[Paper Review] Temporal Logic Task Allocation in Heterogeneous Multi-Robot Systems

Xusheng Luo, Michael M. Zavlanos|arXiv (Cornell University)|Jan 14, 2021

Formal Methods in Verification4 citations

TL;DR

This paper proposes a hierarchical method for optimal task allocation in heterogeneous multi-robot systems using Linear Temporal Logic (LTL) specifications. By pruning and relaxing the Nondeterministic B"uchi Automaton (NBA) to defer constraint checking, it formulates a Mixed Integer Linear Program (MILP) for subtask allocation and solves generalized multi-robot path planning (GMRPP) problems to generate executable, constraint-compliant trajectories. The approach is complete for a broad class of LTL tasks and significantly outperforms existing methods in scalability and solution cost.

ABSTRACT

In this paper, we consider the problem of optimally allocating tasks, expressed as global Linear Temporal Logic (LTL) specifications, to teams of heterogeneous mobile robots. The robots are classified in different types that capture their different capabilities, and each task may require robots of multiple types. The specific robots assigned to each task are immaterial, as long as they are of the desired type. Given a discrete workspace, our goal is to design paths, i.e., sequences of discrete states, for the robots so that the LTL specification is satisfied. To obtain a scalable solution to this complex temporal logic task allocation problem, we propose a hierarchical approach that first allocates specific robots to tasks using the information about the tasks contained in the Nondeterministic Buchi Automaton (NBA) that captures the LTL specification, and then designs low-level executable plans for the robots that respect the high-level assignment. Specifically, we first prune and relax the NBA by removing all negative atomic propositions. This step is motivated by "lazy collision checking" methods in robotics and allows to simplify the planning problem by checking constraint satisfaction only when needed. Then, we extract sequences of subtasks from the relaxed NBA along with their temporal orders, and formulate a Mixed Integer Linear Program (MILP) to allocate these subtasks to the robots. Finally, we define generalized multi-robot path planning problems to obtain low-level executable robot plans that satisfy both the high-level task allocation and the temporal constraints captured by the negative atomic propositions in the original NBA. We show that our method is complete for a subclass of LTL that covers a broad range of tasks and present numerical simulations demonstrating that it can generate paths with lower cost, considerably faster than existing methods.

Motivation & Objective

• To address the computational intractability of optimal LTL task allocation in large-scale heterogeneous multi-robot systems.
• To enable efficient, scalable task allocation where robot types differ in capabilities and tasks require coordinated, temporally ordered actions.
• To decouple high-level task assignment from low-level path planning while preserving temporal logic constraints.
• To ensure completeness and correctness of the solution for a broad subclass of LTL specifications, including co-safe and finite-horizon LTL.
• To demonstrate superior performance in solution cost and computation time compared to existing product-automaton-based methods.

Proposed method

• Prune and relax the Nondeterministic B"uchi Automaton (NBA) by removing negative atomic propositions to simplify initial planning.
• Extract temporal sequences of subtasks from the relaxed NBA, preserving their required order for task execution.
• Formulate a Mixed Integer Linear Program (MILP) inspired by the vehicle routing problem to assign subtasks to robots, respecting temporal ordering and robot type constraints.
• Generate time-stamped task allocation plans for each robot, specifying essential waypoints and visit times.
• Solve a sequence of Generalized Multi-Robot Path Planning (GMRPP) problems to generate executable, collision-free trajectories that satisfy the original NBA’s negative atomic propositions.
• Close robot trajectories by returning robots involved in the last subtask to their initial locations, ensuring specification satisfaction.

Experimental results

Research questions

• RQ1 Can a hierarchical decomposition of the LTL-MRTA problem achieve optimal or near-optimal solutions with significantly reduced computational complexity?
• RQ2 Does deferring the enforcement of negative atomic propositions through NBA relaxation preserve solution correctness and completeness?
• RQ3 How does the proposed method scale in terms of computation time and solution cost compared to product-automaton-based approaches for large-scale multi-robot systems?
• RQ4 To what extent does the method maintain solution quality when the LTL specification involves multiple posets or complex temporal dependencies?
• RQ5 Can the method generate feasible, executable robot paths that satisfy both the temporal logic specification and physical constraints like collision avoidance?

Key findings

• The proposed method is complete for LTL${}^{0}$ specifications and also complete for co-safe LTL and LTL${}_{f}$ when finite-horizon paths exist.
• The method generates paths with lower total cost than existing approaches, as demonstrated in numerical simulations on problems with up to $10^{90}$ states in the product graph.
• In case study I, the first solution for the widest poset achieved a cost within 1.1x of the optimal cost in 35 out of 50 trials, indicating strong suboptimality bounds.
• The method outperforms the approach in [sahin2019multi] in both optimality and scalability, particularly for large workspaces and high robot counts.
• The MILP and GMRPP formulations are computationally efficient, enabling solution times significantly faster than brute-force or product-automaton-based methods.
• The soundness of the method is guaranteed: any path generated by the GMRPP step satisfies the original LTL specification $φ$.