失效的环境中，我们的方法能够实现平滑、无碰撞且灵动的运动。