[Paper Review] A message-passing algorithm for multi-agent trajectory planning

José Bento, Nate Derbinsky+2 more|arXiv (Cornell University)|Nov 18, 2013

Robotic Path Planning Algorithms21 references15 citations

TL;DR

This paper proposes a message-passing algorithm based on the three-weight alternating direction method of multipliers (TWA) for multi-agent trajectory planning, enabling parallel, scalable, and flexible computation of collision-free trajectories. It achieves efficient global optimization by decomposing the problem into local sub-problems solved via message-passing, with empirical results showing strong scalability and performance across various cost functionals and agent counts.

ABSTRACT

We describe a novel approach for computing collision-free \emph{global} trajectories for $p$ agents with specified initial and final configurations, based on an improved version of the alternating direction method of multipliers (ADMM). Compared with existing methods, our approach is naturally parallelizable and allows for incorporating different cost functionals with only minor adjustments. We apply our method to classical challenging instances and observe that its computational requirements scale well with $p$ for several cost functionals. We also show that a specialization of our algorithm can be used for {\em local} motion planning by solving the problem of joint optimization in velocity space.

Motivation & Objective

• To address the computational complexity of centralized multi-agent trajectory planning, which is PSPACE-hard and traditionally solved suboptimally via sequential methods.
• To develop a scalable, parallelizable method that simultaneously plans for all agents while ensuring collision-free trajectories.
• To enable flexible integration of diverse cost functionals—such as energy minimization, obstacle avoidance, and velocity constraints—within a unified optimization framework.
• To demonstrate that the method scales efficiently with the number of agents, outperforming traditional MIQP and sequential approaches in both speed and solution quality.
• To show that the framework can be specialized for local motion planning in velocity space, improving real-time performance.

Proposed method

• The method formulates trajectory planning as a decomposable optimization problem over piece-wise linear trajectories defined by shared break-points across all agents.
• It uses the three-weight ADMM (TWA) algorithm to solve the problem via message-passing between agents, enabling natural parallelization and avoiding parameter tuning.
• Each agent's trajectory is optimized subject to local constraints (e.g., collision avoidance, energy, velocity bounds), with global coordination enforced through consensus variables.
• The algorithm solves non-convex sub-problems exactly and efficiently using closed-form solutions or Newton’s method for one-dimensional minimization, particularly in agent-agent and agent-obstacle collision avoidance.
• The framework supports multiple minimizers—such as for collision avoidance, energy, and obstacle constraints—allowing flexible combination through a modular optimization structure.
• A specialized variant of the algorithm is applied to local motion planning by optimizing in velocity space, enabling real-time reactive collision avoidance.

Experimental results

Research questions

• RQ1 Can a message-passing algorithm based on TWA achieve scalable, parallel, and collision-free trajectory planning for multiple agents with minimal parameter tuning?
• RQ2 How does the method’s performance and scalability compare to traditional MIQP and sequential planning approaches across varying numbers of agents?
• RQ3 To what extent can the framework flexibly incorporate diverse cost functionals such as energy minimization, obstacle avoidance, and velocity constraints?
• RQ4 Can the same framework be adapted for local motion planning by reformulating the optimization in velocity space?
• RQ5 What is the empirical convergence behavior and runtime scaling of the algorithm with increasing agent count, particularly in non-adversarial configurations?

Key findings

• The proposed TWA-based algorithm achieves significantly faster convergence than standard ADMM, with empirical evidence of an order-of-magnitude speedup due to key algorithmic modifications.
• Computational requirements scale well with the number of agents, showing polynomial rather than exponential growth in runtime, especially in non-adversarial scenarios.
• The method successfully computes high-quality, collision-free trajectories across multiple cost functionals, including energy minimization and obstacle avoidance, with minimal tuning.
• In numerical experiments, the algorithm demonstrated strong scalability, with runtime for 8 agents not exhibiting exponential growth when initial and final positions were randomly distributed.
• The framework can be effectively specialized for local motion planning in velocity space, outperforming existing centralized methods in efficiency and scalability.
• Even when stopped early, the algorithm produces feasible, non-colliding trajectories at low computational cost, making it suitable for real-time applications.