Multi-Robot Motion Planning with Dynamics Guided by Multi ... · Title: Multi-Robot Motion Planning with Dynamics Guided by Multi-Agent Search Author: Duong Le, Erion Plaku Subject:

Multi-Robot Motion Planning with Dynamics Guided by Multi-Agent Search∗

Duong Le and Erion PlakuDepartment of Electrical Engineering and Computer Science, Catholic University of America

Washington DC, USA{05le, plaku}@cua.edu

AbstractThis paper presents an effective multi-robot motionplanner that enables each robot to reach its desiredlocation while avoiding collisions with the otherrobots and the obstacles. The approach takes intoaccount the differential constraints imposed by theunderlying dynamics of each robot and generatesdynamically-feasible motions that can be executedin the physical world. The crux of the approach isthe sampling-based expansion of a motion tree inthe continuous state space of all the robots guidedby multi-agent search over a discrete abstraction.Experiments using vehicle models with nonlineardynamics operating in complex environments showsignificant speedups over related work.

1 IntroductionRobotic teams provide a viable venue to enhance automation,increase productivity, and reduce operational costs in trans-portation, exploration, inspection, surveillance, search-and-rescue, and many other applications. In these settings, robotsare often required to move to different locations while avoid-ing collisions with obstacles and other robots.

Multi-robot motion planning is challenging since it is nolonger sufficient to plan the motions of the robots indepen-dently as the plans of one robot can interfere with the plans ofanother robot or even prevent it from reaching the goal. Theplanned motions should also obey the underlying dynamicsof each robot, which impose differential constraints on thevelocity, acceleration, direction, and turning radius. This isessential to ensure that the planned motions can be executedby the robots in the physical world.

Related work has considered various settings for the multi-robot motion-planning problem. In a discrete setting, i.e.,multi-agent pathfinding, each robot is a point with no dynam-ics that in one step can move to an adjacent vertex in a graphabstraction of the world. To solve this NP-hard problem[Yu and LaValle, 2013], decoupled approaches often plan thepaths for each agent one at a time, ensuring that agent i avoids

∗This paper is an abridged version of the paper “CooperativeMulti-Robot Sampling-Based Motion Planning with Dynamics” thatwon the Best Robotics Paper award at ICAPS, 2017.

conflicts with the plans of agents 1, . . . , i − 1 [Silver, 2005;Sturtevant and Buro, 2006; Barer et al., 2014]. Decoupled ap-proaches are fast but suboptimal. To provide optimality, cen-tralized approaches based on A* [Standley and Korf, 2011;Goldenberg et al., 2014; Svancara and Surynek, 2017],M* [Wagner and Choset, 2015], increasing-cost tree search[Sharon et al., 2013], and conflict-based search [Sharon etal., 2015] operate over the composite space of all the agents[Felner et al., 2017]. Solvers have also used auctions [Amiret al., 2015], answer-set programming [Erdem et al., 2013],or rules [Khorshid et al., 2011; Botea and Surynek, 2015].

In a continuous setting, sampling-based motion planninghas often been used. To account for dynamics, RRT [LaValleand Kuffner, 2001], KPIECE [Sucan and Kavraki, 2012],GUST [Plaku, 2015], and others expand a motion tree byadding collision-free and dynamically-feasible trajectories asbranches. For multi-robot problems, sampling-based motionplanners can be used in a decoupled or centralized frame-work. Decoupled approaches plan the motions one robotat a time, treating robots 1, . . . , i − 1 as moving obstacleswhen planning the motions of robot i. Decoupled approacheswork well in open environments but have difficulty find-ing solutions in cluttered environments where coordinationamong robots becomes necessary since the motions plannedfor robots 1, . . . , i− 1 could make it impossible for robot i toreach its goal. Centralized approaches treat the robots as onesystem and operate over the resulting composite state space.They are probabilistically complete, but do not scale well dueto the high dimensionality of the composite state space.

To develop an effective multi-robot motion planner, wepresent a framework that couples the ability of sampling-based motion planning to handle the complexity arising fromgeometric and differential constraints imposed by the obsta-cles and underlying robot dynamics with the ability of multi-agent pathfinding to find non-conflicting routes over discreteabstractions. The underlying idea is to use multi-agent searchas a heuristic to guide sampling-based motion planning as itexpands a motion tree in the composite state space of all therobots. The expansion in the composite state space allows theapproach to obtain probabilistic completeness, which ensuresthat a solution will be found, when it exists, with probabilityapproaching one. Scalability and efficiency are obtained byusing multi-agent search as a heuristic to guide the motion-tree expansion. As with any heuristic, a critical aspect is the

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (IJCAI-18)

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design of an appropriate relaxed problem setting. We obtainthe discrete abstraction by ignoring the vehicle dynamics andusing probabilistic roadmaps to create a network of roads thatcaptures the connectivity of the environment, making it easyto reach each goal from any location.

Multi-agent search and sampling-based motion planningwork in tandem. At each iteration of a core loop, multi-agent search is first invoked to find non-conflicting routesover the discrete abstraction. At a second step, sampling-based motion planning seeks to expand the motion tree alongthe routes. When little or no progress is made, the routesare penalized and the multi-agent search is invoked again tofind alternative routes. This synergistic combination makesit possible to effectively plan collision-free and dynamically-feasible motions that enable each robot to reach its goal. Ex-periments using vehicle models with nonlinear dynamics op-erating in unstructured, obstacle-rich, environments demon-strate the scalability and efficiency of the approach, yieldingsignificant speedups over related work.

2 Problem FormulationEach robot modelMi = 〈Pi,Si,Ai, fi〉 is defined by its ge-ometric shape Pi, state space Si, action space Ai, and under-lying dynamics fi expressed as a set of differential equations

s = fi(s, a), s ∈ Si, a ∈ Ai. (1)

As an example, a vehicle model defines the state in terms ofthe position (x, y), orientation θ, steering angle ψ, and veloc-ity v. The vehicle is controlled by setting the acceleration accand steering rate ω. The motion equations are defined as

x = v cos(θ) cos(ψ), y = v sin(θ) cos(ψ), (2)

θ = v sin(ψ)/L, v = acc, ψ = ω, (3)

where L is the distance from the back to the front wheels.The new state snew ∈ S obtained by applying a control

action a ∈ Ai to a state s ∈ Si is computed by a function

snew ← SIMULATE(s, a, fi, dt), (4)

which numerically integrates fi for one time step dt. Start-ing from s ∈ S and applying actions 〈a1, . . . , a`−1〉 in suc-cession gives rise to a dynamically-feasible trajectory ζ :{1, . . . , `} → Si, where ζ(1)← s and ∀j ∈ {2, . . . , `}:

ζ(j)← SIMULATE(ζ(j − 1), aj−1, fi, dt). (5)

The multi-robot motion-planning problem is defined by

〈W ,O,G1, . . . ,Gn,M1, . . . ,Mn, sinit1 , . . . , sinit

n 〉. (6)

The world W contains obstacles O = {O1, . . . ,Om} andgoals G = {G1, . . . ,Gn}. The objective is to compute trajec-tories ζ1, . . . , ζn, one for each robot, so that ζi : {1, . . . , `} →Si starts at the initial state sinit

i ∈ Si, reaches the goal Gi, andavoids collisions with the obstacles and the other robots.

3 MethodThe approach has several components: (i) a discrete ab-straction based on a relaxed problem setting; (ii) multi-agentsearch over the discrete abstraction to guide sampling-based

Environmentbounding box: W

obstacles: O = {O1, . . . ,Om}

Robot ModelsM = {M1, . . . ,Mn}

Mi = 〈Pi,Si,Ai, fi, siniti ,Gi〉

Discrete AbstractionR1 × . . .×Rn, Ri = (VRi , ERi ,costRi)

Multi-AgentPathfinding

Motion TreeT = (VT , ET )

over S1 × . . .× Sn

Equivalence ClassesΓ = {Γkey1

, . . . ,Γkeyk}

key = 〈c1, . . . , cn〉 ∈ VR1 × . . .× VRn

routes(Γkey) = 〈σ1, . . . , σn〉

Expand Equivalence Classfollow routes σ1, . . . , σn

Select Equivalence ClassargmaxΓ〈c1,...,cn〉∈Γw(Γ〈c1,...,cn〉)

〈c1, . . . , cn〉

roadmaproutes〈σ1, . . . , σn〉

Figure 1: Schematic illustration of the approach.

Figure 2: An example of a roadmap.

motion planning; (iii) expansion of a motion tree in the com-posite state space of all the robots; (iv) using the discrete ab-straction to partition the motion tree into equivalence classes;and (v) interplay of sampling-based motion planning andmulti-agent search to effectively expand the motion tree alongroutes obtained by the multi-agent search. A schematic rep-resentation is shown in Fig. 1.

3.1 Discrete Abstraction via Roadmaps

The discrete abstraction provides a relaxed problem repre-sentation, obtained by ignoring the robot dynamics. Specif-ically, the robot state is reduced to just the position and ori-entation (referred to as configuration) and each robot is al-lowed to translate and rotate in any direction. To solve themulti-robot motion-planning problem in this relaxed setting,a roadmap Ri = (VRi

, ERi) is constructed for each robot

Mi, seeking to make it easy to reach each goal from anylocation in the environment. The roadmap is initially popu-lated by adding collision-free configurations from each goalG1, . . . ,Gn. Leveraging PRM [Kavraki et al., 1996], theroadmap is further populated by sampling collision-free con-figurations and connecting neighboring configurations withcollision-free paths whenever possible, as shown in Fig. 2.Dense roadmaps are preferred in order to provide alternativeroutes along which the robots can move to reach their goals.Robots that have the same shape can use the same roadmap.


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3.2 Multi-Agent Search over the DiscreteAbstraction

MULTIAGENTSEARCH(R1, . . . ,Rn, c1, . . . , cn,G1, . . . ,Gn)searches over the roadmaps R1, . . . ,Rn to compute non-conflicting routes σ1, . . . , σn for each robot such that σi isover Ri, starts at ci, and ends at a roadmap vertex associatedwith the goal Gi. Since in multi-agent search, each robotmoves from one vertex to the next in one step, each roadmapedge is divided into several parts to ensure that the distancefrom one vertex to the next is no more than a user-specifiedstep size. The edge division is done only once after eachroadmap is constructed. Note that the roadmap constructionguarantees that robots will not collide with the obstacles asthey move from one roadmap vertex to the next. Thus, themulti-agent search has to avoid only robot-robot conflicts.The overall approach is agnostic to the inner workings of themulti-agent search, so it can be used in conjunction with anyavailable method that operates over graphs.

3.3 Motion Tree in the Composite State SpaceNote that solutions over the relaxed problem setting do notnecessarily correspond to dynamically-feasible trajectoriessince the discrete abstraction ignores the robot dynamics. Asa result, geometric and differential constraints imposed by theobstacles and the underlying dynamics may make it difficultor impossible for a robot to follow its route σi.

To account for the dynamics, a motion tree T = (VT , ET )is rooted at the initial state 〈sinit

1 , . . . , sinitn 〉 and is incremen-

tally expanded in the composite state space S1 × . . . × Sn.Thus, each vertex v ∈ VT corresponds to a compositestate, denoted by STATE(v), where STATEi(v) represents thestate associated with the i-th robot. By construction, v isadded to VT only if STATE(v) is collision-free, i.e., when therobots are placed according to STATE(v), there is no robot-robot or robot-obstacle collision. To generate a successorv′ from v, some control actions 〈a1, . . . , an〉 are appliedcomponent-wise to STATE(v) and the motion equations ofeach robot are integrated for one time step dt, i.e., ∀i ∈{1, . . . , n} : STATEi(v

′) ← SIMULATE(STATEi(v), ai, fi, dt).When STATE(v′) is not in collision, v′ and the edge (v, v′) areadded to the motion tree T .

The motion tree T is expanded until a vertex vnew is addedwhere each robot has reached its goal, i.e., ∀i ∈ {1, . . . , n} :STATEi(vnew) ∈ Gi. The path 〈v1, . . . , v`〉, with v1 = vinitand v` = vnew, from the root of T to vnew is retrieved andthe solution trajectory for each robot i is constructed as ζi =〈STATEi(v1), . . . , STATEi(v`)〉.

3.4 Partitioning the Motion Tree into EquivalenceClasses Based on the Discrete Abstraction

The motion tree T could be expanded in an uninformed wayby selecting at each iteration a vertex v ∈ VT at random andapplying controls to generate a successor v′ from v. Such anexpansion, however, would have difficulty finding solutions.

In contrast, we introduce multi-agent search to guide themotion-tree expansion. Since multi-agent search operatesover the discrete abstraction, a mapping from the compos-ite state space to the discrete abstraction must be first de-fined. As the discrete abstraction is obtained via roadmaps,

a robot state si ∈ Si is mapped to the nearest configu-ration in the roadmap Ri. In this way, a composite state〈s1, . . . , sn〉 ∈ S1 × . . . × Sn is mapped component-wiseto a composite configuration 〈c1, . . . , cn〉, i.e.,

MAP(〈s1, . . . , sn〉) = 〈c1, . . . , cn〉, (7)

where ci = NEAREST(Ri, si).The approach leverages this mapping to partition the mo-

tion tree T into equivalence classes. The premise is thatvertices in T that provide the same discrete informationshould belong to the same equivalence class. In other words,vertices v, v′ ∈ T belong to the same equivalence classΓ〈c1,...,cn〉 if and only if MAP(STATE(v)) = MAP(STATE(v′)) =〈c1, . . . , cn〉. So, an equivalence class Γ〈c1,...,cn〉 contains allthe vertices in T that map to 〈c1, . . . , cn〉.

The significance of an equivalence class Γ〈c1,...,cn〉 is thatit makes it possible to leverage multi-agent search to computenon-conflicting routes σ1, . . . , σn, where each σi starts at ciand reaches the goal Gi. These routes then serve to guide themotion planner as it seeks to expand the motion tree from ver-tices associated with the equivalance class Γ〈c1,...,cn〉 alongσ1, . . . , σn.

3.5 Putting it all Together: Using Multi-AgentSearch to Guide the Motion-Tree Expansion

The overall approach starts by creating the discrete abstrac-tion, rooting the motion tree T at the initial state, and creat-ing the first equivalence class ΓSTATE(vinit), which contains theroot vertex vinit. Afterwards, the approach enters a core loop,where each iteration consists of the following:• select an equivalence class Γ〈c1,...,cn〉 from Γ;• use multi-agent search over the discrete abstraction to

obtain non-conflicting routes σ1, . . . , σn that reach thegoals starting from c1, . . . , cn; and• expand the motion tree T from vertices in Γ〈c1,...,cn〉

along the routes σ1, . . . , σn.MULTIAGENTSEARCH(R1, . . . ,Rn, c1, . . . , cn,G1, . . . ,Gn)searches the discrete abstraction to obtain non-conflictingroutes 〈σ1, . . . , σn〉, where σi starts at ci and reaches thegoal Gi. To save computation time, the routes are storedwith Γ〈c1,...,cn〉 and retrieved when the multi-agent searchis invoked again for Γ〈c1,...,cn〉. During the selection of anequivalence class, priority is given to equivalence classesassociated with short routes to promote rapid expansionstoward the goals. The expansion adds new vertices to T ,which could result in creating new equivalence classes.When the expansion from Γ〈c1,...,cn〉 fails or has difficultyfollowing the routes σ1, . . . , σn, which could happen dueto the geometric and differential constraints imposed by theobstacles and the underlying robot dynamics, the approachpenalizes Γ〈c1,...,cn〉 in order to promote expansions fromother equivalence classes. This process of selecting andexpanding from an equivalence class is repeated until asolution is found or a runtime limit is reached.

Selecting an Equivalence ClassA weight is defined for each equivalence class and the equiv-alence class with the maximum weight is selected for expan-


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sion. Specifically, the weight for Γ〈c1,...,cn〉 is defined as

w(Γ〈c1,...,cn〉) =αNRSEL(Γ〈c1,...,cn〉)∑n

i=1(COST(σi))2, (8)

where σ1, . . . , σn denote the routes associated withΓ〈c1,...,cn〉, 0 < α < 1, and NRSEL(Γ〈c1,...,cn〉) denotes thenumber of times Γ〈c1,...,cn〉 has been previously selected.

In this way, to promote expansions towards the goals, pref-erence is given to equivalence classes associated with shortroutes. This constitutes the greedy aspect of the selectionprocess. To mitigate the potential pitfalls of a greedy selec-tion, a penalty factor α (0 < α < 1) is introduced, whichreduces the weight of an equivalence class each time it is se-lected. In fact, repeatedly selecting Γ〈c1,...,cn〉 will continueto reduce its weight so that eventually some other equivalenceclass will end up having a greater weight and thus be selectedfor expansion. This is essential to ensure probabilistic com-pleteness and avoid becoming stuck when the motion-tree ex-pansion from Γ〈c1,...,cn〉 repeatedly fails due to the constraintsimposed by the obstacles and the underlying robot dynamics.

Expanding the Motion Tree from the Equivalence ClassAfter selecting an equivalence class Γ〈c1,...,cn〉, the objec-tive becomes to expand the motion tree T from a vertexv ∈ Γ〈c1,...,cn〉 along the routes σ1, . . . , σn associated withΓ〈c1,...,cn〉. The vertex v could be selected at random to pro-mote expansions from different vertices. Attempts are thenmade to expand T from STATE(v) so that each robot i movesfrom STATEi(v) toward the next configuration in σi. Wecan use proportional-integrative-derivative (PID) controllers[Spong et al., 2005] to select controls that would steer eachrobot toward its target, e.g., by turning the wheels and thenmoving toward the target. The expansion from v continuesfor several steps until the target is reached or a robot collideswith an obstacle or another robot. If a new state is in col-lision, the trajectory generation stops and the approach goesback to selecting an equivalence class. Otherwise, the newstate is added to tree T and the partition Γ is updated accord-ingly. If in the new state each robot has reached its goal, thena solution is found and the search terminates successfully.

4 Experiments and ResultsExperiments are conducted using vehicle models with nonlin-ear dynamics operating in complex environments, as shownin Fig. 2 and 3. In many of these scenarios, the robots arerequired to cooperate in order to reach their goals.

The approach is compared to a centralized and a priori-tized version of RRT [LaValle and Kuffner, 2001], denotedby cRRT and pRRT, as RRT is one of the most popularsampling-based motion planners. The approach is also com-pared to a prioritized version of GUST [Plaku, 2015], denotedby pGUST, as GUST has been shown to be significantly fasterthan RRT and other sampling-based motion planners.

Results in Fig. 4 show significant speedups over cRRT,pRRT, and pGUST. As RRT-based planners lack guidance,they have difficulty finding solutions. pGUST is faster thancRRT and pRRT since it uses discrete search to guide themotion-tree expansion. In open scenes (scene 2), where co-operation among the robots is not essential, pGUST is even

scene 2

scene 3

scene 4

scene 5

scene 6

Figure 3: Scenes used in the experiments (scene 1 shown in Fig. 2).Videos of solutions can be found at https://goo.gl/sQ6irb. Figurebest viewed in color and on screen.

1 2 4 6 8 10 1 2 4 6 8 10 1 2 4 6 8 10 3 3 31/81/41/2

1248

163264

scene 1 scene 2 scene 3 s4 s5 s6

abcd

nr. of robots

runt

ime

[s]

Figure 4: Runtime results when comparing (a) our approach to (b)pGUST, (c) pRRT, and (d) cRRT. Runtime measures everythingfrom reading the input file to reporting that a solution is found (in-cluding for our approach the time to build the roadmaps). Missingentries indicate failure by the planner to solve the problem instanceswithin the runtime limit (set to 90s per run). Note in particular thatonly our approach could solve scenes 4, 5, 6 (denoted as s4, s5, s6).

faster than our approach since the robot trajectories do notinterfere with each other. In scenes where cooperation is im-portant, pGUST, as any prioritized approach, has difficultyfinding solutions, even timing out in scenes 4, 5, 6. In con-trast, due to the guidance by multi-agent search, our approachis able to find solutions in a matter of 1-3s.

5 DiscussionThis paper presented an effective multi-robot motion plan-ner that enables each robot to reach its desired location whileavoiding collisions. The approach also took into account thedynamics of each robot and generated dynamically-feasibletrajectories that can be executed in the physical world.

The crux of the approach was coupling the ability ofsampling-based motion planning to handle the complexityarising from obstacles and robot dynamics with the abilityof multi-agent pathfinding to find non-conflicting routes overdiscrete abstractions. The underlying idea was to use multi-agent search over the discrete abstraction as a heuristic toguide sampling-based motion planning as it expands a mo-


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tion tree in the composite state space of all the robots.This work opens up several research directions. Advances

in multi-agent search can significantly improve the efficiencyand scalability of the framework by reducing the runtimeto compute the routes. Moreover, the quality of the routescan also be improved, for example, by requiring multi-agentsearch to compute non-conflicting routes that maximize theseparation among the robots and the clearance from the ob-stacles. Such routes would be easier to follow, which wouldreduce the runtime for the motion-tree expansion. Anotherdirection is to enable the team of robots to perform complextasks given by PDDL or some other high-level formalism.

AcknowledgmentsThis work is supported by NSF IIS1548406.

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Multi-Robot Motion Planning with Dynamics Guided by Multi ... · Title: Multi-Robot Motion Planning with Dynamics Guided by Multi-Agent Search Author: Duong Le, Erion Plaku Subject: