Multi Time Scale Behaviour of The Formation of Multiple Groups of Nonholonomic Wheeled Mobile Robots
MMulti Time Scale Behaviour of The Formation of Multiple Groups ofNonholonomic Wheeled Mobile Robots
Soumic Sarkar and Indra Narayan Kar Abstract — Different geometric patterns and shapes are gen-erated using groups of agents, and this needs formation control.In this paper, Centroid Based Transformation (CBT), has beenapplied to decompose the combined dynamics of nonholonomicWheeled Mobile Robots (WMRs) into three subsystems: intraand inter group shape dynamics, and the dynamics of thecentroid. The intra group shape dynamics can further be parti-tioned into the shape dynamics of each group, giving the notionof multiple group. Thus separate controllers have been designedfor each subsystem. The gains of the controllers are such chosenthat the overall system becomes singularly perturbed system,and different subsystems converge to their desired values atdifferent times. Then multi-time scale convergence analysis hasbeen carried out in this paper. Negative gradient of a potentialbased function has been added to the controller to ensurecollision avoidance among the robots. Simulation results havebeen provided to demonstrate the effectiveness of the proposedcontroller.
I. INTRODUCTIONThe study of the collective behavior of birds, animals,fishes, etc. has not only drawn the attention of biologists,but also of computer scientists and roboticists. Thus severalmethods of cooperative control [13] of multi-agent systemhave evolved, where a single robot is not sufficient to accom-plish the given task, like navigation and foraging of unknownterritory. These methods can broadly be categorized as thebehavior based approach ([1]-[3]), leader follower basedapproach [4]-[5], virtual structure based approach [6]-[9],artificial potential based navigation [10]-[12], graph theoreticmethod [14]-[15], formation shape control [16]-[21]. Amongother works carried out on single group of robots, clusterspace control [38], distance based formation [39], formationcontrol of nonholonomic robots [4], kinematic control [27],and mobile robots subject to wheel slip [40], segregation ofheterogeneous robots [41], are to name a few.The problem associated with the formation control of multi-agent system is that it becomes difficult to accurately positionthe robot within the group, as the number of robots increase.To address this issue shape control and region based shapecontrol have been proposed, such that the robots form adesired shape during movement. The desired shape can beunion or intersection of different geometric shapes. Regionbased shape control has been extended to multiple groupsof robots [22]-[24]. However, the robots can stay anywhereinside the specified region without colliding with each other. S. Sarkar is with Department of Electrical Engineering, Indian Instituteof Technology Delhi, New Delhi-110 016, INDIA soumic4it atgmail.com I.N. Kar is with the Faculty of Electrical Engineering, Indian Institute ofTechnology Delhi, New Delhi-110 016, INDIA ink at iit.ac.in
This means that the position of a robot inside a groupcan be specified and can further be controlled. Thereforethe position of a group of robots inside a larger group ofrobots can also be specified and controlled. Moreover, whenit comes to the control of multiple groups of robots, thereshould be at least one robot to convey the information ofthat group to another group.In an attempt to solve the aforementioned problem, a hierar-chical multi level topology have been proposed, here in thispaper, which is based on the centroid based transformations[16]-[19] for single group of robots. In this architecture, thelarge group of robots has been partitioned into relativelysmall basic units containing three robots. Then the centroidsof each unit are connected to form larger module containingmore robots. Extending the process gives a hierarchical archi-tecture which is a composition of relatively smaller modules.As the construction of this topology involves connecting thecentroid, it is named Centroid Based Topology (CBT). TheCBTs basically capture the constraint relationship amongthe robots. Another advantage of CBT is that it separatesshape variables from the centroid and thus separates theformation shape controller and tracking controller design.As the centroid moves, the entire structure moves main-taining the shape specified by the shape variables. In [42],different CBTs for multiple groups of robots have beenintroduced, to get a modular architecture distinguishable inthe form of intra group shape variables , inter groupshape variables along with centroid. It is to be noted thata centroid based decoupling approach is adopted in [43]to improve the convergence rate of cyclic pursuit schemefor vehicle networks. However, the modularity presented inthis paper allows one to distinguish even the intra groupshape variables of different groups. Thus multi time scaleconvergence behaviour of singular perturbation approach canbe utilized in this modular framework so that convergencetimes of different groups can be selected based on the choiceof the user. Based on this modular structure, a novel feedbackcontrol algorithm is proposed here in this paper. The gains ofthe feedback controller are so selected that the closed loopdynamics becomes singularly perturbed system. Thus themodel of the system reduces to the dynamics of centroid afterthe convergence of intra and inter group shape dynamics.This technique allows us to give importance to the part ofthe formation dynamics, which has to converge earlier thanthe others, based on the choice of the gains of the respectivecontroller. Potential function based controller has also beendesigned to avoid inter robot collision. a r X i v : . [ c s . S Y ] D ec I. PROBLEM FORMULATIONGiven a set of N robots with nonholonomic constraint [25]- [26], given by ¨ p i = A i ( θ i , ˙ θ i ) ˙ p i + B i ( θ i ) u i + C i ( ˙ θ i ) (1)where A i ( θ i , ˙ θ i ) = (cid:20) − sin θ i cos θ i ˙ θ i − sin θ i ˙ θ i cos θ i ˙ θ i sin θ i cos θ i ˙ θ i (cid:21) B i ( θ i ) = (cid:34) cos θ i m r − dR sin θ i Jr cos θ i m r + dR sin θ i Jr sin θ i m r + dR cos θ i Jr sin θ i m r − dR cos θ i Jr (cid:35) C i ( ˙ θ i ) = (cid:20) − d ˙ θ i cos θ i − d ˙ θ i sin θ i (cid:21) u i = [ τ ri , τ li ] T And J ¨ θ = Rr ( τ r − τ l ) where, m is the mass of robot, J = I − m d , I is momentof inertia, R is the distance between left and right wheels, r isthe radius of each wheel, d is the distance from wheel axis tothe center of mass, and θ is the orientation. The positions ofthe robots are described by p i = [ x i , y i ] T , i = 1 , , . . . , N inthe inertial coordinate frame, and u i = [ τ r , τ l ] T is the controltorque input. Then, for a single group of robots, a lineartransformation Φ ∈ R N × N , can be defined, that producesthe following matrix equation [ z T , z T , . . . , z TN − , z Tc ] T = Φ[ p T , p T , . . . , p TN ] T (2)where z i = [ z xi , z yi ] T ∈ R × , i = 1 , , . . . , ( N − arethe shape defining vectors or shape variables in transformedcoordinate, and these vectors define the geometric shape offormation of swarms. Clearly, the transformation Φ generatesshape variables along with the centroid for a single group ofrobots.For multiple groups of robots, these shape variables canbe categorized into two parts. The shape variables whichrepresent the shape of each subgroup, are intra groupshape variables . However, the variables which describethe interconnection among the groups, each group beingconsidered as a single agent, concentrated onto the centroidof that group, are inter group shape variables . Supposethere are m subgroups and each subgroup contains ρ i numberof robots, where i = 1 , , ..., m ( (cid:80) mi =1 ρ i = N , N being thetotal number of robots ) . Then the total number of intra groupshape variables is ρ = (cid:80) mi =1 ( ρ i − , and total number ofinter group shape variables is ( m − . The intra group shapevariables for each subgroup is defined as Z j = [ z Tj , z Tj , . . . , z Tj ( ρ i − ] T where, Z j ∈ R × ρ i − , i, j = 1 , , ..., m and z jk ∈ R × , k = 1 , , ..., ρ i − . Therefore, the intra groups shape vectorsare defined in a compact form as Z s = [ Z T , Z T , · · · , Z Tm ] T The inter group shape variables considering the centroid ofeach group as an agent, is defined as Z r = [ z Tr , z Tr , . . . , z Tr ( m − ] T where, Z r ∈ R × m − and z ri ∈ R × , i = 1 , , ..., ( m − . The geometric center of mass, z c is, defined by z c = 1 N N (cid:88) i =1 p i Using the above definitions for multiple groups of robots,the intra group, inter group shape variable, and centroid canbe written in compact form using a CBT [42] Φ M as [ Z Ts , Z Tr , z Tc ] T = Φ M [ p T , p T , . . . , p TN ] T The detailed description of the matrices Φ and Φ M is given inSection III and IV. Define, the desired intra group shape vari-ables Z sd , the inter group shape variables Z rd , and the de-sired trajectory of the centroid z cd . Let Z = [ Z Ts , Z Tr , z Tc ] T and X = [ p T , p T , . . . , p TN ] T and Z d = [ Z Tsd , Z
Trd , z
Tcd ] T and X d = [ p T d , p T d , . . . , p TNd ] T . The following equation gives thetransformation from Cartesian to the transformed coordinate. Z = Φ M X ; Z d = Φ M X d Therefore, the convergence of Z → Z d as t → ∞ leadsto the convergence of X → X d as t → ∞ as Φ M isnonsingular. However, the objective to devise controllers fordifferent dynamics such that they converge to their desiredvalues at different times. Based on this, the formation controlproblem has been divided into the following sub-problems. Intra Group Formation Control : Given a reference con-stant Z id , determine a control law such that intra group shapevariables Z i ( t ) converges to the desired value as lim t → t i Z i ( t ) → Z id Inter Group Formation Control : Given a reference con-stant Z rd determine a control law such that inter group shapevariable Z r ( t ) converges to the desired value as lim t → t r Z r ( t ) → Z rd Trajectory Tracking : Given a reference time varying tra-jectory z cd ( t ) determine a control law such that the centroid z c ( t ) converges to the desired trajectory as lim t →∞ z c ( t ) → z cd ( t ) where, < t i < t r < t c < t < ∞ , i = 1 , , ..., m .III. GENERAL FORM OF CENTROID BASEDTRANSFORMATIONIn centroid based representation [16]-[19] of formation ofa single group of robots, the centroid is being retained, as itcontains all the positional information of the group of robots.All other vectors ( shape variables ) describe the connectivityrelationship among the robots in the group. However, thegeneral transformation matrix for N robots can be given as Φ = (cid:2) Φ Tr , Φ Tc (cid:3) T here, I is the identity matrix of dimension . The dimen-sion of the matrix Φ r is (2( N − × N ) . The matrix Φ c is (2 × N ) and it captures the information of the coefficients togenerate the centroid vector. As the centroid is to be retained,the last row of the block matrix Φ is given by, Φ c = 1 N (cid:2) I I · · · I (cid:3) ∈ R × N IV. TRANSFORMATION MATRIX FOR MULTIPLEGROUPS OF ROBOTSThis section mainly describes how to generate centroidbased transformation matrices for multiple groups of robots.Fig. 4 depicts nine robots divided in subgroups with three
Fig. 1. Schematic Representation of Multiple Groups of Robots robots in each. The shape variables of the Jacobi transforma-tion applied on the nodes of each subgroup are collectively intra group shape variables. The inter group shape vari-ables can be found applying the Jacobi transformation on thecentroids of subgroups. With this modularity even the intragroup shape variables of each subgroup is identifiable. Thederivation of this example of transformation gives a heuristicunderstanding and some intuitive feeling for the solution ofthe stated problem. The intra group shape variables are Z ⇒ (cid:40) z = √ ( p − p ) z = p − ( p + p ) Z ⇒ (cid:40) z = √ ( p − p ) z = p − ( p + p ) Z ⇒ (cid:40) z = √ ( p − p ) z = p − ( p + p ) Z r ⇒ (cid:40) z r = √ ( µ − µ ) z r = µ − ( µ + µ ) where, µ = ( p + p + p ) , µ = ( p + p + p ) , µ = ( p + p + p ) . A. General Form of The Transformation Matrix for MultipleGroups
Therefore, the general form of the transformation matrixfor multiple groups of robots can be written as follows Φ M = (cid:2) Φ T , Φ T , . . . , Φ Tm , Φ Tr , Φ Tc (cid:3) T Where, Φ , Φ , ..., Φ m are (2( ρ − × N ) , (2( ρ − × N ) , ..., (2( ρ m − × N ) dimensional matrices and m isthe total number of subgroups. The scalar (( ρ i ) − , i =1 , , · · · , m are the number of shape variables required torepresent i th subgroup. Φ r is (2( m − × N ) and Φ c is (2 × N ) . We write the transformation matrix in a morecompact form as Φ M = (cid:104) Φ m T , Φ Tr , Φ Tc (cid:105) T Suppose there are m groups of robots and in the i th group,there are n i number of robots. Then the dimension of thematrix Φ m is ρ + ρ + · · · + ρ m ) − m ) × N . Again Φ m can be written in block diagonal form as Φ m = diag { Φ i } ,where, Φ i denotes the transformation matrix for the i th groupof robots containing ρ i number of robots. The dimension of Φ i is ρ i − × ρ i . B. Shape Variable Generation Algorithm for MultipleGroups of Robots
The following steps describe the generation algorithm forthe shape variables of multiple groups of robots. step 1 : calculate intra group shape variables for each sub-group, i.e., Z , · · · , Z m . step 2 : calculate the centroid of each subgroup, i.e. µ , · · · µ m step 3 : calculate inter group shape variables for the overallgroup assuming each group as an agent. step 4 : calculate the overall centroid. step 5 : write the vectors in matrix form to get the transfor-mation matrix.The algorithm above will generate all the vectors of thetransformation. The coefficients of the vectors collectivelyform the transformation matrix for multiple groups of robots.V. FORMATION CONTROLLER DESIGN ANDSTABILITY ANALYSIS A. Formation Dynamics
The entire formation of N WMRs can be viewed asa deformable body whose shape and movement can bedescribed by vectors in transformed coordinate. Define thenotation A i = A i ( θ i , ˙ θ i ) , B i = B i ( θ i ) and C i = C i ( ˙ θ i ) for i = 1 , , . . . , N , where θ i and ˙ θ i are the orientation andangular speed of the i -th WMR as described in (1).The dynamic equation of N WMRs can be written as ¨ X = A ˙ X + B U + C (3)where, A = diag { A , A , . . . , A N } , B = diag { B , B , . . . , B N } , C = diag { C , C , . . . , C N } nd U = [ u t , u T , ..., u TN ] T . Using the transformation Φ M ,(3) can be written as ¨ Z = P ˙ Z + Q U + R (4)where P = Φ M A Φ − M ; Q = Φ M BU ; R = Φ M C . Withequation (4), controllers are designed in the next section suchthat different dynamics converge to their desired values atdifferent times. B. Asymptotic Stability Analysis of Three-time Scale Singu-larly Perturbed Systems
The asymptotic stability analysis of three-time scale sin-gularly perturbed systems composed of the twice applicationof two-time scale analysis, given in Appendices I and II.There are two ways to address the analysis:
Top-Down and
Bottom-Up . These two approaches logically select theslow and fast dynamics sequentially. The details of thesetwo approaches can be found in [35]-[36]. The
Bottom-Up approach is considered here in this paper to understand thenatural evolution of the multi-time scale singularly perturbedsystems in their own configuration spaces. A generic three-time scale model can be described by ˙ x = f ( x , x , x ) , x ∈ R m (cid:15) ˙ x = f ( x , x , x ) , x ∈ R m (cid:15) (cid:15) ˙ x = f ( x , x , x ) , x ∈ R m (5)The model (29) can be sequentially decomposed into twodifferent two-time scale models. The first two-time scalemodel considers the time scale defined by the stretched timescale t = t(cid:15) (cid:15) , where the reduced (slow) subsystems aredefined by ˙ x = f ( x , x , F ( x , x )) (cid:15) ˙ x = f ( x , x , F ( x , x )) (6)and the boundary layer subsystem is given by (cid:15) (cid:15) dx dt = dx dt = f ( x , x , x ( t )) (7)where in (31), x , x are treated like fix parameters and x ( t ) evolves on its stretched time scale t . F ( x , x ) in(30) represents the quasi-steady-state of the boundary layer(31), when (cid:15) = 0 , that is, f ( x , x , x ) → x = F ( x , x ) . From (30), the slow system can be written as ˙ x = f ( x , F ( x ) , F ( x , F ( x ))) (8)where F ( x ) in (33) is the quasi-steady-state of the bound-ary layer (cid:15) dx dt = dx dt = f ( x , x ( t ) , F ( x , x ( t )) (9)when (cid:15) = 0 , that is, f ( x , x , F ( x , x )) → x = F ( x ) . A Lyapunov Function is constructed initially to sat-isfy the growth requirements of (33) and (9) as in AppendixI and II V ( x , x ) = (1 − d ) V ( x ) + d W ( x , x ) (10) where < d < , and V ( x ) and W ( x , x ) are thechosen Lyapunov function that satisfy the growth require-ments for (33) and (9) respectively. Note that the constructionof V ( x , x ) with the Lyapunov functions for the sub-systems x and x must satisfy the growth requirementson F ( x , x , F ( x , x )) and f ( x , x , F ( x , x )) . Thebound on (cid:15) can be found from the construction of Lyapunovfunction as given in Appendix I.Then for the system of equations (30), define a vector χ = [ x , x ] T and a system of functions F ( χ ) =[ f ( · ) , f ( · )] T . A Lyapunov function V ( χ , x ) is chosenthat satisfies the certain growth requirements for singularlyperturbed system as in Appendix I as V ( χ , x ) = (1 − d ) V ( χ ) + d W ( χ , x ) (11)where < d < and W ( χ , x ) is the chosen Lyapunovfunction that satisfies the growth requirements for (31). C. Three time scale behaviour of multiple groups of robots
The matrix P and R of equation (4), can be written inthe following form P = [ P Ts , P Tr , P Tc ] T ; R = [ R Ts , R Tr , R Tc ] T Therefore, the collective dynamics of (4) can be separatelywritten in the form of intra group shape dynamics ( Z s ) , asfollows ¨ Z s = P s ˙Λ + F s + R s (12)where, Z s = Ψ m X ; F s = Ψ m BU . The inter group shapedynamics ( Z r ) is written as, ¨ Z r = P r ˙Λ + F r + R r (13)where, Z r = Ψ r X ; F r = Ψ r BU . The dynamics of theoverall leader ( z c ) is expressed as, ¨ z c = P c ˙ λ + f c + R c (14)where, z c = Ψ l X ; f l = Ψ c BU .We define the intra group shape error vector as Z se = Z s − Z sd , the inter group shape error vector Z re = Z r − Z rd ,and the tracking error of overall leader z ce = z c − z cd , where Z sd , Z rd , and z cd are the desired intra group, desired intergroup shape variables, and desired trajectory of the overallleader respectively.Define a set of three time instants τ s , τ r , and τ c , such that τ s = τ(cid:15) (cid:15) ; τ r = τ(cid:15) τ s ≤ τ r ≤ τ l ≤ ∞ . The controller is tobe designed such that Z se → , during the interval [ τ , τ s ] , Z re → , during the interval [ τ , τ r ] , z ce → , during theinterval [ τ , τ l ] . Here, Z se is ultra fast variable, Z re is fastvariable, and z ce is slow variable. To achieve the desiredformation and tracking, the following controllers is proposedfor (12)-(14). F s = ν s − P s ˙ Z − R s + ¨ Z sd F r = ν r − P r ˙ Z − R r + ¨ Z rd f l = ν l − P l ˙ z − R l + ¨ z cd (15)where, ν s = − K s Z se − K s ˙ Z se − K sr ˙ Z re − K sc ˙ z ce + ¨ Z sd r = − K r Z re − K r ˙ Z re − K rs ˙ Z se − K rc ˙ z ce + ¨ z rd ν c = − k c z ce − k c ˙ z ce − K cs ˙ Z se − K cr ˙ Z re + ¨ z cd where, K s = K fs ( (cid:15) (cid:15) ) , K s = K fs (cid:15) (cid:15) , K r = K fr (cid:15) , K r = K fr (cid:15) , are controller gain matrices. K fs = k s I N s , K fs = k s I N s and K r = k r I N r , K r = k r I N r , where N s isthe number of intra group shape variables, N r is the numberof inter group shape variables, and k s , k s ∈ R + , k r , k r ∈ R + . K s are coupling gain matrices, where K sr ∈ R N s × N r , K sc ∈ R N s × , K rs ∈ R N r × N s , K rc ∈ R N r × , K cs ∈ R × N r , K cr ∈ R × N r .Using (15), the closed loop error dynamics is given as ¨ Z se = − K s Z se − K s ˙ Z se − K sr ˙ Z re − K sl ˙ z ce ¨ Z re = − K r Z re − K r ˙ Z re − K rs ˙Λ se − K rc ˙ z ce ¨ z ce = − k l z ce − k l ˙ z ce − K cs ˙ Z se − K cr ˙ Z re (16) Theorem 1
Suppose the controllers F s , F r , and f l , asgiven in (15) is designed for system (16). Then The system(16) is exponentially stable for all (cid:15) i ≤ (cid:15) ∗ i , for some small (cid:15) ∗ i , i = 1 , . The analytical upper bounds on (cid:15) i , i = 1 , are derived to establish the stability of whole singularlyperturbed system. Proof:
To write error dynamics define E c = (cid:20) z ce ˙ z ce (cid:21) ; E r = (cid:20) (cid:15) Z re ˙ Z re (cid:21) ; E s = (cid:20) (cid:15) (cid:15) Z se ˙ Z se (cid:21) Hence, the error dynamics of (16) is written in the form ofsingularly perturbed system as follows ˙ E c = (cid:20) − k c − k c (cid:21) E c + (cid:20) − K cs (cid:21) E s + (cid:20) − K cr (cid:21) E r (cid:15) ˙ E r = (cid:20) I − K fr − K fr (cid:21) E r + (cid:15) (cid:18) (cid:20) − K rs (cid:21) E s + (cid:20) − K rc (cid:21) E c (cid:19) (cid:15) (cid:15) ˙ E s = (cid:20) I − K fs − K fs (cid:21) E s + (cid:15) (cid:15) (cid:18) (cid:20) − K sr (cid:21) E r + (cid:20) − K sc (cid:21) E c (cid:19) Sequential application of two time scale results in bottom upapproach sets (cid:15) = (cid:15) = 0 . As a result the slow manifoldsbecome E s = 0 , E r = 0 . The boundary layer systems arederived as follows dE s dτ i = A s E s ; τ i = t(cid:15) (cid:15) (17) dE r dτ r = A r E r ; τ r = t(cid:15) (18)where, A s = (cid:20) I − K fs − K fs (cid:21) and A r = (cid:20) I − K fr − K fr (cid:21) . As the above boundary layer systemsare all linear and time invariant, the exponential stability can be guaranteed if matrices A s and A r are stable. Noticethat these matrices are in companion form. So, there alwaysexist a pair of gain matrices ( K fr , K fr ) and ( K fs , K fs ) in order to ensure the stability of A s and A r .We choose Lyapunov functions for the boundary layer (43)as V s ( E s ) = E Ts P s E s (19)where Q s > such that matrices P s satisfies Lyapunovequations A Ts P s + P Ts A s = − Q s such that ˙ V s ( E s ) ≤ .Similarly for the boundary layer (44), the Lyapunov functionis chosen as V r ( E r ) = E Tr P r E r (20)where Q r > such that matrix P r satisfies Lyapunovequation A Tr P r + P Tr A r = − Q r such that ˙ V r ( E r ) ≤ .With proper choice of the gains k c = k I and k c = k I ,the reduced order slow system ˙ E c = A c E c (21)where, A c = (cid:20) − k c − k c (cid:21) and the Lyapunov function forthe reduced order system can be chosen as V c ( E c ) = E Tc P c E c (22)Hence, the overall system is locally exponentially stable forsmall values of (cid:15) and (cid:15) .To derive the bounds on (cid:15) i , a composite Lyapunov functionof the following form V ( E c , E r ) = (1 − d ) V c ( E l ) + d V r ( E r ) (23)where < d < , is chosen to satisfy the followingcondition ˙ V ( E c , E r ) = − [(1 − d ) E Tc Q c E c − d E Tc A Trc P r E r − d E Tr P r A rc E r + d (cid:15) E Tr Q r E r ]= − (cid:20) E c E r (cid:21) T (cid:20) (1 − d ) Q c − d P r A rc − d A Trc P r d (cid:15) Q r (cid:21) (cid:20) E c E r (cid:21) ≤ − χ T Q (cid:15) χ (24)where χ = [ E Tc , E Tr ] T , A rc = (cid:20) − K rc (cid:21) and Q (cid:15) > .From Q (cid:15) , the bound on (cid:15) can be found using Schur’scompliment for positive definiteness: A > and C − B T A − B > , where, A = (1 − d ) Q c , B = − d P r A rc ,and C = d (cid:15) Q r of the matrix Q (cid:15) . The explicit bound on (cid:15) is (cid:15) < det ( A ) det ( C − B T A − B ) (25)where C = (cid:15) C .We then construct another Lyapunov function of the fol-lowing form to find the composite stability of subsystems χ and E s V ( χ, E s ) = (1 − d ) V ( χ ) + d V s ( E s ) (26)he time derivative of (52) gives the following additionalterms ˙ V ( χ, E s ) = (1 − d ) ˙ V ( χ ) − d (cid:15) (cid:15) E Ts Q s E s + E Tr A sr P s E s + E Tc A Tsc P s E s + E Ts P s A sr E r + E Ts P s A sc E c ≤ − χ T Q (cid:15) χ (27)which leads to the construction of another matrix Q (cid:15) in thesame way Q (cid:15) is derived in (50), and χ = [ E Tc , E Tr , E Ts ] T , A sr = (cid:20) − K sr (cid:21) , and A sc = (cid:20) − K sc (cid:21) . The matrix Q (cid:15) has the following form (1 − d )(1 − d ) Q l d (1 − d ) P r A rc − d A Tsc P s − d (1 − d ) A Trc P r d (cid:15) (1 − d ) Q r − d A Tsr P s − d P s A sc − d P s A sr d (cid:15) (cid:15) Q s (28)Then the composite system is asymptotically stable if Q (cid:15) > . The bound on (cid:15) can be computed from Q (cid:15) using Schur’scompliment for positive definiteness stated above. The ma-trices are A = (1 − d ) Q (cid:15) , B = [ − d P s A sc , − d P s A sr ] T and C = d (cid:15) (cid:15) Q s of the matrix Q (cid:15) . This completes the proofof theorem 1 . (cid:4) D. Asymptotic Stability Analysis of Multi-time Scale Singu-larly Perturbed Systems
The asymptotic stability analysis of multi-time scale singu-larly perturbed systems composed of the repeated applicationof two-time scale analysis, given in Appendices I and II.There are two ways to address the analysis:
Top-Down and
Bottom-Up . These two approaches logically select theslow and fast dynamics sequentially. The details of thesetwo approaches can be found in [35]-[36]. The
Bottom-Up approach is considered here in this paper to understand thenatural evolution of the multi-time scale singularly perturbedsystems in their own configuration spaces. A generic multi-time scale model can be described by ˙ x = f ( x , x , · · · , x n ) , x ∈ R m (cid:15) ˙ x = f ( x , x , · · · , x n ) , x ∈ R m ... ( n − (cid:89) i =1 (cid:15) i ) ˙ x n = f n ( x , x , · · · , x n ) , x n ∈ R m n (29)The model (29) can be sequentially decomposed into ( n − different two-time scale models. The first two-time scalemodel considers the time scale defined by the stretched timescale t n − = t (cid:81) n − i =1 (cid:15) i , where the reduced (slow) subsystem is defined by ˙ x = f ( x , x , · · · , F n ( x , · · · , x n − )) (cid:15) ˙ x = f ( x , x , · · · , x n − , F n ( x , · · · , x n − )) ... ( n − (cid:89) i =1 (cid:15) i ) ˙ x n − = f n − ( x , x , · · · , x n − , F n ( x , · · · , x n − )) (30)and the boundary layer subsystem is given by dx n dt n − = f n ( x , x , · · · , x n ( t n − )) (31)where in (31), x , x , · · · , x n − are treated like fix pa-rameters and x n ( t n − ) evolves on its stretched time scale t n − . F n ( x , · · · , x n − ) in (30) represents the quasi-steady-state of the boundary layer (31), when (cid:15) n − = 0 , that is, f n ( x , x , · · · , x n ) → x n = F n ( x , · · · , x n − ) . Thenfor the system of equations (30), define a vector χ =[ x , · · · , x n − ] T and a system of functions F ( χ , x n ) =[ f ( · ) , · · · , f n − ( · )] T . A Lyapunov function V n − ( χ , x n ) is chosen that satisfies the growth requirements given inAppendix I and II as V n − ( χ , x n ) = (1 − d n − ) V n − ( χ ) + d n − W ( χ , x n ) (32)where < d n − < and W ( χ , x n ) is the chosen Lyapunovfunction that satisfies the growth requirements for (31).Applying the same procedure leads to the construction of V n − ( χ ) = V n − ( χ , x n − ) , χ = [ x , · · · , x n − ] T . Notethat the construction of V n − ( χ , x n − ) with the Lyapunovfunctions for the subsystems χ and x n − must satisfy thegrowth requirements on F ( χ , x n − , F n ( x , · · · , x n − )) and f n − ( χ , x n − , F n ( x , · · · , x n − )) . The bound on (cid:15) n can be found from the construction of Lyapunov function asgiven in Appendix II. Similarly, the bounds on other (cid:15) i s canbe found following the same procedure sequentially until wereach the reduced order slow system ˙ x = f ( x , · · · , F n − ( x , · · · , x n − ,F n ( x , · · · , x n − )) , F n ( x , · · · , x n − )) (33) E. Multi time scale behaviour of multiple groups of robots
The objective of this section is to show that multi timescale convergence of the collective dynamics of (3) can beachieved in singular perturbation framework, depending uponthe selection of gain parameters of the designed controller.The matrix P and R of equation (4) in subsection A, can bewritten as P = [ P T , P T , . . . , P Tm , P Tr , P Tc , ] T , and R =[ R T , R T , . . . , R Tm , R Tr , R Tc , ] T . Therefore, the collectivedynamics of (3) can be separately written in the form of intragroup shape dynamics ( Z i ) , as follows ¨ Z i = P i ˙ Z + F i + R i (34)where, Z i = Φ i X ; F m = Φ i BU , i = 1 , , .., m . The intergroup shape dynamics ( Z r ) is written as, ¨ Z r = P r ˙ Z + F r + R r (35)here, Z r = Φ r X ; F r = Φ r BU . The dynamics of thecentroid ( z c ) is expressed as, ¨ z c = P c ˙ Z + f c + R c (36)where, z c = Φ c X ; f c = Φ c BU .The intra group shape error vectors are defined by Z e = Z − Z d , Z e = Z − Z d , ..., Z me = Z m − Z md , where Z d , Z d , ..., Z md are the desired intra group shape vectors.The inter group shape error vectors are defined as Z re = Z r − Z rd , and the tracking error of centroid is defined as z ce = z c − z cd , where, Z rd and z cd are the desired intergroup shape variables and desired trajectory of the centroidrespectively.Define a set of ( m + 2) time instants t , t , . . . , t m , t r , and t c such that t = t(cid:15) (cid:15) ··· (cid:15) m +1 ; t = t(cid:15) (cid:15) ··· (cid:15) m ; . . . t m = t(cid:15) (cid:15) ; t r = t(cid:15) ; t ≤ t ≤ ... ≤ t m ≤ t r ≤ t c ≤ t as t → ∞ . t is total time of operation, t · · · t r are stretchedtime scales (within which the subsystems must converge),and (cid:15) · · · (cid:15) m +1 are controller gain parameters chosen toachieve different time scale convergence. The controllers areto be designed such that intra group shape error vectors Z ie → , during the interval [ t , t i ] , i = 1 , . . . , m . The intergroup shape error Z re → , during the interval [ t , t r ] , andtracking error z ce → , during the interval [ t , t c ] . To achievethe desired formation the following controllers is proposedfor (34) - (36). F i = ν i − P i ˙ Z − R i + ¨ Z id , i = 1 , , . . . , m.F r = ν r − P r ˙ Z − R r + ¨ Z rd f c = ν c − P c ˙ Z − R c + ¨ Z cd (37)where, ν i = − K i Z ie − K i ˙ Z ie − m (cid:88) j =1 ,j (cid:54) = i K ij ˙ Z je − K ir ˙ Z re − K ic ˙ Z ce + ¨ Z id , i = 1 , , . . . , m.ν r = − K r Z re − K r ˙ Z re − m (cid:88) j =1 K rj ˙ Z je − K rc ˙ Z ce + ¨ Z rd ν c = − k c Z ce − k c ˙ Z ce − m (cid:88) j =1 K cj ˙ Z je − K cr ˙ Z re + ¨ z cd (38)where, K i = K fi ( (cid:81) mi =1 (cid:15) i +1 ) , K i = K fi ( (cid:81) mi =1 (cid:15) i +1 ) , i =1 , .., , m + 1 , K r = K fr (cid:15) , K r = K fr (cid:15) are controller gainmatrices. Suppose, there are N s intra group shape variablesand N r inter group shape variables for m groups of robots.Define a set N s = { N s , N s , ..., N sm } to denote the numberof intra group shape variables in each group. K s are couplinggain matrices, where K ij ∈ R N si × N sj , i, j = 1 , , ..., m and i (cid:54) = j ; K ir ∈ R N si × N r , i = 1 , , ..., m ; K ic ∈ R N si × , i = 1 , , ..., m ; K rj ∈ R N r × N sj , j = 1 , , ..., m ; K rc ∈ R N r × , K cj ∈ R × N sj , j = 1 , , ..., m ; K cr ∈ R × N r . Hence, applying the control law into (4), the closed loop error dynamics are given as follows ¨ Z ie = − K i Z ie − K i ˙ Z ie − m (cid:88) j =1 ,j (cid:54) = i K ij ˙ Z je − K ir ˙ Z re − K ic ˙ Z ce , i = 1 , , . . . , m ¨ Z re = − K r Z re − K r ˙ Z re − m (cid:88) j =1 K rj ˙ Z je − K rc ˙ Z ce ¨ z ce = − k c Z ce − k c ˙ Z ce − m (cid:88) j =1 K cj ˙ Z je − K cr ˙ Z re (39)The main results of this paper for the convergence of errordynamics (39) is stated in the form of the following theorem.As a result, the intra group shape variables Z i , i = 1 , ..., m ,inter group shape variables Z r converge to their desiredvalues in different time scales. Also the centroid z c convergesto the desired trajectory. Theorem 2
Suppose the controllers F , . . . , F m , F r , and f c , as given in (37) is designed for system (39). Then Thesystem (39) is exponentially stable for all (cid:15) i ≤ (cid:15) ∗ i , forsome small (cid:15) ∗ i , i = 1 , , ..., m + 1 . Under mild conditionsstated in Appendix A-E, the analytical upper bounds on (cid:15) i , i = 1 , , ..., m + 1 are derived to establish the stability ofwhole singularly perturbed system. Proof:
The proof consists of two parts. In the first partthe stability of the reduced and boundary layer systems, isanalyzed by a Lyapunov function, which is given by thecomposition of the Lyapunov functions of the slow and thefast systems. The analytical bounds on singularly perturbedparameters (cid:15) i are derived in the second part.To write error dynamics define E c = (cid:20) Z ce ˙ Z ce (cid:21) ; E r = (cid:20) (cid:15) Z re ˙ Z re (cid:21) ; E i = (cid:34) (cid:81) mi =1 (cid:15) i +1 Z ie ˙ Z ie (cid:35) Hence, the error dynamics of the equation (39) is written inthe form of singularly perturbed system as follows ˙ E c = (cid:20) I − k c − k c (cid:21) E c + m (cid:88) i =1 (cid:20) − K ci (cid:21) E i + (cid:20) − K cr (cid:21) E r (40) (cid:15) ˙ E r = (cid:20) I − K fr − K fr (cid:21) E r + (cid:15) (cid:18) m (cid:88) i =1 (cid:20) − K ri (cid:21) E i + (cid:20) − K rc (cid:21) E c (cid:19) (41) m (cid:89) i =1 (cid:15) i +1 ) ˙ E i = (cid:20) I − K fi − K fi (cid:21) E i + ( m (cid:89) i =1 (cid:15) i +1 ) (cid:18) m (cid:88) j =1 ,k (cid:54) = i (cid:20) − K ij (cid:21) E j + (cid:20) − K ir (cid:21) E r + (cid:20) − K ic (cid:21) E c (cid:19) (42)Sequential application of two time scale results in bottom upapproach sets (cid:15) m +1 = (cid:15) m = ... = (cid:15) = 0 . As a result theslow manifolds become E m = 0 , · · · , E = 0 , E r = 0 . Theboundary layer systems are derived as follows dE i dt i = A i E i ; t i = t (cid:81) mi =1 (cid:15) i +1 (43) dE r dt r = A r E r ; t r = t(cid:15) (44)where, A i = (cid:20) I − K fi − K fi (cid:21) , i = 1 , , ..., m and A r = (cid:20) I − K fr − K fr (cid:21) . As the above boundary layer systemsare all linear and time invariant, the exponential stability canbe guaranteed if matrices A i and A r are stable. Notice thatthese matrices are in companion form. So, there always exista pair of gain matrices ( K fr , K fr ) , ( K fm , K fm ) , · · · , ( K f , K f ) in order to ensure the stability of A i and A r .We choose Lyapunov functions for the boundary layer (43)as V i ( E i ) = E Ti P i E i (45)where Q i > such that matrices P i satisfies Lyapunovequations A Ti P i + P Ti A i = − Q i for i = 1 , , ..., m . Similarlyfor the boundary layer (44), the Lyapunov function is chosenas V r ( E r ) = E Tr P r E r (46)where Q r > such that matrix P r satisfies Lyapunovequation A Tr P r + P Tr A r = − Q r . With proper choice of thegains k c = k I and k c = k I , the reduced order slowsystem ˙ E c = A c E c (47)where, A c = (cid:20) − k c − k c (cid:21) and the Lyapunov function forthe reduced order system can be chosen as V c ( E c ) = E Tc P c E c (48)Hence, the overall system is asymptotically stable for smallvalues of (cid:15) , (cid:15) , · · · , (cid:15) m +1 . Remark 1
The computation of the respective bounds on (cid:15) i , i = 1 , is clarified here for the three time scale dynamics of[42]. The same procedure is to be extended to find furtherbounds on (cid:15) i , i = 3 , ..., m for multi-time scale dynamics. Toderive the bounds on (cid:15) , a composite Lyapunov function ofthe following form V ( E c , E r ) = (1 − d ) V c ( E l ) + d V r ( E r ) (49) where < d < , is chosen to satisfy the followingcondition ˙ V ( E c , E r ) = − [(1 − d ) E Tc Q c E c − d E Tc A Trc P r E r − d E Tr P r A rc E r + d (cid:15) E Tr Q r E r ]= − (cid:20) E c E r (cid:21) T (cid:20) (1 − d ) Q c − d P r A rc − d A Trc P r d (cid:15) Q r (cid:21) (cid:20) E c E r (cid:21) ≤ − χ T Q (cid:15) χ (50)where χ = [ E Tc , E Tr ] T , A rc = (cid:20) − K rc (cid:21) and Q (cid:15) > .From Q (cid:15) , the bound on (cid:15) can be found using Schur’scompliment for positive definiteness: A > and C − B T A − B > , where, A = (1 − d ) Q c , B = − d P r A rc ,and C = d (cid:15) Q r of the matrix Q (cid:15) . The explicit bound on (cid:15) is (cid:15) < det ( A ) det ( C − B T A − B ) (51)where C = (cid:15) C .We then construct another Lyapunov function of the fol-lowing form to find the composite stability of subsystems χ and E s V ( χ, E s ) = (1 − d ) V ( χ ) + d V s ( E s ) (52)The time derivative of (52) gives the following additionalterms ˙ V ( χ, E s ) = (1 − d ) ˙ V ( χ ) − d (cid:15) (cid:15) E Ts Q s E s + E Tr A sr P s E s + E Tc A Tsc P s E s + E Ts P s A sr E r + E Ts P s A sc E c ≤ − χ T Q (cid:15) χ (53)which leads to the construction of another matrix Q (cid:15) in thesame way Q (cid:15) is derived in (50), and χ = [ E Tc , E Tr , E Ts ] T , A sr = (cid:20) − K sr (cid:21) , and A sc = (cid:20) − K sc (cid:21) . The matrix Q (cid:15) has the following form (1 − d )(1 − d ) Q l d (1 − d ) P r A rc − d A Tsc P s − d (1 − d ) A Trc P r d (cid:15) (1 − d ) Q r − d A Tsr P s − d P s A sc − d P s A sr d (cid:15) (cid:15) Q s (54)Then the composite system is asymptotically stable if Q (cid:15) > . The bound on (cid:15) can be computed from Q (cid:15) using Schur’scompliment for positive definiteness stated above. The ma-trices are A = (1 − d ) Q (cid:15) , B = [ − d P s A sc , − d P s A sr ] T and C = d (cid:15) (cid:15) Q s of the matrix Q (cid:15) .VI. C OLLISION A VOIDANCE
The controllers of (37) don’t guarantee collision avoidanceamong the robots. Therefore, the barrier-like function of [19]is chosen as a potential function for collision avoidance. Themodified form of the function for the robots i, j ∈ N , i, j =1 , , ..., N is given by V ij ( p i , p j ) = (cid:18) min (cid:26) , (cid:107) q i − q j (cid:107) − R (cid:107) q i − q j (cid:107) − r (cid:27) (cid:19) (55)here R is the radius of sensing and q i , q j represents theposition of i -th and j th robot respectively. r denotes thepermissible distance from the robot i to avoid collision. Thenthe control input for the collision avoidance of i -th robot isthe summation of all potential defined by (55) of the robots j inside the permissible distance r : (cid:53) f i = − n (cid:88) j =1 ,j (cid:54) = i ∂V ij ( p i , p j ) ∂p i T (56)where ∂y∂x is the gradient of a scalar function y (of dependent( x ) and independent variables) with respect to x and (cid:53) f i ∈ R × . Define a matrix (cid:53) F ∈ R N × of control input basedon avoidance potential of all robots i = 1 , , ..., N as (cid:53) F = [ (cid:53) f T , (cid:53) f T , ..., (cid:53) f TN ] T (57)To comply with the solutions of ¨ X = − (cid:53) F under thetransformation Z = Φ M X , define a vector of control inputin the transformed domain as F pot = Φ M (cid:53) F (58) A. Three Time Scale
The vector F pot of (58) is partitioned as F pot =[ F Tpot s , F Tpot r , F Tpot c ] T , where, F pot s ∈ R ρ × , F pot r ∈ R m − × , and F pot c ∈ R × . Then the equations forformation controllers with collision avoidance is given by F s = ν s − P s ˙ Z − R s + ¨ Z sd − k s F pot s F r = ν r − P r ˙ Z − R r + ¨ Z rd − k r F pot r f c = ν c − P c ˙ Z − R c + ¨ Z cd − k c F pot c (59) k s = (cid:15) (cid:15) , k r = (cid:15) , k c = 1 are scalars associated with thepotential terms F pot s , F pot r , F pot c respectively to adjust thegain. Hence, the closed loop dynamics are ¨ Z se = − K s Z se − K s ˙ Z se − K sr ˙ Z re − K sc ˙ z ce − k s F pot s ¨ Z re = − K r Z re − K r ˙ Z re − K rs ˙ Z se − K rc ˙ z ce − k r F pot r ¨ z ce = − k c z ce − k c ˙ z ce − K cs ˙ Z se − K cr ˙ Z re − k c F pot c (60) Theorem 3
The controllers F s , F r , and f c , as given in (15)locally asymptotically stabilize system (16), independently,for all (cid:15) i < (cid:15) ∗ i , i = 1 , and for some small (cid:15) ∗ i , i = 1 , . Asa result, the intra group shape variable Z s , inter group shapevariables Z r , and the centroid z c converge to their desiredvalues as t → ∞ . Proof:
To write error dynamics define E c = (cid:20) z ce ˙ z ce (cid:21) ; E r = (cid:20) (cid:15) Z re ˙ Z re (cid:21) ; E s = (cid:20) (cid:15) (cid:15) Z se ˙ Z se (cid:21) Hence, the error dynamics of (16) is written in the formof singularly perturbed system as follows ˙ E c = (cid:20) − k c − k c (cid:21) E c + (cid:20) − K cs (cid:21) E s + (cid:20) − K cr (cid:21) E r − (cid:20) F pot c (cid:21) (cid:15) ˙ E r = (cid:20) I − K fr − K fr (cid:21) E r + (cid:15) (cid:18) (cid:20) − K rs (cid:21) E s + (cid:20) − K rc (cid:21) E c (cid:19) − (cid:20) F pot r (cid:21) (cid:15) (cid:15) ˙ E s = (cid:20) I − K fs − K fs (cid:21) E s + (cid:15) (cid:15) (cid:18) (cid:20) − K sr (cid:21) E r + (cid:20) − K sc (cid:21) E c (cid:19) − (cid:20) F pot s (cid:21) By setting (cid:15) = (cid:15) = 0 , we’ve the slow man-ifolds E r = (cid:20) I − K fr − K fr (cid:21) − (cid:20) F pot r (cid:21) , E s = (cid:20) I − K fs − K fs (cid:21) − (cid:20) F pot s (cid:21) . And the boundary layer sys-tems are as follows dE r dt r = (cid:20) I − K fr − K fr (cid:21) E r − (cid:20) F pot r (cid:21) ; t r = t(cid:15) dE s dt s = (cid:20) I − K fs − K fs (cid:21) E s − (cid:20) F pot s (cid:21) ; t s = t(cid:15) (cid:15) As the above boundary layer systems are a combination ofa dissipative part and a linear combination of potentiallydecreasing functions, the subsystems will exponentially reachthe trajectory of the gradient of the potential term within thetime τ and τ respectively. So if there is a possibility ofcollision among the robots after τ and τ , the formation willcollapse as the potential terms aren’t time dependent func-tions. Due to the effect of the potential terms in the boundarylayers, the subsystems again reach desired formation whenthe inter robot distance criteria are met.With proper choice of the gains k c and k c , for example, k c = k c = kI where k is a scalar, the reduced order slowsystem ˙ E c = (cid:20) − k c − k c (cid:21) E c + (cid:20) − K cs (cid:21) (cid:20) I − K fs − K fs (cid:21) − (cid:20) F pot s (cid:21) + (cid:20) − K cr (cid:21) (cid:20) I − K fr − K fr (cid:21) − (cid:20) F pot r (cid:21) + (cid:20) F pot c (cid:21) is also exponentially stable because the gains multiplied withthe potential term only add to the total potential. Thus thepotential terms preserve the property of driving away theneighbouring robots to avoid collision. Hence, the overallsystem is stable for small values of (cid:15) , (cid:15) . (cid:4) Remark
It’s necessary for the robots not to collide at thetime of intra group formation or inter group formation ortracking the given trajectory. Hence, the potential term isrequired for the fast system when they reach the boundarylayer, because it’s important to avoid collision even when thesubsystems reach the desired formation. For example, afterthe convergence of intra groups formation, there is a fairpossibility that the formed groups collide at time time ofinter group formation. Hence the potential terms are addedwith adjustable scalar gains to the controllers in (59). . Multi Time Scale
The vector F pot of (58) is partitioned as F pot =[ F Tpot , F Tpot , ..., F Tpot m , F Tpot r , F Tpot c ] T , where, F pot s ∈ R ρ × , F pot r ∈ R m − × , and F pot c ∈ R × . Then (37)is modified as below to assure collision avoidance F i = ν i − P i ˙ Z − R i + ¨ Z id − k i F pot i , i = 1 , , . . . , m.F r = ν r − P r ˙ Z − R r + ¨ Z rd − k r F pot r f c = ν c − P c ˙ Z − R c + ¨ Z cd − k c F pot c (61)The scalars k = (cid:15) (cid:15) , k = (cid:15) (cid:15) (cid:15) , ..., k m = (cid:15) (cid:15) ...(cid:15) m +1 , k r = (cid:15) , k c = 1 are associated with the poten-tial terms F pot , F pot , ..., F pot m , F pot r , F pot c respectively toadjust the performance of the controllers. Hence, the closedloop dynamics are ¨ Z ie = − K i Z ie − K i ˙ Z ie − m (cid:88) j =1 ,j (cid:54) = i K ij ˙ Z je − K ir ˙ Z re − K ic ˙ Z ce − k i F pot i , i = 1 , , . . . , m ¨ Z re = − K r Z re − K r ˙ Z re − m (cid:88) j =1 K rj ˙ Z je − K rc ˙ Z ce − k r F pot r ¨ z ce = − k c Z ce − k c ˙ Z ce − m (cid:88) j =1 K cj ˙ Z je − K cr ˙ Z re − k c F pot c (62) Theorem 4
The controllers F , . . . , F m , F r , and f c , as givenin (61) asymptotically stabilize system (62), for all (cid:15) < (cid:15) ∗ , (cid:15) < (cid:15) ∗ , . . . , (cid:15) m +1 < (cid:15) ∗ m +1 and for some small (cid:15) ∗ , (cid:15) ∗ , . . . , (cid:15) ∗ m +1 . Moreover, the closed loop system is asymptoticallydecoupled. As a result, the intra group shape variable Z , . . . , Z m inter group shape variables Z r converge to theirdesired values in different time scale. Also the centroid z c converges to the desired trajectory. Proof:
To write error dynamics define E c = (cid:20) Z ce ˙ Z ce (cid:21) ; E r = (cid:20) (cid:15) Z re ˙ Z re (cid:21) ; E i = (cid:34) (cid:81) mi =1 (cid:15) i +1 Z ie ˙ Z ie (cid:35) Hence, the error dynamics of the equation (39) is written inthe form of singularly perturbed system as follows ˙ E c = (cid:20) − k c − k c (cid:21) E c + m (cid:88) i =1 (cid:20) − K ci (cid:21) E i + (cid:20) − K cr (cid:21) E r − (cid:20) F pot c (cid:21) (63) (cid:15) ˙ E r = (cid:20) − K fr − K fr (cid:21) E r + (cid:15) (cid:18) m (cid:88) i =1 (cid:20) − K ri (cid:21) E i + (cid:20) − K rc (cid:21) E c (cid:19) − (cid:20) F pot r (cid:21) (64) ( m (cid:89) i =1 (cid:15) i +1 ) ˙ E i = (cid:20) − K fi − K fi (cid:21) E i + ( m (cid:89) i =1 (cid:15) i +1 ) (cid:18) m (cid:88) j =1 ,k (cid:54) = i (cid:20) − K ij (cid:21) E j + (cid:20) − K ir (cid:21) E r + (cid:20) − K ic (cid:21) E c (cid:19) − (cid:20) F pot m (cid:21) (65)For each equation condition 1 and condition 3 of Theorem4 is satisfied. By setting (cid:15) = (cid:15) = ... = (cid:15) m +1 = 0 , we’vethe boundary layer equations as follows dE r dt r = (cid:20) − K fr − K fr (cid:21) E r − (cid:20) F pot r (cid:21) ; t r = t(cid:15) dE i dt i = (cid:20) − K fi − K fi (cid:21) E i − (cid:20) F pot i (cid:21) ; t i = t (cid:81) mi =1 (cid:15) i +1 As the above boundary layer systems are a combination ofa dissipative part and a linear combination of potentiallydecreasing functions, the subsystems will exponentially reachthe trajectory of the gradient of the potential term within thetime τ and τ respectively. So if there is a possibility ofcollision among the robots after τ and τ , the formation willcollapse as the potential terms aren’t time dependent func-tions. Due to the effect of the potential terms in the boundarylayers, the subsystems again reach desired formation whenthe inter robot distance criteria are met.With proper choice of the gains k c = I and k c = I , thereduced order slow system ˙ E c = (cid:20) − k c − k c (cid:21) E c + m (cid:88) i =1 (cid:20) − K ci (cid:21) (cid:20) I − K fi − K fi (cid:21) − (cid:20) F pot i (cid:21) + (cid:20) − K cr (cid:21) (cid:20) I − K fr − K fr (cid:21) − (cid:20) F pot r (cid:21) + (cid:20) F pot c (cid:21) is also asymptotically stable because the gains multipliedwith the potential term only add to the total potential. Thusthe potential terms preserve the property of driving awaythe neighboring robots to avoid collision. Hence, the overallsystem is stable for small values of (cid:15) , (cid:15) , ..., (cid:15) m +1 . (cid:4) Remark 2
It’s necessary for the robots not to collide at thetime of intra group formation or inter group formation ortracking the given trajectory. Hence, the potential term isrequired for the fast system when they reach the boundarylayer, because it’s important to avoid collision even when thesubsystems reach the desired formation. For example, afterthe convergence of intra groups formation, there is a fairpossibility that the formed groups collide at time time ofinter group formation. Hence the potential terms are addedwith adjustable scalar gains to the controllers in (62).VII. SIMULATION RESULTSThe controllers developed in section V-VI have beensimulated on three groups of robots with three robots ineach group. Three time scale convergence being an examplef multi time scale convergence, is demonstrated in thissection with simulation. The controller gain parameters arechosen as K fr = K fr = kI , K f = K f = kI ,where, k = 1 , and (cid:15) = 0 . , (cid:15) = 0 . . The matrices K sr , K sc , K rs , K rc , K cs , K cr are chosen to be all swith appropriate dimension, so that the system becomestightly coupled, although the degree of coupling is left asa choice for the user. All the figures in this section show thetrajectories of the robots moving in formation. The positionsof the robots are marked by ’ (cid:46) ’ and each group contains threerobots marked with red, green and blue color. Potential forceparameters are taken from [19]. The desired trajectory of thecentroid of the formation is kept as z c = [ t ; 30 sin (0 . t )] .The rest of the desired vectors are framed as illustrated inSection IV. The desired shape of individual group is anequilateral triangle and also desired shape of the biggertriangle that tangles all the groups, is equilateral triangle.From Fig. 1, each side of the small triangle and the bigtriangle are b = 7 m and a = 20 m respectively. Theshape variables in the transformed domain, are given as Z sd = [( − . , , (0 , . , ( − . , , (0 , . , ( − . , , (0 , . T , Z rd = [( − . , , (0 , . T . Fig. 2. Formation control using transformation Φ M Fig. 3. Potential force based Formation control using transformation Φ M The initial values of position of robots is respectively asfollows ( x , y ) = ( − , , ( x , y ) = , ( − , , ( x , y ) = ( − , , ( x , y ) = ( − , − , ( x , y ) = (0 , , ( x , y ) =(1 , − , ( x , y ) = (3 , − , ( x , y ) = ( − , − , ( x , y ) = (4 , . In Fig. 2, it is shown that the robotsconverge to the desired formation from the initial conditionsgiven above. Potential force has not been considered for thesimulation in Fig. 2. The convergence of robots to the desiredformation with collision avoidance, is depicted in Fig. 3.Fig. 4 shows the convergence time of the states in thetransformed domain separately (without applying potentialforce). All the intra group shape variables Z . . . Z convergefaster than inter group shape variables Z and Z . It canalso be seen from the figures, that the convergence of thecentroid is the slowest of all. It is evident from Fig. 4 thatthe intra group shape variables converge to desired valueat t = 0 . sec . The inter group shape variables converge attime t = 1 sec and the trajectory of centroid converges to thedesired value at t = 10 sec . Thus convergence of intra groupshape variables are times faster than the convergenceof inter group shape variables. Again, convergence of thetrajectory of centroid is times faster than the convergenceof inter group shape variables. Fig. 4. Plot of intra and inter group shape variables and centroid vs time
VIII. CONCLUSIONIn the paper we propose, an intuitive and simple way ofsolving a complicated formation control problem. For that acentroid based transformation is given for multiple groupsof robots such that a modular architecture results, in theform of intra group, inter group shape variables, and centroid.Thus exploiting the modularity separate controllers have beendesigned for each module of formation. The gains of thefeedback controllers are so selected that the error dynamicsbecome singularly perturbed system and multi time scalebehaviour of the overall system is achieved. Thus the controllaws ensure different time scale convergence of differentgroup of robot dynamics. For collision avoidance, negativegradient of potential function has also been appended theproposed feedback controller. Simulation results shows theperformance of the proposed formation controllers. It is tobe noted that the high gains are to be chosen such that thenput energy does not exceed the maximum capacity of themotor attached to the wheels of the WMR. A new classof problems for solving the group formation of multiplegroups of robots, has been originated due to this formulation,and many potential formation problems can be solved usingthe proposed methodology. Future work entails multiplegroup formation of quadrotor robots with in the singularlyperturbed system framework.A
PPENDIX IA SYMPTOTIC S TABILITY A NALYSIS O F T WO -T IME S CALE S INGULAR P ERTURBATION A UTONOMOUS S YSTEMS
To preserve the integrity of the paper, this section describesthe general formulation for the asymptotic stability analysisfor the two-time scale singular perturbation problems [31],[32]. It is the foundation of the analysis that was conducted todemonstrate the asymptotic stability of the multi-time scalemodel in Section V. C. Consider a nonlinear autonomoussingular perturbed system of the form ˙ x = f ( x, z ) , x ∈ R n , (66) (cid:15) ˙ z = g ( x, z ) , x ∈ R m , (67)which has an isolated equilibrium at the origin ( x = 0 , z =0) . It is assumed throughout the formulation that f and g aresmooth to ensure that for specified initial conditions, system(66) and (67) has a unique solution. The stability of theequilibrium is investigated by examining the reduced (slow)system ˙ x = f ( x, h ( x )) (68)where z = h ( x ) is an associated root of g ( x, z ) and theboundary layer (fast) system dzdτ = g ( x, z ( τ )) , τ = t(cid:15) (69)where x is treated as a fixed parameter and (cid:15) is the parasiticconstant that defines the stretched time scale of the fast sub-system. To prove asymptotic stability, Lyapunov functionsare to exist for reduced and boundary layer systems, whichsatisfy certain growth conditions to be addressed later. Weassume that the following conditions hold for all ( x, y, (cid:15) ) ∈ [ t , ∞ ) × B x × B z × [0 , (cid:15) ] (70)where B x ∈ R n and B z ∈ R m denotes closed sets. Weadd and substract f ( x, h ( x )) to the right-hand side of (66)yielding ˙ x = f ( x, h ( x )) + f ( x, z ) − f ( x, h ( x )) (71)where f ( x, z ) − f ( x, h ( x )) can be viewed as a perturbation ofreduced system (68). It is natural first to look for a Lyapunovfunction candidate for (68) and then to consider the effect ofthe perturbation term f ( x, z ) − f ( x, h ( x )) [31]. A. Assumption (Asymptotic Stability of the Origin)
The origin ( x = 0 , z = 0) is a unique and isolatedequilibrium of (66) and (67), that is, f (0 , and g (0 , (72)moreover, z = h ( x ) is the unique root of g ( x, z, in B x × B z , that is, g ( x, h ( x )) , and there exists a class κ function p ( · ) such that (cid:107) h ( x ) (cid:107)≤ p ( (cid:107) x (cid:107) ) . To studythe asymptotic stability of the equilibrium, Lyapunov func-tion candidates are to be constructed for both reduced andboundary layer systems separately. The respective growthrequirements will be defined separately in Assumptions Band C, whereas the growth requirements that combine bothreduced and boundary layer system requirements, calledinterconnection conditions, will be defined in AssumptionsD and E. B. Assumption (Reduced System Conditions)
There exists a positive-definite Lyapunov function candi-date V ( x ) , that is, < q ( (cid:107) x (cid:107) ) ≤ V ( x ) ≤ q ( (cid:107) x (cid:107) ) (73)for some class κ function q ( · ) and q ( · ) , that satisfies thefollowing inequality ∂V∂x f ( x, h ( x )) ≤ − α ψ ( x ) (74)where ψ ( · ) is a scalar function of vector arguments thatvanishes only when its argument are zero and satisfying that x = 0 is a stable equilibrium of the reduced order system.Condition (74) guarantees that x = 0 is an asymptoticallystable equilibrium of (68). C. Assumption (Boundary Layer System Conditions)
There exists a positive-definite Lyapunov function candi-date W ( x, z ) such that for all ( x, z ) ∈ B x × B z , satisfying < q ( (cid:107) z − h ( x ) (cid:107) ) ≤ W ( x, z ) ≤ q ( (cid:107) z − h ( x ) (cid:107) ) (75)for some class κ function q ( · ) and q ( · ) , that satisfies W ( x, z ) > , ∀ z (cid:54) = h ( x ) and W ( x, h ( x )) = 0 (76)and ∂W∂z g ( x, z ) ≤ − α φ ( z − h ( x )) , α > (77)where W ( x, z ) is a Lyapunov function of boundary layersystem (69) in which x is treated as a fixed parameter and φ ( · ) is a scalar function of vector arguments that vanishesonly when its argument are zero and satisfying that z − h ( x ) is a stable equilibrium of the boundary layer system. . Assumption (First Interconnection Condition) V ( x ) and W ( x, z ) must satisfy the so called intercon-nection conditions. The first interconnection condition isobtained by computing the derivative of V along the solutionof (71), ˙ V = ∂V∂x f ( x, h ( x )) + ∂V∂x [ f ( x, z ) − f ( x, h ( x ))] ≤ α ψ ( x ) + ∂V∂x [ f ( x, z ) − f ( x, h ( x ))] (78)assuming that ∂V∂x [ f ( x, z ) − f ( x, h ( x ))] ≤ β ψ ( x ) φ ( z − h ( x )) (79)so that ˙ V ≤ − α φ ( x ) + β ψ ( x ) φ ( z − h ( x )) (80)Inequality (79) determines the allowed growth of f in z , and α and β are nonnegative constants. E. Assumption (Second Interconnection Conditions)
The second interconnection condition is defined by ∂W∂x f ( x ) ≤ γφ ( z − h ( x )) + β ψ ( x ) φ ( z − h ( x )) , (81)where ψ ( · ) and φ ( · ) are scalar functions of vector argumentsthat vanish only when their arguments are zero, that is, φ ( x ) = 0 if and only if x = 0 . γ and β are nonnegativeconstants.With the Lyapunov function V ( x ) and W ( x, z ) obtained, anew Lyapunov function ν ( x, z ) is considered and defined bythe weighted sum of V ( x ) and W ( x, z ) , ν ( x, z ) = (1 − d ) V ( x ) + dW ( x, z ) (82)for < d < . ν ( x, z ) becomes the Lyapunov functioncandidate for the singular perturbed system (66)-(67). F. Theorem A If x = 0 is an asymptotically stable equilibrium of reducedsystem (68), z = h ( x ) is an asymptotically stable equilibriumof boundary layer system (69) uniformly in x , that is, the (cid:15) − δ definition of Lyapunov stability and the convergence z → h ( x ) are uniform in x [33], and if f ( · , · ) and g ( · , · ) satisfy(74), (77), (80), and (81), then the origin is an asymptoticallystable equilibrium of the singularly perturbed system (66),for sufficiently small (cid:15) [31].A PPENDIX
IIS
ELECTION OF T HE B OUNDS OF THE S TABILITY P ARAMETERS
Calculating the time derivative of ν of (82) along thetrajectory of the full system (66)-(67), we obtain ˙ ν = (1 − d ) ∂V∂x f ( x, h ( x )) + d(cid:15) ∂W∂z g ( x, z )+ (1 − d ) ∂V∂x [ f ( x, z ) − f ( x, h ( x ))] + d ∂W∂x f ( x ) (83) where using the Assumption B-E of Appendix I, we canexpress (83) as ˙ ν ≤ − (1 − d ) α ψ ( x ) + (1 − d ) β ψ ( x ) φ ( z − h ( x )) − d(cid:15) α φ ( z − h ( x )) + dγφ ( z − h ( x ))+ dβ ψ ( x ) φ ( z − h ( x )) = (cid:20) ψ ( x ) φ ( z − h ( x )) (cid:21) T × (cid:20) (1 − d ) α − (1 − d ) β − dβ − (1 − d ) β − dβ d ( α (cid:15) − γ ) (cid:21) × (cid:20) ψ ( x ) φ ( z − h ( x )) (cid:21) (84)The right-hand side of (87) is a quadratic form in thecomparison functions ψ ( x ) and φ ( z − h ( x )) . The quadraticform is negative definite when d (1 − d ) α ( α (cid:15) − γ ) >
14 [(1 − d ) β + dβ ] (85)where α , α , β , β and γ are defined in (80) and (81).Thus, rewriting (85) as (cid:15) < α α α γ + − d ) d [(1 − d ) β + dβ ] ≡ (cid:15) d (86)Inequality (86) shows that for any choice of d , the corre-sponding ν is a Lyapunov function for the singular perturbedsystem (66)-(67) for all (cid:15) satisfying (86). The maximumvalue of (cid:15) d is given by (cid:15) ∗ = α α α γ + β β (87)and occurs for d ∗ = β β + β (88)If all the growth requirements are satisfied, then the originis an asymptotically stable equilibrium of the singularlyperturbed system (66)-(67) for all (cid:15) ∈ (0 , (cid:15) ∗ ) , where (cid:15) ∗ isgiven by (88). Risgiven by (88). R