Nonsingular Formation Control of Cooperative Mobile Robots via Feedback Linearization

Transcription

1 Nonsinglar Formation Control of Cooperative Mobile Robots via Feedback Linearization Erf Yang, Dongbing G, and Hosheng H Department of Compter Science University of Essex Wivenhoe Park, Colchester CO4 3SQ, United Kingdom s: {eyang, dg, hh}@essex.ac.k Abstract This paper addresses the control of a leaderfollower formation where the leader robot has its own target and the follower robots are constrained by the specified formation tasks. The dynamics of the leader robot with nonholonomic constraint is explicitly integrated into the formation system to yield a centralized coordinating controller. As a reslt there is no need to assme the motion of the leader separatively when we develop cooperative formation controllers for coordinating the robots. The feedback linearization is sed to deal with the nonlinear formation control of a team of atonomos mobile robots with nonholonomic constraints. Althogh the nonlinear formation system nder consideration can be exactly linearized by taking advantage of dynamic feedback linearization, there exists strctral singlarity which may pose serios problems in practice. To solve this singlar problem a new formation model for controlling the leader-follower formation in a cooperative manner is developed. This new formation model can be extended to stdying other control and learning isses in mlti-robot systems for both cooperation and noncooperation. The internal dynamics is derived and proven to be globally stable nder the stable linear controller obtained via the partially linearized dynamics. To demonstrate the performance of the developed formation controller, simlation reslts are provided. Index Terms Cooperative control, leader-follower formation, mlti-robot systems, atonomos mobile robots, feedback linearization. I. INTRODUCTION Formation control of mltiple atonomos mobile robots and vehicles has been stdied extensively over the last decade for both theoretic research and practical applications [] [7]. There are many challenging isses in formation control, sch as path planning, trajectory tracking, formation-keeping, collision avoidance, etc. For the control of mltiple mobile robots with challenging nonholonomic constraints the formation itself poses another difficlt constraint in comparison with an individal nonholonomic mobile robot. Crrently there are three basic formation strctres: leaderfollower, behavior-based, and virtal strctre formation. The leader-follower is a very poplar formation strctre and has attracted a considerable research attention. Its application areas inclde aerial, grond, and nderwater vehicle systems [3], [9], [3] [5]. The leader-follower formation can be essentially composed of mltiple atonomos agents. The objective of this research is to develop a centralized formation controller for a team of atonomos mobile robots with nonholonomic constraints. We will focs on tasks in which the leader robot is reqired to move to a desired position while the follower robot is forced to maintain a desired separation with respect to the leader. To achieve this cooperative target, this paper proposes an integration design soltion by sing the poplar inpt-otpt feedback linearization method [8]. The basic idea is to transform a complicated nonlinear control system into a simple, decopling, and linear one sch that the established linear control theory and techniqes can be tilized to controller design. Most previos work in cooperative formation control has focsed on obtaining a stable controller for the follower separatively rather than taking into accont the whole formation system compactly. An explicit disadvantage of these works is that the motion of the leader needs to be assmed as time-invariant or to be planned beforehand [3], [9], [4], [6]. Compared with the existing works on the leaderfollower formation control of mltiple atonomos mobile robots, this stdy explicitly integrates the dynamics of the leader robot with nonholonomic constraint into the formation system. This integration approach can be sed to many cooperative problems to yield the centralized coordinating control laws. A typical application is the problem of handling deformable material cooperatively by mltiple nonholonomic mobile maniplators [6]. It also may be exploited to develop a cooperative mobile transportation platform [9]. Since the leader robot is of the well-known nonholonomic constraint, the famos Brockett s theorem will also apply to the whole formation system. II. FORMATION MODELING OF NONHOLONOMIC MOBILE ROBOTS AND PROBLEM FORMULATION A. Modeling of Leader-Follower Formation The mobile robots considered in this stdy are the poplar wheeled mobile robots of nicycle type, shown in Fig.. The configration of the robot i denoted by q i := (x ei,y ei,θ ei ) T R 3 is 3-D in the Earth fixed inertial coordinate system X Y, where (x ei,y ei ) is the center of the driving wheels of the robot i and θ ei denotes its heading angle measred from the x axis. The kinematics of the robot i are now modeled by ẋ ei = v i cos θ ei, ẏ ei = v i sin θ ei, θ ei = ω i ()

2 Y Center of driving wheels Leader Castor d θ e ( xe, ye) ( xc, yc) Inserting (4), (5) in (3) and combining the leader s dynamics given in () yield the following dynamic model for the leader-follower formation nder consideration in this paper: θ r x cr y cr ẋ e = v cos θ e, ẏ e = v sin θ e, θe = ω ẋ r = v + v cos θ r + ω y r (7) ẏ r = v sin θ r ω x r θ r = ω ω O Fig.. Follower Cooperative formation of two mobile robots where the translational velocity v i and the anglar velocity ω i are the control inpts. Since the wheels are assmed not to slip the model () is of challenging nonholonomic constraint: ẋ ei sin θ ei ẏ ei cos θ ei = () A leader-follower formation is depicted in Fig.. In the following we derive the relative model of leader-follower formation according to [7]. The configration of an individal planar robot can be described by an element of the Lie grop G of rigid motions in R, called SE(). Letg G denote the configration of leader robot, and let g G denote the configration of follower robot. The trajectories of both robots are kinematically modeled as left invariant vector fields on G, described in (). The relative configration of the follower with respect to the leader is denoted by g r G. Ths the dynamics of the relative configration can be derived as follows [7]: ġ r = g r X X g r (3) where X,X G are the Lie algebra associated with the Lie grop G. In the Lie grop SE(), the Lie algebra elements X,X se() are represented as matrices in R 3 3 of the form ω v ω v X = ω X = ω (4) The relative configration g r is given in homogeneos coordinates by cos θ r sin θ r x r g r = sin θ r cos θ r y r (5) where (x r,y r,θ r ) represent the relative position and orientation of the follower with respect to the leader. The transformation relationship between (x r,y r,θ r ) and q i (i =, ) is expressed as x r y r θ r = cos θ e sin θ e sin θ e cos θ e X x e x e y e y e θ e θ e (6) Obviosly the nonholonomic constraint () still applies to the formation system (7). According to the Brockett s theorem, the kinematics model (7) of leader-follower formation is open-loop controllable, bt not stabilizable by pre smooth, time-invariant feedback law. Hence the formation system (7) also has all of the challenges for nonholonomic systems []. Withot loss of generality, the formation model (7) can be written as the following driftless nonlinear control system: ẋ = g (x) + g (x), y = h(x) (8) where x= (x e,y e,θ e,x r,y r,θ r ) T, = (v,ω ) T, = (v,ω ) T. g (x),g (x) are smooth vector fields. h(x) is a smooth fnction vector and defines the otpt y. The definitions of g (x),g (x) can be inferred from (7). B. Problem Statement The control problem in this stdy can be formlated as follows: Problem (Cooperative Formation Control): Given the leader-follower formation system (8), find a continos, smooth feedback formation control law that steers the system in a cooperative manner for the leader robot and the follower robot in task space sch that the follower can keep a relative separation with respect to the leader while the leader is stably approaching to a desired position in the X Y plane. It shold be noted that for the formation control problem, the motion of the leader robot is not needed to be assmed in advance since the control of the leader has also been one part of cooperative formation. In this leader-follower formation each follower robot has a cooperative control law to maintain its relative position with respect to the leader, by which a desired formation for the whole system can be achieved. Ths the formation control task can be transformed into a series of sbtasks of each leader-follower pair in the systems with more than two robots. This paper will jst focs on one pair of leader-follower systems, which shold be easily extended to mlti-pair systems later. III. DYNAMIC FEEDBACK LINEARIZATION AND STRUCTURAL SINGULARITY Crrently feedback linearization is a poplar and simple choice for stabilizing and tracking nonlinear systems, sch as wheeled mobile robots [], etc. The main idea of exact feedback linearization is to transform a complicated nonlinear control system into a relatively simple, decopling, linear one sch that the established linear control theory and techniqes can be exploited to controller design. Compared with the

3 traditional linearized approaches (e.g. Jacobian eqilibrimbased linearization), the exact feedback linearization does not reslt in the information lost of the dynamics of interest. Denoting by L f λ the Lie derivative of a real-valed scalar fnction λ along a vector field f and denoting by r= (r,,r m ) the vector relative degree, the m m decopling matrix ρ(x) is defined as follows ρ(x) = ( r L gi L j f h j (x) ),i,j=,,m (9) m m where the state x= (x,,x n ) T is assmed to belong to an open set U of R n, f(x),g (x),,g m (x) are smooth vector fields, and h (x),,h m (x) are smooth otpt fnctions. Additionally we denote α(x)= r (L r f h (x),, L m f h m (x)) T. Since the distribtion generated by g (x) and g (x) in (8) is not involtive, the necessary condition for fll state feedback linearizability is violated. However, the formation model (7) still can be dynamically linearized by choosing appropriate otpts. The main reslts on the exact feedback linearization of (7) can be smmarized in the following proposition. Proposition : If the linearizing otpt vector of (7) is defined as y= (y,y,y 3,y 4 ) T = (x e,y e,x r,y r ) T, then the nonholonomic formation model (7) cannot be transformed into a linear controllable system by means of static state feedback. However, nder some conditions it is still dynamic feedback linearizable by sing a series of dynamic compensators v = p, ṗ =, ω =, v = 3 () where,, 3 are the defined new control inpts. The forth control inpt is still ω. Proof: According to the definition of exact feedback linearization in [8], the proof is simple and straightforward. The details for the proof will be omitted in this paper. For analyzing the strctral singlarity in the dynamic feedback linearization, we jst give the decopling matrix as follows cos θ e v sin θ e ρ(x) = sin θ e v cos θ e y r cos θ r v sin θ r x r sin θ r v cos θ r Based on Proposition now a nonlinear controller for controlling the formation of nonholonomic mobile robots can be easily designed by taking advantage of the established linear control theory and design methodologies. However, the dynamic integrators v = p and v = 3 reslt in a strctral singlarity for the nonholonomic formation system (7). It can be verified that det [ρ(x)] = v v. Ths the dynamic feedback linearization will break down when v = or v =. This singlarity problem may pose serios difficlties when the formation controller is applied to the prpose of stabilization, sch as parking. In this case the formation target can only be reached approximately. IV. NONSINGULAR FORMATION CONTROL VIA PARTIAL FEEDBACK LINEARIZATION Crrently it seems to be a good choice for the se of castor notion to deal with the strctral singlarities in nonholonomic mobile robots [9], [], [4], [6], []. Inspired by these previos works, in this paper the castor position as shown in Fig. rather than the center of driving wheels is also sed to develop or formation controller. Ths, the original formation model (7) can be changed to a new form in which the decopling matrix will have a well-defined relative degree vector and there does not exist any strctral singlarity. It shold be noted that the formation system in new coordinates will not be exactly feedback linearizable. Feedback linearization abot the castor position was sed in [6] to design l l and l ψ controllers. Lawton et al also sed feedback linearization and the notion of castor position in their work []. In this stdy we extend this approach to the model (7) of leader-follower formation nder consideration first, and then develop its nonsinglar formation controller. A. Model Transformation Let (x ci,y ci ) denote the castor position of robot i. The relation between (x ci,y ci ) and (x ei,y ei ) is given by x ci = x ei + d cos θ ei, y ci = y ei + d sin θ ei () where d represents the distance between the castor and the center of the driving wheels for robot i. Again,let(x cr,y cr ) denote the relative castor position of the follower with respect to the leader. From () we have [ ] [ ] xe x e xcr d(cos θ = e cos θ e ) () y e y e y cr d(sin θ e sin θ e ) Inserting () in (6) gives [ ] [ ][ ] [ ] xr cos θe sin θ = e xcr d cos θr d y r sin θ e cos θ e y cr d sin θ r (3) By sing (3) the formation model (7) can be transformed to the following form: ẋ c = v cos θ e dω sin θ e ẏ c = v sin θ e + dω cos θ e θ e = ω ẋ cr = v cos θ e + v cos(θ r + θ e )+dω sin θ e dω sin(θ r + θ e ) ẏ cr = v sin θ e + v sin(θ r + θ e ) dω cos θ e +dω cos(θ r + θ e ) θ r = ω ω (4) To consider the mechanical dynamics of mobile robots in practice the following eqations are added into the transformed formation model (4): v = F /m, ω = τ /J v = F /m 3, ω = τ /J 4 (5) where F i and τ i are the applied force and torqe to robot i, respectively. m i and J i are the mass and the moment of

4 inertial of robot i, respectively. = (, ) T and = ( 3, 4 ) T are the normalized control inpt vectors for the leader and the follower, respectively. Letting x= (x c,y c,θ e,x cr,y cr,θ r,v,ω,v,ω ) T and =(, ), the formation model (5) can be written as ẋ = f(x) + g(x), y = h(x) (6) where h(x) is a smooth fnction vector and defines the otpt y. The components of vector fields f(x),g(x) can be collected from (4) and (5). B. Formation Controller Design Let (x d c,yc) d and (x d cr,ycr) d denote the desired constant position of the leader s castor, and the desired constant separations of the follower, respectively. Unlike the dynamic extension in Section III, the control design is now based on the castor coordinates rather than the inertial coordinates located at the center of the driving wheels of robot. The main reslt on the design of formation controller in the castor coordinates is smmarized in the following proposition. Proposition : If the otpts are chosen as y= (x c x d c,y c yc,x d cr x d cr,y cr ycr) d T, then the formation system (6) has a globally well-defined decopling matrix and does not need to be extended by adding some integrators in inpt channels. As a reslt the formation system (6) can be feedback linearized partially and there is no strctral singlarity in the process of feedback linearization. Proof: Differentiating y with respect to time then yields ẏ =(ẋ c, ẏ c, ẋ cr, ẏ cr ) T = cos θ e d sin θ e sin θ e d cos θ e cos θ e d sin θ e χ dχ sin θ e d cos θ e χ dχ v ω v ω (7) with χ = cos(θ r + θ e ) and χ = sin(θ r + θ e ). Since none of the control inpts appears in (7), one can differentiate again and give ÿ =(ẍ c, ÿ c, ẍ cr, ÿ cr ) T = cos θ e d sin θ e sin θ e d cos θ e cos θ e d sin θ e χ dχ sin θ e d cos θ e χ dχ 3 4 v ω sin θ e dω cos θ e v ω cos θ e dω sin θ e v ω sin θ e v ω χ + dω cos θ e dω χ v ω cos θ e + v ω χ + dω sin θ e dω χ + (8) Ths, the decopling matrix can be derived from (8): cos θ e d sin θ e ρ(x) = sin θ e d cos θ e cos θ e d sin θ e χ dχ (9) sin θ e d cos θ e χ dχ Since det[ρ(x)] =d () the formation system (6) has a globally well-defined relative degree, i.e., r= (,,, ) T. Conseqently one can design a global formation controller withot velocity singlarity by taking advantage of partial feedback linearization [8]. Toward this end, let s constrct a change of coordinates Φ(x): x (z,ξ), z R 8, ξ R as follows z = x c x d c, z = v cos θ e ω d sin θ e z 3 = y c y d c, z 4 = v sin θ e + ω d cos θ e z 5 = x cr x d cr z 6 = v cos θ e + v χ + dω sin θ e dω χ z 7 = y cr y d cr z 8 = v sin θ e + v χ dω cos θ e + dω χ ξ = θ e, ξ = θ r () sch that the formation system (6) can be transformed into and ż = z, ż = ν, ż 3 = z 4, ż 4 = ν ż 5 = z 6, ż 6 = ν 3, ż 7 = z 8, ż 8 = ν 4 () ξ = ξ (ξ, z, ν), ξ = ξ (ξ, z, ν) (3) where ν =(ν,ν,ν 3,ν 4 ) T is defined as the linear controller to system (). In view of the linearized dynamics () one can design a stabilizing controller ν = Kz for it first and then derive the formation controller as follows = ρ (x)[ α(x) +Kz] (4) where K= (k ij ) 8 8, α(x) can be inferred from (8). It is a simple task to design a linear controller of the form ν = Kz sch that z converges to zero as t. However, sch a linear controller does not necessarily reslt in stabilizing the complete system () and (3). In what follows we need to prove that the controller (4) can also stabilize (3), the socalled internal dynamics. C. Stabilizing the Internal Dynamics The internal dynamics (3) is nobservable and ncontrollable in essence. Hence it mst be garanteed to have the property of stable zero dynamics nder the controller (4). The map Φ(x) is a diffeomorphism, and its inverse can be derived from () and have the following form x c = z + x d c, y c = z 3 + y d c v =(z 3 cos ξ + z 4 sin ξ )/ ω =( z 3 sin ξ + z 4 cos ξ )/(d) x cr = z 5 + x d cr, y cr = z 7 + y d cr, θ e = ξ, θ r = ξ v = z 6 χ 3 + z 8 χ 4 + cos ξ (z 3 cos ξ + z 4 sin ξ )/ +d sin ξ [( z 3 sin ξ + z 4 cos ξ )/(d)] ω =[z 8 χ 3 z 6 χ 4 sin ξ ((z 3 cos ξ + z 4 sin ξ )/)]/d + cos ξ [( z 3 sin ξ + z 4 cos ξ )/(d)] (5) where χ 3 = cos(ξ + ξ ),χ 4 = sin(ξ + ξ ).

5 v Kz Internal dynamics ( ξ, zv, ) ( ξ, zv, ) & ξ = ξ & ξ = ξ z &z= Az+ Bv y= Cz Partially linearized dynamics z y Orientation (rad) Castor position Inertial position x c x c (c) x e x e (e)..4 θ e.6 Orientation (rad) Castor position Inertial position y c y c (d) y e y e (f) θ e.5.5 Φ () z Φ( x) Fig.. + ρ ( x) α( x) &x= f( x) + g( x) y= h( x) Original formation dynamics Design process of the formation controller via feedback linearization Inserting (5) in (4) yields the internal dynamics as follows ξ =( z 3 sin ξ + z 4 cos ξ )/(d) ξ =[z 8 χ 3 z 6 χ 4 sin ξ ((z 3 cos ξ + z 4 sin ξ )/)]/d +(cos ξ )[( z 3 sin ξ + z 4 cos ξ )/(d)] (6) Applying the stabilizing controller ν = Kz to () implies that z converges to zero as t. Ths, the zero dynamics is fond by setting z = = z 8 =. As a reslt ξ =, ξ = (7) This shows that the zero dynamics is also stable nder the control of ν = Kz. It shold be noted that the zero dynamics is jst stable rather than asymptotically stable. In fact, we are only interested in controlling the position and separations of the robotic formation and do not reqire that the zero dynamics is asymptotically stable in this stdy. The whole design procedre of the nonsinglar formation controller by sing partial feedback linearization and stable zero dynamics is depicted in Fig.. In the figre A, B, C can be inferred from (6) and the defined otpts. Additionally it is also noted that there is actally no need to always choose the castor point. Any control point not on the wheel axis works as well withot singlarities in the static inpt-otpt feedback linearization law. Hence, the reslts presented above can be extended in a straightforward way to the case of other otpt position rather than the castor point. V. SIMULATION RESULTS To illstrate the effectiveness of the nonsinglar formation controller obtained via feedback linearization, in this section we provide some simlation reslts. The desired position for the leader and the desired separations for the follower were given by (, ) and (, ), respectively. In the simlation the distance d was set to be.m. x y Fig. 3. Time histories of state variables for point-approaching and formationkeeping control via the nonsinglar formation controller Linear velocity (m/s) Anglar velocity (rad/s) Anglar velocity (rad/s) (c) Fig. 4. Time histories of velocities for point-approaching and formationkeeping control via the nonsinglar formation controller The simlation was performed with the following linear controller ν =.3z +.z,ν =.45z 4 +.5z 3 (8) ν 3 =.35z 6 +.3z 5,ν 4 =.z 8 +.z 7 and the initial state x c () =.5m,y c () =.5m,θ e () =.rad x cr () =.3m,y cr () =.6m,θ r () =.rad (9) v () = v () =.m/s,ω () = ω () =.rad/s The time histories of state variables, control inpts, and otpts are given in Figs The formation trajectories for both the castor and the center of the driving wheels were shown in Fig. 7. The good performance for controlling the formation with the developed nonsinglar control law can be obviosly observed from Figs The otpts of the formation system asymptotically converge to the desired vales as time increases, as shown in Fig. 6. VI. CONCLUSIONS In this stdy feedback linearization has been exploited to the nonlinear formation control for coordinating the mltiple mobile robots with challenging nonholonomic constraints. For the original formation model sed in this stdy, it has been shown that there exists serios strctral singlarity thogh it can be exactly feedback linearized via dynamic extension. To obtain a nonsinglar formation controller for the considered formation system, a new formation model which explicitly and compactly integrates the dynamics of v v ω ω

6 Control inpt (m/s ) Control inpt (N/kg.m) Control inpt (m/s ) x (c) Control inpt (N/kg.m) (d) y c Castor trajectoy Leader Follower x c y e Center trajectory Leader Follower x e Fig. 5. Time histories of control inpts for point-approaching and formationkeeping control via the nonsinglar formation controller Otpt y.5 Actal.5 5 (c).5 Otpt y 3.5 Actal 5 Otpt y Otpt y Actal 5 (d) Actal 5 Fig. 6. Time histories of otpts for point-approaching and formationkeeping control via the nonsinglar formation controller the leader robot was firstly developed in this stdy by sing the castor position rather than the center of driving wheels. As a reslt the new formation model has a well-defined relative degree vector and can be feedback linearized partially. Next, the internal dynamics was derived and proven to be globally stable nder a stable linear controller designed throgh the partially linearized dynamics. Hopeflly, this new formation model can be sed as a research platform to stdy other control and learning isses in mlti-robot systems for both cooperation and noncooperation. Finally, the performance of the nonsinglar formation controller was demonstrated by the simlation reslts. ACKNOWLEDGMENT This research is fnded by the UK Engineering and Physical Sciences Research Concil (EPSRC) nder grant GR/S4558/. The athors also thank the reviewers for their constrctive comments. REFERENCES [] J. A. Marshall, M. E. Brocke, and B. A. Francis, Formations of vehicles in cyclic prsit, IEEE Transactions on Atomatic Control, vol. 49, no., pp , November 4. [] J. A. Fax and R. M. Mrray, Information flow and cooperative control of vehicle formations, IEEE Transactions on Atomatic Control, vol. 49, no. 9, pp , September 4. [3] E. Yang, Y. Masko, and T. Mita, Dal-controller approach to threedimensional atonomos formation control, Jornal of Gidance, Control, and Dynamics, vol. 7, no. 3, pp , May-Jne 4. 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