This paper presents a novel approach for collision avoidance in optimal and model predictive control, in which the environment is represented by a large number of points and the robot as a union of padded polygons. The conditions that none of the points shall collide with the robot can be written in terms of an infinite number of constraints per obstacle point. We show that the resulting semi-infinite programming (SIP) optimal control problem (OCP) can be efficiently tackled through a combination of two methods: local reduction and an external active-set method. Specifically, this involves iteratively identifying the closest point obstacles, determining the lower-level distance minimizer among all feasible robot shape parameters, and solving the upper-level finitely-constrained subproblems. In addition, this paper addresses robust collision avoidance in the presence of ellipsoidal state uncertainties. Enforcing constraint satisfaction over all possible uncertainty realizations extends the dimension of constraint infiniteness. The infinitely many constraints arising from translational uncertainty are handled by local reduction together with the robot shape parameterization, while rotational uncertainty is addressed via a backoff reformulation. A controller implemented based on the proposed method is demonstrated on a real-world robot running at 20Hz, enabling fast and collision-free navigation in tight spaces. An application to 3D collision avoidance is also demonstrated in simulation.

翻译：本文提出了一种在最优与模型预测控制中实现避碰的新方法，其中环境由大量点表示，机器人则表示为加厚多边形的并集。所有点不与机器人发生碰撞的条件可表述为每个障碍物点对应的无限约束。我们证明，通过结合局部缩减法与外部有效集法两种技术，可有效求解由此产生的半无限规划最优控制问题。具体而言，该方法通过迭代识别最近点障碍物、确定所有可行机器人形状参数中的下层距离最小化器，并求解上层有限约束子问题来实现。此外，本文还研究了椭球状态不确定性下的鲁棒避碰问题。对所有可能不确定性实现施加约束满足将扩展约束无穷维度的范围。针对平移不确定性产生的无限约束，采用局部缩减法与机器人形状参数化相结合的处理方式；而旋转不确定性则通过回退重构方法解决。基于所提方法实现的控制器已在真实机器人上以20Hz频率运行，实现狭窄空间内的快速无碰撞导航。同时，通过仿真验证了该方法在三维避碰中的应用效果。