In this paper, we focus on non-conservative obstacle avoidance between robots with control affine dynamics with strictly convex and polytopic shapes. The core challenge for this obstacle avoidance problem is that the minimum distance between strictly convex regions or polytopes is generally implicit and non-smooth, such that distance constraints cannot be enforced directly in the optimization problem. To handle this challenge, we employ non-smooth control barrier functions to reformulate the avoidance problem in the dual space, with the positivity of the minimum distance between robots equivalently expressed using a quadratic program. Our approach is proven to guarantee system safety. We theoretically analyze the smoothness properties of the minimum distance quadratic program and its KKT conditions. We validate our approach by demonstrating computationally-efficient obstacle avoidance for multi-agent robotic systems with strictly convex and polytopic shapes. To our best knowledge, this is the first time a real-time QP problem can be formulated for general non-conservative avoidance between strictly convex shapes and polytopes.

翻译：本文聚焦于具有控制仿射动力学的机器人（其形状为严格凸或多面体）之间的非保守避障问题。该避障问题的核心难点在于，严格凸区域或多面体之间的最小距离通常隐含且非光滑，导致距离约束无法直接施加于优化问题中。为应对这一挑战，我们采用非光滑控制障碍函数，将对偶空间中的避障问题重新表述，通过二次规划等价地表示机器人间最小距离的正定性。所提方法可证明保证系统安全性。我们从理论上分析了最小距离二次规划的平滑性质及其KKT条件。通过展示多智能体机器人系统（具有严格凸及多面体形状）的计算高效避障，验证了我们的方法。据我们所知，这是首次能够为严格凸形状与多面体间的通用非保守避障问题构建实时二次规划问题。