Cooperative Target Tracking Control of Multiple Robots

Target Tracking Control of Multiple RobotsZongyao Wang,and Dongbing Gu,Senior Member,IEEEAbstract—This paper focuses on the problem of moving tar-get tracking with a group of mobile robots.Each robot in the group has a pan/tilt camera to detect the target and has limited communication capability to communicate with neighbor robots. The problem is solved by separating it into two parts.One part is the estimation of target position and another is theflocking control of multiple robots moving toward the estimated position. In the target estimation part,we propose to use a novel distributed Kalmanfilter to estimate the target position.The distributed Kalmanfilter is deduced based on a standard Kalmanfilter by modeling the neighbor’s information as one of measurements.In the motion control part,a distributedflocking algorithm is devel-oped.It is used to track the estimated target and avoid collision.In both parts,only local communication between neighbor robots is required.Finally,the tracking algorithms are simulated with2-D and3-D robots to verify their performance.The simulation results provide afirm conclusion that the proposed algorithms are able to track a moving target.A group of real ground mobile robots is used to test the proposed algorithm.The experiment results show that multiple robots are able to cooperate to track the target under the proposed algorithms and the tracking result outperforms the result produced by individual robots without cooperation.Index Terms—Cooperative tracking control,distributed Kalmanfilter,distributed processing,flocking control.I.I NTRODUCTIONT HE collective and cooperation behavior of living beings, such asflocks of bird,shoals offish,herds of wildebeest, has certain advantages,including avoiding predators,increas-ing the chance offinding food,saving energy,etc.Inspired by biological behaviors,multi-robot systems have became an active research area in the past few years.The multi-robot system can be defined as a robot group in which the robots can communicate with others and can cooperatively implement different kinds of tasks.Multi-robot systems have the potential in numerous applications,such as military surveillance,space exploration,cooperative classification,rescue and search,etc. In this paper,we will investigate a multi-robot system,in which each robot is equipped with a local vision camera to detect a moving target and tracks the target.The neighborManuscript received February1,2010;revised September10,2010and February3,2011;accepted April9,2011.Date of publication April21,2011; date of current version March30,2012.Z.Wang is with the Global institute of Management and Economics, Dongbei University of Finance and Economics,Dalian116025,China(e-mail: qianxilong@).D.Gu is with the School of Computer Science and Electronic Engineering, University of Essex,CO43SQ Colchester,U.K.(e-mail:dgu@). Color versions of one or more of thefigures in this paper are available online at .Digital Object Identifier10.1109/TIE.2011.2146211robots interact with each other to share target information.All the robots in the group gradually reach to an agreement about the target position.Finally,all the robots use the estimated target position as a tracking destination to control their motions toflock cohesively.A multi-robot system can be considered as a mobile sensor network.When a sensor network is used to track a moving object,the task of local data aggregation in the network presents a new set of challenges[1]–[8].Although the sensor network covers large spatial area,the moving target only can be viewed by a small number of sensor nodes while most sensor nodes cannot observe the moving target[9]–[11].Cooperative esti-mation of the target position between neighbor sensor nodes and coordinated motion control of multiple robots toward the estimated target position are the challenging issues investigated in this paper.The target tracking control problem is normally solved by separating it into two parts.One part is the estimation of target position and another is theflocking control of multiple robots moving toward the estimated position.In the target estimation part,variants of distributed Kalmanfilters(DKF) have been developed[12]–[14].In these algorithms,multiple measurements and their covariance matrices in the information form of DKF can be expressed as the sum of measurements and covariance matrices of individual sensors.The consensus algorithm developed in[12],[13]can be used to approximate the sums and implement the Kalmanfilter in a distributed way. The main advantage of these DKF s is that the estimates in all sensor nodes will eventually converge to the estimate of centralized Kalmanfilter.However,they are no longer optimal in each step in terms of minimizing estimated error due to the heuristic use of consensus iteration inside of the algorithm, i.e.,the error covariance of these DKF s is an approximation to the optimal error covariance.In the case discussed in this paper,only small number of sensor nodes can detect the moving targets.Most nodes do not have direct measurements and have to use the results of motion model prediction as their estimates. The error covariance approximation could deteriorate the esti-mation result of the entire group.There are also other ad hoc applications of consensus algo-rithms in Kalmanfilters.In[15],a consensus Kalmanfilter with purpose of optimizing two design parameters,the Kalman gain and the consensus matrix,was proposed.In[16],a distributed adaptive algorithm was designed and analyzed.The presence of measurement noises and of packet losses was considered.Each node of the network locally computes adaptive weights that guarantee to minimize the estimation error variance.Decentral-ized conditions on the weights,which ensure the stability of the0278-0046/$26.00©2011IEEEestimates throughout the overall network,were also considered. In[4],the problem was solved by communicating estimates between neighbors and then forming a weighted average as the new estimate.The weights were optimized to yield a small estimation error covariance.The minimization can be done offline,thus allowing only estimates to be communicated. This paper focuses on the Kalmanfilter tracking problem where only part of sensor nodes can observe the moving target. The difference with aforementioned approaches is that the neighbor’s information is modeled as additional measurement in this paper.This modeling strategy results in an implicit consensus algorithm to track the target cooperatively for those robot which can detect the target.For the robots which cannot detect the target,they simply use the neighbor’s information to update its estimated result so that the estimates of small number of robots can be propagated over the entire network through local communication.By modeling the measurements and estimates from neighbors as measurement equations in a local sensor node,a DKF is deduced from the standard Kalmanfilter.The proposed DKF is able to propagate the local information over the entire network.With the estimated target position,mobile robots are required to move toward to the target.Or we can say they are required to move as aflocking group to track the target.We develop aflocking algorithm for the robots to track the target.The flocking algorithm includes a cohesion control component and a separation control component.The cohesion control can keep the group cohesive,and consequently,each robot can communi-cate with at least one neighbors.The key information to achieve the cohesion control is the use of estimated target position as the tracking target for each robot.The separation control is to avoid collision between neighbor robots.When the robots move close enough,they use the potential function approach to separate each other.A collision avoiding potential function is developed for the separation control.The recent work in[17]employed path following strategy for formation control,in[18]employed sliding-mode strategy for formation control,and in[19]em-ployed a vision-guided control for two roboticfishes to track a target.They are different with the proposed work in this paper. The outline of this paper is arranged in the following order. Section II introduces a DKF with a setting of sensor networks. Section III develops the proposed DKF.Section IV describes theflocking algorithm.Section V presents simulation results of 2-D and3-D robots.A group of three real robots is tested and the results are provided in Section VI.A conclusion is given in Section VII.II.K ALMAN F ILTER W ITH M ULTIPLE S ENSORSThe Kalmanfilter is used to estimate the state of a dynamic system based on the sensor measurement.The sensor mea-surement is related to the estimated state,the measurement model,and the system model.The Kalmanfilter is an optimal estimator in the case of Gaussian noise.In the following,a standard Kalmanfilter with multiple sensors is revisited.This is a centralized Kalmanfilter(CKF)where all sensors send their readings to a centralized unit to execute a standard Kalmanfilter to estimate the target position.Consider a discrete-time linear time-invariant system defined by a system equationx t+1=Ax t+Bw t(1) where x t is the system state and w t is the system disturbance. w t is assumed to be Gaussian random vector and its covariancematrix Q is time invariant and Q≥0.The subscript t is the time index.Assuming N robots are used,each of which has a sensor and the measurement is y i t at time t.For robot i,the measurement equation is defined asy i t=C i x t+v i t(2) where v i t is the measurement noise and is assumed to be Gaussian random vector with a time-invariant covariance ma-trix R i>0.Multiple measurements are stacked into a measurement vec-tor and the compact measurement equation isy t=Cx t+v t(3) wherey t=⎡⎢⎣y1t...y N t⎤⎥⎦,C=⎡⎣C1...C N⎤⎦,v t=⎡⎢⎣v1t...v N t⎤⎥⎦.R is the covariance matrix of v tR=diag[R1,...,R N].Following[20],the standard Kalmanfilter consists of two steps.The time update step in the Kalmanfilter¯x t+1=Aˆx t¯P t+1=AP t A T+BQB T.(4) And the measurement update step in the Kalmanfilterˆx t+1=¯x t+1+K t+1(y t+1−C¯x t+1)P t+1=¯P t+1−K t+1C¯P t+1(5) where thefilter gain isK t+1=¯P t+1C T(C¯P t+1C T+R)−1andˆx0and P0are initial state and covariance.In the following presentation,we call(¯x t+1,¯P t+1)as the predicted result and (ˆx t+1,P t+1)as the estimated result.The evolution of covariance in the above two steps can be integrated into a Ricatti difference equation(RDE)P t+1=AP t A T+BQB T−AP t C T(CP t C T+R)−1CP t A T.(6) For Q≥0,R>0,and P0≥0,if[A,C]is detectable and [A,Q1/2]is stabilizable,the following results are available:thefollowing algebraic Ricatti equation(ARE)exists a unique nonnegative definite symmetrical solution P∞P∞=AP∞A T+BQB T−AP∞C T(CP∞C T+R)−1CP∞A T.(7) The iteration of RDE converges to the solution of ARE. The property of[A,C]is detectable guarantees that nonneg-ative symmetric P t is bounded.The property of[A,Q1/2]is stabilizable guarantees that P t is asymptotic stable.If the covariance¯P t increases,i.e.,the predicted state¯x t becomes worse,it will change thefilter gain accordingly and result in an estimated state which relies more on the measure-ment than the predicted state.If the measurement covariance R increases,i.e.,the measurement y t becomes worse,it will change thefilter gain accordingly and result in an estimated state which relies less on the measurement result than the predicted state.We will use the informationfilter form[21]to deduce the new DKF.The time update step of the CKF in the informa-tion form is the same as in(4).The measurement update step of the CKF in the information form can be written asˆx t+1=¯x t+1+¯P t+1Ni=1(C i)T(R i)−1y i t+1−C i¯x t+1(P t+1)−1=(¯P t+1)−1+Ni=1(C i)T(R i)−1C i.(8) III.DKFIt is not scalable and robust for a robot to receive the measurements from all other robots in order to use a CKF to estimate the result.A DKF advocates each robot only communicates with its neighbors to exchange information and executes DKF using its own measurement and neighbor’s information.In our proposed DKF,the exchanged information between neighbors includes the sensor measurement y i t and the predicted result(¯x i t+1,¯P i t+1).The target state equation is still the same as in(1)in our proposed DKF.However,the measurement equation needs to be reformulated to include the neighbor’s information.The local sensor measurement equation is the same as in (2).The predicted state in time update stepˆx j t(j∈N i)from one of its neighbors is viewed as a measurement for robot i.An additional measurement equation is required for this measurementz ij t=¯x j t.This measurement equation can be further written asz ij t=x t+u ij t(9) where u ij t=¯x j t−x t can be viewed as the measurement noisewith a covarianceS ij t=E¯x j t−x tT¯x j t−x t=¯P j t.By combining the sensor measurement equation(2)and the information from neighbors(9),thefinal measurement equation of robot i can be written asz i t=H i x t+u i t(10) where z i t=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣y i ty i1t...y iN itz i1t...z iN it⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,H i=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣C iC i1...C iN iI...I⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,and u i t=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣v i tv i1t...v iN itu i1t...u iN i t⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦witha covarianceU i=diag[R i,R i1,...,R iN i,S i1,...,S iN i]=diagR i,R i1,...,R iN i,¯P i1t,...,¯P iN it.The measurement updating step is deduced from(8)as follows:ˆx i t+1=¯x i t+1+¯P i t+1(H i)T(U i)−1z i t+1−H i¯x i t+1=¯x i t+1+¯P i t+1j∈N i∪i(C j)T(R j)−1×y j t+1−C j¯x i t+1+¯P i t+1j∈N i¯P jt+1−1ׯx j t+1−¯x i t+1P i t+1−1=¯P i t+1−1+(H i)T(U i)−1H i=¯P i t+1−1+j∈N i∪i(C j)T(R j)−1C j+j∈N i¯P jt+1−1.(11)There are three terms inˆx i t+1.Thefirst term¯x i t+1is the predicted state.The second term¯P i t+1j∈N i∪i(C j)T(R j)−1 (y j t+1−C j¯x i t+1)is the contributions from local measure-ment y i t+1and neighbor measurements y j t+1.The third term¯P it+1j∈N i(¯P j t+1)−1(¯x j t+1−¯x i t+1)is the difference between local predicted state and neighbor predicted state.It is the third term that plays a consensus role,i.e., it reduces the estimation difference between neighbor ro-bots.This consensus term is weighted with an adaptive gain ¯P it+1j∈N i(¯P j t+1)−1.If the neighbor’s predicted state is not good,the predicted covariance¯P j t+1is large and the adaptive gain is small.Thus,the corresponding consensus term is small. If the neighbor’s predicted state is very good,the predicted covariance¯P j t+1is small and the adaptive gain is large.Thus, the corresponding consensus term is large.The inverse covariance (P i t +1)−1also includes three terms.The first term (¯P i t +1)−1is the local prediction.The second term j ∈N i ∪i (C j )T (R j )−1C jis the contributions from local andneighbor covariances.The third term j ∈N i (¯P j t +1)−1is the neighbor’s predicted covariances.The addition of the third term can reduce the local estimated covariance P i t +1.If there is no local measurement,but there is at least one of measurements or predicted states from neighbors,then [A,H i ]is detectable.The conclusions about the RDE and RDE of the CKF described in the previous section are true for the proposed DKF .Thus,the robots which cannot detect the target can still maintain a stable estimated result.The DKF algorithm is summarized in the following table:Algorithm 1Distributed Kalman FilterInitialize P i0,ˆx i 0,t =0Repeat:Time update step:updates ¯x i t +1and ¯P i t +1using (4)Communication step:sends (y i t +1,C i ,R i ,¯x i t +1,¯P i t +1)to its neighbors receives (y j t +1,C j ,R j ,¯x j t +1,¯P j t +1)from its neighborsMeasurement update step:updates ˆx i t +1and P it +1using (11)EndIV .D ISTRIBUTED F LOCKING A LGORITHMIn this section,we will use the estimated target position as a tracking destination and design a flocking controller to enable the entire group to track the target.The artificial potential function approach is used to design the flocking controller.The robot model we used is a kinematic model˙q i =p i(12)where q i is the robot state vector in 2-D or 3-D space and p i isthe robot control vector in 2-D or 3-D space.The first component in the flocking controller is tracking control.The estimated target position plays a key role in this component.When all the robots move toward the target,they will have potential to converge to the target and achieve the cohesive property.The cohesive potential function of robot i isdefined as H c ( q i −ˆxi )=(1/2) q i −ˆx i 2,where q i −ˆx i is the Euclidian norm of q i −ˆxi .In the flocking control,adjacent robots are required to keep a specific distance.If the distance between adjacent robots is too small,they attempt to separate.The robot communica-tion range is limited,the interaction will disappear between neighbors when their distance is larger than the communication range.Based on these requirements,the second component in the flocking controller is separating control.Let H s (r ij )denote the separating potential function between robot i and j where r ij = q i −q j .The potential function of robot i should be nonnegative.We use ∇H s (r ij )to denote the gradient of H s (r ij )with respect to q i.Fig.1.Fuzzy input membershipsets.Putting two control components together,the controller of robot i is designed asp i=−j∈N i∇H s(r ij)−k(q i−ˆx i)(14)where k is a positive gain.Finally,the DKF andflocking control algorithm are summarized in the following table:Algorithm2Distributed Tracking AlgorithmInitialize P i0,ˆx i0,q i0,t=0Loops:Communication step:Robot i sends q i t and(y i t+1,C i,R i,¯x i t+1,¯P i t+1)to its neighborsRobot i receives q j t and(y j t+1,C j,R j,¯x j t+1,¯P j t+1) from its neighborsEstimation step:Robot i estimates the stateˆx i t+1and covariance P i t+1using DKFControl step:Robot i computes p i t using(14)EndV.S IMULATIONSIn our simulations,the pan/tilt camera is simulated with a sensing range of3m.The view of the camera is40◦and it can pan from-170◦to170◦to increase the view.The pan speed of the camera is0.8rad/s.The view of the camera could be blocked by neighbors or obstacles.The camera will not stop scanning unless it detects a target.Also,Gaussian noise is added to the measurement.A.2-D Flocking TrackingIn thefirst simulation,the target moved along a sine-shape trajectory.The60robots were randomly placed in a7m×7m area.The robot communication range was selected as C=1.2m and the expected distance between robots was D= 1m.The robots used the DKF to estimate the position of the target and used theflocking controller toflock and track the estimated target.Fig.3shows the tracking trajectories.The flocking network of thefinal simulation stage is shown in Fig.3 as well.We can see that almost all the individuals maintained the specific distance with their neighbors.An obstacle was placed near the target trajectory.It can be seen that the trajectories of theflocking robots were slightly changed due to the obstacle avoidance maneuvers.But the flocking robots can still keep cohesive and pass through the obstacle.The process of60robots avoiding the obstacle is shown in Fig.4.Fig.4(a)shows theflocking system was moving to the obstacle.At t=115s,theflocking system just encountered the obstacle.Some robots detected the obstacle and moved around it.In Fig.4(b),one of the robots was pushed away from theflocking system and it came back in Fig.4(c).Fig.3.Sixty robots tracking a sinetrajectory.Fig. 4.Sixty robots tracking the target and avoiding the obstacle.(a)t=107s;(b)t=115s;(c)t=126s;(d)t=140s.Finally,theflocking system passed the obstacle and continuedthe tracking.Fig.5shows the estimated results of all the robots.The solidline denotes the true trajectory of the target and the dotted linesare the estimated results of the robots.At the beginning,therobots did not reach to an agreement about the target position.Particularly at thefirst wave crest of the sine trajectory,the dif-ferences between the estimates were enormous.Gradually,allthe estimates asymptotically approximated to a single trajectorywhich was close to the true trajectory.B.3-D Flocking TrackingIn a3-D space,a pan/tilt camera with the sensing range3m issimulated.In the x–y plane,the view of the camera is40◦.Thecamera can rotate360◦around z-axis with a speed0.8rad/s.The camera can detect the target with a tilt view of−15◦∼15◦. The60robots were randomly placed in a7m×7m×7marea.The robot communication range and the expected distancebetween robots were the same as that in the2-D simulation.Fig.6shows the robot trajectories.Fig.5.Targetestimation.Fig.6.Sixty robots track a target in3-Dspace.Fig.7.Target estimates of the60robots in3-D space.The results of the target estimation are shown in Fig.7.The solid line denotes the true trajectory of the target.The dashed lines are the estimated results of the robots.The estimated trajectories are very close to the trueone.Fig.8.Sixty robots track the target and avoid the obstacles.(a)t=20s;(b)t=40s;(c)t=60s;(d)t=80s;(e)t=100s;(f)t=110s;(g)t=115s;and(h)t=120s.In the second simulation,we tested the capability of obstacle avoidance.The process of60robots avoiding the obstacle is shown in Fig.8.The solid line in Fig.8(a)is the trajectory of the target.The two cylinders were placed in the arena to act as the obstacles.The video was taken from t=20s. Fig.8(a)shows that theflocking system kept cohesive and moved to the obstacle.Theflocking system encountered the first obstacle at t=40s[Fig.8(b)].Some robots detected the obstacle and moved around it.In Fig.8(c),theflocking pattern was changed due to the potential force from thefirst obstacle. Fig.8(d)shows that theflocking system encountered the second obstacle and was“nipped”by the two obstacles.Because the gap between two obstacles was not wide enough to let the entire flocking system pass through,some of the robots were pushed away from theflocking system andflew around the obstacles [Fig.8(e)and(f)].In Fig.8(g),the robots rejoined together and theflocking system became cohesive again.Finally,the flocking system passed through the obstacles and continued the tracking[Fig.8(h)].During the tracking,no robot collided with the obstacles.Fig.9.Minimum distance between the robots.To prove that there was no collision,the minimal distance among the robots was calculated and shown in Fig.9.During the whole process,the minimal distance kept a stable level around0.9m and no collision happened at any time.There were two small valleys at t=60s and t=110s which were the moments of encountering thefirst and second obstacle.VI.E XPERIMENTSA group of three wifibots was used to test the proposed algo-rithm in our robotic arena of University of Essex.The wifibots are networked robots which can use wireless communication to exchange information with neighbors.The TCP/IP protocol was used to implement the communication between robots. A3-D racking system(VICON)was used to provide robot positions to each of the robots.It also provides the ground true target trajectory for analysis.The on-board camera can pan from−170to170◦to increase the viewfield.A feedback control loop was used to control the camera pan angle.The control objective was to keep the target in the middle of the image.The target was also a wifibot and detected by using the on-board pan/tilt camera of other three tracking robots.To simplify the vision processing,the target wifibot was equipped with a red marker.The on-board camera only detected the red marker by using color threshold techniques.The image obtained by the on-board camera was in RGB(red,green,blue)space.Because an image in HSV(hue,saturation,value)space is more sensitive to colors,the RGB image was converted to the HSV image.It is known that different colors have different hue degrees in HSV images.A band-passfilter was used tofilter the red color.For the purpose of de-noise,an erosion algorithm was applied to erase the high-frequency component in the image.Finally,we got an image with the target only.The mean position of all the pixels of target was calculated.To measure the distance to the target,the width of the target image was also calculated during the process.Tofind the relationship between the distance and the width of the target image,we recorded the distances from the robot to the target and also recorded the corresponding width of thetarget Fig.10.Target estimation of three robots withoutcooperation.Fig.11.Target estimation of three robots with cooperation.image.It was found that the relationship between the distance and the image width can befitted to a logarithm functiond ir=−590log(α/195)+300whereαis the width of the target image.In thefirst test,three robots used the standard Kalmanfilter, i.e.,they did not exchange their information.Fig.10shows the estimated results of the target position of all three robots. It can be seen that the differences between three estimates were enormous.Due to the disagreement,it made theflocking control more difficult.During the testing,theflocking network easily collapsed because robots tracked“different targets”.In the second test,three robot used the developed DKF, i.e.,they exchanged their information for tracking cooperation. Fig.11shows the estimated results of the target position of all three robots.The solid line denotes the true trajectory of the target;the dashed lines are the estimated results of threeand(d)t=100s.Fig.13.Robot trajectories.pared with thefirst testing result,the disagreement of estimates was reduced.All the estimates asymptotically approximated to a single trajectory which was close to the true trajectory.Fig.12shows the video snapshots at time t=0s,t=37, t=72,and t=100s to demonstrate theflocking behavior. At the beginning,three robots were placed randomly in the robot arena.The target shown in Fig.12(a)was moving along a predefined trajectory.Three robots used the cameras and the DKF to estimate the target position.At t=37s,three robots had reached to an agreement about the target position,the robots cooperatively tracked the target,and all the three robots were moving to the estimated position.After t=72s,three robots gradually caught up with the target.The trajectories of three robots are shown in Fig.13.The solid line represents thetrajectory of the target.The other three dashed lines are theFig.16.Camera detectability(a)individual robots and(b)cooperative robots. Fig.16(a)and cooperative robots in Fig.16(b).It demonstrates a clear improvement of the view coverage with the proposed algorithm.VII.C ONCLUSIONThis paper presents a cooperative target tracking and coor-dinated control approach to multi-robot systems.A DKF is developed to estimate the target position cooperatively.Mul-tiple robots provide a wider view of the system and therefore improve the estimation result.The proposed DKF is deduced from the standard Kalmanfilter and only requires the local communication between neighbor robots.Aflocking algorithm based on the estimated target position is used to control the robot group to track the target and keep the group connected. Substantial work has been done to implement the algorithms in real robots.A group of three robots was used to test the algorithm performance.The experiment results show that the proposed DKF can provide the estimated result of target position with an accuracy of average250mm in our experiment. 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