Collision-Free Trajectory Optimization in Cluttered Environments with Sums-of-Squares Programming

In this work, we propose a trajectory optimization approach for robot navigation in cluttered 3D environments. We represent the robot's geometry as a semialgebraic set defined by polynomial inequalities such that robots with general shapes can be suitably characterized. To address the robot navigation task in obstacle-dense environments, we exploit the free space directly to construct a sequence of free regions, and allocate each waypoint on the trajectory to a specific region. Then, we incorporate a uniform scaling factor for each free region, and formulate a Sums-of-Squares (SOS) optimization problem that renders the containment relationship between the robot and the free space computationally tractable. The SOS optimization problem is further reformulated to a semidefinite program (SDP), and the collision-free constraints are shown to be equivalent to limiting the scaling factor along the entire trajectory. In this context, the robot at a specific configuration is tailored to stay within the free region. Next, to solve the trajectory optimization problem with the proposed safety constraints (which are implicitly dependent on the robot configurations), we derive the analytical solution to the gradient of the minimum scaling factor with respect to the robot configuration. As a result, this seamlessly facilitates the use of gradient-based methods in efficient solving of the trajectory optimization problem. Through a series of simulations and real-world experiments, the proposed trajectory optimization approach is validated in various challenging scenarios, and the results demonstrate its effectiveness in generating collision-free trajectories in dense and intricate environments populated with obstacles.

Geometry-Aware Safety-Critical Local Reactive Controller for Robot Navigation in Unknown and Cluttered Environments

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.