Two-dimensional Decompositions of High-dimensional Configurations for Efficient Multi-vehicle Coordination at Intelligent Intersections

For multi-vehicle complex traffic scenarios in shared spaces such as intelligent intersections, safe coordination and trajectory planning is challenging due to computational complexity. To meet this challenge, we introduce a computationally efficient method for generating collision-free trajectories along predefined vehicle paths. We reformulate a constrained minimum-time trajectory planning problem as a problem in a high-dimensional configuration space, where conflict zones are modeled by high-dimensional polyhedra constructed from two-dimensional rectangles. Still, in such a formulation, as the number of vehicles involved increases, the computational complexity increases significantly. To address this, we propose two algorithms for near-optimal local optimization that significantly reduce the computational complexity by decomposing the high-dimensional problem into a sequence of 2D graph search problems. The resulting trajectories are then incorporated into a Nonlinear Model Predictive Control (NMPC) framework to ensure safe and smooth vehicle motion. We furthermore show in numerical evaluation that this approach significantly outperforms existing MILP-based time-scheduling; both in terms of objective-value and computational time.

💡 Research Summary

This paper presents a novel and computationally efficient framework for safe and cooperative trajectory planning of multiple Connected Autonomous Vehicles (CAVs) at unsignalized, intelligent intersections. The core challenge addressed is the combinatorial explosion of complexity when coordinating N vehicles, which traditional methods like Mixed-Integer Linear Programming (MILP) struggle with as N grows.

The authors’ key innovation is a dimensionality reduction strategy. Instead of solving the optimal coordination problem directly in the high-dimensional joint configuration space of all vehicles (where dimensions correspond to each vehicle’s progress along its predefined path), they decompose it into a sequence of manageable two-dimensional (2D) sub-problems. In this configuration space, potential collision zones between any two vehicles are represented as high-dimensional axis-aligned rectangles (hyper-rectangles). The critical observation is that the optimal path for two vehicles within such a 2D subspace, with rectangular obstacles, will pass through the corners or boundaries of these rectangles. Furthermore, since vehicles are assumed to only move forward along their paths, the visibility graph in each 2D subspace becomes a Directed Acyclic Graph (DAG), allowing for very efficient shortest-path search with complexity O(r^2), where r is the number of conflict regions.

The overall coordination path for N vehicles is constructed by sequentially solving these 2D sub-problems according to a specific order or permutation of vehicle pairings. The paper investigates two distinct algorithms for determining this order, analyzing their computational complexity (related to the number of permutations considered) and their impact on solution quality. The final output is a path in the (N-1)-dimensional configuration space, represented as a time-scaled unit vector whose components define the relative velocity profile for each vehicle along its path.

This planned high-level trajectory is then passed to a low-level tracking controller. The paper employs a completely decoupled Nonlinear Model Predictive Control (NMPC) scheme. Each vehicle has its own NMPC controller based on a kinematic bicycle model. The controller’s objective is to track the assigned velocity profile from the planner while respecting individual vehicle dynamics and constraints (e.g., acceleration, steering limits), ensuring smooth and dynamically feasible motion.

Numerical evaluations demonstrate the significant advantages of the proposed approach. Compared to a baseline method that uses MILP for high-level scheduling, the decomposition-based planners achieve markedly lower total crossing times (better objective values) and, most importantly, drastically reduced computation times. This speedup becomes increasingly pronounced as the number of vehicles increases, showcasing the method’s scalability and potential for real-time application in complex multi-vehicle intersection scenarios. The work effectively bridges high-level combinatorial coordination and low-level dynamic control through a coherent optimization framework.