Source-linked AI summary
Finding a needle in an exponential haystack: Discrete RRT for exploration of implicit roadmaps in multi-robot motion planning
Kiril Solovey, Oren Salzman, Dan Halperin
TL;DR
Multi-robot motion planning becomes difficult when complete methods face exponential growth in degrees of freedom and tightly coupled robots. The paper combines an implicit tensor-product roadmap with dRRT, which explores the graph using limited neighborhood information. MRdRRT solves highly coupled problems up to 60 degrees of freedom and is reported to be at least ten times faster than known existing algorithms.
Problem
Complete multi-robot planners can require exponential time as the number of robots and degrees of freedom increase, while existing methods face challenges in tightly coupled scenarios.
Method
MRdRRT maintains an implicit composite tensor-product roadmap and applies dRRT, an RRT adaptation that considers only partial neighborhood information during graph exploration.
Results
MRdRRT solved highly coupled scenarios with up to 60 degrees of freedom and was at least ten times faster than known existing algorithms.
Takeaways & Limitations
The framework provides a practical way to explore only a small portion of a high-dimensional composite configuration space while planning for tightly coupled robots.
Takeaways & Limitations
The completeness result for MRdRRT currently requires modifying dRRT by exposing an unexposed edge through brute force, and an alternative proof is still sought.
Abstract
from arXiv · showhide
We present a sampling-based framework for multi-robot motion planning which combines an implicit representation of a roadmap with a novel approach for pathfinding in geometrically embedded graphs tailored for our setting. Our pathfinding algorithm, discrete-RRT (dRRT), is an adaptation of the celebrated RRT algorithm for the discrete case of a graph, and it enables a rapid exploration of the high-dimensional configuration space by carefully walking through an implicit representation of a tensor product of roadmaps for the individual robots. We demonstrate our approach experimentally on scenarios of up to 60 degrees of freedom where our algorithm is faster by a factor of at least ten when compared to existing algorithms that we are aware of.
1 Introduction
The paper addresses continuous, labeled multi-robot motion planning, where robots must move between assigned starts and targets without collisions. Existing complete methods can be exponential in robot count, while sampling-based and decoupled approaches trade generality or efficiency; MRdRRT combines an implicit composite roadmap with dRRT to address tightly coupled, high-dimensional scenarios.
- Continuous multi-robot planning seeks collision-free paths from each robot’s start position to its assigned target in a shared workspace.
- Complete multi-robot planners guarantee finding a solution when one exists but can require exponential time in the number of robots because of high dimensionality.
- Decoupled planners improve efficiency by solving robots’ problems separately, but typically work only for restricted problem sets.
- Composite-roadmap sampling methods can become impractical because the roadmap grows exponentially with the number of robots.
- MRdRRT maintains an implicit composite roadmap and uses dRRT to inspect only one neighbor at each step while exploring high-dimensional configuration space.
- MRdRRT solved highly coupled problems with up to 60 degrees of freedom and was at least ten times faster than known existing algorithms.
2 Composite roadmaps for multi-robot motion planning
The composite roadmap combines individual robots’ PRM roadmaps into a graph whose vertices are collision-free joint placements. Implicit representation avoids materializing the exponentially large graph, while M* and dRRT differ in how much neighborhood information they explore.
- A composite roadmap is built from individual PRM roadmaps, with each robot’s start and target in the same connected component.
- Its vertices represent combinations of individual robot configurations that are pairwise collision-free.
- Composite roadmap construction: Cartesian-product edges move one robot while the others remain fixed, subject to collision avoidance with stationed robots.
- Composite roadmap construction: Tensor-product edges connect joint configurations when every robot follows an edge in its individual roadmap and the resulting motion remains collision-free.
- Implicit pathfinding: The composite graph can contain O(n^m) vertices, making standard full-graph pathfinding unsuitable.
- Implicit pathfinding: M* dynamically explores lower-dimensional coupled spaces and is effective with low coupling, but its running time rises sharply as coupling increases because it considers many neighbors.
3 Discrete RRT
dRRT adapts RRT to explore an implicitly represented, geometrically embedded graph using random samples and local neighbor queries. It grows a tree through existing graph vertices, attempts target connection, and has a high-probability vertex-revelation guarantee under general-position assumptions.
- 3 Discrete RRT: dRRT explores discrete graphs embedded in Euclidean space rather than generating new vertices and edges in continuous space.The graph is represented implicitly, so neighbor information is retrieved through an oracle.
- 3.1 Oracle to query the implicit graph: The direction oracle selects the neighbor whose outgoing direction forms the smallest angle with the direction toward a query point.This provides partial neighbor information without requiring exhaustive consideration of all neighbors.
- 3.2 Description of dRRT: The algorithm grows a tree from the start, repeatedly expands toward random samples, and stops when it connects to the target.A solution path is then retrieved by concatenating the tree path with the target connection.
- 3.2 Description of dRRT: Each expansion samples qrand, selects the nearest tree node qnear, and adds the graph neighbor qnew whose direction best matches the sample.Because vertices and edges must come from the graph, expansion cannot necessarily proceed exactly toward the random point.
- 3.4 Probabilistic completeness of dRRT: With high probability, dRRT eventually reveals every vertex of a connected graph embedded in [0, 1]^d when vertices satisfy the general-position assumption.The proof uses positive-measure intersections of Voronoi regions; connecting the target is handled separately through a local connector.
4 Multi-robot motion planning with dRRT
MRdRRT adapts dRRT to an implicitly represented composite roadmap by selecting coordinated individual-robot neighbors, rejecting colliding candidates, and using a prioritized local connector. Its probabilistic-completeness proof remains unfinished as roadmap sizes grow, although a finite-graph modification restores completeness.
- Algorithm: MRdRRT applies dRRT to a composite roadmap embedded in the joint configuration space of multiple robots.The section also implements the direction oracle and a local connector using only the individual roadmap representations.
- Oracle OD: The direction oracle independently selects each robot’s neighbor toward its query sample, combines these choices, and returns the candidate only if the joint edge is collision-free.Otherwise, the sample is ignored and another sample is drawn during expansion.
- Local connector: The local connector assigns individual paths and seeks a robot ordering that avoids collisions while higher-priority robots reach their targets.A directed priority graph determines success: an acyclic graph yields a solution, whereas a cycle causes failure.
- Local connector: The connector was chosen for low running-time cost and outperformed M* with bounded coupling in the authors’ experiments.The comparison concerns the local connector component rather than the full planning framework.
- Completeness: Probabilistic completeness still requires proving composite-roadmap convergence and extending Theorem 1 as Voronoi cells shrink with increasing graph size.Because the composite roadmap is finite, exposing unexposed edges can make dRRT complete, though the authors seek a proof without altering the algorithm.
5 Experimental results
Experiments compare MRdRRT with RRT and recursive M* for polyhedral multi-robot motion planning. MRdRRT solved every tested attempt, while RRT failed across scenarios and M* struggled especially when coordination was substantial; substantially larger robot counts remained unsolved by MRdRRT.
- Setup: MRdRRT was evaluated against RRT and recursive M* on polyhedral robots translating and rotating among polyhedral obstacles.The comparison used an inflated M* variant with relaxed optimality guarantees because dRRT does not optimize solution quality.
- Setup: Each algorithm was run 10 times per scenario using a generic composite-roadmap framework with PQP collision detection and FLANN nearest-neighbor queries.The reported table includes roadmap construction, visited vertices, running time, success rate, and MRdRRT phase durations.
- Results: MRdRRT solved all scenarios in all 10 attempts, whereas RRT solved none and M* achieved at most 40% success on the first three scenarios.RRT exceeded memory limits, while M* often exhausted memory or exceeded the prescribed time limit.
- Results: When M* solved one of the first three scenarios, it explored 2.5 to 10 times as many vertices as dRRT.On the fourth scenario, MRdRRT and M* produced comparable results.
- Limitations: MRdRRT could not solve scenarios containing substantially more robots than those used experimentally, motivating a stronger local connector.This is an explicit scope boundary of the reported experiments.
6 Discussion
The discussion attributes MRdRRT’s benefits to implicit composite roadmaps and minimalistic, Voronoi-biased exploration. These choices avoid repeated obstacle checks and help dRRT handle tightly coupled robots, while M* remains better for sparse coordination.
- Benefits: MRdRRT combines implicitly represented roadmaps with an efficient pathfinding method for multi-robot motion planning.The framework’s central benefit is integrating the roadmap representation with dRRT search.
- Benefits: Precomputed individual roadmaps eliminate additional robot-obstacle collision checks when querying the composite roadmap.This matters because checking robot-obstacle collisions is often costlier than checking collisions among robots.
- Comparison: M* performs very well when only a small subset of robots coordinate, but substantial coordination forces it to consider exponentially many neighbors.dRRT instead advances through the roadmap in small steps using a minimalistic search.
- Search strategy: Voronoi bias drives dRRT toward unknown roadmap regions while limiting redundant exploration near already explored vertices.This is useful because nearby composite-roadmap vertices can represent essentially the same robot conformation.
7 Future work
Future work targets solution-quality guarantees by adapting ideas from RRT*, while the current framework deliberately finds some solution and reuses precomputed individual-roadmap information to avoid costly checks.
- Optimality: The current framework seeks some solution rather than providing solution-quality guarantees.The authors’ immediate goal is to adapt RRT*-like rewiring to the discrete setting.
- Optimality: Discrete rewiring could revise tree connections to improve previously examined paths, but it is unclear whether this would produce optimal paths.This uncertainty marks the main open issue in the proposed optimality direction.
- Framework: The framework reuses precomputed individual-roadmap information and avoids robot-obstacle collision checks by constraining robots to those roadmaps.This is the stated mechanism behind its efficiency in the discussed setting.