COMPETITIVE DYNAMICS FOR BEHAVIORCOORDINATION IN A JOINTTRANSPORTATION TASK 1

Estela Bicho, Luis Louro, ∗,2 Rui Soares ∗∗,3

Wolfram Erlhagen ∗∗∗,4

∗Dept of Industrial Electronics, University of Minho∗∗ Escola Superior de Gestao e Tecnologia de Felgueiras∗∗∗Dept of Mathematics for S&T, University of Minho

Portugal

Abstract: We address the problem of coordinating two non-holonomic mobilerobots that move in formation while transporting a long payload. A competitivedynamics is introduced that gradually controls the activation and deactivationof individual behaviors. This process introduces (asymmetrical) hysteresis duringbehavioral switching. As a result behavioral oscillations, due to noisy information,are eliminated. Results in indoor environments show that if parameter values arechosen within reasonable ranges then, in spite of noise in the robots communi-cation and sensors, the overall robotic system works quite well even in clutteredenvironments. The robots overt behavior is stable and smooth.

Keywords: motion coordination, mobile robots, payload/object transportation,collision avoidance, competitive dynamics, attractor dynamics.

1. INTRODUCTION

Joint transportation is a frequent task in many ofour daily activities (Fig. 1) and as we probablyhave experienced ourselves motion coordinationwith our partner, which is essential for the successof the task, is not easy. The problem is even harderwhen we do not have a complete view of theenvironment we are moving in. Needless to saythat in many applications it would be very usefulif teams of autonomous robots could be used totransport long payloads. Specially in hazardous

1 Project supported in part through grantsPOSI/SRI/38051/2001 from the Portuguese Foundationfor Science and Technology and EU-Projet JAST - JointAction Science and Technology (FP6-IST2, Proj. nr.003747)2 {estela.bicho,llouro}@dei.uminho.pt3 rsoares@estgf.ipp.pt4 wolfram.erlhagen@mct.uminho.pt

Fig. 1. Examples of joint transportation tasks

environments, places not yet reachable to humans(e.g. Mars), large distribution centers and largeindustrial plants. Thus many researchers and en-gineers working on cooperative robotics devotetheir efforts to this challenge(Ahmadabadi and

Nakano, 2001; Asahiro et al., 2001; Chaimowiczet al., 2001; Kosuge et al., 2000; Zaerpoor et al.,2005; Wang et al., 2004a; Wang et al., 2004b; Za-erpoor et al., 2005). Despite the efforts there arestill many problems to be solved.

There are two general approaches for controllingmultiple robots transporting an object: (1)- cen-tralized control schemes and (2)- decentralizedcontrol schemes. The limited success of (1) ismainly due to communication costs, (2) is betterbut there is a major difficulty: the precise controland coordination of the robots is very difficult.From the point of view of a cooperating robot theenvironment, which consists of the manipulatedobject, the other robots and the world scenario(static or dynamic), exhibits complex dynamicbehavior. The problem is exacerbated when theenvironment is unknown and no path is given.

In our approach we use leader-follower decentral-ized control strategies, and particular to our work,we use non-linear dynamical systems as a designand theoretical tool to design and implement thecontrol architectures (Soares and Bicho, 2002; Bi-cho et al., 2004; Soares et al., 2007). The controlarchitecture of each robot is structured in terms ofelementary behaviors. Each behavior is modeledas a contribution to the vector field of the dy-namical systems that generate the time course ofthe reference values to the control variables (head-ing direction and path velocity). Sensorial and/orcommunicated information is used to control acti-vation variables that determine which componentterm of the vector field must dominate the dy-namics.

In our previous work a non-linear static modelwas used to control the activation variables, i.e.their current values where made a function ofthe instantaneous values of the input variables.This raised, sometimes, the problem of oscillationsduring behavioral switch, due to limited sensorsrange and noise in the sensorial and/or com-municated information. To overcame this draw-back, we introduced a competitive dynamics, asreported in this paper, that gradually controlsthe activation and deactivation of individual be-haviors. This process introduces (asymmetrical)hysteresis during behavioral switching. As a resultthe behavioral oscillations are eliminated and therobots overt behavior is smooth. For other appli-cations in robotics using competitive dynamics see(Steinhage, 1998; Steinhage and Bergner, 1998;Large et al., 1999; Schoner and Santos, 2001).

We report tests with a team of two mobilerobots. Results in indoor environments show thatif parameter values are chosen within reasonableranges then, in spite of noise in the robots com-munication and sensors, the overall robotic systemworks quite well even in cluttered environments.

The rest of the paper is structured as follows:next section presents the robots and task con-straints. In section 3 we describe the behavioraldynamics for each robot in the team. The compet-itive dynamics for behavioral switching is definedin section 4. Implementation details and resultsobtained from real experiments are presented insection 5. The paper ends with section 6 withconclusions and an outlook for future work.

2. TASK CONSTRAINTS AND ROBOTTEAM

The control and coordination of the two robots isbased on the following ideas:

i) The leader robot holds an extremity of theobject and moves from an initial position to a finaltarget destination while avoiding sensed obstacles.

ii) The follower robot “helps” the leader to carrythe long object from the initial position to thefinal destination. This implies that this robothas to steer so that it keeps at all times anappropriate orientation and distance to the leader(i.e. an adequate formation). By default the robotsmust transport the object side by side (i.e. lineformation). However, due to the obstacles thismight not be possible. When obstructions aresensed the follower must drive in “transition”and/or column formation. Once it is possible itmust return to the line formation again (Figure 2).

Fig. 2. By default the robots must transport the objectkeeping a line formation. When due to encounteredobstacles that is not possible the follower robot mustchange its direction of navigation appropriately asillustrated.

iii) Each robot has a free rotational joint coupledto a free prismatic joint (see Figure 3). These areused to support the object and provide importantinformation to the robot (see Figure 4): a) fromthe current angle of the free rotational joint thefollower knows the direction, ψleader, at which the

leader is as seen from its current position andwith respect to the external reference axis x ; b)displacements (∆d) of the object are measured inthe free prismatic joint.

Fig. 3. We built a base for the object that consists of a360 deg rotational free joint and a prismatic free jointcoupled as illustrated here. The prismatic joint is anensemble of a linear slider and moving cart, with 30cm travel distance. The prismatic joint is connectedto a linear incremental encoder which measures thedistance of the cart to the reset position (i.e. ∆d).To measure ψleader, the rotational joint integrates arotational absolute position encoder.

Fig. 4. The robots are tightly coupled through a rotaryand a prismatic free joints as depicted. Each mobilerobot consists of a cylindrical platform with twolateral motorized wheels and a passive rear casterwheel. Each platform has a single board computersystem based on a 586DX4 processor operating at133 MHz, equipped with 4 Mbytes of DRAM and 8Mbytes of FLASH memory. All programming, controland computation are done on-board. The two lateralwheels are each driven by DC brushless servo-motor,each separately controlled by electronic circuitry thatguarantees accurate control of rotation speed. Eachrobot is equipped with nine infra-red sensors, whichare used to measure distance to obstructions atthe directions at which they are pointing in space.Their signal is uncalibrated as it depends on surfacereflectivity. The angular range over which distancesare averaged is about 30 deg. They are mounted ona ring centered on the robot’s axis. The sensors arearranged such that their sensitive cones just touch,thus completely covering the forward semi-circle. Thedistance range was set to 80 cm.

iv) The maximum allowed displacement of theobject is ±15 cm (i.e. ∆dmax = ±15) otherwisethe object falls down.

v) The leader sends to the follower only itscurrent path velocity and heading direction. Thefollower sends to the leader the measured value∆d.

3. ATTRACTOR DYNAMICS FOR OBJECTTRANSPORTATION

To model the robots’ behavior we use their head-ing direction, φr (0 ≤ φr ≤ 2π rad), with respectto an arbitrary but fixed world axis and pathvelocity, vr. Behavior is generated by continuouslyproviding values to these variables, which controlthe robot’s wheels. The time course of each ofthese variables is obtained from solutions of dy-namical systems. The attractor solutions (asymp-totically stable states) dominate these solutionsby design. In the present system, the behavioraldynamics of heading direction, φr(t), and pathvelocity, vr(t),(r = leader, follower) are non-lineardynamical systems defined as differential equa-tions

φr = fr(φr,parameters) (1)

vr = gr(vr, parameters). (2)

where the vector fields, fr(φr, parameters) andgr(vr, parameters), consist of a number of addi-tive contributions that express task constraints orelementary behaviors.

The leader ’s heading direction and path velocitydynamics has been previously defined and evalu-ated:

φleader = fobs(φleader) + ftar(φleader) + fstoch (3)

vleader = gobs(vleader) + gtar(vleader)(4)

These vector fields have two components, indexedby obs and tar, that represent the obstacle andtarget contributions, respectively. The stochasticforce guarantees escape from unstable fixed pointsin the heading direction dynamics (for details(Bicho et al., 2000; Bicho, 2000)).The complete behavioral dynamics of the fol-lower ’s heading direction and path velocity is alsogoverned by a non-linear dynamical system whichhas been previously defined (see (Soares and Bi-cho, 2002; Soares et al., 2007)):

φfollower =| Sline | fline(φfollower) (5)

+ | Stran | ftran(φfollower)+ | Scol | fcol(φfollower)

+fstoch

vfollower =| Sline | gline(vfollower) (6)

+ | Stran | gtran(vfollower)+ | Scol | gcol(vfollower)

where the indexes line, tran and col refer to thecomponents of the vector fields that model thebehaviors line, transition and column formationsrespectively. Sline, Stran and Scol are mutuallyexclusive activation variables that, depending onthe sensorial information acquired by the dis-tance sensors mounted on the follower robot andthe current heading direction of the leader robot

(i.e. φleader(t)), determine which component termof the vector field must dominate the dynam-ics. In our previous work these variables wereimplemented as a non-linear static model. Thisraised the problem of oscillations during behav-ioral switch. To overcame this drawback we makethe activation variables dynamic as explainednext.

4. COMPETITIVE DYNAMICS FORBEHAVIOR COORDINATION

The control over activation and deactivation of theactivation variables is obtained using a competi-tive dynamics (see e.g. (Schoner and Dose, 1992;Steinhage, 1998; Steinhage and Bergner, 1998;Schoner and Santos, 2001) for other applications):

Sline = αlineSline − |αline|S3line −

Sline(βl,tS2tran + βl,cS

2col) + fstoch

Stran = αtranStran − |αtran|S3tran −

Stran(βt,lS2line + βt,cS

2col) + fstoch (7)

Scol = αcolScol − |αcol|S3col −

Scol(βc,tS2tran + βc,lS

2line) + fstoch

This dynamical system has (Sline, Stran, Sline) ={(0, 0, 0), (±1, 0, 0), (0,±1, 0), (0, 0,±1)} as fixedpoints. Their stability depends on parameters αi

(i=line, tran, col).

In the absence of sensed obstacles the terms fline

and gline must dominate the vector fields in Eqs.5and 6, respectively. Thus (Sline, Stran, Sline) =(±1, 0, 0) must be made asymptotically stablefixed points of the competitive dynamics. This canbe achieved guaranteing that αline > 0, αtran < 0and αcol < 0 when obstacles are not sensed ortheir repulsion is weak.

Conversely, when obstructions are detected andthe difference between the direction ψleader andφleader is larger than θ the robots must drive intransition formation, so the terms ftran and gtran

must dominate the vector fields. This requiresthat states (Sline, Stran, Scol) = (0,±1, 0) must beasymptotically stable fixed points. This happenswhen αline < 0, αtran > 0 and αcol < 0.

Else, if obstructions are detected but the differ-ence between the direction ψleader and φleader issmaller than θ we want the robots to move in col-umn formation, thus the states (Sline, Stran, Scol) =(0, 0,±1) must be now asymptotically stable fixedpoints of the competitive dynamics. Achieved bymaking αline < 0, αtran < 0 and αcol > 0.

αline, αtran and αcol determine the relaxation rateto the states (±1, 0, 0), (0,±1, 0) and (0, 0,±1).

βij permits to control the inhibition of Sj over Si

(i,j=line, tran, col, and (i 6= j)).

A function that indicates if obstacle contribu-tions are present is the potential function ofthe obstacle avoidance dynamics for this robot,Uobs(φfollower), which has been defined in (Bichoand Schoner, 1997; Bicho et al., 2000). Has shown,positive values of this potential function indicatethat the robot’s heading direction is in a repulsionzone of sufficient strength. Conversely, negativevalues of the potential indicate that the headingdirection is outside the repulsion range or repul-sion is very weak. Applying a sigmoidal thresholdfunction,with a large slope, to the potential we geta function that ranges from −1 to 1:

αpot(φfollower) = 2 arctan[cs Uobs(φfollower)]/π (8)

Also, applying a sigmoidal threshold function,with a large slope, to the difference | ψleader −φleader | −θ:

α∆ = 2 arctan[cs (| ψleader − φleader | −θ)]/π (9)

we can write the following quasi-boolean functions(note that these are mutually exclusive) thatindicate which behavior is the desired:

γline = 1− αpot(1 + αpot)/2 (10)

γtran = αpot(1 + αpot)(1 + α∆)/4 (11)

γcol = αpot(1 + αpot)(1− α∆)/4 (12)

We have realized, through experiments with theplatforms, that the performance of the roboticsystem is improved when for the same sensoryconditions the behavioral switch from transitionformation or column formation to line formationoccurs later in time and is slower then the reverse(i.e. αline < αtran, αcol).

Finally, setting

αline = α1 · (−1)(1−γline) , α1 > 0 (13)

αtran = α2 · (−1)(1−γtran) , α2 > 0 (14)

αcol = α3 · (−1)(1−γcol) , α3 > 0 (15)

in the competitive dynamics guarantees that theadequate activation variable relaxes to ±1 (witha relaxation rate determined by αi, i = line, tran,col) while the other two relax to 0.

5. IMPLEMENTATION AND RESULTS

The complete dynamic architectures, includingthe attractor dynamics for object transportationand the competitive dynamics for behavior co-ordination, were implemented and evaluated on

the robots. In the implementation all differen-tial equations are integrated numerically usingthe forward Euler method. Sensory informationis acquired once per computation cycle. The cy-cle(step) time is measured and is approximately50 ms for the leader and 60 ms for follower. Asthe time step must be smaller than the fastest re-laxation time on the system, this imposes minimaltime scales on the entire dynamical architectures.Thus the computational cycle time is the limitingfactor for determining the relaxation times of thedynamics in real time units and thus for the over-all speed at which the robots’ behavior evolves.Because the systems operate close to attractorsof known stability, the time scales, or reverselythe relaxation rates in, can be set as a functionof the computation cycle and thus guarantee thenumerical stability. The rate of change of headingdirection obtained from the dynamics of headingdirection, i.e. Eqs. 3 and 5, directly specifies theangular velocity of the robots for rotation aroundtheir center. The path velocity (obtained from thenumerical integration of Eqs. 4 and 6) specifies theaverage rotation speed of both wheels. Together,the rotation speeds of both wheels are computedand sent as set points to the velocity servos of thetwo motors.

We filmed the robotic system in task scenarioswhere the two robots transport a long object incluttered indoor environments. As may be seen onthe videos the robots’ overt behavior is smooth.This is due to how the competitive dynamicspermits sensorial and communicated informationto affect in a graded fashion, and with hysteresis,behavioral switch.

Figure 5 illustrates, in one of the tests,the robots’behavior through a sequence of video images. Fig-ure 6 shows the activation variables’ time series.The situation is a scenario testing the ability tocarry the object while simultaneously coping withsituations where obstacle avoidance is in conflictwith the robots’ task. Initially, the robots areplaced as indicated in Panel A. The leader alwaysmoves toward the target while avoiding collisions.The follower steers so as to avoid collisions and tohelp the leader carrying the object. The followerstarts by steering so as to keep a line formationand steering on the left side of the leader (Panels A- B). When it senses the obstacle it turns right andsteers so that it follows the leader and simultane-ously avoids collisions with the obstacles, i.e. tran-sition formation (Panels C - F). Then the leaderenters into the narrow passage (Panel G). Duringthe movement through this narrow passage thefollower keeps a column formation with the leader(Panels G - J). Once it is possible (Panel K) thefollower tries to drive again in line formation withthe leader until the final target position is reached(Panels L - N).

Fig. 5. Sequence of video images illustrates themotion of the robots while transporting anobject in a cluttered environment. The robotsmove smoothly and around the obstacles.

6. CONCLUSION AND FUTURE WORK

We have shown how a competitive dynamics maybe used to coordinate two non-holonomic mobilerobots that move in formation while transportinga long payload in cluttered indoor environments.Results have demonstrated that robots’ overt be-havior is stable and smooth. Near future work isconcerned with object transportation by teamsof mobile manipulators and human-robot jointtransportation.

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

1.2

time (s)

line formation transition formation column formation line formation

transitionformation

line formation |S

i|

Fig. 6. Time series of |Sline| (red full line), |Stran|(blue doted line) and |Scol| (green dashedline). Active behavior is indicated. βl,t = 10,βl,c = βt,l = βt,c = βc,t = βc,l = 0.8 andα1 = 3, α2 =α3 = 4 and θ = 5deg.

ACKNOWLEDGMENT

We would like to thank Toni Machado, NzojiHipolito, Eliana Silva and Andre Domingos fortheir help in many ways.

REFERENCES

Ahmadabadi, M N and E Nakano (2001). A con-straint and move approach to distributedobject manipulation. IEEE Transactions onRobotics and Automation 17(2), 157–172.

Asahiro, Y, E C Chang, A Mali, I Suzuki andM Yamashita (2001). A distributed laddertransportation algorithm for two robots in acorridor. In: Proc. IEEE Int. Conf. Roboticsand Automation. ICRA 2001. pp. 3016–3021.

Bicho, E (2000). Dynamic Approach to Behavior-Based Robotics: design, specification, analy-sis, simulation and implementation. ShakerVerlag. Aachen. ISBN 3-8265-7462-1.

Bicho, E and G Schoner (1997). The dynamic ap-proach to autonomous robotics demonstratedon a low-level vehicle platform. Robotics andAutonomous Systems (21), 23–35.

Bicho, E, L Louro and W Erlhagen (2004). Coor-dinated transportation with minimal explicitcommunication between robots. In: Proc. of5th IFAC/Euron Symposium on IntelligentAutonomous Vehicles. IAV 2004. Istituto Su-perior Tcnico, Portugal.

Bicho, E, P Mallet and G Schoner (2000). Targetrepresentation on an autonomous vehicle withlow-level sensors. The International Journalof Robotics Research 19(5), 424–447.

Chaimowicz, L, T Sugar, V Kumar and M Cam-posa (2001). An architecture for tightly cou-pled multi-robot cooperation. In: Proc. IEEEInt. Conf. Robotics and Automation. ICRA2001. pp. 2292–2297.

Kosuge, K, H Takeda and Y Hirata (2000). Decen-tralized motion control of two tracked mobilerobots transporting a single object in coordi-nation based on function allocation concept.

In: in Aro Workshop 2000 on Intelligent Sys-tems, Australian National University, Can-berra. Australia.

Large, E W, H I Christensen and R Bajcy (1999).Scaling the dynamic approach to path plan-ning and control: Competition amoung be-havioral constraints. The International Jour-nal of Robotics Research 18(1), 37–58.

Schoner, G and C Santos (2001). Control of move-ment time and sequential action through at-tractor dynamics: A simulation study demon-strating object interception and coordination.In: in ISIE 2001: International Symposium onIntelligent Robotic Systems. Toulouse.

Schoner, G and M Dose (1992). A dynamical sys-tems approach to task-level system integra-tion used to plan and control autonomous ve-hicle motion. Robotics and Autonomous Sys-tems 10, 253–267.

Soares, R and E Bicho (2002). Using attractordynamics to generate decentralized motioncontrol of two mobile robots transporting along object in coordination. In: Proc. of theworkshop on Cooperative Robotics, in IROS2002: 2002 IEEE/RSJ Int. Conf. on Intel-ligent Robots and Systems. EPFL Lausanne,Switzerland.

Soares, R, E Bicho, T Machado and W Erlha-gen (2007). Object transportation by multiplemobile robots controlled by attractor dynam-ics: theory and implementation. In: Proc. ofthe 2007 IEEE/RSJ Int. Conf. on IntelligentRobots and Systems. IROS 2007. San Diego,CA, USA. pp. 937–944.

Steinhage, A (1998). Dynamical Systems for theGeneration of Navigation Behavior. ShakerVerlag. Aachen.

Steinhage, A and T Bergner (1998). Dynamicalsystems for the behavioral organization of ananthropomorphic mobile robot. In: Proc. ofthe Fifth International Conference on Simu-lation of Adaptive Behavior. From Animals toAnimats 5. pp. 147–152.

Wang, Z, Y Hirata and K Kosuge (2004a). Controla rigid caging formation for cooperative ob-ject transportation by multiple mobile robots.In: Proc. IEEE Int. Conf. on Robotics andAutomation. ICRA 2004. pp. 1580–1585.

Wang, Z., Y. Takano, Y. Hirata and K. Kosuge(2004b). A pushing leader based decentralizedcontrol method for cooperative object trans-portation. in Proc. IEEE Int. Conf. Intelli-gent Robots and Systems 1, 1035–1040.

Zaerpoor, A, M Ahmadabadi, M Baruni andZ Wang (2005). Distributed object trans-portation on a desired path based on con-strain and move strategy. Robotics and Au-tonomous Systems (50), 115–128.