Collision-Free Trajectory Optimization in Cluttered Environments Using 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 nav...

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Bibliographic Details
Published in:arXiv.org 2024-08
Main Authors: Li, Yulin, Zheng, Chunxin, Chen, Kai, Xie, Yusen, Tang, Xindong, Michael Yu Wang, Ma, Jun
Format: Article
Language:English
Subjects:
Barriers
Collision avoidance
Collision dynamics
Configurations
Exact solutions
Navigation
Obstacle avoidance
Optimization
Polynomials
Robots
Scaling factors
Trajectory optimization
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Summary: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. Our code is available at: https://github.com/lyl00/minimum_scaling_free_region
ISSN:2331-8422