V*: An Efficient Motion Planning Algorithm for Autonomous Vehicles

Autonomous vehicle navigation in structured environments requires planners capable of generating time-optimal, collision-free trajectories that satisfy dynamic and kinematic constraints. We introduce V*, a graph-based motion planner that represents speed and direction as explicit state variables within a discretised space-time-velocity lattice. Unlike traditional methods that decouple spatial search from dynamic feasibility or rely on post-hoc smoothing, V* integrates both motion dimensions directly into graph construction through dynamic graph generation during search expansion. To manage the complexity of high-dimensional search, we employ a hexagonal discretisation strategy and provide formal mathematical proofs establishing optimal waypoint spacing and minimal node redundancy under constrained heading transitions for velocity-aware motion planning. We develop a mathematical formulation for transient steering dynamics in the kinematic bicycle model, modelling steering angle convergence with exponential behaviour, and deriving the relationship for convergence rate parameters. This theoretical foundation, combined with geometric pruning strategies that eliminate expansions leading to infeasible steering configurations, enables V* to evaluate dynamically admissible manoeuvres, ensuring each trajectory is physically realisable without further refinement. We further demonstrate V*'s performance in simulation studies with cluttered and dynamic environments involving moving obstacles, showing its ability to avoid conflicts, yield proactively, and generate safe, efficient trajectories with temporal reasoning capabilities for waiting behaviours and dynamic coordination.