R ESEARCH ARTICLE

doi: 10.2306/scienceasia1513-1874.2010.36.333

ScienceAsia 36 (2010): 333–341

Collaborative nonlinear model-predictive motionplanning and control of mobile transport robots for ahighly flexible production systemSuparchoek Wangmanaopituka,∗, Holger Voosb, Waree Kongprawechnona

a School of Information, Computer and Communication Technology,Sirindhorn International Institute of Technology, Thammasat University, P.O. Box 22,Thammasat-Rangsit Post Office Pathum Thani 12121, Thailand

b Automatic Control Laboratory, Research Unit in Engineering, University of Luxembourg,Campus Kirchberg, 6 rue Coudenhove-Kalergi, L-1359, Luxembourg-Kirchberg

∗Corresponding author, e-mail: suparchoek w@yahoo.comReceived 24 May 2010Accepted 12 Oct 2010

ABSTRACT: This study is based on a new approach for an advanced microproduction system or highly flexible productionsystems where all necessary production and assembly processes are connected in a very flexible way using autonomousmobile transport and handling robots. Each robot has to follow its planned paths while avoiding collisions with other robots.In addition, problem-specific constraints for a defined microproduction system, such as limitations of the velocity andaccelerations of the robots, have to be fulfilled. This paper focuses on a two-level model predictive optimizing approach.On a global long-term level, simple dynamic models of the robots are used to compute optimal paths under differentialconstraints where a safety distance between all robots is achieved. Since many uncertainties and unforeseen events couldoccur, all robots also use a nonlinear model predictive control approach on a local real-time level. This control approachsolves the path following and the collision avoidance problems in parallel, while also taking into account differentialconstraints of the single robots.

KEYWORDS: mobile robots cooperation, model-predictive control, path following and collision avoidance

INTRODUCTION

In a multi-robot system, several mobile autonomousrobots are used to act together to achieve overall com-mon goals. Possible areas of application are flexiblemanufacturing environments, and here, an industrialexample of a flexible microproduction system hasbeen introduced1. The production of microsystemssuch as micromotors, micropumps, microgears, andmicroscale optical devices needs special methods,due to their special requirements and characteristics.These products most often are produced only in smallquantities with special customer needs. Hence theentire manufacturing process for miniaturized andmicro-structured parts must be optimized in order todevelop opportunities for cost-effective and flexiblemicroproduction. During the last few years there hasbeen considerable progress in new manufacturing orassembly processes for single microcomponents suchas shafts, toothed wheels, and lenses. However, thereis hardly any solution for the automated production ofoverall microproducts from these components. One

Fig. 1 Structure of the proposed microproduction system.

main reason is that microproducts require their ownspecialized sequence of manufacturing and assemblysteps and humans still play an important role in theprocesses.

This paper focuses on the development of sucha highly flexible automated microproduction system.

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It is assumed that the system includes all necessarymanufacturing and assembly processes in the form ofsuitable stationary machine tools as shown in Fig. 1.In order to form a highly adaptable and reconfigurableoverall microproduction process, these machine toolswill be interconnected by autonomous mobile robots.The multi-robot transport infrastructure allows forvery flexible and even parallel interconnection of thedifferent stationary microproduction machine tools. Itis assumed that in the first solution the workpiecesare fixed on palette systems for transport in order toavoid the handling of extremely small workpieces bythe robots. The autonomous mobile robots have onemain important task which is the transportation of thepalettes in the right sequence between the differentmachine tools without any collision, and thus formingan adaptable and reconfigurable interconnection ofsingle processes.

Motion planning for mobile robots is one of thefundamental and most intensively studied roboticstasks (see, e.g., Refs. 2–6 for overviews). In thispaper, the main contribution is on motion planningfor multiple robots7–11, and the considered obstaclesare dynamic (namely the other robots). An adaptedversion of prioritized planning on a global long-termlevel, for planning rough collision free paths definedby waypoints for all the robots, will be applied. Thisapproach fits well to the underlying transportationproblem: if any robot starts its transportation task, itcan be assumed that the already moving robots havea higher priority. Therefore, the considered robotcomputes its own collision free path with the helpof a model predictive approach taking the alreadydetermined paths of the other prioritized robots asfixed.

This approach then has to be extended to in-clude differential constraints. In order to simplifythe algorithms, this approach only considers velocityconstraints on the global long-term planning leveland more detailed differential constraints on the localreal-time control level. For global motion planning,the velocities of the robots are considered as beingconstant but limited between two waypoints. Planningunder differential constraints also has been intensivelystudied2. One useful approach is the discretizationof the constraints by using a simplified discrete-timemodel of the robotic motion. In this paper, theresult of the global long-term decoupled planningunder simplified differential constraints is a priorityrelationship between the robots and a set of collision-free waypoints for all robots, from the start to thegoal location, with a fixed limited velocity for eachway-segment between two waypoints. However, in

reality, uncertainties and unforeseen events during theexecution of the plans can have a strong influence onthe overall resulting motion of the robots. Problemsof this type are also intensively studied in the litera-ture2, and solutions are obtained using probabilisticplanning or dynamic re-planning on the global long-term level. In this contribution, those problems arenot solved on the global long-term planning level, butare combined with the solution of the path followingproblem, and therefore solved on a local, real-timemotion control level. Here, the solution of the long-term motion planning can be interpreted as a set ofpaths that must be followed by the robots with a‘desired’ velocity on the respective path segments.If these conditions are perfectly fulfilled, this wouldresult in collision free paths.

Therefore, all robots are continuously combiningthe task of path following with collision avoidanceunder detailed differential constraints on the localreal-time motion control level. The main task for eachrobot is to follow the specified path with the desiredvelocity, while continuously checking for any possiblecollision. This is done with the help of the blackboardand the knowledge about all current locations ofthe robots, and therefore, in a collaborative fashion.The problem of motion control, like path following,has also been investigated. One promising approachwhich motivates the proposed solution is based onmodel predictive control12–16 for the path followingor tracking problem17, 18, since it offers a natural wayto include differential constraints. In addition, thiscontribution extends a nonlinear model predictive pathfollowing algorithm with collision avoidance to a veryefficient overall approach.

MATERIALS AND METHODS

The multi-robot system and experimental testbed

The proposed microproduction system has some spe-cial characteristics with respect to the group of mobileautonomous robots. As a manufacturing facility, itwill be an indoor environment with a defined struc-ture, i.e., the machine tools and any other objectsare stationary at fixed positions with free flat spacein between for the navigation of the mobile robots.Since this manufacturing infrastructure is fixed, it isassumed that a Cartesian map of the environment isdefined and available to each single robot. The onlymoving objects within this environment then are therobots and any human workers.

For the localization of robots and human workers,a global vision-based positioning system is assumed.That leads to the advantage that the single robots must

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Fig. 2 The experimental testbed with global positioningsystem.

not each be equipped with their own localization sys-tem like in the simultaneous localization and mappingproblem, based on visual information provided bycamera or laser-scanning systems. In addition, themoving human workers can be detected outside ofany sensorial range of the robots. For the design ofthe positioning system, a network of camera systemsmounted at the top of the manufacturing facility isassumed, where the single cameras are distributed ina way that the full environment is covered by thefield of view of these visual sensors. Each camerais equipped with an image processing system that isable to detect the actual position and orientation of anymobile robot and human worker in the respective fieldof view (Fig. 2).

In order to uniquely identify the mobile robots,these are equipped with a distinct visual pattern fixedon the robot chassis. Each distinct pattern is associ-ated with a unique ID-number for the identificationof the respective mobile robot. The position andorientation of each robot, clearly identified by the ID-number, is posted at an electronic blackboard. Inaddition, the position and orientation of any humanworkers are also posted at the blackboard without aunique ID since it would be unrealistic to assumeany fixed identifiable pattern for the human workers.

Each robot is able to access the blackboard and toread the information of all currently available mov-ing objects. This access to the blackboard will berealized using wireless communication. The robotsare also equipped with a camera system which ismainly intended for the control of the manipulationtask. During transportation this camera system witha limited field of view in the direction of movementis used for a local emergency stop mechanism. Thisindependent mechanism leads to an increased levelof safety in case of any malfunction of the globalpositioning or the communication system.

Distributed Navigation

Within the multi-robot system, one main task is thefulfilment of the transportation orders assigned to thesingle robots. As mentioned before, these ordersare defined by the start position and the destinationposition of the respective robots.

The overall navigation problem is structured asfollows. First, each robot plans its individual optimalpath according to its given transportation task. How-ever, looking at the multi-robot system, this individualoptimal planning might lead to paths that include col-lision points of two or even several robots, which leadsto a non-optimal solution from an overall perspec-tive. One possible solution would be a coordinateddetailed path planning algorithm on the multi-robotlevel which would lead to optimal individual pathsunder the constraints that collision points are avoided.

This paper proposes a two-level distributed ap-proach. First, all robots use a local long term planningalgorithm for the calculation of individual optimalpaths. This algorithm is based on a grid-map of theenvironment and a computation of the shortest pathon the grid using efficient algorithms. It also doesnot include any velocity or acceleration constraintson the robots. The single robots then publish theirindividual optimal paths on a blackboard. All robotscan access this blackboard and are looking for anypoints where more than two robots can meet. In thosecases, the involved robots form a group and solvetheir problems in a way that priorities are given tothe robots. The path of the robot with highest priorityremains unchanged. The robots with lower priorityhave their local paths recalculated while the formercollision grid point is blocked for them within the nextshortest path calculation.

After this recalculation, the result would be a setof rough grid map based paths of the robots, whereonly collision points of a maximum of two robotsoccur. These situations are then resolved on a locallevel using a model-predictive approach as described

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R

L

b

Fig. 3 Model of a mobile robot with differential drive.

in the following section.

Mathematical problem description

Mobile transport robots are robots with twodifferential-drive wheels on one common axis and onecaster wheel (Fig. 3). Robots with this configurationhave a restricted mobility in the sideways directionand thus have an underlying nonholonomic property.The posture, i.e., position and orientation of the robotin a Cartesian x-y-coordinate system is described bythe kinematic equations:

x = v cos θ, y = v sin θ, θ = ω, (1)

where v is the heading velocity, θ is the heading angle,i.e., the angle between the x-axis and the axis of therobot, and ω is the angular velocity of the robot. Usinga differential drive, the two input variables v and ω aregenerated via the two wheel velocities vR and vL ofthe right and the left wheel, respectively. If slippagecan be neglected(

vω

)=(

0.5 0.51/b −1/b

)(vRvL

). (2)

The length b is the wheel base of the robot. However,if the desired values of v and ω are computed, thecorresponding values of vR and vL can be calculated.Therefore, this equation can be inverted.

It is assumed that each robot has to follow thepreviously calculated path, given by straight pathsegments between waypoints on the grid map. Thepath following problem of a single robot under con-sideration (which will be called Robot 1) is depictedin Fig. 4 and describes the task of following the givenpath while the forward velocity vR1 of the robot is notpart of the control problem. It is more suitable to workwith a path coordinate system where d is the currentorthogonal distance between the robot and the path,and s is the distance travelled along the path directionstarting from the last waypoint. The waypoint iis defined by its position vector ri = (xi, yi) inCartesian space and rR1 = (xR1, yR1) is the current

R1R1xR1R1y

r

r

Fig. 4 The path following problem of a single robot.

position vector of Robot 1. The orientation of the pathsegment between the neighbouring waypoints ri andrj is given by the angle ϕij (Fig. 4).

The vector rij is the vector that points along thecurrent path segment and is given by rij = rj − ri.Using this vector and the current position of the robot,the orthogonal distance between robot and currentpath segment can be calculated from

d =|rij × (rR1 − ri)|

|rij |. (3)

For the description of the path following problem,it is more suitable to describe the movement of theconsidered Robot 1 with regard to the path coordinatesystem in the form

sR1 = vR1 cos(θR1 − ϕij),

dR1 = vR1 sin(θR1 − ϕij).(4)

However, while following the desired path from way-point to waypoint, the robots also have to avoid colli-sions. As previously described, the distributed globalpath planning algorithm results in situations where theconsidered Robot 1 can only meet a second robot,called Robot 2. When two robots are in a collisionsituation, the ‘right-before-left’ rule is followed, i.e.,the robot that comes from the right side relative to thecurrent orientation of the two robots has priority andthe other robot (coming from the left) is responsiblefor the collision avoidance (Fig. 5).

The distance R between the two robots withcurrent local position vectors rR1 = (xR1, yR1) and

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R1R1xR1R1y

R2

R2

R2y

R2x

Fig. 5 The engagement geometry of two mobile robots.

rR2 = (xR2, yR2) is defined as:

R = |rR1 − rR2|=√

(xR1 − xR2)2 + (yR1 − yR2)2. (5)

Collision avoidance means that the distance R mustnever be smaller than a defined security threshold Rs

defining the constraint

R > Rs, ∀t. (6)

The security threshold must be defined with respectto the geometry of the involved robots. The mi-croproduction environment results in some additionalapplication-specific constraints. Since the robots haveto transport extremely small parts in palette systems,which should not be shaken too much, the accelera-tions in the travel direction (aR1x ) and perpendicularto the travel direction (aR1y ) must be limited, as wellas the velocities and turning rates:

− aR1y,max < aR1y = vR1ωR1 < aR1y,max,

− aR1x ,max < aR1x = vR1 < aR1x ,max,

− ωR1,max < ωR1 < ωR1,max,

− vR1,max < vR1 < vR1,max.

(7)

The task of Robot 1 now consists in following the de-sired path defined by (4) while keeping the constraintsgiven by the kinematic equations (1), the constraintsadded by the collision-avoidance problem (5), (6)

and the problem-specific constraints (7). Therefore,this approach directly combines these three differentand partially contradicting tasks of path followingand collision avoidance under the problem-specificconstraints in contrast to other existing solutions.The problem is solved by a model-predictive controlapproach as described in the following section.

Model-predictive path following and collisionavoidance

In order to derive the model-predictive approach,a discrete-time version of the underlying dynamicmodel will be developed. The dynamic equations thatdescribe the kinematics of any robot n, n ∈ {1, 2} aswell as the dynamics with respect to the desired pathare obtained from (1) and (4):

xRn = vRn cos θRn,

yRn = vRn sin θRn,

θRn = ωRn,

sRn = vRn cos(θRn − ϕij),

dRn = vRn sin(θRn − ϕij).

(8)

The vector of state variables xRn =[xRn, yRn, θRn, sRn, dRn], and the vectoruRn = [vRn, ωRn] are defined as the vector ofinput variables with respect to robot n.

From (7) the problem-specific constraints cantherefore be written as

− aRny,max < u1,Rnu2,Rn < aRny,max,

− aRnx,max < u1,Rn < aRnx,max,

− ωRn,max < u2,Rn < ωRn,max,

− vRn,max < u1,Rn < vRn,max,

(9)

where ui,Rn is component i of the vector uRn. Thecollision avoidance constraint between Robot 1 andRobot 2 is given by (6).

By applying the Euler approximation to the dif-ferential quotient with time interval ∆T , the set ofdifferential equations (8) is converted into a set ofalgebraic equations (using the notation of the inputand state variables). The conversion of the firstdifferential equation in (8) is

x1,Rn(k + 1)− x1,Rn(k)−∆T (u1,Rn(k) cosx3,Rn(k)) = 0, (10)

where k denotes a discrete time step, and xRn(k) anduRn(k) denote the discrete-time vectors of state andinput variables.

Assume that at t = 0 (and hence k = 0)Robot 1 and Robot 2 have the initial vectors of state

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variables xR1(0) and xR2(0) and both robots haveto follow a path with given current path angles ϕij,1

and ϕij,2, respectively. The proposed algorithm thenworks as follows. For a given time horizon of Ktime steps, those trajectories of input and state vectors,XRn = [xRn(1), . . . ,xRn(K + 1)] and URn =[uRn(0), . . . ,uRn(K)], have to be calculated in away that the two robots travel along their paths whilekeeping all constraints. This can be done via theminimization of the objective function

JRn(URn,XRn) =K+1∑k=1

(dRn(k))2 − (sRn(k))2

=K+1∑k=1

(x5,Rn(k))2 − (x4,Rn(k))2.

(11)

The first term is included to minimize the distanceto the path while the second term is included inorder to move the robot forward along the path. Theset of constraints with respect to the dynamics ofthe robot (8) after discrete-time formulation (10) cangenerally be formulated as a set of equality constraintswith the vector function gRn(URn,XRn) = 0. Theproblem-specific constraints (9) can be given as aset of inequality constraints with the vector functionhRn(URn) < 0. The constraints describing the col-lision avoidance problem (5), (6) between Robot 1and Robot 2 can finally be formulated as a set ofinequality constraints defined by the vector functioncR1,R2(xR1,xR2) < 0. It has to be taken into accountthat this set of inequality constraints depends on thestate variables of both robots. However, since Robot 2has priority, it can optimize its own path followingproblem over the horizon of K time steps withouttaking into account the collision avoidance problemby solving the nonlinear static optimization problem,

min{UR2,xR2}

JR2(UR2,xR2)

s.t. gR2(UR2,xR2) = 0,hR2(UR2) < 0. (12)

The results are the sets of optimal input and corre-sponding vectors of state variables over the horizondenoted by U∗R2 and x∗R2. Robot 1 now has to followits path while avoiding collisions with Robot 2, whichis assumed to be on its optimal path defined by x∗R2. Inthe collaborative approach as proposed in this paper,it is assumed that Robot 2 communicates this plannedoptimal path to Robot 1. With the information aboutthe future behaviour of Robot 2 given by x∗R2, Robot 1now solves the following nonlinear static optimization

problem:

min{UR1,xR1}

JR1(UR1,xR1)

s.t. gR1(UR1,xR1) = 0,hR1(UR1) < 0,

cR1,R2(xR1,x∗R2) < 0. (13)

After the calculation of the trajectories of optimalvectors of input variables U∗R1 and U∗R2, only theoptimal steering commands u∗R1(0) and u∗R2(0) forthe current time step are realized, and the overallprocedure starts again in the next time step. Thismeans that the steering commands of the two robotsare always calculated from model-based predictionsof the future trajectories, but the calculated futuretrajectories are not fully implemented. The reason forthat approach is the possibility of disturbances of thestate variables that can occur in the next time step.Thus the overall scheme is a model-predictive controlalgorithm, but realized by communicating robots. Thefull procedure can be summarized as follows:

1. The current discrete time is set to k = 0.Both robots receive the current posturevectors (xR1(0), yR1(0), θR1(0)) and(xR2(0), yR2(0), θR2(0)) from the globallocalization system.

2. Both robots determine the current distancedR1(0), dR2(0) to the respective paths with thehelp of (3) and the internally stored global pathsgiven by waypoints. The initial value of s is setto sR1(0) = sR2(0) = 0.

3. Robot 2 solves (12) with the initial values andobtains the optimal trajectories U∗R2 and x∗R2 forthe time horizon of K time steps.

4. Robot 2 communicates the optimal trajectory ofthe state variables x∗R2 to Robot 1.

5. Robot 1 receives x∗R2 and solves the combinedpath following/collision avoidance problem (13)to obtain the optimal trajectories U∗R1 and x∗R1

for the time horizon of K time steps.

6. Both robots realize the optimal steering com-mands u∗R1(0) and u∗R2(0) for the current timestep. Then they proceed with Step 1 again.

This model predictive motion control approachwas implemented in both simulation environments, aswell as in the previously described testbed. For theimplementation of the model predictive approach, thespecial multiple shooting based dynamic optimizationpackage MUSCOD-II19 was applied.

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Fig. 6 Local real-time motion control.

RESULTS

The results of the model predictive approach as de-picted in Fig. 6 are promising and underline its effi-ciency. Robot 1 first tries to minimize the deviationfrom the desired path. Then, it has to start avoidingthe approaching Robot 2. That results in a devia-tion from the desired path of Robot 1 again. AfterRobot 2 has passed, Robot 1 is again approachingthe desired path. Fig. 6 also shows that the collisionavoidance constraints are always fulfilled. Also, thesecurity threshold has been limited. The result canbe interpreted as the best compromise between pathfollowing and collision avoidance while additionallykeeping the differential constraints. In addition,for microproduction-specific constraints, the accelera-tions both in travel direction (aR1x ) and perpendicularto the travel direction (aR1y ), have been limited, aswell as the velocities and turning rates.

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

x(m)

y(m

)

robot 1robot 2robot 3

E E

E

S

S

S

Fig. 7 Global long-term motion planning. S: start; E: goallocation; markers: calculated waypoints.

Moreover, the concept of the global long-termmotion planning has been simulated as well (Fig. 7).Three robots are considered in an x-y-coordinatesystem. The robots start at the same time after pri-oritization where Robot 1 has the highest priority, andRobot 3 has the lowest. The result of the decoupledprioritized planning is demonstrated.

It is clear that Robot 1 with the highest prioritytravels directly from start to the goal location, keepingthe velocity constraints. Robot 2 then has to take thispath of Robot 1 into account and to plan a path wherethe distance between these two robots is always largerthan 3 m. Finally, Robot 3 has the lowest priority andhas to adapt its path to the two other already computedpaths of Robots 1 and 2. Also in this case, the obtainedpath of Robot 3 keeps a distance of a least 3 m betweenitself and the other two robots. Additional results areshown in Figs. 8 and 9.

DISCUSSION

While tremendous progress in the area of nonlinearmodel predictive control (NMPC) has been made inthe recent years, there is the concern that only a smallnumber of the existing NMPC schemes can be appliedin real applications. This concern is mostly basedon the high computational demand often related toNMPC. Hence computation speed issues have beenaddressed in this approach as well. In this proposedmicroproduction system, the transport robots are sup-posed to move with low acceleration and decelerationduring the transportation because they have to carrymicroproducts which are very sensitive. There are alsoa lot of study groups (e.g., Refs. 19, 20) investigatingthis area to improve the performance of the NMPC for

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340 ScienceAsia 36 (2010)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.50.5

0.75

1

1.25

1.5

1.75

2

x(m)

y(m

)

robot1

robot2

desired pathof robot1

Fig. 8 Result from different scenarios.

0.5 1 1.5 2 2.5 3

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

x(m)

y(m

)

robot1robot2

desiredpath of robot1

Fig. 9 Another result with lower safety distance.

other applications.This two-level approach works properly for the

proposed microproduction system, because the Mi-crosystems fabrication laboratory is mostly a big cleanroom, and there are no constraints from the envi-ronment such as tables, chairs or others obstacles.In other applications, some practical problems couldoccur, for example, a dead-lock situation in the caseof a narrow path, and so on. Then, these well-known problems require an extra approach21, e.g.,robot coordination and simultaneous scheduling.

From Figs. 6, 8 and 9, instead of applying thetraffic rule (right before left), Robot 1 has to deviatefrom its desired path in order to avoid hard breaking,and keep the acceleration constraints at the same time.Hence Robot 1 always moves with the maximumspeed. The safety distance always has to be under thesetpoint value as well.

The location and orientation of human workers isknown by all robots because a global vision basedpositioning system has been used in the proposedsystem. Hence the possible movement and trajectoriesof the human workers can be estimated. As mentionedbefore, the limited speed of transport robots is quitelow during the handling tasks, so the proposed algo-rithm will have enough time.

There are a lot of possibilities to use a team ofmobile robots within industries in the future. Futurework comprises the integration of the algorithms in therobots using embedded solutions for higher accuracyand efficiency to enable them to run in real time.While the implemented solution always converged inthe simulation and experimental tests, proof of thestability of the described approach is currently beinginvestigated. Aspects of stability are also discussed inRef. 15.

Acknowledgements: The authors would like to thankthe School of Information, Computer and CommunicationTechnology, Sirindhorn International Institute of Technol-ogy, Thammasat University and the Laboratory for Mo-bile Robotics and Mechatronics, Hochschule Ravensburg-Weingarten University of Applied Sciences for providinginspiration and great support. The authors would like alsoto give special thanks to the Thai Khem Kaeng stimulusprogramme for financial support.

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