A Convex Obstacle Avoidance Formulation

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.

Model Predictive Control (MPC) is widely used for autonomous-vehicle (AV) motion planning, but its real-time applicability is often limited by the need for accurate models and online solution of nonlinear, nonconvex optimization problems in dynamic road environments. Actor-critic reinforcement learning offers a promising alternative for online policy generation, yet its policy-learning process often lacks explicit control-theoretic structure. This article proposes a learning predictive control (LPC) framework with deep Koopman operators for efficient real-time motion planning under nonconvex constraints. To address nonlinear and uncertain vehicle dynamics, a deep-Koopman-based predictor is used to lift the system into an interpretable linear observable space in a data-driven manner. Unlike traditional MPC, which computes open-loop control sequences, the proposed LPC framework yields a closed-loop state-feedback policy within each prediction interval through receding-horizon actor-critic learning. To ensure safety under nonconvex environmental constraints, LPC constructs convex local surrogate representations of obstacles and defines corresponding potential-field functions. These functions and their gradients are directly embedded into the actor-critic structure, enabling efficient, safety-aware policy learning. Extensive simulations and real-world experiments on the HongQi-EHS3 platform demonstrate favorable performance in diverse obstacle-avoidance scenarios in terms of safety, computational efficiency, and driving comfort, compared with benchmark methods such as CBF-MPC and LMPCC.

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.