Robust Leader-follower Formation Control of Mobile Robots Based on a Second Order Kinematics Model

Transcription

1 Vol. 33, No. 9 ACTA AUTOMATICA SINICA September, 27 Robust Leader-follower Formation Control of Mobile Robots Based on a Second Order Kinematics Model LIU Si-Cai 1, 2 TAN Da-Long 1 LIU Guang-Jun 3 Abstract In tis paper, we study te problem of modeling and controlling leader-follower formation of mobile robots. First, a novel kinematics model for leader-follower robot formation is formulated based on te relative motion states between te robots and te local motion of te follower robot. Using tis model, te relative centripetal and Coriolis accelerations between robots are computed directly by measuring te relative and local motion sensors, and utilized to linearize te nonlinear system equations. A formation controller, consisting of a feedback linearization part and a sliding mode compensator, is designed to stabilize te overall system including te internal dynamics. Te control gains are determined by solving a robustness inequality and assumed to satisfy a cooperative protocol tat guarantees te stability of te zero dynamics of te formation system. Te proposed controller generates te commanded acceleration for te follower robot and makes te formation control system robust to te effect of unmeasured acceleration of te leader robot. Furtermore, a robust adaptive controller is developed to deal wit parametric uncertainty in te system. Simulation and experimental results ave demonstrated te effectiveness of te proposed control metod. Key words Leader-follower robot formation, formation control, robust control, robust adaptive control 1 Introduction Over te last decade or so, various control tecniques ave been applied to te formation control design of mobile robots, suc as te leader-follower approac 1 4, virtual leader approac 5, 6, beavior-based approac 7 1, and bio-inspiration approac Te leader-follower formation control of mobile robots, one of te main approaces in tis field, as been studied by many researcers. In a robot formation wit leader-follower configuration, one or more robots are selected as leaders, and tese are responsible for guiding te formation, and te rest of robots are controlled to follow te leaders and named follower robots. Te control objective is to make te follower robots track te leaders wit some prescribed offsets. Desai et al. 1 3 presented a feedback linearization control metod for te formation of nonolonomic mobile robots using te leader-follower approac. In 1 3, te absolute velocity of te leader robot in te local coordinates of te follower robot is treated as a necessary exogenous input for te formation tracking controller. However, te absolute velocities of te leader cannot be measured directly by a local sensor carried by te follower robot. Vidal et al. 4, 15 proposed a formation control approac for nonolonomic mobile robots using motion segmentation and visual servoing tecniques. In 4, 15, te problem of distributed formation control in te configuration space was translated into separate visual servoing tasks in te image plane of a central-panoramic camera. Te motion of te leader robot needed by te tracking controller was estimated by te follower robot troug comparing te optical flows of two pixels, are corresponding to a point on te leader and te oter to a static point in te environment. Ciem and Cervera 16 presented a vision-based robot formation approac using te properties of Bézier curves. Te global linear velocity of te leader robot was assumed to be constant and known to te follower robot, and only te angular velocity of te follower robot needs to be computed using te curvature of Bézier curves between te two Received August 1, 26; in revised form January 17, 27 Supported by National Hig Tecnology Researc and Development Program of Cina 863 Program21AA Robotics Laboratory, Senyang Institute of Automation, Cinese Academy of Sciences, Senyang 1116, P. R. Cina 2. Graduate University of Cinese Academy of Sciences, Beijing 149, P. R. Cina 3. Department of Aerospace Engineering, Ryerson University, Toronto M5B 2K3, Canada DOI: 1.136/aas robots. In 17, a dual-mode model predictive controller was proposed for te leader-follower robot formation. Te formation controller was designed based on te kinematics model and te feedback linearization controller given in 1 3. In 18, a sliding mode controller was designed for te leader-follower robot formation based on te kinematics model given in 2. Te formation control law in tis paper requires te absolute velocity and te absolute acceleration of te leader robot, and as it is well konwn, in practive, it is difficult to measure or estimate in practice te absolute acceleration of te leader robot. Very recently, in 19, a leader-follower formation controller was proposed utilizing only te relative positions between robots, in wic te derivatives of te relative positions were estimated using a ig-gain observer. A iggain discontinuous auxiliary control term was added to te control law to stabilize te resulting second order error dynamics in a globally uniformly ultimately bounded manner. Te control gains of te discontinuous term were determined by te magnitude of te absolute acceleration of te leader robot and its derivative. In leader-follower formation control, te most widely used control tecnique is feedback linearization based on te kinematics model of te system. In tis study, we focus on te problem of leader-follower robot formation control wit relative motion sensors and develop a control metod tat does not require a global positioning system or absolute motion sensors. Te first part of tis paper presents a second order kinematics model for leader-follower robot formation, wic is derived in terms of te relative motion states between te robots and te local motion of te follower robot. Te formulation of te proposed model is inspired by te first order kinematics model presented in 1 3, were te model is described in terms of absolute motion states of te robots. Based on te proposed kinematics model, feedback linearization is implemented to acieve te objective of formation control wit relative motion states. However, tere is a difficulty in te implementation of feedback linearization. Te difficulty is related to te existence of unstable zero dynamics 2. In practice, te feedback linearization cannot be used if te zero dynamics is unstable. In tis study, a stability condition for te zero dynamics of leader-follower robot formation system is derived troug constraining te motion of te leader robot and te formation configuration. Te stability condition can also be treated as te coopera-

2 948 ACTA AUTOMATICA SINICA Vol. 33 tive protocol of formation control between robots. As discussed later in te paper, te internal dynamics of leader-follower robot formation system is only locally stable. Wen te internal state is in an unstable region, te formation system suffers an increasing perturbation from internal dynamics. To suppress suc a perturbation, a robustness inequality is introduced to design te feedback gain. Under te proposed control sceme, te formation tracking error approaces zero even wen te internal state is in an unstable region. On te oter and, te convergence of tracking error also forces te internal state to enter te stable region of te internal dynamics. Furtermore, a sliding mode controller is integrated to te feedback linearization controller to compensate for te uncertainty associated wit te unknown absolute acceleration of leader robot. Te Lyapunov metod is used to stabilize te overall system. In te proposed approac, te commanded acceleration for te follower robot is determined by te control law. However, it is normally not practical for a mobile robot to track an acceleration signal. Instead, in implementation, te acceleration inputs tat are determined by te control law are translated into velocity variations by multiplying tem wit te control period. Hence, te reference velocities for te follower robot, as commonly used in current control metods, can be obtained by adding te velocity variation to te current velocity. As a furter improvement of te proposed robust formation controller, a robust adaptive control sceme is also developed based on te work in 21, 22. Te controller consists of an adaptive control part tat serves to deal wit parametric uncertainty and a robust control part to compensate for te uncertainty associated wit te absolute acceleration of te leader robot. Tis paper is organized as follows. In Section 2, te proposed formation kinematics model is derived. Te proposed control scemes and robust adaptive control sceme for leader-follower robot formation system are presented in Sections 3 and 4, respectively. Simulations and experimental results are used to illustrate te effectiveness of te proposed control metods in Section 5 and 6, respectively. Finally, conclusions are drawn in Section 7. 2 Formation model formulation In tis section, we formulate te kinematics model of te leader and follower robots in formation. Te goal of te formation control is to make te follower robot track te leader robot wit desired relative distance and bearing between te robots. As indicated in Fig. 1, te follower robot R 2 follows te leader robot R 1 wit a relative distance l 12 and a relative bearing ϕ 12 between te two robots, and a relative motion sensor is mounted at point C on te follower robot R 2. Te kinematics equations of te robots are given by te following equations ẋci = ẏ ci cos θi d sin θ i sin θ i d cos θ i vi ω i θ i = ω i were x ci y ci are te coordinates of te center of mass P c in te world coordinates system, and θ i is te eading angle of te robot. As sown in te Fig. 1, v i and ω i are te linear and angular velocities of te robot R i, and teir scalar representation are given by v i and ω i, for i = 1, 2, respectively. P o is te intersection of te axis of symmetry 1 wit te driving weel axis; d is te distance from te center of mass to point P o; is te distance from te reference point C to point P o. Te velocity of point C in te local coordinates of robot R 2 is given by V 2 = v 2 ω 2 T. Set θ 12 = θ 1 θ 2. Ten, θ 12 is te relative orientation between R 1 and R 2, and θ v2 = π ϕ 12 θ 12 is te relative bearing between velocity v 2 and line l 12. Fig. 1 Two robots using leader-follower controller 2.1 Model formulation of te leader-follower robots system Similar to 1 3, te kinematics of te leader and follower robots in formation can be described by l12 v1 v2 Ml 12, θ v2 Nl 12, θ 12, θ v2 = ω 1 ω 2 ϕ 12 were te matrices M and N are defined as cos θv2 l M = 12 sin θ v2 sin θ v2 / l 12 cos θ v2 / cos θ12 l N = 12 sin θ v2 sin θ 12/ l 12 cos θ v2 / In 2, te formation motion states ave been projected to te local frame of te follower robot R 2. By solving 2, te absolute velocity of te leader robot can be presented by te formation motion states and te absolute velocity of te follower robot N M 1 l12 v2 v1 = 3 ω 2 ω 1 ϕ 12 Differentiating 2 and incorporating 3 yields v2 l12 = M ω 2 ϕ 12 M l12 + C ϕ 12 θ 12 l12 ϕ 12 δ1 ω 2 + l 12/l 12 δ 2 ω2 v 2/ 2 + were te matrix C and te vector δ are defined as θ C = v2 sin θ v2 θ v2 l 12 cos θ v2 l 12 sin θ v2 θ v2 cos θ v2 / θ v2 l 12 sin θ v2 l 12 cos θ v2 / 4

3 No. 9 LIU Si-Cai et al.: Robust Leader-follower Formation Control of Mobile Robots Based on 949 δ = δ1 = M δ 2 v 1 cos ϕ 12 v 1 sin ϕ 12/l 12 ω 1 were θ 12 = ω 1 ω 2 and θ v2 = ϕ 12 θ 12 ave been used in 4. Denoting te state variables by q = l 12 ϕ 12 T and q = l 12 ϕ 12 T, and te input of te robot formation system by u = v 2 ω 2 T, we can rewrite model 4 in te following form u = Mq, θ 12 q + Cq, q, θ 12, θ 12 q + Gq, q, θ 12, θ 12, v 2, ω 2 + δ v 1, ω 1, q, q 5 were te function Gq, q, θ 12, θ 12, v 2, ω 2 is defined as G = θ ω2 l12 ϕ M ω v 2 6 2/ l 12/l 12 5 represents te input output relation for te leaderfollower robot formation system, were te outputs are te relative distance and te relative bearing and te relative velocities between robots. Te inputs of te system are te absolute accelerations of te follower robot in local coordinates. Te relative motion states q and q and θ 12 in 5 can be measured by te relative motion sensors fixed on te follower robot. Te second order kinematics model 5 forms a nonlinear and multivariable system, wic allows us to do more rigorous analysis of te performance of control system, and to design robust nonlinear control laws tat guarantees te global stability and tracking of more complex trajectories tan tose for te one order kinematics model. Remark 1. Using model 5, te relative centripetal and Coriolis accelerations given by C q + G in 5 can be computed directly in te ligt of outputs of te relative motion sensors and te local motion sensor fixed on te follower robot. Furtermore, tese relative accelerations can be utilized to linearize te nonlinear leader-follower formation system, as discussed later. Remark 2. Te last term δ in model 5 represents te effect of te absolute acceleration of leader robot, wic is difficult to accurately measure or estimate due to te limitation of te motion sensors and treated as model uncertainty of te system. 2.2 Formulation of internal dynamics Using 3, te internal dynamics 2 of te leader-follower robot formation system is given by θ 12 = ω 1 ω 2 = v1 sin θ12 l 12 sin θ v2 ϕ 12l 12 cos θ v2 + ω 11 + l12 cos θv 2 = A θ sin θ 12 α + ω 1 1 l 12 sin θ v2 ϕ 12l 12 cos θ v2 7 were te amplitude A θ and te pase α are defined as A θ = k A 2 + k B 2 and α = arctan k B/k A for k A = v 1 ω 1l 12 sin ϕ 12/ and k B = ω 1l 12 cos ϕ 12/, respectively. Let q d = l12 d ϕ d 12 T and q d = l 12 d ϕ d 12 T be te desired relative formation motion states between two robots. Ten, te formation tracking errors are represented by q = l 12 l d 12 ϕ 12 ϕ d 12 T, q = l 12 l d 12 ϕ 12 ϕ d 12 T Te zero dynamics of te leader-follower robots system can be obtained by constraining all te tracking errors to zero and assuming l d 12 and ϕ d 12 to be zero. Pysically, tis means tat te robots in formation are bot keeping desired relative positions and moving at a constant linear velocity and angular velocity in te plane. Te zero dynamics 2 is given by θ 12 = A θ sin θ 12 α + ω 1 8 To make system 8 to ave a stable equilibrium in te region θ 12 α π/2, te following condition sould be satisfied A θ > ω 1 9 for any ω 1 and v 1. If ω 1 =, te stable equilibrium of system 8 is located at te origin. If condition 9 is not satisfied, te sign of θ12 will remain positive or negative wen te robots are in te desired formation configuration. In tis case, te value of θ 12 will increase or decrease continuously all te time, wic means tat te zero dynamics is unstable. Condition 9 ensures te stability of te zero dynamics by limiting te range of te leader s motion and te class of te desired formation configuration. Once condition 9 is satisfied, te relative angular velocity will converge to zero wen te tracking errors converge to zero. In te case were condition 9 is not establised, te solutions of 7 imply a continuous relative rotation motion between robots, and te relative angular velocity of te motion is periodically varying. Assumption 1. For te velocity v 1 ω 1 T of te leader robot, te following condition olds A θ > ω 1 in te region q q T Q, were Q is strictly positive. Assumption 1 can be viewed as a cooperation protocol between te leader and te follower robot, wic provides a stability margin for te zero dynamics. 2.3 Reformulation of internal dynamics Under Assumption 1, te internal dynamics is reformulated in terms of tracking errors in tis section. Troug reformulation, we derive a bound for te time derivative of te square of te internal tracking error. Tis result will be used to prove te stability of te closed-loop system in te following section. Wen te formation system is in steady state, te internal dynamics is given by β = A θ sin β + ω 1 1 l d 12 sin θ v2 ϕ d 12l d 12 cos θ v2 1 were β is te solution of te stable internal dynamics. Make te following state transformation and set te function θ 12 = θ 12 α β Γ θ 12, q = 1 sin θv2 l 12 cos θ v2 q 11

4 95 ACTA AUTOMATICA SINICA Vol. 33 Subtract 7 from 1. Ten te internal dynamics of system 2 can be rewritten as θ 12 = A θ sin θ + β sin β + Γ θ12, q 12 Lemma 1. If te states l 12 and θ 12 are bounded by l12 L and θ12 Θ, were L and Θ are positive constants, te internal state in 12 satisfies te following inequality θ 12 θ12 ρ θ θ ρ1 q 1 θ12 + ρ 2 q 2 θ12 13 were ρ θ = 2A θ /3 min{cos β +.5 θ 12} for β +.5 θ 12 <.5π, ρ 1 = 1/, and ρ 2 = L/. Proof. Lemma 1 can be directly proved as follows. Internal dynamics 12 can be rewritten as θ 12 = A θ sin θ12 + β sin β + Γ θ12, q = 2A θ cos β + θ 12 2 sin θ Γ θ 12, q 14 Multiplying 14 by θ 12, we get θ 12 θ12 = 2A θ cos β + θ 12 2 θ 12 sin θ θ 12Γ θ 12, q 15 were θ 12 sin.5 θ 12 and sin.5 θ12 θ12 /3 for θ12.5π, and θ 12Γ θ 12, q = 1 θ 12 sin θv2, l 12 cos θ v2 q 1 θ12 q 1 + L θ12 q 2 Ten, te following inequality olds θ 12 θ12 < ρ θ θ θ12 q 1 + L θ12 q 2 16 for ρ θ = 2A θ /3 min{cos β +.5 θ 12} and θ12.5π. Tus, Lemma 1 is proved. Lemma 1 sows tat te internal dynamics becomes stable wen te perturbation Γ θ 12, q vanises at te origin. Te stability can also be seen from 14. If te tracking error q q T converges so tat te following inequality is satisfied, 2A θ cos β + θ 12 2 Γ θ12, q 17 ten te internal tracking error θ 12 approaces zero. An effective metod to stabilize internal dynamics 14 is to force te tracking error q q T approac origin sufficiently fast to satisfy 17 for system 5, wic will be discussed in te following section. 3 Robust formation control In tis section, a robust controller is designed to stabilize te system in te presence of modeling uncertainty. Te objective is to develop a control law to determine u = v 2 ω 2 T for te leader-follower formation system suc tat te follower robot tracks te leader robot wit a given formation configuration q d = l d 12 ϕ d 12 T. Te proposed control sceme consists of a nominal part designed based on te nominal model of te system witout te perturbation of modeling uncertainty and a sliding mode robust compensator to stabilize te overall system in te presence of uncertainty. Te overall control is defined as u = M q r + C q + G Mη sgns 18 }{{}}{{} u u 1 were u is te nominal control, and u 1 is a sliding mode compensator. Teir design is described in te following. Te new reference acceleration vector q r is formed by sifting te desired acceleration q d according to te position error q and velocity error q. q r = q d Λ 2 q Λ 1 q 19 were Λ i = diag λ i1 λ i2 for i = 1, 2 are symmetric positive definite matrices. Tus, te reference trajectory is expressed in terms of te tracking errors. Te component of te constant vector η satisfy η i > M 1 δ i + η i, were η i is a strictly positive constant. Te nominal control u in 18 leads to a linear closedloop equation of te nominal system q = Λ 2 q Λ 1 q 2 and control law 18 results in te actual closed-loop system of 5 in te form of q = q r η sgns M 1 δ 21 Define te tracking errors as z i = q T i q i for i = 1, 2 and θ 12 = θ 12 α β were β is te stable solution of zero dynamics 8 under Assumption 1. Ten, te closed-loop equation can be written as ż1 A1 z1 = ż 2 A 2 z 2 were η sgns + M 1 δ 1 η sgns + M 1 δ 2 22a θ 12 = 2k θ cos β + θ 12 2 sin θ Γ θ 12, q 22b A i = 1, i = 1, 2 λ i1 λ i2 Let P i be te symmetric positive definite solution of te following Lyapunov equation A T i P i + P ia i = Q i, i = 1, 2 23 were Q 1 and Q 2 are symmetric positive definite matrices. Let λ minq i denote te smallest eigenvalue of te matrix Q i. Te stability of te resultant closed-loop 22 can be proved using te Lyapunov teory. Teorem 1. If Assumption 1 olds, closed-loop system 22 under control law 18 is asymptotically stable at te origin wit te variable s i in 18 selected as s i = z T i P i 2, i = 1, 2 24

5 No. 9 LIU Si-Cai et al.: Robust Leader-follower Formation Control of Mobile Robots Based on 951 were z T i P i 2 denotes te second element of te vector z T i P i, and Λ 1 and Λ 2 are designed suc tat te following inequality olds 2λ minq i ρ θ ρ 2 i >, i = 1, 2 25 Proof. Consider te composite Lyapunov function candidate V t = 1 2 z T i P iz i θ T θ 26 Differentiating 26 wit time along te solutions of 22, substituting z T i P i wit s i from 24, and using inequality 13, we ave V t = 1 2 żz T i P iz i + z T i P iżz i + θ T θ = { 1 2 zt i Q iz i } z T i P i η i sgns i + M 1 + δ θ T θ i.5 λ minq i zi 2 ρ θ θ ρ zi i θ12 η i s i = zi θ12.5λminq i.5ρ i.5ρ i.5ρ θ zi θ12 η i s i 27 From te last equation of 27, we ave V t < provided tat λ minq iρ θ ρ 2 i > for i = 1, 2, and z 1 z 2 θ12. Tus, Teorem 1 is proved. Te stability of te closed-loop system is guaranteed troug te design of feedback gains in u, wic satisfies robustness inequality 25 by solving Lyapunov equation 23. In 27, te perturbation from te internal dynamics is suppressed by decreasing formation tracking errors. Tus, even te internal dynamics is not in its stable region, te wole closed-loop system is still stable. Te proper selection of te control gains in u makes te wole system robust to te internal perturbations. Te gain ρ θ in 25 is determined by te configuration of te formation. As it can be noted, a larger ρ θ leads to a smaller λ minq i, equivalently, a lower gain of te controller. Tus, robustness inequality 25 provides a tradeoff between te feedback gains and te nature of formation commands tat can be tracked. Te control cattering introduced by te switcing term u 1 can be treated using a saturation function suc as te one described in Robust adaptive formation control In Section 3, we made an assumption tat te parameter of te follower robot is exactly known. However, te rotation center of a mobile robot in practice is not strictly on te axis of symmetry as in teory, wic is due to te error of macining and installation of robot weels. On te oter and, it is difficult to measure te actual rotation center wen te mobile robot is in motion. In addition, te calibration error of visual measurement system can also cause uncertainty in determining te value of. Tus, te accurate value of may be unavailable under certain situations. In tis section, we take te uncertainty in into account and propose an adaptive control metod to improve te control performance. Inspired by 21 and 22, we rewrite te nominal part of controller 18 in a decomposed form: an uncertainty free part denoted by u NA and te second part contains te parameter uncertainty and is written in a form of vector product described by Y 2 2H 2 1, as sown in te following equation. { lr 12 v2 ω 2 were u NA = were un = M l12 C ϕ 12 ϕ r 12 } l12 ϕ 12 + ω 2 + l 12/l 12 θ 12 dω2 v 2/d = u NA + Y 2 2H ω2 θ12, Y = y 22, H = 1/ u n = l r 12 + ϕ 12l 12 ϕ 12 + θ 12 ω 2 cos θ v2 ϕ r 12l 12 + l 122 ϕ 12 + θ 12 ω 2 sin θ v2 y 22 = l r 12 ϕ 12l 12 ϕ 12 + θ 12 + ω 2 sin θ v2 + v 2 θ12 ϕ r 12l 12 + l 122 ϕ 12 + θ 12 + ω 2 cos θ v2 and q r = q d Λ 2 q Λ 1 q. Te vector q r is te same as defined in 19. Te vector H contains te parameters to be estimated by an adaptive estimation algoritm. In te following, te vector ĤH will be used to denote te estimation of H, and te estimation error is presented by te vector H = H ĤH. Similarly, te estimation error of te matrix M is given by M = M ˆM = ĥ ĥ sin θ v2 l 12 cos θ v2, 29 were ˆM is te estimation of te matrix M. Clearly, if ĥ =, we ave M =. Te uncertainty of M is only associated wit te estimation error of or, equivalently, H. Now, take te uncertainty of H into account and revise 28 in te form of û = ˆM q r + Ĉ q + Ĝ = u NA + Y 2 2 ĤH were te matrixes Ĉ and Ĝ denote te estimations of C and G, respectively. 3 is te controller for nominal part of system 2 wen te system suffers from te parameter uncertainty of. Based on te above discussion, a robust adaptive controller for te leader-follower robots system can be designed as u = u NA + Y ĤH ˆMη sgns 31a

6 952 ACTA AUTOMATICA SINICA Vol. 33 ĤH = Γ ˆM 1 Y T zt 1 P 1 2 H from 31b, we get 31b z T 2 P 2 2 V t.5λ minq i z i 2 ρ θ θ were Γ is a symmetric positive definite matrix, and a constant η i is cosen suc tat te following inequalities old true η i > ˆM 1 M q + δ i + η i, i = 1, 2 32 were η 1 and η 2 are positive constants and q = q r q, and s i = z T i P i 2 for i = 1, 2. Controller 31 is composed of two parts: te first part is te output equation of te dynamical controller system, wic as te same structure as control law 18 in Section 3 and provides te control action to te follower robot; te second part is te dynamical part, wic adaptively estimates te vector H needed by te first part. Te resulting closed-loop system using control law 31 is given by żz 1 żz 2 = A1 A 2 z1 z 2 ˆM 1Y H 1 + η 1 sgns 1 + ˆM 1 M q + δ 1 ˆM 1Y H 2 + η 2 sgns 2 + ˆM 1 M q + δ 2 33a ρ zi i θ12 η i s i z T i P i 2 ˆM 1 Y H i + H T Γ 1 H = zi θ12.5λminq i.5ρ i.5ρ i.5ρ θ z i θ12 η i s i 36 From 36, we ave V t < provided tat λ minq iρ θ ρ 2 i > for i = 1, 2, and z 1 z 2 θ12 T. Tus, Teorem 2 is proved. Under te adaptive control law, te effect from te uncertainty of parameter as been compensated. As a result, te amplitude of switcing term in 31a will increase due to te term M q in Simulations We will now illustrate te effectiveness of te proposed control scemes by studying te following examples. Te formation configuration is defined as θ 12 = 2k θ cos β + θ 12 2 sin θ Γ θ 12, q 33b Te stability of closed-loop system 33 is proved by te Lyapunov teory in te following. Teorem 2. If Assumption 1 olds, te closed-loop system 33 under controller 31 is asymptotically stable at te origin, provided tat te variable s i in 31a is set at s i = z T i P i 2 for i = 1, 2, and Λ 1 and Λ 2 are cosen suc tat inequality 25 olds. Proof. Select te Lyapunov function candidate q d = l d 12 ϕ d 12 T = 4 2π/3 T Te actual value of parameter in te simulations is set at = 1 mm. Te trajectory of te leader robot is defined by an ellipse curve in te first pase, and ten te trajectory is defined by a circle after point A, as depicted in Fig. 2. Te trajectory of te leader robot is smoot wit uniform linear and angular velocities and accelerations except for point A. V t = 1 2 z T i P iz i + θ T θ + H T Γ 1 H 34 Differentiating 34 wit respect to time along te solutions of system 33, we ave V t = 1 2 żz T i P iz i + z T i P iżz i + θ T θ + H T Γ 1 H = z T i P i zt 1 P 1 2 { 1 2 zt i Q iz i z T 2 P 2 2 η i sgns i + ˆM 1 M q + δ i T } ˆM 1 Y H + θ T θ + H T Γ 1 H 35 Using inequalities 13 and 25 to 35, and substituting Fig. 2 Te trajectories of te two robots Te control parameters are selected as λ 11 = λ 21 = 3, λ 12 = λ 22 = 2, η 1 = 15, η 2 =.6 By solving te Lyapunov function A T i P i + P ia i = I, te

7 No. 9 LIU Si-Cai et al.: Robust Leader-follower Formation Control of Mobile Robots Based on 953 matrixes P 1 and P 2 are given by P 1 = P 2 = Ten, te sliding variables in 18 are determined as s 1 = z T 1 P 1 2 =.5l 12 l d l 12 s 2 = z T 2 P 2 2 =.5ϕ 12 ϕ d ϕ 12 Under controller 18, te trajectories of te two robots are depicted in Fig. 2. In te presence of initial formation tracking errors, te formation states are sown in Fig. 3. Bot te relative distance and te relative bearing converge to teir desired values, and te relative orientation remains bounded in te wole process. At point A, te motion mode of te leader robot switces from te ellipse cure to a circle, and at te same time, te relative orientation increases to a new steady state to adapt to te new motion mode of te leader robot. Te corresponding acceleration commands are sown in Fig. 4. Bot of te commanded accelerations approac zero wen te robots converge to te desired configuration. Te peaks appearing on te curves at 45t second are due to te disturbance from te discontinuous motion of te leader robot at point A. In implementation, te acceleration inputs are translated into velocity variations by multiplying tem wit te control period. Hence, te reference velocities for te follower robot can be obtained by adding te velocity variations to te current velocity. Adaptive robust controller 31 is also tested. Te control parameters are cosen as λ 11 = λ 21 = 3, λ 12 = λ 22 = 2, η 1 = 16, η 2 =.8 By comparative simulation tests, te gain for te adaptive estimator is designed as Γ = diag τ 1 τ 2 = diag 1.2 Fig. 4 Fig. 3 Te formation states Te commanded linear and angular accelerations It sould be noted tat ig estimator gain will make te estimator excessively sensitive to te system tracking errors. For instance, ig value of τ 2 will introduce control cattering into te commanded angular acceleration of te follower robot. Te initial estimate of parameter is set at ĥ = 15 mm, wereas its actual value is 1 mm. To sow te effectiveness of te proposed controller under te parametric uncertainty in, ere, we make te initial error as large as 5 %. In practice, te initial estimation error may be likely smaller tan 5 %. Under controller 31, te trajectories of te two robots are depicted in Fig. 5. To test te proposed approac in a more rigorous situation, te trajectory of te leader robot is defined by a more complex curve composed of several ellipse and circle curves. Te distance from te starting point of te follower robot to te leader robot is also larger tan te corresponding distance in te first test. Te corresponding formation states are sown in Fig. 6. Bot te relative distance and te relative bearing converge to teir desired values, and te parametric uncertainty in is effectively compensated by te adaptive controller. In contrast to Fig. 3, more severe oscillation appears in te internal state curve, but te oscillation as little impact on te formation tracking errors, as sown in Fig. 6. Tis sows tat te formation system is robust to te perturbation from tat te internal dynamics. Fig. 5 Te trajectories of te two robots by using adaptive controller 31 Fig. 6 Te system states by using adaptive controller 31 Te values of θ 12 and θ v2 are sown in Fig. 7. After te initial pase, tey ave te same amplitudes but wit opposite signs, and tis verifies tat ϕ 12 approaces zero on

8 954 ACTA AUTOMATICA SINICA Vol. 33 one side since θ v2 = ϕ 12 θ 12. Te estimation of parameter vector H = 1, 2 T is sown in Fig. 8. Altoug te estimates of 1 and 2 do not converge to teir actual values, te tracking formation errors still approac zero, as sown in Fig. 6, and an explanation for tis penomenon can be found in 17. Te output accelerations of te controller 18 are depicted in Fig. 9. Wen te leader robot moves along te irregular trajectory sown in Fig. 5, te follower robot continually regulates its commanded accelerations to track te leader robot wit desired formation. After te initial pase, te commanded linear acceleration varies approximately wit a fixed period to adapt to te motion of te leader robot. 6 Experiments Experimental investigation as been conducted using tree nonolonomic mobile robots, as sown in Fig. 1. A vision system transplanted from a robot-soccer platform is used to measure te relative positions between mobile robots. Te vision system recognizes eac robot by te color marker adibited on te top of eac robot, as sown in Fig. 1. A virtual vision sensor is fixed on eac follower robot. Te virtual sensor translates te relative positions obtained by te actual vision system into te relative motion states from te view of te follower robot. Fig. 11 a Fig. 1 Tree robots formation moving Te trajectories of te leader robot R 1 and te follower robot R 2 Fig. 7 Te derivatives of angular θ 12 and angular θ v2 by using adaptive controller 31 Fig. 11 b Te relative distance between robot R 1 and robot R 2 Fig. 8 Te estimation of H by using adaptive controller 31 Fig. 11 c Te relative bearing between robot R 1 and robot R 2 Fig. 9 Te commanded linear and angular accelerations Fig. 11 d Te relative orientation between robot R 1 and robot R 2

9 No. 9 LIU Si-Cai et al.: Robust Leader-follower Formation Control of Mobile Robots Based on 955 Te leader robot in te group is controlled by a joystick and runs an arbitrary trajectory, as sown in Fig. 11a. Eac follower robot in te group is controlled by decentralized leader-follower controller 18 wit te desired configuration q d = l12 d ϕ d 12 T = 4cm 2π/3 T. Under te proposed formation controller 18, te trajectories of te two follower robots, as recorded by te vision system, are depicted in Fig. 11a. Te corresponding formation states are sown in Fig. 11b and Fig. 11c. Bot of te tracking errors are limited wen te formation is in te steady state. Wen te trajectory of te leader robot is not smoot, te rotation motion of te leader robot varies wit ig frequency. As a result, te perturbations from te internal dynamics becomes larger, as sown in Fig. 11d. As depicted in Fig. 11b and Fig. 11c, in tis stage, bot of te formation tracking errors remain bounded and are influenced little by te varying of internal state, sowing tat te formation system is robust to te perturbation from te internal dynamics. 7 Conclusions In tis paper, we ave derived a second order kinematics model for leader-follower formation of mobile robots. Based on tis model, a robust controller is proposed to control te leader-follower formation using only te relative measurement of te motion states between robots. Te proposed controller does not need global sensor for formation control, and makes te closed-loop formation system robust to te perturbations from internal dynamics and te uncertainty associated wit te absolute acceleration of te leader robot. Furtermore, a robust adaptive controller is developed, as an upgrading version of te robust controller, to deal wit te parameter uncertainty associated wit te rotation center of te follower robots. Simulation and experimental results ave demonstrated te effectiveness of te proposed metods. References 1 Desai J P, Ostrowski J, Kumar V. Controlling formations of multiple mobile robots. In: Proceedings of 1998 IEEE International Conference on Robotics and Automation. IEEE, Desai J P, Ostrowski J, Kumar V. Modeling and control of formations of nonolonomic mobile robots. IEEE Transactions on Robotics and Automation, 21, 176: Das K, Fierro R, Kumar V, Ostrowski J P, Spletzer J, Taylor C J. A vision-based formation control framework. IEEE Transactions on Robotics and Automation, 22, 185: Vidal R, Sakernia O, Sastry S. Distributed formation control wit omnidirectional vision-based motion segmentation and visual servoing. IEEE Robotics & Automation Magazine, 24, 1114: Ogren P, Egerstedt M, Hu X. A control Lyapunov function approac to multi-agent coordination. IEEE Transactions on Robotics and Automation, 22, 185: Egerstedt M, Hu X, Stotsky A. 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In: Proceedings of te 26 American Control Conference. Minneapolis, IEEE, Isidori A. Nonlinear Control Systems: An Introduction. New York: Springer-Verlag, Liu G, Goldenberg A A. Uncertainty decomposition-based robust control of robot manipulators. IEEE Transactions on Control Systems Tecnology, 1996, 44: Slotine J J E, Li W. Applied Nonlinear Control. New Jersey: Prentice Hall, Kalil H. Nonlinear Systems. New Jersey: Prentice Hall, 22 LIU Si-Cai P. D. candidate at Senyang Institute of Automation, Cinese Academy of Sciences. He received is bacelor degree from Wuan University of Tecnology in 21. His researc interest covers mobile robot modeling and control. Corresponding autor of tis paper. liusicai@sia.ac.cn TAN Da-Long Professor at Senyang Institute of Automation, Cinese Academy of Sciences. He received is bacelor degree from Tsingua University in His researc interest covers autonomous robots and applications of AI, and robotics to te industry and space probe. dltan@sia.cn LIU Guang-Jun Tenured associate professor at Department of Aerospace Engineering, Ryerson University, Canada. He received is bacelor degree from University of Science and Tecnology of Cina in 1984, master degree from Senyang Institute of Automation, Cinese Academy of Sciences in 1987, and P. D. degree from University of Toronto in 1996, respectively. His researc interest covers control systems, robotics, mecatronics, and aircraft systems. gjliu@ryerson.ca