Optimization-based methods are widely used for computing fast, diverse solutions for complex tasks such as collision-free movement or planning in the presence of contacts. However, most of these methods require enforcing non-penetration constraints between objects, resulting in a non-trivial and computationally expensive problem. This makes the use of optimization-based methods for planning and control challenging. In this paper, we present a method to efficiently enforce non-penetration of sets while performing optimization over their configuration, which is directly applicable to problems like collision-aware trajectory optimization. We introduce novel differentiable conditions with analytic expressions to achieve this. To enforce non-collision between non-smooth bodies using these conditions, we introduce a method to approximate polytopes as smooth semi-algebraic sets. We present several numerical experiments to demonstrate the performance of the proposed method and compare the performance with other baseline methods recently proposed in the literature.

翻译：基于优化的方法被广泛用于计算复杂任务（如无碰撞运动或接触存在下的规划）的快速、多样化解决方案。然而，大多数此类方法需要在物体之间强制执行非穿透约束，这导致问题变得非平凡且计算成本高昂，使得基于优化的方法在规划与控制中的应用面临挑战。本文提出一种方法，能在对集合的配置进行优化的同时高效地强制执行非穿透性，该方法可直接应用于碰撞感知轨迹优化等问题。为实现这一目标，我们引入了具有解析表达式的新型可微条件。为利用这些条件在非光滑体之间强制执行非碰撞，我们提出了一种将多面体近似为光滑半代数集的方法。我们通过多个数值实验展示了所提方法的性能，并与文献中近期提出的其他基线方法进行了性能比较。