Motion Planning for Autonomous Vehicles using Optimization over Graphs of Convex Sets

2026-05-13

AI Summary

This work addresses the challenge of generating real-time, collision-free, and dynamically feasible trajectories for autonomous vehicles in complex environments by proposing a structured optimization approach based on Graphs of Convex Sets (GCS). The method models free space as a GCS and integrates Bézier curve path parameterization with polynomial time scaling within each convex region, embedding trajectory constraints under a simplified bicycle model and linear tire assumptions. By reformulating the nonlinear optimal control problem as a graph-based optimization that preserves convexity through continuous relaxation, the approach effectively unifies geometric and dynamic constraints. Experimental results demonstrate that the generated trajectories achieve efficient static obstacle avoidance and lane changes in CommonRoad scenarios, attaining solution accuracy comparable to nonlinear programming while significantly improving computational efficiency and reducing sensitivity to initial conditions.

Abstract

Motion planning for autonomous vehicles requires generating collision-free and dynamically feasible trajectories in complex environments under real-time constraints. While nonlinear optimal control formulations provide high-fidelity solutions, they are computationally demanding and sensitive to initialization, whereas geometric planning methods scale well but often decouple path selection from trajectory optimization. This paper studies the extent to which optimization over Graphs of Convex Sets (GCS) can approximate solutions of nonlinear optimal control problems in the context of autonomous driving. The free space is represented as a finite union of convex regions organized as a directed graph, allowing nonconvex geometry to be handled through discrete connectivity decisions while maintaining convex trajectory constraints within each region. Vehicle motion is parameterized using Bezier curves for the spatial path and a polynomial time-scaling function for temporal evolution. Under small-slip and linear tire assumptions, a simplified dynamic bicycle model enables approximate enforcement of dynamic feasibility through convex constraints on trajectory derivatives. The approach is evaluated in CommonRoad scenarios involving static obstacle avoidance and lane-changing maneuvers, and is compared against a nonlinear discrete-time optimal control formulation. The results indicate that the GCS-based method generates collision-free and dynamically consistent trajectories that closely match those obtained from the nonlinear program, while exhibiting improved computational efficiency and reduced sensitivity to initialization. These findings suggest that GCS provides a structured approximation of nonlinear motion planning problems, capturing dominant geometric and dynamic effects while preserving convexity in the continuous relaxation.

Problem

• Motion Planning
• Autonomous Vehicles
• Nonlinear Optimal Control
• Dynamic Feasibility
• Collision-Free Trajectories

Innovation

• Graphs of Convex Sets
• motion planning
• autonomous vehicles
• convex optimization
• Bezier curves