Robot team control: A geometric approach

Robotics and Autonomous Systems 53 (2005) 59–71

Robot team control: A geometric approach

Joao Sequeira∗, M. Isabel Ribeiro

Instituto Superior Tecnico/Institute for Systems and Robotics, Av Rovisco Pais 1, 1049-001 Lisboa, Portugal

Received 1 March 2005; received in revised form 7 March 2005; accepted 31 May 2005Available online 22 August 2005

Abstract

This paper describes the control of robot teams in the framework of Hilbert spaces. The paper focus are the intrinsic propertiesof robot control architectures, namely the conditions under which a generic mission can be successfully executed.

The proposed paradigm develops in two levels: (i) single robot control supported on a monotonic and non-expansive projectionmap defined over some behavioral space such as the robot configuration space or the velocity space, and (ii) team control supportedon a supervision scheme over a set of neighboring relations among the teammates, accounting for their relative motion.

Each robot monitors its own neighboring relations for relevant changes and adapts its motion to the objectives of the teamusing a finite state automaton supervisor.

Simulation results on teams of 2D holonomic and cart robots are presented.© 2005 Elsevier B.V. All rights reserved.

Keywords: Robot control; Co-operative control; Geometric approaches

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. Introduction

This paper presents an approach to the controlf robot teams focused on the intrinsic properties ofontrol architectures. These properties result fromeneral principles on robot motion obtained usingasic geometry tools for Hilbert space and lead tokeleton control algorithms. The control structure atach robot is separated in (i) single robot control in the

ramework of differential inclusions, (ii) team controln a discrete event systems framework. Therefore, theverall system resembles a hybrid system.

∗ Corresponding author. Fax: +351 218418291.E-mail addresses: [email protected] (J. Sequeira),

[email protected] (M.I. Ribeiro)

In the framework of dynamical systems, theithrobot in a team withn members, moving in a spaiQ = iq, is represented by a dynamic systemiq(t) =fi(iq(t), iu(t)), with initial condition iq(0) = iq0,i = 1, . . . , n, t being the time andiu ∈ iU the controvector.1 When considered isolated from the restthe team, the synthesis of theui that makes the robfollow a reference path or move towards a refereconfiguration is a classical robot control problemhas been widely studied[6,9,14,17]. In the last yearthere has been an increasing interest in multiple rproblems, e.g., spacecraft and military formatio

1 Without loosing generality, this model can simply be assumethe differential kinematics.

921-8890/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.robot.2005.05.002

60 J. Sequeira, M.I. Ribeiro / Robotics and Autonomous Systems 53 (2005) 59–71

for which techniques ranging from control theory toartificial intelligence were proposed in the literature.

In [12] a behavior based architecture for fault toler-ant cooperating robots was proposed. This three levelarchitecture is based on the subsumption architecture.The highest level models the motivation of a robotusing performance measures such as the impatience(the attitude of the robot towards the teammates)and acquiescence (the attitude towards itself). Theintermediate level contains sets of specific behaviorsactivated by the motivational behaviors. Each of thesesets emphasizes a particular global robot behavior,e.g., find a location using a methodical behavior or awander behavior. The lowest level contains the basiscompetence layers for the robot to survive.

Motor schemas (primitive motion strategies) wereused in[3] to control teams of cart and car-like vehiclesaiming at moving in formation. The formations consid-ered were defined either using the distances betweena robot and its neighbors, between a robot and a teamleader or between a robot and some characteristic pointof the formation, e.g. the center of mass. Formationspecific motor schemas allow each robot to computeits motion direction so that no formation break occurs.

In [4] the formation control problem was decoupledinto the planning of a reference path to be followed bya virtual leader and a tracking problem to be handledby the real robots in the team. The formation is de-fined by the kernel of a convex map such as the sum ofquadratic errors between the position of each robot and

nth

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position. The free space the team is allowed to use toavoid formation breakings is obtained by superimpos-ing this region on the obstacles in the environment.

The approach presented in this paper differs fromthe ones in the current literature in that it separates thecontrol problem into a layer that depends exclusivelyon the properties of the behavioral spaces of the robots,i.e., on the intrinsic properties of the space used toformulate the control problem, being independentof the particular robot, and a layer that handles theproblem-dependent information.

A large variety of single robot missions neitherrequire exact path following nor that the robot reachesa specific configuration. Instead, the robot is requiredto move within some bounded region in the freeconfiguration space and/or to reach a goal region ora specific configuration. This problem, of relevancein behavioral robotics, has been considered within theViability Theory framework[1].

Similar considerations apply to teams of robotsoperating either under tight or loose constraints on thedistance among the team members. For example, toavoid that the distance among team members growsabove a pre-specified limit, the team may be requiredto span a compact region in the workspace whilemoving towards the goal (a synthesis problem) or onemay be interested in determining the region spannedby the team during a mission (an analysis problem).

In the team control problem considered in this papertheith robot in a team must reach a goal set,Ki ⊂ iQ.

hese-ringme

obot

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its reference trajectory. The formation control is giveby a steepest descent technique on the map definingformation.

Formations defined by smooth functions of the reative distances of fully actuated spacecrafts were aconsidered in[8]. The overall system is divided intoan average system, that captures the average motiothe team, and a shape system that captures the relavelocities among the robots and hence the formatpattern. PD-like laws are used to make both the averand shape systems follow their reference trajectorie

In [10] a formation is defined by a set of vectordefined after the relative positions of the robots in tteam. The formation control problem is formulatedthe control of a set of double integrators, acting unda leader-follower strategy. A compactness assumpton the control space of each robot allows the definitiof a compact uncertainty region around its curre

e

fe

This set can be a priori defined to account for tmission specifications and modified either as a conquence of the data acquired by on-board sensors duthe mission or as requested by the team motion. In sosense, this set defines a reference behavior for the rto follow during the assigned mission.

The team control paradigm proposed in this paencompasses (i) the definition of the goal setsKi to ac-count for the specific mission specifications (ii) a cotrol strategy that will drive theith robot towardsKi,and (iii) a negotiation procedure, handled by a finstate automaton (FSA) at each robot, to adapt thetion strategy to the requirements imposed by the teneighboring constraints during the motion towardsKi.

Once defined the goal setKi, and in the absence omotion constraints, the set∆Ki (

iq) = k − iq∀k ∈ Kidefines the motion directions that drive theithrobot directly towardsKi. In general, given the

J. Sequeira, M.I. Ribeiro / Robotics and Autonomous Systems 53 (2005) 59–71 61

constraints imposed by the robot model, only asubset of ∆Ki (

iq) is feasible. This subset resultsfrom the intersection between the feasible velocities,fi(iq, iU) ≡ fi(iq, iu), iu ∈ iU, defined by therobot dynamics, and the desired velocities,∆Ki (

iq),defined from the mission goals. The correspondingcontrol problem can be expressed in the framework ofdifferential inclusions byiq ∈ fi(iq, iU) ∩ ∆Ki (

iq).The existence of solutions to differential inclusions

requires regularity conditions, namely various types ofcontinuity or semicontinuity for the mapFi : iQfi(iq, iU) ∩ ∆Ki (

iq) [2]. In addition, convexFi val-ues are also required by standard theoretical results onthe existence of solutions to such problems[16]. Formost problems in robotics, convexifying procedurescan be applied to theFi values if necessary, at a costof a slight reduction/enlargement of the set of allowedmotion directions. Given the dependence ofFi on theenvironment (through∆Ki ), these conditions are sel-dom verified. Furthermore, the co-domain values ofFi

are often the empty set, meaning that the robot can-not reach a desired motion direction. In this situationfi(iq, iU) and∆Ki (

iq) must be made to converge toeach other. The main focus of the paper lies in this lastproblem,2 i.e., what motion direction to choose whenFi = ∅.

The organization of the paper is as follows.Section2 presents sufficient conditions for the conver-gence of the relative motion of two sets which is usedto solve the single robot control problem modelled

in

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ne

the two sets converging to 0. Common behavioralspaces such as the configuration space and the velocityspace of real mobile robots are Hilbert spaces andhence the metric induced by an inner product can beused to define a distance between sets in such behav-ioral spaces. In particular, the following definition forthe distance between two setsA andB is used.

Definition 1 (Distance between subsets). LetA andBbe two subsets of a normed space. The distance betweenA andB is defined as

d(A, B) = minxa∈A,xb∈B

‖xa − xb‖, (1)

where‖ · ‖ stands for a norm. Although commonly re-ferred as a distance[7] (1) is not a metric. The distancebetween a pointxa ∈ A and a setB is defined by makingA a singleton.

The following theorem defines a projection map ofa point onto a set. See[2] for a demonstration.

Theorem 1 (Best approximation theorem).Let G bea closed convex subset of a Hilbert space, X. There is amap π : X → G and a unique element πG(x) verifying

∀x∈X, ‖x − πG(x)‖ = miny∈G

‖x − y‖.

The mapπ : X → G projectingX into the goal setGis called abest approximation projector (b.a.p.) when-

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by a differential inclusion. Section3 details thecontrol structure for single robots using the resultsSection2. Section4 expands the structure to multiplerobots. Section5 presents simulation examples for 2Dholonomic and cart robot teams. Section6 concludesthe paper and points directions for further work.

2. Motion in Hilbert spaces

The existence of a solution for the control probleformulated in Section1 requires the convergencebetween two velocity sets. This convergence canexpressed by having a measure of the distance betw

2 The convexity of theFi values whenFi = ∅ can be readily veri-fied given the convexity of the robots velocity model and of the co∆Ki

.

n

ever the following two conditions are verified

‖πG(x) − πG(y)‖ ≤ ‖x − y‖ (non-expansivity),

〈πG(x) − πG(y), x − y〉 ≥ 0 (monotonicity) (2)

with 〈·, ·〉 standing for an inner product and‖ · ‖ for thecorresponding induced norm.

The generation of a trajectory converging towara convex set can be derived using some basic toolsgeometry. The following lemma states sufficient coditions for this convergence to occur, provided that tadequate control is chosen. The demonstration follofrom the use of a b.a.p. defined fromTheorem 1and theconditions(2), and from Schwarz inequality (see[15]for details).

Lemma 1 (Point-to-set convergence).Let X be aHilbert space, G ⊂ X a convex set, x ∈ X\G andπG(x) a b.a.p. of a point x onto G. Furthermore, let


Fig. 1. Interpretation ofLemma 1whenX is a 2D Euclidean space.

xn = x(tn) be a subsequence of a path x(t), outside G.Choose xn+1 such that the sequence defined by

λn = 〈xn+1 − xn, πG(xn+1) − xn+1〉, (3)

verifies,

∃α>0, ∃N>0 : ∀n>N, d(xn, G) < αλn ∧ λn > 0.

Then limn→∞ d(xn, G) = 0.

Fig. 1 illustrates an intuitive interpretation ofLemma 1whenX is a 2D Euclidean space. The innerproduct,λn in (3), represents a rate at which the dis-tance toG decreases by choosing the pointxn+1 whenthe system is atxn (assuming that controls driving thesystem betweenxn andxn+1 exist).

Whenxn+1 → xn, λn becomes

λ = 〈x, T ∗G(x)〉, (4)

where T ∗G(x) represents the conjugate cone toG at

πG(x), i.e., the set of vectors orthogonal to the spacetangent to the border ofG at πG(x). Lemma 1has adiscrete nature, asxn stands for a point sampled attimetn. The continuous time formulation for the lemmais straightforward to obtain, withλ > 0 and requiringthat the decreasing rate ofλ be smaller than that ofd(x, G).

If the control problem is defined by algebraic ex-pressions, the discrete convergence lemma can bused. This is the case when the robot is modelled

t-isedd-

ism

For non-algebraic robot models the continuous ver-sion ofLemma 1can also be used. This is the case whenthe robot is modelled by a differential system and thegoal set is expressed in the configuration space.

3. Control of single robots

The control of a robot, considered as an isolatedagent, is based on expression(3) or in its equivalent(4), the continuous time version ofLemma 1. The con-ditions in the lemma indicate that controls leading toλn > 0 (orλ > 0) should be preferred. However, sucha greedy approach may lead to points in the null spaceof λn (which is precisely what happens whenλn con-verges to 0 faster than the distance to the set). A typ-ical situation arises due to the loss of degrees of free-dom in the control vector, i.e., one or more controls donot cause any progress in the mission even though therobot may be moving. This is reflected in ˙qn+1 = qn, orqn+1 − qn ⊥ π∆K (qn+1) − qn+1. Such situations arehandled by a FSA that modifies the motion strategywheneverλn approaches 0.

Table 1summarizes the control algorithm for singlerobots,3 with the posterior superscripts denoting theidentification of the robot. The control space of theith robot is partitioned in three regions,iU = iU1 ∪iU2 ∪ iU3, where each of theiU1, iU2, iU3 contains,respectively, the controls leading toiλn > 0, iλn = 0

iredheof

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by an algebraic model (e.g., as the direct kinemaics model of a serial manipulator) and the goal setdefined in the configuration space. Robots modellby differential systems can also be controlled accoring to the discrete lemma if the control problemset on the velocity space (thus making the problealgebraic).

e

andiλn < 0.The data used to define the goal set may be acqu

through onboard sensors and is seldom convex. Talgorithm starts by extracting a convex goal set outthis data to account for the conditions inLemma 1.Extracting a convex set out of an acquired rawK canbe made using computational geometry techniques.instance, assuming a simple polygonal region, polygconvexifying techniques[5] or the convex hull[13] canbe used. An alternative technique is to decomposeraw data into Dirichlet regions, choosing one amothem (see[11] for an introduction to the convex partitioning of simple polygons).

3 Throughout the paper the discrete version of the lemma is useillustrate several ideas. For the sake of simplicity, references tocontinuous version of the lemma are often omitted. Neverthelethere is no lack of generality as the arguments used with the discversion can be easily stated for the continuous version.


Table 1Single robot control algorithm

if necessary, convexify the goal set∆K(iqn)computeiU1 = iu ∈ iU : iλn > 0if iU1 = ∅ chooseiu = argiU1

max(iλn)otherwise

computeiU2 = iu ∈ iU : iλn = 0compute the partitioniU2 = iU21 ∪ iU22 ∪ iU23

whereiU21 = iu ∈ iU2 : iqn+1 − iqn = 0iU22 = iu ∈ iU2 : iqn+1 − iqn ⊥ π∆K

(iqn+1) − iqn+1iU23 = iu ∈ iU2 : π∆K

(iqn+1) − iqn+1 = 0if iU23 = ∅ then choose anyiu ∈ iU23

otherwisecomputeiU3 = iU\iU1 ∪ iU2chooseiu ∈ iU21 ∪ iU22 ∪ iU3

repeat until goal is reached

Controls iniU1 guarantee that the robot approachesits goal. Choosing the control that maximizes thevalue of iλn corresponds to the maximization ofthe rate at which the distance between robot andgoal decreases. Controls iniU21 guarantee that thedistance between the robot and its goal (measuredby ‖π∆K (iqn+1) − iqn+1‖) does not increase and,depending on the specific robot, may even decrease.Controls in iU23 guarantee that the robot reaches itsgoal. Under controls iniU3 the distance between therobot and its goal increases. Nevertheless, they maystill be useful to maneuver the robot to a point thatsimplifies the reaching of the goal.

Computing theiU1,2,3 regions of the control spacemay require the use of exhaustive search techniques toevaluate the projection map and avoid solving inversedynamics problems, e.g., computingiu from iq. Inthe simulation experiments presented in Section5the control space is discretized into a small numberof admissible values thus allowing an exhaustivesearch.

The liveness properties of the algorithm inTable 1express the ability of the algorithm to successfullydrive the robot towards the mission goal. The algo-rithm has no deadlocks because each state has anoutput transition. However, the choice of specificcontrols in iU21 ∪ iU22 ∪ iU3 depends on the robotcapabilities and/or mission specifications. If notproperly chosen these may lead to a livelock situationi ce ofm

4. Control of robot teams

The paper focus a particular class of robot teams,namely that where the configuration of a team, denotedby 1q, . . . , nq, determines a set of distance relationsamong the team members. These express, for exam-ple, how much the distance between a robot and itsneighbors is allowed to vary with the motion of theteam.4

A useful extension is obtained if the distance be-tween theith robot and a teammate is taken alonga subset of directions in the configuration space.This extension allows, for instance, the distinctionbetween teammatesbehind or in front of the ithrobot. These extended distance relations define re-gions around each robot where its neighbors are lo-cated and hence are called positional neighboringrelations.

Formally, a neighboring relationship between theithrobot and its neighbor teammates can be expressed by aset valued map,Ri : iQ iQ, where each value in theco-domain,Ri(iq), defines a region where the neighborteammates are located when roboti is at iq. Thoughnot in the scope of this paper, further extensions arepossible, accounting for example for the time and thevelocities of the robots.

Neighboring relations holding for the entire durationof a mission are topological invariants.5 For the purposeof this paper, a team of robots satisfying such a set ofneighboring conditions is called a formation. For a teamo ali eV .I edb te ofnf

e-l d

onb cest lp

si .,c fort

n which the robot may endlessly repeat a sequenaneuvers.

f robots moving in a 2D surface, a practical topologicnvariant to define formations can be obtained from thoronoi diagram of the set of positions of the robotsn this case, the neighboring relations may be definy imposing that all the Voronoi polygons, defined aach instant, preserve the contact with a fixed seteighbor polygons.Fig. 2 shows an example of such

ormation.For practical purposes, the class of neighboring r

ations leading to useful formations is usually restricte

4 When these relations are exact, for example if the relative positietween robots is to be kept constant, the control of the team redu

o a set ofn (one per robot) standard trajectory following controroblems.5 A topological invariant is a mathematical object that maintain

ts qualities between any two initial and final configurations (e.gonnectedness) under the motion in the space of team trajectorieshe assigned mission.


Fig. 2. A formation defined over the set of Voronoi polygons.

to the translations that bound the inter-robot distances(measured in the workspace). Among the practicalcases of formations in this class are some soccer andmilitary tactical movements.Table 2summarizes theteam control algorithm, implemented as a FSA at eachrobot.

The negotiation state inTable 2handles the viola-tion of the neighboring conditions based on the infor-mation exchanged among the teammates. In general,these are mission and robot specific. For example, anegotiation may simply be defined asstop the robot ifRij = ∅, for any teammate j, otherwise move to goal,and hence involves querying the teammates about theirconfigurations. In this case, the algorithm inTable 2isa straightforward two-state FSA. Complex negotiationalgorithms may require arbitrarily complex automata.

The FSA in Tables 1 and 2suggest that a largeclass of formation control problems share the sameskeleton structure. It must be emphasized that ensuringthat the overall control algorithm leads each robotoperating in a team to the successful completion of itsmission requires that conditions for theiλn sequencessimilar to those inLemma 1be verified by each singlerobot.

Table 2Team control algorithm

given the neighboring relations for theith robot,Ri : iQ iQ

computeRij = Ri(iq) ∩ jq, j = i;if R = ∅, for any teammatej, then

5. Experimental results

This section presents simulation results on teamcontrol, illustrating the ideas outlined along the pa-per. Two experiments are presented, both with teamsof size 3. In both experiments, the mission assigned tothe team is to move through a sequence of goal regionsdefined in the configuration space. The mission ends assoon as one of the team members reaches its final goalregion.

In the first experiment the team is composed by2D holonomic robots moving in the plane, each ofwhich controlled according to the discrete conditionin Lemma 1. In the second experiment the team iscomposed of cart robots, each of which controlledaccording to the continuous version ofLemma 1.Two versions of this experiment, differing in theneighboring conditions, are presented. For the sakeof simplicity, the robots are dimensionless in bothexperiments.

The goal setsKi are balls of radius 0.1 centeredin a priori defined locations whilst the sets of desiredvelocities are defined as∆Ki (

iq) = k − iq ∀k ∈ Ki.The computation ofiλn (for the discrete ver-

sion of the lemma) andiλ (for the continu-ous version) values relies on an exhaustive searchover a small cardinality control set, thus avoid-ing having to solve inverse dynamics problems.To speed up the search process, the control spacein both experiments is restricted to(v, ω) ≡ V ×Ω

cr esa ort eto ncef

π

a

i

I

T

ij

negotiate with teammates;otherwise, proceed to goal;

, with V = −0.1, −0.05, 0, 0.05, 0.1 and Ω =−0.1, −0.05, 0, 0.05, 0.1. For the 2D holonomiobots,V andΩ stand, respectively, for the velocitilong thex- andy-axis of the world reference frame. F

he cart robots,V andΩ stand, respectively, for the sf linear and angular velocities in the robot refere

rame. The b.a.p. map at instanttn is given by

∆K (iqn) = −iqn + 0.1(iqn + iqn)/‖iqn + iqn‖,

nd, from(3),

λn = 〈π∆K (iqn) − iqn+1,iqn+1 − iqn〉.

n the second experiment,

∗K(iq) = α(qK − iq)/‖qK − iq‖


Fig. 3. Structure of the FSA composed from the single and teamcontrol FSAs.

with α an arbitrary positive constant andqK the centerof K. Expression(4) is given by

iλ = 〈[cos(iθ)iv, sin(iθ)iv, iω], T ∗K(iq)〉.

The composition of the FSA inTables 1 and 2re-sults in a FSA with the structure shown inFig. 3. Thestructure is similar for both experiments, differing onlyin the definition of the events firing the transitions be-tween states.

The core states arecontinue and negotiate. Afterthe initialization procedures each robot reaches thestatecontinue, remaining there as long as no violationof the neighboring conditions occurs (triggeringe2)and its current goal is not yet reached (triggeringe5).The controls used are those defined by the algorithmin Table 1. In practical terms, the statecontinueaggregates a set of states handling all the single robocontrol motion strategies, i.e., the specific motionstrategies generated by controls in each of theiU1,2,3regions.

Whenever any of the neighboring relations isviolated, e2 is triggered and the control is trans-ferred to the statenegotiate where it remains untilthe violation disappears (triggeringe3). In thiscase the controls inTable 1 are overridden byspecific (problem-dependent) controls. For the two

experiments in this section the robots stop whenin the negotiate state, implicitly assuming that theteammates will take the adequate action, bring-ing to an end any violations of the neighboringconditions.

When a robot reaches its current goal set,e5 istriggered and the control jumps to statesynchronize.The robot waits in this state for the teammates to reachtheir goal sets before choosing another goal set (ifany is still left in the mission sequence) by triggeringe1 after what it jumps tocontinue to proceed to thegoal.

The FSA inFig. 3 does not use all the conditionsin Lemma 1, namely it does not ensure that theiλn

converges to 0 slower thand(iqn, K). Therefore, thesuccess of a mission with arbitrary goal sets cannot beguaranteed.

5.1. Team of 2D holonomic robots

The objective of the mission is to have a team of2D holonomic robots reaching the sequence of goalsets, with centers defined inFig. 4a, while preservinga reference formation. This reference formation is de-fined from the initial configuration of the robots andthe neighboring relations between theith robot and theteammates defined by

Ri = q ∈ iQ : 〈iqgoal − iq, q − iq〉 ≥ 0 ∧

eisft

s

ott.

t

0.1 < d(iq, jq) < 0.5 ∪ q ∈ iQ

: 〈iqgoal − iq, q − iq〉 < 0.This relationship defines a circular sector, in thdirection of the goal set, where any teammateallowed to stay. The whole region in the back othis sector is also an allowed region. Any robolocated outsideRi violates this neighboring condi-tion, forcing this robot to enter a negotiation withall the teammates. While negotiating a robot istopped.

The algorithm inTable 1is exemplified inTable 3.Fig. 4b shows the trajectories of the robots, with the

goal sets shown as wide circles. The triangles in the plindicate the relative positions at the same time instanThe mission is successfully completed.

The trajectories are mainly in straight line dueto the holonomy of the robots (a straightforward


Fig. 4. Mission definition and results for the 2D holonomic robot team.

consequence for controls inU1) and the motion strat-egy adopted when in thenegotiate state. For the mostpart of the mission each of the robots acts as an isolatedagent as no neighboring relations are violated. Thespikes inFig. 4c and d, around samples 60 and 120,

illustrate the reaching of the goal sets, with the robotschanging their objectives and waiting for the teammatesto reach their owns. When entering the neighborhoodof the final goal sets they are forced to shorten thedistances among them, resulting in the violation of


Table 3Controller for the single 2D holonomic robot

(the goal velocity sets are always convex)computeiU1 = iu ∈ iU : iλ > 0;if iU1 = ∅ chooseiu = argiU1

max(iλ)otherwise

computeiU2 = iu ∈ iU : iλ = 0;compute the partitioniU2 = iU21 ∪ iU22 ∪ iU23

if iU23 = ∅ choose anyiu ∈ iU23

otherwiseif iU21 ∪ iU22 = ∅

choose anyiu ∈ iU21 ∪ iU22

otherwise signal errorrepeat until goal is reached

the neighboring conditions and in the correspondingnegotiations. The closest robot to the final goal sethas no neighboring conditions violated and henceproceeds to its goal. The remaining two wait for theleader to go away to have their neighboring conditionssatisfied and hence proceed also to their goals. This isclearly visible inFig. 4c and e starting around sample140.

5.2. Team of cart robots

The goal of the mission is to have the cart robotsreaching the goal sets, defined in the configurationspace, while preserving a reference formation.Table 4shows the centers of the goal sets. The neighboring re-lations for theith robot are defined, in the configurationspace, by

Ri = q ∈ iQ : d(iq, q) ≤ 0.5 ∪ q ∈ iQ : d(iq, q)

> 0.5 ∧ d(iq, iqgoal)

< d(jq, jqgoal) ∀j = i.The first subset in the above relationship defines a re-gion around the robot (within a distance of 0.5) wherethe relationship is considered verified if at least a team-mate is located there. This ensures that if the robots stay

Table 4Mission definition for experiment 2

Robot 2 Robot 3

(−0.7, −0.7, 0) (−0.6, −0.8, 0)(−0.56, 0.73, π/2) (−0.79, 0.45, π/2)(0.22, 0.48, 0) (0.48, 0.76, 0)

Table 5Controller for the single cart robot

(the goal velocity sets are always convex)let iU1 = iu ∈ iU : iλ > 0;if iU1 = ∅ chooseiu = argiU1

max(iλ)otherwise

let iU2 = iu ∈ iU : iλ = 0;with iU2 = iU21 ∪ iU22 ∪ iU23

if iU23 = ∅ choose anyiu ∈ iU23

otherwiseif iU21 ∪ iU22 = ∅

let align = (ix − ixgoal) cos(iθ) + (iy − iygoal) sin(iθ)if align < 1e−2

chooseiu = (ivcte, 0)elseif‖(ix, iy) − (ixgoal,

iygoal)‖ < 1e−2

chooseiu = (0, iωcte)otherwise signal error

repeat until goal is reached

close to each other no special negotiation is requiredand each of them proceeds towards its own goal. Ifthe team is splitting apart and the roboti is being leftbehind then the relationship is also verified and therobot proceeds towards its current goal. This condi-tion ensures that the team does not span arbitrary largeregions while moving. While negotiating a robot isstopped.

The control of each single robot follows the al-gorithm in Table 5. The controls to be applied wheniU1 = ∅ are divided into two of the basic maneuversa cart robot can perform: pure translations and purerotations.

The mission was successfully completed with theplots in Fig. 5 showing similar behavior by all therobots in the team.Fig. 5a and c shows the result-ing trajectories in the 3DC-space and in thexy plane,with the goal sets shown as shaded regions in the 3Dtrajectories plot. The triangles inFig. 5c indicate therelative positions of the robots at the same instant oftime.

The trajectory plots show intense maneuvering asa result of the nonholonomy of the robots which also

Robot 1

qstart (−0.8, −0.6, 0)qgoal1 (−0.31, 0.31, π/2)qgoal2 (0.76, 0.47, 0)qgoal3 (0.5, −0.5, −π/2)
(0.5, −0.5, −π/2) (0.5, −0.5, −π/2)


Fig. 5. Results of the first experiment with the cart team.


Fig. 6. Results of the second experiment with the cart team.


leads to the intense negotiations shown inFig. 5b. Thisagrees with the behavior of theλ curves, with the spikesindicating the motion strategy changes induced by thenegotiation state at each robot and the following de-scending slopes indicating the motion under thecon-tinue state.

When negotiating, each robot is stopped. Definingneighboring conditions based only in the inter-robotdistances would result in a blocking strategy, i.e., ifall the robots violate simultaneously the neighboringcondition the team ends in a deadlock. Includingconditions on the distance to the goal set allows thefarthest robot to move out of the blocking formationheading towards its goal acting as a leading robot.As this robot approaches its goal, another robotyet blocked is freed and starts moving towards itsown goal set, thus releasing the formation from theblocking condition. This effect can be seen inFig. 5band f. The time spent in the negotiation state agreeswith the time when inter-robot distance is above0.5.

The purpose of this experiment is not to demonstrateany obstacle avoidance capabilities. Also, situationswhere the team ends in a deadlock, with all membersnegotiating, are easy to create.

Fig. 6 replicates the experiment when the neigh-boring conditions are similar to those used in theexperiment in Section5.1, with 0.15 < d(iq, q) < 0.3.These bounds were carefully chosen to avoid allowing

ein

topdt

re

neer

unconstrained motion. The FSA handles situationswhere controls verifying the sufficient conditionsand violations of the neighboring relations amongteammates cannot be found.

The simulation examples presented are relativelysimple, aiming at illustrating that formations of holo-nomic and nonholonomic robots share the same con-trol structure. The specific neighboring relations con-sidered reduce to the definition of regions around therobots where teammates must be located.

Further work includes (i) the design of motionstrategies to ensure that theλ values decrease slowerthan the distance between the robot and a goal set,(ii) the study of the formation transient behavior,namely the behavior of the team when a formationis broken and how long does it take for the team torecover. This additional open issue amounts to thestudy of equilibria in the proposed hybrid structureand hence results from hybrid systems theory andgeneralized Lyapunov functions are likely to beused.

Acknowledgements

This work was supported by the FCT projectPOSI/SRI/40999/2001 and Programa Operacional So-ciedade de Informac¸ao (POSI) in the frame of QCAIII.

,

-6–

-–

s

l,

ece,

the team enter a deadlock, with all members in thnegotiation state. The results are similar to thoseFig. 5, with the trajectories differing in the velocity ofexecution.

6. Conclusions

This paper presented a geometric approachthe control of robots, supported on a projection madefined from basic properties of Hilbert spaces ansufficient conditions for the convergence of the roboconfiguration to a goal subset in theC-space.

The approach encompasses a two-level structuwith: (i) an algorithm to partition the control space and(ii) a FSA to choose which element of this partitionprovides the controls at each time instant. The partitioalgorithm uses the sufficient conditions stated in thpaper for the successful execution of a mission und

References

[1] J.P. Aubin, Viability Theory, Birkhauser, 1991.[2] J.P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag

1984.[3] T. Balch, R. Arkin, Behavior-based formation control for multi

robot teams, IEEE Trans. Robotics Automat. 14 (1999) 92939.

[4] M. Egerstrdt, X. Hu, Formation constrained multi-agent control, IEEE Trans. Robotics Automat. 17 (6) (2001) 947951.

[5] B. Grunbaum, How to convexify a polygon, Geocombinatoric5 (1995) 24–30.

[6] O. Khatib, S. Quinlan, D. Williams, Robot planning and controRobotics Auton. Syst. 21 (1997) 249–261.

[7] E. Kreysig, Functional Analysis, Wiley, 1978.[8] D. Lee, P. Li, Formation and maneuver control of multipl

spacecraft, Proceedings of the American Control ConferenDenver, CO, June 4–6, 2003.


[9] R. Murray, S. Sastry, Nonholonomic motion planning: steer-ing using sinusoids, IEEE Trans. Automat. Contr. 38 (5)(1993).

[10] P. Ogren, N. Leonard, Obstacle avoidance in formation, Pro-ceedings of the IEEE International Conference Robotics andAutomation, 2003.

[11] J. O’Rourke, Computational Geometry in C, 2nd ed., Cam-bridge University Press, 1998.

[12] L.E. Parker, ALLIANCE: an architecture for fault tolerant mul-tirobot cooperation, IEEE Trans. Robotics Automat. 14 (2)(1998) 220–240.

[13] F. Preparata, M. Shamos, Computational Geometry: An Intro-duction, Springer-Verlag, 1985.

[14] C. Samson, Path following and time-varying feedback stabiliza-tion of a wheeled mobile robot, Proceedings of the ICARCV’92,Singapore, September 1992.

[15] J. Sequeira, M.I. Ribeiro, Geometric control of robotteams, Technical Report RT-602-03, Institute for Systemsand Robotics/Instituto Superior Tecnico, Lisboa, Portugal,2003.

[16] G. Smirnov, Introduction to the Theory of Differential Inclu-sions, Graduate Studies in Mathematics, vol. 41, AmericanMathematical Society, 2002.

[17] O. Sordalen, Feedback control of nonholonomic mobile robots,PhD Thesis, Department of Engineering Cybernetics, The Nor-wegian Institute of Technology, N-7034 Trondheim, Norway,1993.

J. Sequeira received his Licenciatura,M.Sc. and Ph.D. degrees in electrical andcomputer engineering from the InstitutoSuperior Tecnico, Technical University ofLisbon, in 1987, 1991 and 1999, respec-tively. He is an assistant professor at In-stituto Superior Tecnico since 1991 anda researcher at Institute for Systems andRobotics (ISR). His research interests in-clude behavior based robotics, human–robot interaction and cooperative robotics.

M. Isabel Ribeiro received her Licen-ciatura, M.Sc. and Ph.D. degrees in elec-trical engineering from the Instituto Supe-rior Tecnico, Technical University of Lis-bon, Portugal in 1978, 1983 and 1988, re-spectively. From 1978 she has had teachingand R&D responsibilities at Instituto Supe-rior Tecnico where she is now a full pro-fessor. She is a member of the Institute forSystems and Robotics (ISR), being respon-sible for the Mobile Robotics Laboratory.

Her research interests include navigation of mobile robots, cooper-ative robotics, search and rescue robotics, world representation andmapping. She is a senior member of IEEE and of Ordem dos Engen-heiros, the Portuguese Association of Engineers.

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Robot team control: A geometric approach