Maintaining Team Coherenceunder the Velocity Obstacle Framework

Andrew KimmelUniversity of Nevada, Reno1664 N. Virginia St., MS171

Reno, NV 89557akimmel@cse.unr.edu

Andrew DobsonUniversity of Nevada, Reno1664 N. Virginia St., MS171

Reno, NV 89557dobsona@cse.unr.edu

Kostas BekrisUniversity of Nevada, Reno1664 N. Virginia St., MS171

Reno, NV 89557bekris@cse.unr.edu

ABSTRACTMany multi-agent applications may involve a notion of spa-tial coherence. For instance, simulations of virtual agents of-ten need to model a coherent group or crowd. Alternatively,robots may prefer to stay within a pre-specified communica-tion range. This paper proposes an extension of a decentral-ized, reactive collision avoidance framework, which definesobstacles in the velocity space, known as Velocity Obsta-cles (VOs), for coherent groups of agents. The extension,referred to in this work as a Loss of Communication Ob-stacle (LOCO), aims to maintain proximity among agents byimposing constraints in the velocity space and restricting theset of feasible controls. If the introduction of LOCOs resultsin a problem that is too restrictive, then the proximity con-straints are relaxed in order to maintain collision avoidance.If agents break their proximity constraints, a method is ap-plied to reconnect them. The approach is fast and integratesnicely with the Velocity Obstacle framework. It yields im-proved coherence for groups of robots connected through aninput constraint graph that are moving with constant veloc-ity. Simulated environments involving a single team movingamong static obstacles, as well as multiple teams operatingin the same environment, are considered in the experimentsand evaluated for collisions, computational cost and prox-imity constraint maintenance. The experiments show thatimproved coherence is achieved while maintaining collisionavoidance, at a small computational cost and path qualitydegradation.

Categories and Subject DescriptorsI.2.11 [Distributed Artificial Intelligence]: Multiagentsystems

General TermsAlgorithms, Design, Experimentation

KeywordsAAMAS proceedings, multi-agent, collision avoidance, teamcoherence, communication constraints

Appears in: Proceedings of the 11th International Confer-ence on Autonomous Agents and Multiagent Systems (AA-MAS 2012), Conitzer, Winikoff, Padgham, and van der Hoek (eds.),June, 4–8, 2012, Valencia, Spain.Copyright c© 2012, International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved.

Figure 1: Two agents navigating around a static ob-stacle: (a) the agents split to reach their goals with-out collisions, while in (b) the agents move togetheras a coherent team. The second behavior must beachieved in a decentralized manner.

1. INTRODUCTIONMany practical applications of decentralized collision avoid-

ance may involve a secondary objective, where teams ofagents need to maintain a certain level of coherence. Co-herence often implies that the agents should remain withina certain distance of one another. In games and simulations,the agents may need to remain within a certain distance be-cause of implied social interactions, or because they need toreach their destination together so as to be more effective incompleting an objective at their goal. For instance, a teamof agents in a game will be more effective in attacking an en-emy if all the units move together against the opponent anddo not split into multiple groups. In mobile sensor networks,robots may have to respect radial communication limits.

The Velocity Obstacle (VO) formulation [8, 27, 22] is aframework for reactive collision avoidance. It is fast as it op-erates directly in the velocity space of each agent. It is also adecentralized approach as each agent reasons independentlyabout its controls, as long as it can compute the position andvelocity of its neighbors. Current work in Velocity Obstaclesdoes not directly address the issue of coherence. Considerthe situation in Figure 1, where two agents are required toavoid an obstacle and reach their desired destination. Anapplication of the basic VO framework may result in the twoagents splitting and passing the obstacle from opposite ends.It would be desirable, however, for the agents to select, ina decentralized manner, a single direction to follow so as toavoid the obstacle and reach their goal while maintaining

coherence. The decentralized nature of the solution will beespecially helpful in robotic applications because no commu-nication will be required between robots. It will also provideimproved scalability in simulations and games.

This paper proposes a method for maintaining team co-herence within the VO framework in a decentralized manner.The desired coherence for a problem can be defined as agraph of dependencies between agents. An edge in the graphimplies that the corresponding agents should remain withina predefined distance as they move towards their goal. Giventhis input graph, an agent constructs an additional obstaclein the velocity space for each neighbor with which it wantsto maintain connectivity. This construction is referred to inthis work as a Loss of Communication Obstacle (LOCO).

Building on top of the VO framework allows LOCOs to focuson coherence maintenance, rather than obstacle avoidance.In order to construct the LOCO, the assumption is that aneighbor will maintain its current control, as in the origi-nal VO framework. Then, a LOCO defines the set of velocitiesthat will lead the two agents to be separated beyond a de-sired distance within a certain time horizon. A scheme isproposed for integrating information from the multiple VOsdefined for collision avoidance and the LOCOs defined for co-herence maintenance. If the set of velocities that satisfyboth constraints is not empty for a satisfactory time hori-zon, then the velocity in this set that brings the agent closerto its goal is chosen. A valid velocity implies that, giventhe neighbor does not change control, following this velocitywill not lead the agent into a collision or violation of prox-imity constraints for the given horizon. If the set of validvelocities is empty, then the objective of maintaining coher-ence is dropped in favor of guaranteeing collision avoidance,which should be satisfiable, unless oscillations appear in theselected controls of agents. If two agents that are supposedto retain proximity end up violating the distance constraint,the proposed method makes them move in a direction thatwill allow them to reconnect.

This paper describes the LOCO method and provides sim-ulations which show that, by reasoning about distance con-straints, it is possible to solve decentralized collision avoid-ance problems while improving the coherence of a team. Theapproach is built on top of and compared against a formula-tion of velocity obstacles proposed in the literature for teamsof agents that execute the same protocol, referred to as Re-ciprocal Velocity Obstacles [27, 22]. The experiments con-sider disk-shaped agents that move with a constant speedin a holonomic manner, i.e., the agents can freely chooseto follow any direction instantaneously. First, a series ofstatic environments are tested with a single team of agentsnavigating while maintaining coherence. Different types offormations between the teams are considered. Then, testsfor multiple teams of agents navigating in the same environ-ment while maintaining coherence are performed.

The organization of the paper is as follows. First, in Sec-tion 2, the relevant work to this problem is reviewed. Sec-tion 3 provides a formal description of the problem as wellas the applicable notation used throughout the manuscript.The Velocity Obstacle framework is outlined in Section 4together with the proposed approach for maintaining teamcoherence. Section 5 describes relevant results on variousexperimental setups. Lastly, Section 6 concludes the paperwith a discussion of the technique and possible future work.

2. BACKGROUND

2.1 Virtual Agent ApplicationsThe need to move multiple agents as a coherent team

arises in many virtual agent applications [25, 18], rangingfrom crowd simulation [17], to pedestrian behavior analy-sis [16, 19], to shepherding and flocking behaviors [15, 29].Many methods make use of “steering behaviors”, with theobjective of having agents navigate in a life-like and im-provisational manner [20]. These steering behaviors can becombined to achieve higher-level goals, such as “get to thegoal while avoiding obstacles”or“join a group of characters”.Similar is the objective of the social force model [10].

2.2 Coupled Multi-Robot Path PlanningThere is also extensive literature on motion coordination

and collision avoidance in robotics. The multi-robot pathplanning problem can be approached by either a coupled ap-proach or a decoupled one [12, 13]. The coupled approachplans for the composite robot, which has as many degreesof freedom as the sum of degrees of freedom of each individ-ual robot. Integrated with complete/optimal planners, thecoupled algorithm achieves completeness/optimality. Nev-ertheless, it becomes intractable due to its exponential de-pendency on the number of degrees of freedom.

2.3 Decoupled Multi-Robot Path PlanningDecoupled approaches plan for each agent individually.

In prioritized schemes, paths are computed sequentially andhigh-priority agents are treated as moving obstacles by low-priority ones [6]. Searching the space of priorities can assistin performance [4]. Such decoupled planners tend to prunestates in which higher priority agents allow lower priorityones to progress, which may eliminate the only viable solu-tions. Search-based decoupled approaches consider dynamicprioritization and windowed search [21], as well as spatialabstraction for improved multi-agent heuristic computation[24, 30]. In particular, mobile robotic sensor networks re-quire that robots move while maintaining communication.Techniques which attempt to tackle this issue have to bal-ance a trade-off between centralized and decentralized plan-ning. There are techniques that create networks of robotsto compute plans in a centralized manner across distributedsystems [5] or in a fully decentralized manner [3].

2.4 FormationsThere is a significant amount of work on formation con-

trol, which is a way of moving multiple agents as a coherentteam. One direction is to use a virtual rigid body struc-ture to define the shape of a formation, and then plan forthis rigid body [14]. Other techniques attempt to have moreflexible structures, where interactions between robots aremodeled as flexible joints [1]. An alternative is to first gen-erate a feasible trajectory for the group’s leader accordingto its constraints and then use feedback controllers for thefollowers [7, 2].

2.5 Reactive Obstacle AvoidanceMany techniques attempt to solve the problem of colli-

sion avoidance using reactive methods. One technique usedin robotics is the Dynamic Window approach, which oper-ates directly in the velocity space of a robot, reasoning overthe achievable velocities within a small time interval [9]. An

Figure 2: Agents a and b share a proximity con-straint dprox. Agent a moves with velocity va where‖v‖ = s and can sense all agents within dsense.

alternative approach, known as the Velocity Obstacle (VO),assumes that neighboring agents will keep following theircurrent control. Based on this assumption, it defines conicregions in velocity space, which are invalid to follow, as theylead to a collision with the neighbor at some time in thefuture [8]. If the future trajectory of other robots is known,non-linear VOs can be constructed [11]. The basic VO for-mulation can result in oscillations in motion when multipleagents execute the same algorithm. The reciprocal natureof other robots can be taken into account in order to avoidthese oscillations, which leads to the definition of ReciprocalVelocity Obstacles [27]. Using this idea of reciprocity, therobots can attempt to optimally steer out of collision courseswith other robots using an extension of this framework calledOptimal Reciprocal Collision Avoidance (ORCA) [26]. Thistechnique was extended to 3D cases using simple-airplanesystems [23]. Further work extends the VO formulation towork with acceleration constraints as well as many kinemat-ically and dynamically constrained systems [28].

2.6 ContributionThis work uses a reactive technique to define new obsta-

cles in the velocity space, called Loss of Communication Ob-stacles (LOCO). LOCOs are computed quickly and, when inte-grated with Velocity Obstacles, allow robots to reactivelyavoid static and dynamic obstacles while maintaining a bet-ter sense of coherence. The new technique does not imposecommunication requirements, yet maintains connectivity ina decentralized manner. The approach still requires thatrobots are able to sense the position and velocity of otherrobots in the scene, as well as the position of static obstacles.

3. PROBLEM SETUPConsider n planar, holonomic disks moving with constant

speed s. Let the set of all agents beA. Each disk agent a ∈ Ahas radius ra and can instantaneously move with a velocityvector va that has magnitude s. Agents are assumed to becapable of sensing the position and velocity of other agentsin the environment within a sensing radius dsense. Further-more, the agents have available a map M of the environmentthat includes the static obstacles. The configuration spacefor each agent is Q = R2, and it can be partitioned into two

sets, Qfree and Qobst, where Qfree represents the obstaclefree part of the space, and Qobst is the part of the spacewith obstacles. Each agent follows a trajectory qa(t), wheret is time. Initially an agent is located at a configurationqa(0) = qinita and has a goal location qgoala .

Consider the distance d(a, b, t) between agents a and b attime t (Figure 2). If d(a, b, t) < ra + rb, then agents a andb are said to be in collision at time t. Collisions with staticobstacles occur when qa(t) ∈ Qobst. An input graph G(A,E)is provided that specifies which agents need to be in closeproximity. The vertices of graph G correspond to the setof agents A and an edge (a, b) implies that agents a and bmust satisfy d(a, b, t) ≤ dprox, where dprox is a proximityconstraint.

The objective is for the agents to move from qinit to qgoal

without any collisions with obstacles or among them, whilesatisfying as much as possible the proximity constraints spec-ified in the input graph G. More formally, all agents shouldfollow trajectories qa(t) for 0 ≤ t ≤ tfinal, so that ∀a ∈ A :• qa(0) = qinita ,• qa(tfinal) = qgoala ,• qa(t) ∈ Qfree, ∀t ∈ [0, tfinal],• and ∀b : ra + rb < d(a, b, t) < D, where

- D = dprox, if ∃(a, b) ∈ E of graph G,- and D =∞ otherwise.

4. APPROACH

4.1 Velocity Obstacle FrameworkVelocity obstacles are defined in the relative velocity space

of two agents. V O∞a|b can be geometrically constructed as

in Figure 3. Agent a’s geometry is reduced to a point byperforming the Minkowski sum of agent a and agent b: a⊕b.Then, tangent lines to the Minkowski sum disk a ⊕ b areconstructed from agent a. These tangent lines bound a conicregion. This region represents the space of all velocities vaof agent a that would eventually lead into collisions withagent b assuming that b has zero velocity. Given that agentb has a velocity vb, the conic region needs to be translatedby the vector vb. This construction assumes an infinite timehorizon. In practice, it is often helpful to truncate the VO

based on a finite time horizon τ . This VO represents allvelocities va, which will lead into a collision within timet ≤ τ , given that agent b keeps moving with velocity vb.

This work adopts a modification of the basic VO frame-work, which deals with the case that the two agents arereciprocating [27]. Reciprocal Velocity Obstacles (RVOs) arecreated by translating the VO according to a weighted aver-age of the agents’ velocities, (α · vA) + ((1−α) · vB) where αis a parameter representing the level of reciprocity betweenagents A and B. If both agents are equally reciprocal, thenα = 0.5.

Given this framework, an algorithm for calculating validvelocities for decentralized collision avoidance can be de-fined. Agent a will have a set of reachable velocities andthe task is to select one such velocity, which is collision-free.For holonomic disk agents moving at constant speed s, theset of reachable velocities would be Vreach = {v|‖v‖ = s}.Let the set of velocities which are invalid according to allthe VOs for neighbors of a be defined as follows: Vinv ={v| ∃ VOa|b s.t. v ∈ VOa|b, b ∈ A, a 6= b}. Then, the set offeasible velocities which are reachable but not in the invalidset is defined as Vfeas = Vreach \ Vinv. The selected veloc-

Figure 3: Construction of V O∞a|b for an infinite time

horizon. The Minkowski sum of a and b, a ⊕ b isused to define a cone in velocity space, which is thentranslated by vb. The shaded region represents allvelocities va, which lead a into collision with b.

Figure 4: Construction of LOCOτa|b and VVCτa|b. The cir-cular region represents the viable velocities to main-tain d(a, b, t) ≤ dprox for time horizon τ . The shadedregion outside the disk represents invalid velocitiesfor agent a.

ity should be feasible, v ∈ Vfeas, and typically minimizes ametric relative to the preferred velocity vprefa , e.g., a velocityvector that points to the goal. More details will be providedregarding the specific velocity selection scheme used in thiswork, in section 4.4.

4.2 Loss of Communication ObstaclesThe proposed approach extends the VO framework by cre-

ating new obstacles in the velocity space. These obstaclesaim to prevent loss of communication, or more generally,to satisfy proximity constraints between agents in the formd(a, b, t) ≤ dprox for agents a and b for at least a finite timehorizon τ . The LOCO imposed by agent b on agent a for ahorizon τ , denoted as LOCOτa|b, is the set

LOCOτa|b = {v| ∀ t ∈ [0, τ ] : d(a, b, t) ≤ dprox},

under the assumption that agent b follows its current veloc-ity vb for at least time τ .

Assume agents a, b ∈ A for which (a, b) ∈ E of the inputgraph G. The relative position of agent b for agent a will be

denoted as qab = qb− qa. If the relative position at time t isq(ab)(t), then at time t + τ the relative position of the twoagents is going to be:

qab(t+ τ) = q(ab)(t) + τ ∗ (Vb − Va).

For the two agents to be able to communicate at time t+ τ ,it has to be that:

(qXab(t+ τ))2 + (qYab(t+ τ))2 ≤ d2prox ⇒

(qXab(t)+τ∗(V Xb −V Xa ))2+(qYab(t)+τ∗(V Yb −V Ya ))2 ≤ d2prox ⇒

(qXab(t)

τ+ V Xb − V Xa )2 + (

qYab(t)

τ+ V Yb − V Ya )2 ≤ dprox

2

τ2⇒

(V Xa −qXab(t)

τ− V Xb )2 + (V Ya −

qYab(t)

τ− V Yb )2 ≤ (

dproxτ

)2

The last expression implies that the velocity Va of agent ahas to be within a circle with center ( qab

τ+ Vb) and radius

dproxτ

. This circle will be referred to as the valid velocitycircle (VVCτa|b) and is the complement of the LOCOτa|b. Figure4 gives an example of a VVC circle.

An agent a, however, may have multiple neighbors leadingto the definition of multiple LOCOs and VVCs. A choice that ismade for simplicity is to consider the same time horizon τ forthe definition of all the LOCOs for all the neighbors. Then,the set of velocities va that will not allow a to maintainconnectivity with at least one neighbor b in the graph G isthe union of individual LOCOs:

LOCOτa =

⋃∀b | ∃(a,b)∈E

LOCOτa|b

There are two complications arising from this definition.Firstly, it is not straightforward to compute the longesthorizon for which this union is not the entire plane, i.e.,the longest horizon for which the intersection of VVCs is notempty. The problem is that both the centers and the radiiof the VVCs are changing for different time horizons. Sec-ondly, the resulting region is rather complex to describe,as it corresponds to multiple circle intersections. This rep-resentation can impose significant computational overheadwhen the LOCOs are integrated with VOs.

Instead of computing the exact intersection of VVCs, thispaper proposes a conservative approximation that is easierto represent and beneficial for computational purposes. Theapproximation of the valid set of velocities corresponds to acircle inside the intersection of VVCs. Figure 5 illustrates theprocedure for two and three neighbors. Given two circleswith centers Cb and Cc and radii rb and rc, the inscribedcircle of their intersection has the following radius and cen-ter:

rbc =rb + rc − ||Cb, Cc||

2

Cbc = Cb + (rb − rbc)Cc − Cb||Cb, Cc||

The procedure works in an incremental manner. First itcomputes the inscribed circle (Cbc, rbc) of the intersection oftwo VVCs, and then computes the inscribed circle of (Cbc, rbc)with another VVC and so on.

Figure 5: The conservative approximation of VVCτa inthe velocity space of agent a for two (left) and three(right) neighbors. The white circle corresponds tovelocities that are guaranteed to maintain connec-tivity with the neighbors for time τ .

4.3 Integration of VOs and LOCOsThe final step for computing the LOCOτa is to select a suit-

able time horizon. A tuning approach over consecutive sim-ulation steps is used. Given the horizon τ from the previoustime step, the LOCOτa is computed as in Figure 5. Then theset Vvalid = Vreach \LOCOτa is computed, which considers thereachable controls that do not violate the LOCO constraints.This operation can be done in an efficient manner especiallyfor systems with constant speed s, as it corresponds to a cir-cle to circle intersection. In the general case for systems withvarying velocity there are still computational advantages, asthe circular representation of the VVC greatly increases thespeed in which Vvalid can be computed. Once Vvalid is avail-able for a given τ , the measure of |Vvalid| is compared againsta predefined threshold |Vthresh|. If |Vvalid| < |Vthresh|, thenτ is decreased and Vvalid is recomputed. A smaller τ impliesthat a bigger set of valid velocities for proximity mainte-nance will be returned that provides guarantees for a shorterhorizon. Alternatively, if |Vvalid| > |Vthresh|, then τ is in-creased. This will return a smaller set of valid velocitiesbut will provide guarantees for a longer horizon. This tun-ing process continues until Vvalid comes close to Vthresh, butthis takes place over multiple simulation steps and adaptson the fly to changes in the relative configuration of agents.The effects of tuning τ can be seen in Figure 6.

Once LOCOτa has been computed, it must then be includedin the list of constraints in order to correctly compute Vfeas,which is now redefined to be Vfeas = Vvalid \ Vinv, as inFigure 6. Sometimes, the additional constraints imposedby LOCOs, cause Vfeas to become empty. In this situation,the LOCO constraints are ignored, as attempting to maintaincommunication may be perilous to the agent’s safety.

4.4 Velocity SelectionEach agent has a preferred velocity it would like to fol-

low, denoted as vprefa for agent a. Ignoring the proximityconstraints and in obstacle-free environment, the preferredvelocity should be in the direction of the goal configurationvgoal = qgoala − qa. If there are obstacles, however, settingthe goal velocity in the same manner can lead the agentsinto local minima, causing the agent to become stuck be-hind the obstacle. This work avoids this issue by computinga discrete wave-front function in environments with obsta-cles. The goal velocity is computed as the direction to the

Figure 6: An example of how changing the horizonaffects the set of feasible controls. The left image hasa larger value for τ while the right has a smaller valueof τ . Larger values of τ provide stronger guaranteesfor communication maintenance, but make finding afeasible control more difficult.

cell with the minimum distance to the goal within a 3 × 3region from the current cell of the agent.

In order to account for agents violating proximity con-straints, a weighted velocity selection scheme is employedthat takes neighbors into account. For some agent a, let dibe the distance from agent a to another agent i and Xi bethe state of agent i, where agent i is part of the proxim-ity graph of agent a. Then, for n agents in the proximitygraph of agent a, the average weighted configuration can becomputed as:

qavg =

∑ni

didprox

qi∑ni

didprox

Then, davg = ||qa − qavg|| is the distance between agent aand qavg. Let vavg = qavg − qa be the vector pointing toqavg, and let vgoal be the vector towards the goal computedthrough the wavefront. Then, agent a’s preferred velocity iscomputed as follows:

vprefa =davgdprox

vavg +dprox − davg

dproxvgoal

Thus, agents which are farther away from their proximityconstraints (i.e., have violated constraints) will be inclined toshorten this distance, whereas agents that have not violatedany proximity constraints move towards their goals.

Once vprefa is computed, it can be checked for validitygiven the set Vfeas. If vprefa is feasible, then agent a willuse it. In the case that vprefa is not in Vfeas, a differentfeasible control must be computed. In the general case, theregion Vfeas defines an area in velocity space which must besearched to find a control. The control found is the velocityv ∈ V afeas which minimizes distance between v and vprefa .

The distance metric used in this approach is the Euclideandistance in the velocity space. Other metrics for finding thedistance between velocities in this space are possible [28]. Alist of intersections of the boundaries of the RVOs with V areachis generated and a rotational sweep algorithm determineswhich points are valid. These points define the boundariesof the V afeas regions. They are checked to find which oneof them minimizes the distance to the preferred velocity :

(a) PACHINKO environment (b) WEDGE environment (c) CROSSROADS environment (d) Random environment

Figure 7: The environments on which the experiments were executed

(a) (b) (c) (d)

Figure 8: Examples of input proximity graphs used in the experiments.

minv∈V afeas

[d(v, vprefa )]. In the general case, V afeas will be a

non-convex region or possibly disjoint non-convex regions inthe velocity space where a variant of the simplex algorithmcan be used to find the optimum.

In the event that there is no valid velocity available, theagent will select its current velocity. The reasoning behindthis is that in the V O framework, agents assume that theirneighbors will keep using their current control. Thus, choicesthat keep the current control are preferable for this scheme.

5. RESULTSThe approach was implemented using a simulation soft-

ware platform. Experiments were run on computers with a3.06 GHz Intel Core 2 Duo processor and 4GB of RAM. Theexperiments are organized in the following manner. First,experiments were conducted using a single team of agentsmoving in an environment with obstacles. Then, experi-ments with multiple teams were run both in an obstacle-free world and in an environment with obstacles. A teamof agents corresponds to a connected component of the in-put graph G. The experiments conducted compare the LOCO

formulation against RVOs. The reason for comparing againstRVOs, rather than other formation techniques, is due to thedecentralized nature of the LOCO algorithm, which requiresno additional information outside the VO framework. In ad-dition, LOCOs utilize the notion of coherence, which is moreabstract in nature than formations. Thus, RVOs are the mostrelevant technique to experiment against. Each experimentwas measured with regards to the following metrics:• total number of collisions during the entire experiment,• computation time per frame,• total time to solve the problem,• average ratio of respected proximity links per frame,• and number of successful runs.

For each variation of the environment, input graph G andalgorithm, there were 10 runs executed. Each run had a ran-dom initial qinit and goal configuration qgoal, which, satisfiedthe proximity link in the graph G.

5.1 Single TeamFor the single team scenarios, two different environments

were tested with varying numbers of agents. The PACHINKO

environment uses a series of walls with small gaps (Figure7(a)), and a team of 10 agents was considered in this case.The proximity graph imposed on the team is shown in Figure8(a). The WEDGE environment (Figure 7(b)) attempts to splitthe agents into two lanes. The proximity graph imposed onthe team is shown in Figure 8(b).

Table 1: Results for a single team.

Table 1 shows that there is a significant improvement interms of the percentage of links maintained by the LOCO-based approach relative to RVOs. Another interesting statis-tic to examine is the coherence of agents over time, shownin Figure 9, which the LOCO-based approach also improves.This comes at the cost of a slightly increased computationalcost and increased time taken by the algorithm to bring allof the agents to their goals. An example of the paths takenby 24 agents in the PACHINKO environment is shown in Figure10. Both approaches were able to solve all of the problemsand without any collision, with the exception of one experi-ment for the WEDGE environment, where a single collision wasreported.

Figure 9: Coherence over time for 24 agents in thewedge grid experiment.

Figure 10: An example of the paths taken by 24agents in the PACHINKO environment for the RVO (left)and LOCO (right) approach.

5.2 Multiple TeamsThe performance of LOCOs was evaluated against RVOs in

a scenario involving four teams of agents navigating in anobstacle-free environment as shown in Figure 7(c) (CROSS-ROADS) and for the connectivity graph for each team shownin Figure 8(c). There were four agents in each team. Thelast setup involved again four teams of agents, each one ofwhich had six agents connected as in Figure 8(d). This timethe environment involved a random set of rectangular obsta-cles as in Figure 7(d). For each of the 10 runs the rectangleswere randomly placed. The results are shown in Table 2.

In both the random and crossroads examples, LOCOs out-perform RVOs as far as connections are concerned, thoughoftentimes LOCOs take more time to solve the specified prob-lem. On different problems, both approaches can take someextra time to complete the problem for two different rea-sons. The VOs take extra time as agents will sometimes getcaught on obstacles or are pushed away from their goals byagents on other teams. LOCOs may take extra time as theyspend effort navigating agents in densely-packed situationsthat occur at the center of the environment as they crossover. Furthermore, LOCOs exhibit a flocking behavior whichsometimes causes agents to overshoot their goals.

Table 2: Results for multiple teams.

In the random obstacle environment, LOCOs imposed avery small computational overhead. The crossroads scenarioresulted in a more significant increase, because all agentsmeet in the center of the environment nearly at the samepoint in time. This increases computational cost, as LOCOsmust tune their horizon and reconnect broken proximitiesmore frequently. In both environments, LOCOs and RVOs re-sulted in no collisions, with the exception of one run of RVOsin the random environment. The main focus of these re-sults, however, is the improvement in maintained links. TheLOCO-based approach maintained 95% of the proximity links,which was an improvement over RVOs. LOCOs cause a smalldegradation in path quality, which is measured in the num-ber of steps it took for agents to solve the environment. Inaddition to this, LOCO failed to solve one of the randomlygenerated environments, because agents tend to spread tothe edge of their proximity constraints, which may causeproblems when agents move around obstacles. Overall, theresults seem to validate the initial approach, as the goal ofLOCOs is to maintain safety while providing better proximitymaintenance is experimentally supported.

6. DISCUSSIONThis work provides the formal definition for a new kind of

obstacle in the velocity space of moving agents, referred to asa Loss of Communication Obstacle (LOCO), which correspondto proximity constraints with neighboring agents. Theseobstacles can be easily computed and integrated into theexisting framework of Velocity Obstacles for decentralizedcollision avoidance. Additionally, an approach for tuningthe time horizon parameter for these obstacles over multiplesimulation steps is provided. These additional constraintsincrease the overall coherence of teams of agents while navi-gating through environments with static obstacles and othermoving bodies. LOCOs can be dropped if it is determined thatit is too difficult to maintain proximity without jeopardizingsafety. The implementation of the technique shows improvedcoherence for agents who share communication links withoutsacrificing safety at a small computational overhead.

A natural extension is to consider more challenging sys-tems, including kinematically and dynamically constrainedsystems. Adapting LOCOs to work in these cases is possible,as there have already been extensions on using VOs withdynamics. Since these extensions are in an orthogonal di-rection to the LOCO algorithm, it is easy so as to achievedecentralized coherence maintenance for dynamic systems.Further extending the work to applications of real robots willrequire introducing a method for handling sensor errors, aswell creating a more robust reconnection strategy built onthis error model. One drawback of reactive techniques isthat they may get stuck in local minima. It is interesting tobetter study the integration with the wavefront approach oranother global planner so as to guarantee that agents willmake progress towards their goal while guaranteeing safetyand connectivity. A global planner can also improve theperformance of the reconnection strategy. Currently, the di-rection to the agent with which connectivity has been lost

ignores the existence of local obstacles. Another interestingdirection is to study reciprocity tuning. More constrainedagents, such as those who create a bridge for two otherwisedisconnected agents, may be less able to reciprocate thanothers. Further reducing the computational overhead of thetechnique would also have great benefits, since larger-scaletests could be performed and have practical application ar-eas, as in crowd simulation and video games.

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