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STyLuS*: A Temporal Logic Optimal Control Synthesis Algorithm for Large-Scale Multi-Robot Systems
Yiannis Kantaros, Michael M. Zavlanos
TL;DR
Large multi-robot LTL planning becomes difficult when explicit product automata grow exponentially. STyLuS* uses biased sampling guided by feasible shortest paths in the Büchi automaton, retaining formal guarantees and enabling synthesis at scales beyond existing methods.
Problem
Explicit graph search over product automata becomes intractable as the number of robots or the NBA size increases, limiting optimal LTL planning for large multi-robot systems.
Method
STyLuS* incrementally builds a tree approximating the product automaton while biasing samples toward feasible NBA transitions on shortest paths to accepting states.
Results
The algorithm is probabilistically complete, asymptotically optimal, and exponentially convergent, and experiments show synthesis for product automata with hundreds of orders of magnitude more states than state-of-the-art methods can manipulate.
Takeaways & Limitations
STyLuS* provides optimal motion plans for very large multi-robot systems beyond the scale handled by existing optimal synthesis methods and off-the-shelf model checkers.
Abstract
from arXiv · showhide
This paper proposes a new highly scalable and asymptotically optimal control synthesis algorithm from linear temporal logic specifications, called $\text{STyLuS}^{*}$ for large-Scale optimal Temporal Logic Synthesis, that is designed to solve complex temporal planning problems in large-scale multi-robot systems. Existing planning approaches with temporal logic specifications rely on graph search techniques applied to a product automaton constructed among the robots. In our previous work, we have proposed a more tractable sampling-based algorithm that builds incrementally trees that approximate the state-space and transitions of the synchronous product automaton and does not require sophisticated graph search techniques. Here, we extend our previous work by introducing bias in the sampling process which is guided by transitions in the B$\ddot{\text{u}}$chi automaton that belong to the shortest path to the accepting states. This allows us to synthesize optimal motion plans from product automata with hundreds of orders of magnitude more states than those that existing optimal control synthesis methods or off-the-shelf model checkers can manipulate. We show that $\text{STyLuS}^{*}$ is probabilistically complete and asymptotically optimal and has exponential convergence rate. This is the first time that convergence rate results are provided for sampling-based optimal control synthesis methods. We provide simulation results that show that $\text{STyLuS}^{*}$ can synthesize optimal motion plans for very large multi-robot systems which is impossible using state-of-the-art methods.
I. INTRODUCTION
The paper addresses optimal LTL control synthesis for collaborative multi-robot tasks, where explicit product-automaton search becomes intractable as system size grows. It introduces STyLuS*, a biased sampling-based extension designed for scalable synthesis with completeness, optimality, and convergence guarantees.
- Motivation: Product-automaton state spaces grow exponentially with robot count and NBA size, making graph-search and A*-type methods intractable for large systems.Existing approaches construct a synchronous product among robot transition systems and the NBA before searching for optimal plans.
- Proposed approach: STyLuS* extends incremental tree-based synthesis by biasing samples toward NBA transitions on shortest feasible paths to accepting states.The method first prunes infeasible NBA transitions and then uses an NBA distance metric to guide sampling.
- Scope: STyLuS* is designed for complex global LTL tasks in large-scale multi-robot systems rather than only point-to-point navigation.Related sampling-based methods described in the paper focus on simpler navigation tasks or require arbitrary graph abstractions.
- Guarantees: The algorithm preserves probabilistic completeness and asymptotic optimality while adding exponential convergence-rate guarantees.The paper presents these guarantees as an extension beyond the preliminary work.
- Problem formulation: Robot mobility is modeled with weighted transition systems whose states represent locations and whose transitions carry travel costs.The robots' transition systems are synchronously combined into a product transition system before incorporating the LTL automaton.
A. A Solution to Problem 1
The paper solves optimal LTL synthesis by incrementally constructing a directed tree that approximates the product Büchi automaton instead of explicitly searching its full graph. Feasible NBA transitions and shortest-path distances provide the structure used for this synthesis.
- Scalability: The approach avoids explicit arbitrary-graph representation of the full product automaton, targeting the scalability problem of conventional graph search.The product automaton is represented approximately through sampled tree nodes and edges.
- Tree-based synthesis: The method incrementally constructs a directed tree that approximates the product Büchi automaton's state space and transitions.The tree stores sampled product states, transitions between them, and the cost of reaching each node from the root.
- Plan construction: The algorithm translates and prunes the NBA, builds prefix and suffix parts, and combines them into an optimal prefix-suffix plan satisfying the LTL specification.The synthesized plan has the form τ = τ_pre[τ_suf]^ω and satisfies the specification.
- NBA guidance: The distance function d assigns the minimum number of feasible NBA transitions between states, using unit-weight shortest paths in the pruned NBA graph.This distance later guides sampling toward accepting states.
- Feasibility: Feasible NBA transitions are those enabled by symbols that can be generated by the product transition system.Robot-level feasibility excludes symbols requiring a robot to occupy multiple disjoint regions simultaneously.
B. Construction of Optimal Prefix Parts
Optimal prefix construction selects feasible accepting NBA states and incrementally grows a product-state tree toward them. Sampling is biased toward tree nodes and successor states that reduce NBA distance to the selected accepting state.
- Goal selection: The prefix tree targets a randomly selected feasible NBA final state reachable from an initial state and supporting a suffix loop.If no such final state exists, the construction stops without detecting a prefix-suffix plan.
- Biased sampling: The sampling distribution selects minimum-distance tree nodes with probability p_rand greater than 0.5, favoring states closer to the accepting target.The set D_min contains product states whose NBA components have minimum distance to the selected final state.
- Guarantees: The biased density satisfies the assumptions needed for probabilistic completeness and asymptotic optimality, while the bias improves scalability in experiments.The distribution may vary across iterations as long as the required assumptions and bias properties are preserved.
- Successor construction: Each sampled product state combines a reachable robot configuration with an NBA state selected through feasible transitions that decrease or minimize the target distance.The procedure constructs reachable NBA successors, identifies minimum-distance candidates, and samples among transitions that move toward the target.
1 Pick a state qrand
The sampling step first selects an existing tree state, then generates reachable robot states and compatible NBA states to extend the product tree. Its distributions favor states and successors that move toward feasible accepting states.
- Pick a state qrand: The generated robot states are paired with candidate NBA states, and each resulting product state is added only if it is new and has a valid tree parent.Existing states trigger rewiring intended to reduce tree costs.
- Pick a state qrand: The robot-state sampling density may change across iterations but must preserve assumptions for completeness and optimality while favoring shortest-path successors.The paper identifies this bias as a source of improved scalability.
2) Construction of Paths:
The prefix construction builds trees toward feasible accepting states, then extracts paths to form candidate prefix plans.
- The goal-state set collects tree states lying in the prefix goal region for subsequent plan extraction.
- Prefix plans are obtained by tracing parent nodes from each accepting tree state back to the root.
- Sampling is biased toward product states containing feasible final states of the Büchi automaton.
- After feasible final states are detected or a preset iteration limit is reached, sampling can switch to unbiased exploration.
C. Construction of Optimal Suffix Parts
Suffix construction builds a loop from each prefix endpoint back to itself, using biased sampling to reach the root and then identifying a feasible cycle.
- Each suffix plan starts and ends at the same prefix endpoint, producing the repeating loop required by the prefix-suffix structure.
- Suffix-tree construction uses a root at the prefix endpoint and is repeated for every prefix plan, in parallel when possible.
- If the root already lies in the suffix goal region, the suffix tree consists of the root and a zero-cost self-loop.
- Sampling is biased toward the root so that a loop around the prefix endpoint can be detected.
- Once a suffix candidate is found, unbiased sampling can resume to explore the product state space in all directions.
D. Construction of Optimal Discrete Plans
The algorithm combines prefix and suffix candidates into LTL-satisfying plans, selects the lowest-cost plan, and establishes completeness and asymptotic optimality under biased sampling.
- The algorithm selects the motion plan with the smallest cost among plans satisfying the LTL specification.
- The tree representation requires O(|V_T|) memory because it contains |V_T|−1 edges, compared with O(|V_P| + |E_P|) for the explicit product graph.
- The sampling densities preserve the assumptions needed to inherit probabilistic completeness and asymptotic optimality from the prior sampling method.
- The biased process can change its target state or switch to uniform sampling while maintaining the required lower-bound conditions.
- STyLuS* finds with probability 1 an optimal LTL-satisfying motion plan as prefix and suffix iterations grow without bound.
B. Rate of Convergence
STyLuS* is shown to converge exponentially to feasible and optimal solutions under a prefix-suffix solution assumption. The analysis models sampling successes probabilistically and uses Chernoff bounds to establish the convergence guarantees.
- The analysis first establishes exponential convergence to a feasible solution before extending the result to the optimal path.
- The proof represents sampling progress with Bernoulli variables whose sum follows a Poisson binomial distribution.
- Chernoff bounds provide a lower bound on the probability of accumulating enough sampling successes to detect a goal state.
- Biasing toward shorter paths can accelerate detection, but states favored by the bias may be infeasible, so probabilities near one should be avoided.
- STyLuS* converges exponentially fast to the optimal solution of Problem 1 when a prefix-suffix solution exists.
- Biasing toward selected paths preserves exponential convergence provided the sampling assumptions hold, even when the bias changes or becomes temporarily unbiased.
V. NUMERICAL EXPERIMENTS
The numerical experiments evaluate STyLuS* scalability and feasibility through case studies using increasingly large transition systems. The section also describes reachability-based sampling and proposition-based formula reduction used to control computational cost.
- The experiments assess STyLuS* scalability against state-of-the-art methods using MATLAB case studies on a 2.2GHz Intel Core i7 computer with 4GB RAM.
- Reducing atomic propositions to Boolean formulas can produce smaller LTL formulas and lower the cost of constructing the NBA and feasible-symbol sets.
- The experiments report iterations and constructed-tree sizes required to find the first prefix and suffix parts of feasible plans.
A. Completeness, Optimality, and Scalability
STyLuS* is evaluated for probabilistic completeness, scalability, sampling-density effects, and asymptotic optimality across increasingly large multi-robot planning problems. It remains usable where model checkers become impractical and reaches lower-cost plans in the reported small benchmark.
- Probabilistic Completeness & Scalability: As robot count or transition-system size increased, STyLuS* outperformed NuSMV and nuXmv in runtime and problem size, while model checkers failed on denser large instances.For N = 1 and |Qi| = 10^4 with average degree 42, both NuSMV and nuXmv failed to build the model, while SPIN also exceeded memory requirements.
- Probabilistic Completeness & Scalability: STyLuS* handled a team of 200 robots with |Qi| = 10^4 states per robot and a PBA of approximately 10^801 states in almost 35 minutes.The corresponding LTL formulas contained thousands of logical operators and atomic propositions.
- Transition-System Sparsity: For N = 10 robots and |Qi| = 1000 states, denser transition systems required fewer iterations but initially increased total runtime for the first feasible prefix-suffix plan.The runtime eventually decreased for fully connected systems because only a few iterations were needed.
- Switching between sampling densities: Biased sampling produced lower-cost initial plans than uniform sampling, reducing prefix cost from 415.48 to 314.58 meters versus 823.96 to 706.18 meters within 5.5 hours.The biased run detected 156 final states and found the first after 0.42 seconds, whereas uniform sampling detected 4 and found the first after 1.34 hours.
- Asymptotic Optimality: STyLuS* found a 5-meter optimal prefix, compared with 8.8824 meters from NuSMV and nuXmv, in the two-robot asymptotic-optimality case study.The benchmark product automaton had 6,144 states, allowing comparison with existing optimal synthesis methods.
B. Scalability for larger and denser NBA
With a larger Büchi automaton, STyLuS* required more iterations and runtime for first-plan detection, but retained its scalability advantage as robot count or transition-system size grew. The experiments also show that self-loops materially reduce STyLuS*'s planning effort.
- Larger and denser NBA: The larger NBA increased STyLuS*'s iterations and runtime for detecting the first feasible plan compared with the corresponding problems in Table I.The larger automaton used |QB| = 59, with 6 feasible final states and 884 transitions.
- Larger and denser NBA: As robot count or transition-system size increased under the larger NBA, STyLuS* outperformed NuSMV and nuXmv in runtime and the size of state-space handled.For few robots and small transition systems, both model checkers could find the first feasible plan faster.
- Self-loops in wTSs: Removing self-loops increased iterations from 33 to 533 for the prefix and from 33 to 991 for the suffix in the order-10^21 problem.The corresponding runtimes rose from 0.7 to 7.1 seconds and from 0.4 to 10.6 seconds, respectively.
C. Comparison with off-the-shelf model checkers: Summary
STyLuS* outperforms NuSMV and nuXmv on large, dense transition systems, while uniquely providing exponentially convergent optimal-plan detection and supporting collision-avoidance constraints.
- STyLuS* outperforms NuSMV and nuXmv for large, dense transition systems, regardless of self-loops or NBA size.
- NuSMV and nuXmv find feasible paths, whereas STyLuS* detects the optimal plan with probability converging exponentially to 1.
- STyLuS* checks collision-avoidance safety properties during sampling, which NuSMV and nuXmv cannot handle as input constraints.
- As the required inter-robot distance R increases, STyLuS* runtime increases because more samples are rejected.
D. The effect of biased sampling
Biased sampling steers exploration toward feasible final states and shortest paths in the Büchi automaton. It can detect shorter feasible prefixes and build smaller trees, although its advantage depends on the target state and automaton structure.
- The experiments compare STyLuS* against uniform sampling on a planning problem small enough to support both sampling strategies.
- Both biased and unbiased sampling probabilities converge to 1 as npre max increases, consistent with the theoretical guarantee.
- The study estimates path-completion and final-state success probabilities over 100 experiments for different prefix-sampling budgets and target final states.The analysis compares biased and unbiased densities for feasible prefixes connected to selected final states.
- Sampling bias toward a selected feasible final state does not universally improve success probability; for NBA final state 13, uniform sampling performs better.The passage identifies this as problem-specific behavior dependent on the NBA structure.
- Biased sampling detects feasible prefix paths with smaller K than uniform sampling in the Figure 12 experiments.The comparison uses 100 experiments, varies npre max, and deactivates rewiring.
- For npre max = 1000, biased sampling produces trees averaging 2000 nodes versus 4000 with uniform sampling.Uniform sampling explores more product-automaton states, while biased sampling favors neighborhoods of shortest paths to final states.
APPENDIX A PROOFS OF PROPOSITIONS
The appendix proves that the sampling functions preserve positive probability of reaching relevant product-automaton states, establishing the conditions needed for the algorithm’s guarantees.
- Several proof conditions hold directly by the definitions of the random and new-sample functions, while the remaining condition follows from the positive bounds.
- The proofs establish a strictly positive lower bound on sampling probabilities for relevant states across successive iterations.For the product space, the bound includes (min(ϵ, 1 − ϵ))^N / |Q_PTS|.
- The component-level proof shows that each robot’s sampling function remains bounded away from zero on the relevant transition-system regions.
- These positive bounds imply an infinite sequence of future samples with nonzero probability for reaching fixed product-automaton nodes.