This work proposes a safety-critical local reactive controller that enables the robot to navigate in unknown and cluttered environments. In particular, the trajectory tracking task is formulated as a constrained polynomial optimization problem. Then, safety constraints are imposed on the control variables invoking the notion of polynomial positivity certificates in conjunction with their Sum-of-Squares (SOS) approximation, thereby confining the robot motion inside the locally extracted convex free region. It is noteworthy that, in the process of devising the proposed safety constraints, the geometry of the robot can be approximated using any shape that can be characterized with a set of polynomial functions. The optimization problem is further convexified into a semidefinite program (SDP) leveraging truncated multi-sequences (tms) and moment relaxation, which favorably facilitates the effective use of off-the-shelf conic programming solvers, such that real-time performance is attainable. Various robot navigation tasks are investigated to demonstrate the effectiveness of the proposed approach in terms of safety and tracking performance.

翻译：本工作提出了一种安全关键型局部反应式控制器，使机器人能够在未知且杂乱的环境中完成导航任务。具体而言，轨迹跟踪问题被构建为带约束的多项式优化问题。随后，通过引入多项式正性证书概念及其平方和（SOS）近似，在控制变量上施加安全约束，从而将机器人运动限制在局部提取的凸自由区域内。值得注意的是，在设计所提出的安全约束过程中，机器人的几何形状可采用任意能通过一组多项式函数表征的几何体进行近似。该优化问题进一步利用截断多序列（tms）与矩松弛方法凸化为半定规划（SDP），这有利于有效利用现成的锥规划求解器，从而实现实时性能。通过多种机器人导航任务验证了所提方法在安全性和跟踪性能方面的有效性。