Developing controllers for obstacle avoidance between polytopes is a challenging and necessary problem for navigation in tight spaces. Traditional approaches can only formulate the obstacle avoidance problem as an offline optimization problem. To address these challenges, we propose a duality-based safety-critical optimal control using nonsmooth control barrier functions for obstacle avoidance between polytopes, which can be solved in real-time with a QP-based optimization problem. A dual optimization problem is introduced to represent the minimum distance between polytopes and the Lagrangian function for the dual form is applied to construct a control barrier function. We validate the obstacle avoidance with the proposed dual formulation for L-shaped (sofa-shaped) controlled robot in a corridor environment. To the best of our knowledge, this is the first time that real-time tight obstacle avoidance with non-conservative maneuvers is achieved on a moving sofa (piano) problem with nonlinear dynamics.

翻译：开发控制器,以避免在顶端形成障碍,这是紧凑空间航行的一个棘手而必要的问题。传统方法只能将避免障碍的问题发展成为离线优化问题。为了应对这些挑战,我们提议使用非移动控制屏障功能进行基于双重性的安全临界最佳控制,以避免在顶端之间形成障碍,这可以通过基于QP的优化优化问题实时解决。在建立控制屏障功能时,引入了双重优化问题,以代表多顶和拉格朗江功能之间的最小距离。我们用在走廊环境中为L型(sofa形)受控制的机器人提议的双重配方来验证避免障碍。据我们所知,这是第一次在非保守操纵的沙发(piana)上,在非线性动态的移动沙发(piana)问题上实时地紧紧地避免障碍。