Scalable Asymptotically-Optimal Multi-Robot Motion Planning

Scalable Asymptotically-Optimal Multi-Robot Motion Planning

Abstract

Discovering high-quality paths for multi-robot problems can be achieved, in principle, through asymptotically-optimal data structures in the composite space of all robots, such as a sampling-based roadmap or a tree. The hardness of motion planning, however, which depends exponentially on the number of robots, renders the explicit construction of such structures impractical. This work proposes a scalable, sampling-based planner for coupled multi-robot problems that provides desirable path-quality guarantees. The proposed  is an informed, asymptotically-optimal extension of a prior method , which introduced the idea of building roadmaps for each robot and implicitly searching the tensor product of these structures in the composite space. The paper describes the conditions for convergence to optimal paths in multi-robot problems. Moreover, simulated experiments indicate  converges to high-quality paths and scales to higher numbers of robots where various alternatives fail. It can also be used on high-dimensional challenges, such as planning for robot manipulators.

I Introduction and Prior Work

Many multi-robot planning applications [22, 4, 3] require high-dimensional platforms to simultaneously move in a shared workspace, where high-quality paths must be computed quickly as sensing input is updated, as in Fig. 1. Preprocessing given knowledge of the static scene can help the online computation of high-quality paths. Sampling-based roadmaps can help with such high-dimensional challenges and provide primitives for preprocessing a static scene [8, 9]. These methods converge to optimal solutions given sufficient density [7], i.e., at least edges are needed for a roadmap with vertices, while near-optimal solutions are achieved after finite computation time [2, 6].

Fig. 1: Simultaneous planning for multiple high-dimensional systems is a difficult, motivating challenge for this work.

Naïvely constructing a sampling-based roadmap or tree in the robots’ composite configuration space provides asymptotic optimality but does not scale well. In particular, memory requirements depend exponentially on the problem’s dimension [15]. The alternative is decoupled planing, where paths for robots are computed independently and then coordinated [10]. These methods, however, typically lack completeness and optimality guarantees. Hybrid approaches can achieve optimal decoupling to retain guarantees [21]. The problem is more complex when the robots exhibit non-trivial dynamics [12]. Collision avoidance or control methods can scale to large numbers of robots, but typically lack global path quality guarantees [20, 19].

The previously proposed  approach [16] is a scalable sampling-based approach, which is probabilistically complete. It searches an implicit tensor product roadmap of roadmaps constructued for each robot individually [18]. The current work proposes and shows that it is an efficient asymptotically optimal variant of the prior method. Simulations show the method practically generates high-quality paths while scaling to complex, high-dimensional problems, where alternatives fail. 1

Ii Problem Setup and Notation

Consider a shared workspace with robots, each operating in configuration space for . Let be each robot’s free space, where it is free from collision with the static scene, and is the forbidden space for robot . The composite -space is the Cartesian product of each robot’s -space. A composite configuration is an -tuple of robot configurations. For two distinct robots , denote by the set of configurations where and collide. Then, the composite free space consists of configurations subject to:

  • for every ;

  • for every .

Each requires robots to not collide with obstacles, and each pair to not collide with each other. The composite forbidden space is defined as .

Given , where , a trajectory is a continuous curve in , such that , where the robots move simultaneously. is an -tuple of robot paths such that .

The objective is to find a trajectory which minimizes a cost function . The analysis assumes the cost is the sum of robot path lengths, i.e., , where denotes the standard arc length of . The arguments also work for .2 Section IV shows sufficient conditions for  to converge to optimal trajectories over the cost function .

Iii Methods for Composite Space Planning

For a fixed , define for every robot the PRM roadmap constructed over , such that with connection radius . Then, is the tensor product roadmap in space (for an illustration, see Figure 2). Formally, is the Cartesian product of the nodes from each roadmap . For two vertices the edge set contains edge if for every it is that or .3

Fig. 2: An illustration of a two-robot tensor product roadmap between roadmaps and . Two nodes in the tensor-product roadmap share an edge if all the individual robot configurations share an edge in the individual robot roadmaps.

As shown in Algorithm 1, grows a tree over , rooted at the start configuration and initializes path (line 1). The method stores the node added each iteration (Line 2), as part of an informed process to guide the expansion of  towards the goal. The method iteratively expands given a time budget (Line 3), as detailed by Algorithm 2, storing the newly added node (Line 4). After expansion, the method traces the path which connects the source with the target (Line 5). If such a path is found, it is stored in if it improves upon the cost of the previous solution (Lines 6, 7). Finally, the best path found is returned (Line 8).

1 , ;
2 ;
3 while  do
4       ;
5       ;
6       if  and  then
7             ;
8            
return
Algorithm 1

The expansion step is given in Alg. 2. The default initial step of the method is given in Lines 1-4, i.e., when no is passed (Line 1), which corresponds to an exploration step similar to RRT: a random sample  is generated in  (Line 2), its nearest neighbor  in is found (Line 3) and the oracle function returns the implicit graph node  that is a neighbor of on the implicit graph in the direction of (Line 4). If a , however, is provided (Line 5)—which happens when the last iteration managed to generate a node closer to the goal relative to its parent—then the is greedily generated so as to be a neighbor of in the direction of the goal (Line 6).

1 if  then
2       ;
3       ;
4       ;
5      
6else
7       ;
8      
9;
10 ;
11 if  then
12       ;
13      
14if   then
15       ;
16       ;
17      
18else
19       ;
20      
21for   do
22       if  and  then
23             ;
24            
25if   then
26       ;
27      
28else
29       ;
30      
Algorithm 2

In either case, the method next finds neighbors , which are adjacent to  in and have also been added to (Line 7). Among , the best node is chosen, for which the local path is collision-free and that the total path cost to is minimized (Line 8). If no such parent can be found (Line 9), the expansion fails and no node is returned (Line 10). Then, if is not in , it is added (Lines 11-13). Otherwise, if it exists, the tree is rewired so as to contain edge , and the cost of the ’s sub-tree (if any) is updated (Lines 14, 15). Then, for all nodes in (Line 16), the method tests should be rewired through to reach this neighbor. Given that is collision-free and is of lower cost than the existing path to (Line 17), the tree is rewired to make be the parent of (line 18).

Finally, if in this iteration the heuristic value of is lower than its parent node (line 19), the method returns (Line 20), causing the next iteration to greedily expand . Otherwise, is returned so as to do an exploration step. Note that the approach is implemented with helpful branch-and-bound pruning after an initial solution is found, though this is not reflected in the algorithmics.

 is determined via an oracle function. Using this oracle function and a simple rewiring scheme is sufficient for showing asymptotic optimality for (see Section IV). The oracle function for a two-robot case is illustrated in Figure 3. First, let be the ray from configuration terminating at . Then, denote as the minimum angle between and . When  is drawn in , its nearest neighbor  in is found. Then, project the points  and  into each robot space , i.e., ignore the configurations of other robots.

Fig. 3: (A) The method reasons over all neighbors of so as to minimize the angle . (B) finds graph vertex by minimizing angle . (C,D) and are projected into each robot’s -space so as to find nodes and , respectively, which minimize angle .

The method separately searches the single-robot roadmaps to discover . Denote . For every robot , let be the neighborhood of , and identify . The oracle function returns node .

Fig. 4: (A) The Voronoi region of vertex is shown where if  is drawn, vertex is selected for expansion. (B) When  lies in the directional Voronoi region , the expand step expands to . (C) Thus, when  is drawn within volume , the method will generate  via .

As in the standard as well as in , the approach has a Voronoi-bias property  [11]. It is, however, slightly more involved to observe as shown in Figure 4. To generate an edge , random sample  must be drawn within the Voronoi cell of , denoted (A) and in the general direction of , denoted (B). The intersection of these two volumes is the volume to be sampled generate  via .

Iv Analysis

In this section, the theoretical properties of  are examined, beginning with a study of the asymptotic convergence of the implicit roadmap to containing a path in whose cost converges to the optimum. Then, it is shown  eventually discovers the shortest path in , and that the combination of these two facts proves the asymptotic optimality of .

For simplicity, the analysis is restricted to the setting of robots operating in Euclidean space, i.e. is a -dimensional Euclidean hypercube for fixed . 4 Additionally, the analysis is restricted to the specific cost function of total distance, i.e., . Discussion on lifting these restrictions is provided in Section VI.

Iv-a Optimal Convergence of

For each robot, an asymptotically optimal  roadmap is constructed having samples and using a connection radius necessary for asymptotic convergence to the optimum [7]. By the nature of sampling-based algorithms, each graph cannot converge to the true optimum with finite computation, as such a solution may have clearance of exactly . Instead, this work focuses on the notion of a robust optimum 5, showing that the tensor product roamdap converges to this value.

Definition 1

A trajectory is robust if there exists a fixed such that for every it holds that , where denotes the standard Euclidean distance.

Definition 2

A value which denotes a path cost is robust if for every fixed there exists a robust path such that . The robust optimum, denoted by , is the infimum over all such values.

For any fixed , and a specific instance of constructed from roadmaps, having samples each, denote by the shortest path from to over .

Definition 3

is asymptotically optimal (AO) if for every fixed it holds that a.a.s.6, where the probability is over all the instantiations of with samples for each PRM.

Using this definition, the following theorem is proven. Recall that denotes the dimension of a single-robot configuration space.

Theorem 1

is AO when

where is any constant larger than .

Remark. Note that  was developed in [6, Theorem 4.1], and guarantees AO of for a single robot. The proof technique described in that work will be one of the ingredients used to prove Theorem 1. 7

By definition of , for any given there exists robust trajectory , and fixed , such that the cost of is at most and for every it holds that . Next, it is shown that contains a trajectory such that

(1)

a.a.s.. This immediately implies that , which will finish the proof of Theorem 1.

Thus, it remains to show that there exists a trajectory on which satisfies Equation 1 a.a.s.. As a first step, it will be shown that the robustness of in the composite space implies robustness in the single-robot setting, i.e., robustness along .

For define the forbidden space parameterized by as

Claim 1

For every robot , , and , .

{proof}

Fix a robot , and fix some and a configuration . Next, define the following composite configuration

Note that it differs from only in the -th robot’s configuration. By the robustness of it follows that

The result of claim 1 is that the paths are robust in the sense that there is sufficient clearance for the individual robots to not collide with each other given a fixed location of a single robot. A Lemma is derived using proof techniques from the literature [6], and it implies every contains a single-robot path that converges to

Lemma 1

For every robot , constructed with samples and a connection radius contains a path with the following attributes a.a.s.:

  • , ;

  • ;

  • , s.t. .

{proof}

The first property (i) follows from the fact that are directly added to . The rest follows from the proof of Theorem 4.1 in [6], which is applicable here since .

Lemma 1 also implies that contains a path in , that represents robot-to-obstacle collision-free motions, and minimizes the multi-robot metric cost. In particular, define , where are obtained from Lemma 1. Then

However, it is not clear whether this ensures the existence of a path where robot-robot collisions are avoided. That is, although , it might be the case that . Next it is shown that can be reparametrized to induce a composite-space path whose image is fully contained in , with length equivalent to .

For each robot , denote by the chain of vertices traversed by . For every assign a timestamp of the closest configuration along , i.e.,

Also, define and denote by the ordered list of , according to the timestamp values. Now, for every , define a global timestamp function which assigns to each global timestamp in a single-robot configuration from . It thus specifies in which vertex robot resides at time . For , let be the largest index such that . Then simply assign . From property (iii) in Lemma 1 and Claim 1 it follows that no robot-robot collisions are induced by the reparametrization, concluding the proof of Theorem 1.

Iv-B Asymptotic Optimality of

Finally,  is shown to be AO. Denote by the time budget in Algorithm 1, i.e., the number of iterations of the loop. Denote by the solution returned by for and .

Theorem 2

If then for every fixed it holds that

Since is AO (Theorem 1), it suffices to show that for any fixed , and a fixed instance of , defined over PRMs with samples each, eventually (as tends to infinity), finds the optimal trajectory over . This can be shown using the properties of a Markov chain with absorbing states [5, Theorem 11.3]. While a full proof is omitted here, the high-level idea is similar to what is presented in previous work [16, Theorem 3], and expanded upon in Appendix A. By restricting the states of the Markov chain to being the graph vertices along the optimal path, setting the target vertex to be an absorbing vertex, and showing that the probability of transitioning along any edge in this path is nonzero (i.e. the probability is proportional to ), then the probability that this process does not reach the target state along the optimal path converges to as the number of  iterations tends to infinity. The final step is to show that the above statements hold when both and tend to . A proof for this phenomenon can be found in [16, Theorem 6].

V Experimental Validation

This section provides an experimental evaluation of  by demonstrating practical convergence, scalability, and applicability to dual-arm manipulation. The approach and alternatives are executed on a cluster with Intel(R) Xeon(R) CPU E5-4650 @ 2.70GHz processors, and 128GB of RAM. 8

Fig. 5: The 2D environment where the 2 disk robots operate.

2 Disk Robots among 2D Polygons: This base-case test involves disks () of radius , in a region, as in Figure 5. The disks have to swap positions between and . This is a setup where it is possible to compute the explicit roadmap, which is not practical in more involved scenarios. In particular,  is tested against: a) running on the implicit tensor roadmap (referred to as “Implicit ”) defined over the same individual roadmaps with nodes each as those used by ; and b) an explicitly constructed  roadmap with nodes in the composite space.

Results are shown in Figure 6.  converges to the optimal path over , similar to the one discovered by Implicit , while quickly finding an initial solution of high quality. Furthermore, the implicit tensor product roadmap is of comparable quality to the explicitly constructed roadmap.

Fig. 6: Average solution cost over iterations. Data averaged over roadmap pairs. (solid line) converges to the optimal path through (dashed line).

Table I presents running times.  and implicit  construct -sized roadmaps (row 3), which are faster to construct than the  roadmap in (row 1).  becomes very costly as increases. For , the explicit roadmap contains vertices, taking GB of RAM to store, which was the upper limit for the machine used. When the roadmap can be constructed, it is quicker to query (row 2).  quickly returns an initial solution (row 5), and converges within of the optimum length (row 6) well before Implicit  returns a solution as increases (row 4). The next benchmark further emphasizes this point.

Number of nodes: = 50 100 200
-PRM* construction 3.427 13.293 69.551
-PRM* query 0.002 0.004 0.023
2 -size PRM* construction 0.1351 0.274 0.558
Implicit A* search over 0.684 2.497 10.184
 over (initial) 0.343 0.257 0.358
 over (converged) 3.497 4.418 5.429
TABLE I: Construction and query times (SECs) for 2 disk robots.

Many Disk Robots among 2D Polygons: In the same environment as above, the number of robots is increased to evaluate scalability. Each robot starts on the perimeter of the environment and is tasked with reaching the opposite side. An roadmap is constructed for every robot. It quickly becomes intractable to construct a  roadmap in the composite space of many robots.

Fig. 7: Data averaged over runs. (Top): Relative solution cost and success ratio of ,  and  for increasing . : average iteration and variance for initial solution (top of box), and solution cost and variance after iterations (bottom). Similar results for . Single data point for  (no quality improvement after first solution). (Bottom): Solution costs over time.

Figure 7 shows the inability of alternatives to compete with  in scalability. Solution costs are normalized by an optimistic estimate of the path cost for each case, which is the sum of the optimal solutions for each robot, disregarding robot-robot interactions. Implicit  fails to return solutions even for 3 robots. Directly executing  in the composite space fails to do so for . The original  method (without the informed search component) starts suffering in success ratio for and returns worse solutions than . The average solution times for  may decrease as increases but this is due to the decreasing success ratio, i.e.,  begins to only succeed at easy problems.

Fig. 8: (Top):  is run for a dual-arm manipulator to go from its home position (above) to a reaching configuration (below) and achieves perfect success ratio as increases. (Bottom):  solution quality over time. Here, larger roadmaps provide benefits in terms of running time and solution quality.

Dual-arm manipulator: This test shows the benefits of  when planing for two -dimensional arms. Figure 8 shows that  fails to return solutions within iterations. Using small roadmaps is also insufficient for this problem. Both  and Implicit  require larger roadmaps to begin succeeding. But with , Implicit  always fails, while  maintains a success ratio. As expected, roadmaps of increasing size result in higher quality path. The informed nature of  also allows to find initial solutions fast, which together with the branch-and-bound primitive allows for good convergence.

Vi Discussion

This work studies the asymptotic optimality of sampling-based multi-robot planning over implicit structures. The objective is to efficiently search the composite space of multi-robot systems while also achieving formal guarantees. This requires construction of asymptotically optimal individual robot roadmaps and appropriately searching their tensor product. Performance is further improved by informed search in the composite space.

These results may extend to more complex settings involving kinodynamic constraints by relying on recent work [14, 13]. Furthermore, the analysis may be valid for different cost functions other than total distance. The tools presented here can prove useful in the context of simultaneous task and motion planning for multiple robots, especially manipulators [1].

-a Proof of Convergence

This appendix examines the result of Theorem 2, and formally proves the convergence of the  tree toward containing all optimal paths.

Lemma 2 (Optimal Tree Convergence of )

Consider an arbitrary optimal path originating from and ending at , then let be the event such that after iterations of , the search tree contains the optimal path up to segment . Then,

Proof. This property will be proven using a theorem from Markov chain literature [5, Theorem 11.3]. Specifically, the properties of absorbing Markov chains can be exploited to show that will eventually contain the optimal path over for a given query. An absorbing Markov chain is one such that there is some subset of states in which the transition matrix only allows that state to transition to itself.

The proof follows by showing that the method can be described as an absorbing Markov chain, where the target state of a query is represented as an absorbing state in a Markov chain, re-stated here.

Theorem 3 (Thm 11.3 in Grinstead & Snell)

In an absorbing Markov chain, the probability that the process will be absorbed is 1 (i.e., as ), where is the transition submatrix for all non-absorbing states.

There are two steps to using this proof. First, that the search can be cast as an absorbing Markov chain, and second, that the probability of transition for from each state to the next in this chain is nonzero (i.e. that each state can eventually be connected to the target).

For query , let the sequence of length represent the vertices of corresponding to the optimal path through the graph which connects these points, where corresponds to the target vertex, and furthermore, let be an absorbing state. Theorem 3 operates under the assumption that each vertex is connected to an absorbing state ( in this case).

Then, let the transition probability for each state have two values, one for each state transitioning to itself, which corresponds to the search expanding along some other arbitrary path. The other value is a transition probability from to . This corresponds to the method sampling within the volume .

Then, as the second step, it must be shown that this volume has a positive probability of being sampled in each iteration. It is sufficient then to argue that . Fortunately, for any finite , previous work has already shown that this is the case given general position assumptions [16, Lemma 2].

Given these results, the is cast as an absorbing Markov chain which satisfies the assumptions of 3, and therefore, the matrix . This implies that the optimal path to the goal has been expanded in the tree, and therefore

-B More Experimental Data

This appendix presents additional experimental data omitted from Section V.

2-Robot Benchmark

Fig. 9: 2-Robot convergence data over time.

For the two-robot benchmark, additional data is presented in Figure 9. Here, the data presented in Figure 6 is shown again over time instead of over iterations.

Fig. 10: -robot solution times for varying .

R-Robot Benchmark

For the -robot benchmark, additional data is presented in Figure 10, showing query resolution times for the various methods.

Fig. 11: (Top): Convergence rate and success ratio over the minimal -node roadmap (Bottom): Solution cost over time when using the minimal roadmap.

To emphasize the lack of scalability for alternate methods, additional experiments were run in this setup using a minimal roadmap. The tests use a -node roadmap for each robot as illustrated in Figure 12. Each roadmap is constructed with slight perturbations to the nodes within the shaded regions indicated in the figure.

Fig. 12: Minimal graph for the -robot case.

The data for this modified benchmark (shown in Figure 11) indicate that even using a very small roadmap does not allow alternate methods to scale. While the methods scale better, Implict  does time out for , and  times out for .

Manipulator Benchmark

For the dual-arm manipulator benchmark, additional data is presented in Figure 13. Here, the data of Figure 8 is shown over iterations instead of over time.

Fig. 13: Motoman benchmark solution quality over iterations.

Footnotes

  1. Additional material is provided in the appendices of this paper as well as the accompanying video of the original MRS submission.
  2. The types of distances the arguments hold are more general, but proofs for alternative metrics are left as future work.
  3. Notice this slight difference from  [16] so as to allow edges where some robots remain motionless while others move.
  4. For simplicity, it is assumed that all the robots have the same number of degrees of freedom .
  5. Note that the given definition of robust optimum is similar to that in previous work [17].
  6. Let be random variables in some probability space and let be an event depending on . We say that occurs asymptotically almost surely (a.a.s.) if .
  7. Note that  can be refined to incorporate the proportion of , which would reduce this expression.
  8. Additional data are provided in Appendix -B.

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