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.
Path planning is a fundamental component of autonomous vehicles, where achieving safe, comfortable, and dynamically feasible paths while ensuring computational efficiency remains a significant challenge. This paper presents a sequential path planning framework in which a rough path obtained from graph search is explicitly exploited to guide a Model Predictive Control (MPC)-based path refinement. A rough path is first obtained via Dijkstra search on a discretized grid and is then used to construct a spatially varying convex lateral safety corridor that explicitly captures obstacle avoidance constraints, transforming discrete obstacle avoidance decisions into continuous feasibility constraints for optimization. Within this corridor, an MPC problem is formulated to refine the path, enabling efficient optimization while maintaining path smoothness by penalizing the third-order spatial derivative of the lateral offset over a prediction horizon. The proposed algorithm is evaluated in multiple overtaking scenarios on both straight and curved roads, including cases with single and multiple target vehicles, using high-fidelity environment simulations (i.e., CarMaker). Compared with the previous study, which used polynomial fitting and a quadratic programming method, the proposed approach consistently achieves lower lateral acceleration, curvature, and jerk while reducing computational cost by 28.08% on straight roads and 29.52% on curved roads. These results demonstrate that exploiting graph-search structure within an MPC formulation provides an effective balance between path smoothness and computational efficiency for autonomous vehicles in structured driving environments.
Autonomous driving requires reliable collision avoidance in dynamic environments. Nonlinear Model Predictive Controllers (NMPCs) are suitable for this task, but struggle in time-critical scenarios requiring high frequency. To meet this demand, optimization problems are often simplified via linearization, narrowing the horizon window, or reduced temporal nodes, each compromising accuracy or reliability. This work presents the first general convex obstacle avoidance formulation, enabled by a novel approach to integrating logic. This facilitates the incorporation of an obstacle avoidance formulation into convex MPC schemes, enabling a convex optimization framework with substantially improved computational efficiency relative to conventional nonconvex methods. A key property of the formulation is that obstacle avoidance remains effective even when obstacles lie outside the prediction horizon, allowing shorter horizons for real-time deployment. In scenarios where nonconvex formulations are unavoidable, the proposed method meets or exceeds the performance of representative nonconvex alternatives. The method is evaluated in autonomous vehicle applications, where system dynamics are highly nonlinear.
Motion planning problems such as collision-free navigation and contact-rich manipulation can be naturally formulated as optimization problems that couple discrete decisions with continuous trajectories. The Graphs of Convex Sets (GCS) framework offers a practical solution to these problems. It represents discrete decisions as nodes of a graph and encodes continuous trajectories in the edges connecting them. However, the resulting optimization subproblems can become computationally prohibitive for online replanning. In this work, we propose a learning-based strategy to mitigate this limitation. Specifically, we replace the costly convex relaxation step required by nominal GCS with a single forward pass through a Graph Attention Network that predicts a set of highly probable candidate paths through the graph. A lightweight ranking network then orders these candidates by their estimated trajectory cost. Evaluating them in this order, we terminate our search early while still recovering a near-optimal motion plan. We validate the resulting pipeline across diverse robotic tasks, including collision-free motion planning for a 3D quadrotor and a 7-DoF manipulator, and planning through contact for planar pushing. Across both convex and non-convex cost and constraint settings, our approach yields up to two orders of magnitude speedup over nominal GCS while maintaining a 100% success rate, at the cost of some suboptimality in the recovered solutions. Code implementations and video demonstrations can be found at https://neural-gcs.github.io/.