论文2G精华版本 ICRA 2018 files 1646

A Failure Tolerant Approach to Synchronous Formation Control of Mobile Robots under Communication Delays Zhe Liu Hesheng Wang Senior Member IEEE Lingyun Xu Yun Hui Liu Fellow IEEE Junguo Lu and Weidong Chen Member IEEE Abstract Robot malfunction is inevitable in practical ap plications of the robot formation control due to uncontrolled crashing system malfunction or communication loss In this paper we study the synchronous formation control problem in the presence of robot malfunctions Our main idea is to improve the network connectivity and motion synchronism of the robot formation through a series of topology switchings and robot replacements Firstly the synchronous formation control method is introduced which enables the robots to tracking their desired trajectories while keeping predefi ned formation shapes Secondly a recursive switched topology control strategy is proposed to restore the formation shape as well as to improve the network connectivity and motion synchronism in the pres ence of robot malfunctions Thirdly the convergence analysis of the proposed control system is presented and a suffi cient condition is obtained under an average dwell time scheme What s more the proposed approach is fully distributed and the communication delays between neighboring robots also have been taken into consideration Simulation results demonstrate the effectiveness of the proposed approach I INTRODUCTION Mobile robots formation control has been studied exten sively and classical formation control approaches include the behaviour based approach the virtual structure method and the leader follower strategy 1 2 In recent years the synchronous formation control and trajectory tracking approach 2 4 has received a lot of research interests due to its advantages of the simple control structure high motion coordination performance and strong robustness In 2 by introducing the cross coupling technique mobile robots can maintain desired formation shapes during the whole trajectory tracking period instead of only achieving a desired formation shape after the convergence of each robot s position error In 3 a coordinated path tracking control method is established for mobile robots with non holonomic constraints The formation coordination is achieved by syn chronizing the path parameters of all the robots In 4 the synchronized formation control problem with sampled data has been studied and the robustness of the synchronization mechanism in the presence of external disturbances has been tested This work is supported in part by the Natural Science Foundation of China under Grant U1613218 the Hong Kong Research Grants Council under Grant 414813 and 14204814 and CUHK VC Discretionary Fund under grant 4930745 Z Liu L Xu and Y H Liu are with the Department of Mechanical and Automation Engineering Chinese University of Hong Kong Hong Kong H Wang J Lu and W Chen are with the Department of Automation Shanghai Jiao Tong University China Corresponding author Hesheng Wang wanghesheng In practical application scenarios the mobile robot forma tion may work in hazardous environments and it is inevitable that the robot failure may occur as a result of uncontrolled crashing system malfunction and communication loss 5 6 The robot failure will break the desired formation shape reduce the network connectivity and degrade the motion synchronization performance of the robot formation 7 There are some related works in self organization re searches of swarm robot systems To name a few in 8 a decentralized algorithm is proposed to detect disconnections maintain connectivity and coverage over a given area In 9 10 a gradient based algorithm is established to formulate the global coordinate system and the self reconfi guration problem of a large scale swarm has been investigated How ever these algorithms suffer from long term robot move ments and large topology changes which are not suitable for formation control of mobile robots In formation control area there are also some robot malfunction tolerant algorithms 2 5 6 However these algorithms mainly focused on monitoring the real time performance of each robot and detecting system malfunctions 5 6 When robot failure occurs they usually isolate the failed robot or treat the failed robot as a virtual robot which does not affect the formation performance 2 6 Recently in 7 a recursive topology switching based algorithm is presented to resolve the robot malfunction problem in formation generation and maintenance missions However this work has not taken into account the dynamic characteristic of the mobile robot for mation thus it will be diffi cult to guarantee the convergence of robot trajectory tracking error during the dynamic motion process In this paper we study the robot malfunction tolerant formation control and trajectory tracking problem under communication delays The contributions of this paper in clude Firstly in existing synchronous formation control and trajectory tracking researches this paper is the fi rst to investigate the failure tolerant problem from the perspective of maintaining network connectivity and improving motion synchronism Secondly aiming for practical applications we take the communication delays into consideration and present a fully distributed controller to achieve the robot malfunction tolerant formation control task Thirdly a suf fi cient condition is obtained through the Lyapunov based stability analysis approach and under the average dwell time scheme The proposed condition guarantees the exponential convergence of the trajectory tracking and formation control system 2018 IEEE International Conference on Robotics and Automation ICRA May 21 25 2018 Brisbane Australia 978 1 5386 3080 8 18 31 00 2018 IEEE1661 II SYSTEMFORMULATION A System Modeling Similar to 2 11 we consider the following robot dynamics Mi qi i 1 where Mi diag mi mi represents the inertia matrix and qi xiyi T R2denotes the position of robot i iis the control input The topology coupling matrix of a network with N robots can be described as LN N lij 2 where lij 1 means that there is a topology connection communication link between robot j and robot i otherwise lij 0 The diagonal elements of L satisfy lii j6 i lij 3 lii also represents the degree of robot i The neighbor set Ni of robot i is defi ned as Ni Rj lij 1 j 6 i j 1 2 N 4 In synchronous formation control approaches the expected formation shape is accomplished by synchronizing the mo tion of each robot during the whole task period So motion synchronism or synchronizability is a crucial performance index According to 12 13 the second largest eigenvalue 2of L can be used as an index to evaluate the network connectivity as well as the motion synchronization perfor mance of the robot formation The smaller 2represents the better synchronization performance which means the faster convergence of the synchronous formation control system and the stronger robustness to external disturbances In this paper we also take the time varying communication delays into consideration in system design Without loss of generality we assume that the delay d t satisfi es 0 d t d d t d 5 B Synchronous Formation Control and Trajectory Tracking According to 2 11 the expected trajectory of robot i in the formation can be described by the time varying boundary curve of the desired formation shape S S 0 qd i t Ai t C t Bi t 6 where Ai t contains the trajectory parameters corresponding to robot i C t denotes the uniform parameters of the robot formation shape and the desired trajectory and Bi t repre sents the additional parameters The trajectory description 6 implies that each robot should satisfy the following synchronization constraint A 1 i qi t Bi t C t A 1 j qj t Bj t 7 And since the desired position qd i t also satisfi es the above constraint i e A 1 i qd i t Bi t C t A 1j qdj t Bj t 8 then the synchronization goal can be defi ned as 2 i 6 j ciei cjej 9 where ei qi qd i and ci t A 1 i t Incorporating the topology coupling matrix 2 3 the synchronization error of robot i can be defi ned as i j6 i lijk ciei cjej 10 where k represents the inner coupling matrix If the posi tion error eiand the synchronization error iconverge to zero simultaneously then robots can track their expected trajectories and in the meantime synchronize their motions between each other maintain expected formations i e the synchronous formation control task is completed C Preliminaries The following lemmas will be used in the mathematical derivation in the rest of this paper Lemma 1 15 Let x t x1 t x2 t xn t T if Gx t 0 as t then xi xj i j 1 2 n i 6 j where G 1 1 1 1 1 1 n 1 n Lemma 2 15 The topology coupling matrix L defi ned in 2 3 satisfi es GL HG where H GLJ J 111 1 011 1 1 11 00 01 000 0 n 1 n 1 and G is defi ned in Lemma 1 III FAILURE TOLERANTSYNCHRONOUSFORMATION CONTROL ANDTRAJECTORYTRACKING A Synchronous Formation Controller Design By defi ning Ei t ci t ei t and taking delay d t into account synchronization error iin 10 can be revised as i j6 i lijk Ei t d t Ej t d t 11 Then we defi ne a coupled error as the following integral form i t Ei t Z t 0 kEEi s ds Z t 0 i s ds 12 Then we design the following distributed control law for each robot i Mic 1 i i ci qi c 1 i kd i c 1 i kp i 13 where kd and kp are positive defi nite parameter matrices 1662 Introducing the proposed control law into the robot dy namics 1 we have Mic 1 i i c 1 i kd i c 1 i kp i 0 14 Since Mi diag mi mi we have ciM 1 i c 1 i M 1 i then the above equation can be rewritten as i M 1 i kd i M 1 i kp i 15 which represents the closed loop dynamics of the coupled error iand contains the delayed synchronization error i Formula 15 implies that the coupled error iof each robot converges to zero exponentially which further guarantees the exponential convergence of the position error eiand synchronization error iof each robot B Topology Switching Strategy When robot malfunction occurs the corresponding topol ogy connections will be broken as a result 2of the coupling matrix L will increases In order to maintain the synchronization performance of the robot formation the network connectivity should be restored by switching the network topology and replacing the failed robot with other substitute robots According to 7 we fi nd that switching the network topology and replacing the failed robot by the robot with the lowest degree in the network will lead to the smallest 2 In order to facilitate the fully distributed implementation we design a recursive switched topology control based equivalent approach to accomplish the above replacement process Firstly in Nfof the failed robot f use the robot with the lowest degree denoted by robot i to replace the failed robot if more than one robot has the same lowest degree choose one of them randomly Secondly in Ni use the robot with the lowest degree to replace robot i Then further repeat the replacement process and switch the network topology recursively until the robot with the lowest degree in the network has switched its topology and replaced one of its neighbors The above recursive robot replacement and topology switching process achieves the same results with directly switching the network topology by replacing the failed robot with the robot which has the lowest degree Remark 1 The robot replacement and topology switching process mentioned above can be accomplished in a fully distributed manner more details can be found in our previ ous work 7 Introducing robot replacements and topology switchings into synchronous formation control will seriously affect the stability of the control system The new challenges can be summarized as 1 How to analyze the stability of the control system during topology switchings 2 Under what condition the position error and synchronization error of each robot can be guaranteed to converge to zero exponentially in the presence of topology switches and communication delays These questions will be answered in this paper We introduce the average dwell time scheme in this paper to guarantee the stability of the proposed control system When robot malfunction occurs in t0 assume that it needs K steps to accomplish the robot replacement and topology switching process i e there are K 1 different topologies during the recursive switching process Assume that the switching instant of the kthtopology is tk which means that in t tk tk 1 the robot network is under the kthtopology Defi ne Lk lk ij as the corresponding coupling matrix of the kthtopology the synchronization error i in 11 can be revised as i j6 i lk ijk Ei t d t Ej t d t t tk tk 1 16 where i 1 2 N 1 represents the rest N 1 robots For any t t0 let N t0 t denotes the number of topology switchings during t0 t Then Ta is defi ned as the average dwell time which satisfi es N t0 t t t0 Ta 17 7 10 9 8 4 5 62 3 1 Synchronous Formation Control and Trajectory Tracking Topology Switching Time Dwell TimeDwell Time Robot Malfunction 7 10 9 8 4 62 3 1 7 10 9 8 4 62 3 1 7 10 9 8 4 62 3 1 7 10 8 4 9 62 3 1 7 10 8 4 9 62 3 1 7 10 8 4 9 62 3 1 Fig 1 An example of the proposed failure tolerant approach Fig 1 shows an example of the proposed switching strat egy In order to maintain the synchronism and connectivity of the robot formation after robot 5 fails there are two steps robot replacement and topology switching i e robot 9 replaces robot 5 switches its topology connections and changes its desired trajectory to that of robot 5 fi rstly and then robot 10 replaces robot 9 There is an enough dwell time between each two consecutive topology switching instants This guarantees the convergence of the position error and synchronization error of each robot during robot replacements and topology switchings C Convergence Analysis We analyze the stability of the proposed control system by using the Lyapunov based method and the switching control theory Theorem 1 If there exists a given constant 0 and matrices Pk 0 Qk 0 Rk 0 such that k k 11 k 12 0 k T 12 k 22 k 23 0 k T 23 k 33 ln 19 where 1 satisfi es Pk Pl Qk Ql Rk Rl k l 1 2 K 20 Proof From 15 if we design a Lyapunov function as V i iT i TiM 1 i kp i 21 one can easily conclude that V i 2 iTM 1 i kd i 0 Since V iis a common Lyapunov function for each topology Lk and is continuous in each switching instant tk this implies that under the proposed control law 13 the coupled error iand its derivative iof each robot converge to zero exponentially even in the presence of topology switchings and communication delays From 11 12 we have i t Ei t kEEi j6 i lk ijk Ei t d t Ej t d t Since iconverges to zero exponentially we have Ei t kEEi j6 i lk ijk Ei t d t Ej t d t 22 Defi ne E ET 1 ET2 ETN T kE IN 1 kE Lk Lk k E t kEE t LkE t d t 23 Design the following piecewise Lyapunov candidate V t Vk t V1k t V2k t V3k t t tk tk 1 24 V1k t ET GTPk GE V2k t Z t t d t e s t ET s GTQk GE s ds V3k t Z 0 d Z t t e s t ET s GTRk G E s dsd where G G I2 G is defi ned in Lemma 1 with n N 1 t tk tk 1 from 5 23 we have V1k 2ET GTPk G kEE 2ET GTPk G LkE t d t From Lemma 2 we have G Lk Hk G and since G kE kE G we have V1k V1k 2ET GTPk kE GE 2ET GTPk Hk GE t d t ET GTPk GE From 5 we have V2k V2k ET GTQk GE 1 d t e d t ET t d t GTQk GE t d t V2k ET GTQk GE 1 d e dET t d t GTQk GE t d t By using the Jensen s Inequality 4 we have V3k V3k d ET GTRk G E e d Z t d t t d ET GTRk G E e d Z t t d t ET GTRk G E V3k d ET GTRk G E e d d GE t d t GE t d T Rk GE t d t GE t d e d d GE t GE t d t T Rk GE t GE t d t Then we have Vk Vk T t k t 25 where t ET t GT ET t d t GT ET t d GT T S ince k 0 we have Vk Vk Tamin From the simulation results in Fig 2 and 3 we can fi nd that although each robot begins with large initial position errors the position error and synchronization error of each robot converge quickly the robot network tracks the desired sine like trajectory and maintains the expected triangle for mations simultaneously When t t0 5s robot 5 fails In order to restore the network connectivity and maintain the motion synchronism robot 9 and 10 are selected to replace the failed robot When t t0 Ta 7 75s robot 9 switches its topology builds new connections with robot 2 3 4 and disconnects the connections with robot 10 and begins to track the desired trajectory of robot 5 changes its desired trajectory to that of robot 5 When t t0 2Ta 10 5s robot 10 switches its topology builds new connections with robot 9 8 and begins to track the previous desired trajectory of robot 9 From Fig 2 and 3 we can fi nd that after the two switching instants the proposed synchronous formation controller guarantees the fast convergence of the position error and synchronization error of each robot What s more Fig 3 shows that although only robot 9 and 10 suffer from 1665 020406080100120 15 10 5 0 5 10 15 x m y m t 2 5s t 5s t 7 75s t 12 5s t 20st 17 5s t 15s t 10 5s Fig 2 Robot trajectories and formation shapes the brown lines are real robot trajectories the blue shapes are desired formations while the red ones are real formations 05101520 time s 4 2 0 2 4 6 position error in x axis m 05101520 time s 3 2 1 0 1 2 3 position error in y axis m robot 10 robot 9 robot 9 robot 10 a Position errors 05101520 time s 2 1 5 1 0 5 0 0 5 1 synchronization error in x axis m 05101520 time s 1 25 1 0 75 0 5 0 25 0 0 25 0 5 0 75 synchronization error in y axis m robot 9 robot 10 robot 9 robot 10 b Synchronization errors Fig 3 Simulation results large position errors due to the switching of the desired trajectory other robots also respond to the errors and slow down to wait robot 9 and 10 thus avoiding large syn chronization errors during the topology switching process Finally from Fig 2 and 3 we can fi nd that the position error and synchronization error of each robot converge to zero successfully which validate the effectiveness of the proposed approach V CONCLUSION In this paper we present a robot malfunction tolerant approach to solve the synchronous formation control and trajectory tracking problem of mobile robots under commu nicat