Distributed Formation Control of Multi-Robot Systems: A Fixed-Time Behavioral Approach
DDistributed Formation Control of Multi-Robot Systems:A Fixed-Time Behavioral Approach
Ning Zhou, Xiaodong Cheng, Yuanqing Xia, Yanjun Liu
Abstract — This paper investigates a distributed formationcontrol problem for networked robots, with the global objectiveof achieving predefined time-varying formations in an environ-ment with obstacles. A novel fixed-time behavioral approach isproposed to tackle the problem, where a global formation taskis divided into two local prioritized subtasks, and each of themleads to a desired velocity that can achieve the individual taskin a fixed time. Then, two desired velocities are combined viathe framework of the null-space-based behavioral projection,leading to a desired merged velocity that guarantees the fixed-time convergence of task errors. Finally, the effectiveness of theproposed control method is demonstrated by simulation results.
I. I
NTRODUCTION
Due to broad applications of multi-robot systems in e.g.,search and rescue missions, natural resource monitoring, andoutdoor industrial operations such as fault diagnosis andrepair, control of multi-robot systems has attracted increasingattention from the systems and control domain [1], [2].However, in many practical applications, autonomous robotsare deployed in a dynamical environment to perform multipleparallel tasks, for example, to maintain the desired formationand avoid moving obstacles at the same time. An efficientand reliable control scheme for operating this kind of systemsposes a challenge in our domain.To achieve multi-mission control problems, the so-calledbehavioral approach is resorted [3]–[5]. In this approach,a comprehensive control task is decomposed into multiplesmaller and simpler subtasks, characterized by behavioralfunctions , and each of them provides a motion command.Merging these commands through a certain method thenleads to the eventual control law. One of the commonlyused merging methods is the
Null-Space-Based Behavioral (NSB) approach [5], [6]. This approach is used to handle thesituation when some tasks, or behaviors, conflict with eachother, e.g. a team of robots are supposed to form certainformation when they also have to avoid obstacles appearingon their way. With the NSB approach, the behaviors are
The work was supported in part by the National Natural Science Founda-tion of China under Grant 61603095, Grant 61972093, Grant 61720106010,and Grant 61973147. The work of Yuanqing Xia was supported in part bythe Science and Technology on Space Intelligent Control Laboratory underGrant KGJZDSYS-2018-05.Ning Zhou is with School of Electrical Engineering, Hebei Uni-versity of Science and Technology, Shijiazhuang 050018, China. [email protected] .Xiaodong Cheng is with Department of Electrical Engineering, Eind-hoven University of Technology, 5600 MB Eindhoven, The Netherlands. [email protected] .Yuanqing Xia is with the School of Automation, Beijing Institute ofTechnology, Beijing 100081, China. xia [email protected] .Yanjun Liu is with the College of Science, Liaoning University ofTechnology, Jinzhou 121001, China. [email protected] . prioritized, where the lower prioritized tasks are projectedto the null space of higher-priority tasks, guaranteeing thatthey do not contradict the higher ones. The NSB allowsthe entire networked systems to exhibit the robustness withrespect to eventually conflicting tasks/behaviors [7], and itshows great potential in real-world applications [4], [8],[9]. Recently, the behavioral approach has been extendedto a decentralized/distributed manner [4], [10], in whichcontrollers are localized in each autonomous robot and canonly acquire information from its neighboring robots. Adecentralized framework of behavioral approaches was firstlygiven in [4], but the theoretical guarantee of the convergenceof behavior errors is lacking. A distributed formation controlmethod for multi-robot systems using NSB is providedin [10], which results in the asymptotic stability of theclosed-loop system. However, this method only focuses on atriangular formation control problem. Inspired by the worksin [4], [10], we investigate a distributed NSB control schemefor formation control of multi-robot systems. Differently, weconsider obstacles in the environment and use a distributedestimator [11] to enable a distributed control scheme.The major difference between the current work and the ex-isting approaches is that we propose a fixed-time behavioralapproach, which provides faster convergence speed and bet-ter control precision, in comparison to the asymptotic results[8], [10], [11], and initial-state-independent convergence timecompared with the finite-time approaches [8], [11]. This ismotivated by the need for multi-robot applications requiringa fast convergence speed and a high control accuracy, e.g.,cooperative robotic imaging. Note that the fixed-time control,which is firstly presented in [12], in general, has a fastconvergence rate, high-precision control performance, anddisturbance rejection properties [13]. In this paper, we willcombine the behavioral approach and fixed-time control toachieve multiple tasks of networked robots in a fixed time.To the best of our knowledge, such a problem has not beenaddressed in the literature so far. To solve this problem,we introduce the behavior functions of collision avoidanceand cooperative formation, respectively, which lead to twovelocity commands using the fixed-time design and inversekinematics method. These two commands are merged inpriority, via the null-space-based behavioral projection, togive a desired velocity for each agent that can be computedbased on only local information. Using a distributed fixed-time estimator, a fixed-time behavioral approach is designedand implemented in a distributed manner. The theoreticalproof of the fixed-time convergence of task errors is pro-vided. The developed fixed-time behavioral approach can a r X i v : . [ m a t h . O C ] A ug andle time-varying formations in a distributed frameworkand guarantee collision/obstacle avoidance, and it is not onlylimited to some certain triangle-based formations in an idealenvironment.The rest of the paper is organized as follows: Section IIrecaps the concept of fixed-time control and formulates theproblem; Section III presents the main result of this paper,which provides a fixed-time behavioral control scheme formulti-robot systems; The simulation result is provided inSection IV, and concluding remarks are made in Section V. Notation:
The set of real numbers is denoted by R . For avector or matrix, (cid:107) · (cid:107) denotes its Euclidean norm. The i -thelement of a vector v is denoted by v i . The n -dimensionalvector whose elements are all 1 is denoted by 1l n ∈ R n . Theinvolution operation without loss of the number’s sign isrepresented by x [ p ] : = | x | p sgn ( x ) , x , p ∈ R . sgn ( · ) is the signfunction that returns −
1, 0 or 1.II. P
RELIMINARIES AND P ROBLEM S ETTING
A. Fixed-Time Stability
Consider a nonlinear system˙ x ( t ) = f ( t , x ) , x ( ) = x , (1)with x ( t ) ∈ R n and the nonlinear function f ( t , x ) . If f ( t , x ) is discontinuous, the solutions of (1) are Filippov . Supposethe origin is an equilibrium point of (1), then the fixed-timestability is defined as follows.
Definition 1: [12] The origin x = globallyfixed-time stable if it is globally asymptotically stable andany solution x ( t , x ) of (1) reaches x = t = T ( x ) and remains there for all t ≥ T ( x ) , where T ( x ) is globally bounded by some number T max ∈ R > .Notice that in the concept of the fixed-time stability, thesettling (convergence) time T ( x ) is always bounded inde-pendent of the initial condition x . Furthermore, the fixed-time stability of the nonlinear system (1) can be characterizedby the following lemma. Lemma 1: [12], [14] If there exists a continuous radiallyunbounded and positive definite function V : R n → R > suchthat V ( x ) = x =
0, and any solution x ( t , x ) of (1) satisfies ˙ V ( x ) ≤ − η V k ( x ) − η V k ( x ) , (2)or ˙ V ( x ) ≤ − ( η V k ( x ) + η V k ( x )) k , (3)where η , η , k , k , k , k , k ∈ R > with k >
1, 0 < k < k k >
1, and k k <
1, then the origin of (1) is globallyfixed-time stable and the settling time function T can beestimated by T ≤ T max : = η ( k − ) + η ( − k ) , or T ≤ T max : = η k ( k k − ) + η k ( − k k ) , where T max is independent on the initial condition x ( ) . B. Problem Setting
Consider a network of n mobile robots, and each of themhas the following dynamics:˙ p i ( t ) = v i ( t ) , p i ( t ) = [ x i ( t ) , y i ( t ) , z i ( t )] (cid:62) , (4)where i = , , · · · , n . The positions and velocities of theoverall system are represented by two stacked vectors p ( t ) = [ p ( t ) , . . . , p n ( t )] (cid:62) , v ( t ) = [ v ( t ) , . . . , v n ( t )] (cid:62) . (5)The robots are interconnected via a communication net-work, whose topology is an undirected graph G = ( V , E ) ,with V = { , , · · · , n } the set of nodes and E ⊆ V × V theset of edges. An edge ( i , j ) ∈ E if and only if there exists aninformation exchange between robots i and j , and the weightof ( i , j ) , denoted by a i j ∈ R ≥ , represents the communicationstrength. The Laplacian matrix L of G is thereby defined as [ L ] i j = − a i j if i (cid:54) = j , and [ L ] ii = ∑ nj = a i j otherwise.In this paper, we resort to the virtual leader approach[15] to achieve the time-varying formation of the robots. Inthis scheme, a virtual leader is specified as a time-varyingreference and the robots are designed to maintain desiredoffsets with respect to the position of the virtual leader p ( t ) .Define B = diag { b , . . . , b n } ∈ R n × n , where b i > i , and b i = H : = L + B .Assuming that G is strongly connected, and at least one robotcan acquire information from the leader, then we obtain that H is positive definite.Given a desired relative position p i between the robot i and the leader and d ∈ R > the radius of circular repulsivezone of each robot, the control objective in this paper is todesign a distributed fixed-time controller for each robot i such that both targets (cid:107) p i ( t ) − p ( t ) − p i (cid:107) = (cid:107) p i ( t ) − p oi ( t ) (cid:107) ≥ d are achieved for all t ≥ T i , where T i > p oi denotes theposition of the nearest obstacle of the robot i .III. F IXED -T IME B EHAVIORAL C ONTROL
Two types of behaviors are considered in this paper,namely, the collision avoidance behavior and cooperativeformation behavior, which yield two desired velocities. Bothvelocities guarantee fixed-time convergence, and then theyare prioritized and combined as a merged velocity thatguarantees the fixed-time convergence of the global task.
A. Collision Avoidance Behavior
The primary concern for a team of robots to perform tasksin an environment with obstacles is to avoid collision withthese obstacles as well with the other robots. This is referredto as collision avoidance behavior. We define ρ io : R → R > as a behavior function for every individual robot: ρ io : = (cid:107) p i − p oi (cid:107) , (6)with p oi : R > → R the position of the nearest obstacle(or teammate) of the robot i . Then the behavior-dependentacobian matrixes are given as J io = ∂ ρ io ∂ p i = ( p i − p oi ) (cid:62) , J oi = ∂ ρ io ∂ p oi = − ( p i − p oi ) (cid:62) , (7)with the right pseudoinverse J † io = ( p i − p oi ) / (cid:107) p i − p oi (cid:107) . Wedefine a circular repulsive zone around each obstacle/robot,with the coordinates of the object as the center and d ∈ R > as the radius. Then, we define the CAB task error as˜ ρ io : = ρ od − ρ io , with ρ od : = d , (8)from which the desired collision-avoidance behavioral veloc-ity ˙ x io : R ≥ → R is designed for the agent i as v io = J † io (cid:104) λ io ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] − J oi v oi (cid:105) , (9)where β , β , r , r , r ∈ R > are design parameters with r r > r r < r ∈ ( , ) , and v oi is velocity of the nearestobstacle/teammate of robot i .The following lemma shows that the desired velocity in(9) guarantees collision avoidance for each agent. Lemma 2:
Consider the collision avoidance behaviorfunction (6). Suppose that each agent i ∈ V is driven by thedesired velocity (9). If (cid:107) p i ( ) − p oi ( ) (cid:107) ≤ d , then for any˜ ρ io ( ) ∈ R , there exists a settling time T i , o > (cid:107) p i ( t ) − p oi ( t ) (cid:107) ≥ d , ∀ t ≥ T i , o . Proof:
Consider a Lyapunov function as follows V i , o = γ i , o ˜ ρ io , where γ i , o > V i , o yields˙ V i , o = − γ i , o ˜ ρ io ( J io v io + J oi v oi )= − γ i , o ˜ ρ io J io J † io [ λ io ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] − J oi v oi ] − γ i , o ˜ ρ io J oi v oi = − (cid:34) γ io V r r + r i , o + γ io V r r + r i , o (cid:35) r , with the two positive scalars γ io = ( γ i , o λ i , o ) r ( γ i , o ) r r + r β , γ io = ( γ i , o λ i , o ) r ( γ i , o ) r r + r β . From Lemma 1, we have ˜ ρ io converges to 0 in a fixed time T i , o ≤ ( γ io ) r ( r r − ) + ( γ io ) r ( − r r ) , irrespective of the initial conditon ˜ ρ io ( ) . B. Fixed-Time State Estimator and Cooperative Behavior
This section considers the cooperative formation behavior,where all the robots are moving towards a predefined forma-tion following to a virtual leader. More specifically, we aimto achieve, for each robot i , (cid:107) p i ( t ) − p ( t ) − p i (cid:107) = t ≥ T i , with T i a fixed settling time.To realize our scheme in a distributed manner, we firstdesign a fixed-time sliding mode estimator ˆ p i for each robot i to estimate the leader’s information from its neighbors.˙ˆ p i ( t ) = − K η i ( t ) − K η i ( t ) − K sgn ( η i ( t )) , (10) η i ( t ) = n ∑ j = a i j ( ¯ p i ( t ) − ¯ p j ( t )) [ r r ] , a i = b i , η i ( t ) = n ∑ j = a i j ( ¯ p i ( t ) − ¯ p j ( t )) [ r r ] , η i ( t ) = sgn ( n ∑ j = a i j ( ¯ p i ( t ) − ¯ p j ( t ))) , where K , K , K ∈ R > are gains to be designed,¯ p i ( t ) = ˆ p i ( t ) − p ( t ) , ¯ p j ( t ) = ˆ p j ( t ) − p ( t ) , ¯ p ( t ) =[ ¯ p (cid:62) ( t ) , . . . , ¯ p (cid:62) n ( t )] (cid:62) , r , r , r , r ∈ R > are design parameterswith r r > r r <
1. The virtual leader p , a i j , and b i aredefined in Section II-B. Assumption 1:
Suppose that the supremum of the leader’svelocity is bounded as K ≥ sup t ≥ (cid:107) ˙ p ( t ) (cid:107) ∞ , where K ∈ R > is the gain of estimator (10). Lemma 3:
If Assumption 1 is satisfied, then the estimator(10) converges to the leader’s state p ( t ) in a fixed time, i.e.,for any ( ˆ p i ( ) , p ( )) ∈ R × R , there exists T E > p i ( t ) ≡ p ( t ) , ∀ t ≥ T E . Proof:
Consider a Lyapunov candidate V E ( ¯ p ) : =
12 ¯ p (cid:62) ( H ⊗ I ) ¯ p , where ¯ p ( t ) = [ ¯ p (cid:62) ( t ) , . . . , ¯ p (cid:62) n ( t )] (cid:62) , ¯ p i ( t ) = ˆ p i ( t ) − p ( t ) . Thetime derivative of V E is computed as˙ V E = ¯ p (cid:62) ( H ⊗ I )([ ˙ˆ p (cid:62) ( t ) , . . . , ˙ˆ p (cid:62) n ( t )] (cid:62) − n ⊗ ˙ p )= ¯ p (cid:62) ( H ⊗ I )[ − K ( H ¯ p ) [ r r ] − K ( H ¯ p ) [ r r ] − K sgn ( H ¯ p ) − n ⊗ ˙ p ] ≤ − K ¯ p (cid:62) ( H ⊗ I )(( H ⊗ I ) ¯ p ) [ r r ] − K ¯ p (cid:62) ( H ⊗ I )(( H ⊗ I ) × ¯ p ) [ r r ] − ( K − sup t ≥ (cid:107) ˙ p ( t ) (cid:107) ∞ ) (cid:107) − K ¯ p (cid:62) ( H ⊗ I ) ¯ p (cid:107) ≤ − κ V ι E − κ V ι E , (11)where κ : = K ( λ ( H ) λ min ( H ) ) r r + r , κ : = K ( λ ( H ) λ min ( H ) ) r r + r , ι : = r + r r , ι : = r + r r , and λ min ( H ) and λ max ( H ) denote theminimum and maxmum eigenvalues of H , respectively. Itfollows from Lemma 1 that for all ( ˆ p i ( ) , p ( )) ∈ R × R ,there exists T E : = κ ( ι − ) + κ ( − ι ) such that ˆ p i ( t ) ≡ p ( t ) for any t ≥ T E .Using the distributed estimator in (10), we define thecooperative behavior function of the robot i as ρ i f : = (cid:107) p i − ˆ p i − p i (cid:107) . (12)with a desired relative position p i from the robot i to thevirtual leader. The task error is defined as˜ ρ f : = [ ˜ ρ f , . . . , ˜ ρ n f ] (cid:62) , with ˜ ρ i f : = − ρ i f (13)he Jacobian matrices related to (12) are defined as J ˆ f = blkdiag (cid:26) ∂ ρ f ∂ ˆ p , . . . , ∂ ρ n f ∂ ˆ p n (cid:27) ∈ R n × n , J = blkdiag (cid:26) ∂ ρ f ∂ p , . . . , ∂ ρ n f ∂ p n (cid:27) ∈ R n × n , J f = blkdiag (cid:26) ∂ ρ f ∂ p , . . . , ∂ ρ n f ∂ p n (cid:27) ∈ R n × n , with J † f = blkdiag (cid:26) ( p − ˆ p − p ) (cid:107) p − ˆ p − p (cid:107) , . . . , ( p n − ˆ p n − p n ) (cid:107) p n − ˆ p n − p n (cid:107) (cid:27) where ∂ρ if ∂ p i = ( p i − ˆ p i − p i ) (cid:62) , ∂ρ if ∂ ˆ p i = ∂ρ if ∂ p i = − ( p i − ˆ p i − p ) (cid:62) . Based on these, we design the desired cooperativebehavior velocity v f as follows: v f =[ v (cid:62) f , . . . , v (cid:62) n f ] (cid:62) , with (14) v i f : = J † i f [ λ f ( β ˜ ρ [ r ] i f + β ˜ ρ [ r ] i f ) [ r ] − J i ˆ f ˙ˆ p i − J i ˙ p i ] . The gain λ f ∈ R > , and β , β , r , r , r ∈ R > are designparameters satisfying r r > r r < r ∈ ( , ) . Notethat v f can be presented in a compact form as v f = J † f [ Λ f ( β ˜ ρ [ r ] f + β ˜ ρ [ r ] f ) [ r ] − J ˆ f ˙ˆ p − J v ∗ ] , (15)where Λ f = λ f I ∈ R n × n , ˙ˆ p = [ ˙ˆ p (cid:62) , . . . , ˙ˆ p (cid:62) n ] (cid:62) , and v ∗ : =[ ˙ p , . . . , ˙ p n ] (cid:62) . With the distributed estimator (10) and thedesired velocity in (15), we have the following lemma. Lemma 4:
Consider the cooperative behavior error ˜ ρ f in(13). If each robot i ∈ V is driven by the desired velocityin (14), then, for any initial condition ˜ ρ f ( ) , there exists asettling time T f > (cid:107) p i − ˆ p i − p i (cid:107) = ∀ t ≥ T f . Proof:
This lemma can be proved using the similarprocedure as the proof of Lemma 2. However, we use adifferent Lyapunov function V f : = γ f ˜ ρ (cid:62) f ˜ ρ f , where γ f > V f with respect to time then leads to˙ V f = − γ f ˜ ρ (cid:62) f ˙˜ ρ f = − γ f ˜ ρ (cid:62) f ( J f v f + J ˆ f ˙ˆ p + J v ∗ )= − γ f ˜ ρ (cid:62) f J f J † f [( β ˜ ρ [ r ] f + β ˜ ρ [ r ] f ) [ r ] − J ˆ f ˙ˆ p − J v ∗ ] − γ f ˜ ρ (cid:62) f J ˆ f ˙ˆ p − γ f ˜ ρ (cid:62) f J v ∗ = − γ f λ f ˜ ρ (cid:62) f ( β ˜ ρ [ r ] f + β ˜ ρ [ r ] f ) [ r ] . The rest of this proof follows similarly as the proof ofLemma 2, and details are omitted to conserve space.
C. Merged Desired Velocities
In this section, we design a desired velocity by mergingthe two behaviors in the above two sections. This merging istaken using the null-space-based behavioral approach, wherethe collision avoidance behavior is given a higher priority.Specifically, we determine the desired velocity as follows: v id = v io + ( I − J † io J io ) v i f , ∀ i ∈ V . (16)where v io and v i f are given in (9) and (14). With this mergedvelocity, we prove that each robot can achieve both taskssimultaneously within a fixed settling time. Fig. 1. The autonomous robots network.
Theorem 1:
Consider the collision avoidance behavior in(6) and the cooperative behavior in (12). If each robot i ∈ V is driven by the merged desired velocity in (16), then forany ( ˜ ρ io ( ) , ˜ ρ i f ( )) , there exists a settling time T i such that (cid:107) p i − p oi (cid:107) ≥ d , and (cid:107) p i − ˆ p i − p i (cid:107) = , ∀ t ≥ T i .Note that the proof of the theorem is not just a simplecombination of the conclusions of Lemma 2 and Lemma 4.When there is no conflict between the two tasks, we have J f J † io =
0, which means that two tasks in the velocity spaceare orthogonal and thus the fixed-time properties can beproved independently. However, if J f J † io (cid:54) =
0, i.e., the tasksare conflicting, then the proof becomes nontrivial. We presentthe detailed proof in the Appendix.
Remark 1:
To adjust the convergence time of trackingerrors, we can tune the parameters in (16) according to theformula of settling time T i . For example, the larger values of β and β , or the smaller values of γ i , o and γ f , will lead toa faster convergence speed.IV. S IMULATION R ESULTS
Consider a multi-agent systems connected by an undi-rected network as shown in Fig. 1, which contains 6 followersand a virtual leader in a 3-dimensional space. The graph G associated with the communication network is unweighted,i.e., a i j = a ji = i and j , and b i = i canobtain the information from the leader.The trajectory of leader is p = [ ( . t ) , ( . t ) , . t ] (cid:62) . The initial positions of the robots are p =[ , , ] (cid:62) m, p = [ , + √ , ] (cid:62) m, p = [ − , + √ , ] (cid:62) m, p = [ − , − , ] (cid:62) m, p = [ − , − −√ , ] (cid:62) m, p = [ , − −√ , ] (cid:62) m. The positions of environmental obstacles are O = [ , , ] (cid:62) m, O = [ , − , ] (cid:62) m, O = [ − . , , ] (cid:62) m, O = [ , − , ] (cid:62) m. The radius of its repulsive zone in(8) is d = p =[ √ + . ( . t ) , , ] (cid:62) m, p = [ . ( . t ) , , ] (cid:62) m, p = [ . ( . t ) − √ , , ] (cid:62) m, p = [ . ( . t ) − √ , − , ] (cid:62) m, p = [ . ( . t ) , − , ] (cid:62) m, p = [ √ + . ( . t ) , − , ] (cid:62) m. The design parameters are selectedas β = β = . r = . r = . r = . K = . K = . K = r = r = r = r = ∼ ∼ ∼ ∼ ime [s] [ m ] k p − p − p kk p − p − p kk p − p − p kk p − p − p kk p − p − p kk p − p − p k Fig. 2. The cooperative tracking behavior ofrobots.
Time [s] [ m ] k p − p o kk p − p o kk p − p o kk p − p o kk p − p o kk p − p o k Fig. 3. The distances between robots andobstacles. x [m] y [m] z [ m ] leaderrobot 1robot 2robot 3robot 4robot 5robot 6leaderrobot 1robot 2robot 3robot 4robot 5robot 6obstacle 1obstacle 2obstacle 3obstacle 4 Fig. 4. Trajectories of the six robots in the environ-ment with four obstacles. and force the robots to move away from obstacles, resultingin a deviation from their desired trajectories for formationtask. Fig. 4 then shows the trajectories of the six robots inthe formation. The simulation result shows that the proposedalgorithm is effective in time-varying formation problem ofa team of robots in an environment with obstacles.V. C
ONCLUSIONS
This paper has proposed a novel fixed-time behavioralcontrol method, which can be applied to distributed time-varying formation control of networked robots in an environ-ment with obstacles. Using the null-space-based projection,the collision avoidance task and cooperative formation taskare combined, leading to a desired driving velocity for eachrobot to achieve a time-varying formation in a fixed-timeconvergence while avoiding collisions and obstacles. Thesimulation result has shown the effectiveness of this method.The future work will further discuss the adverse effects fromconstraints of input.A
PPENDIX : P
ROOF OF T HEOREM Proof.
There are three cases to be discussed.
Case A : If | ˜ ρ io ( t ) | > | ˜ ρ i f ( t ) | =
0, then accordingto Lemma 2, there exists a settling time T i , o > (cid:107) p i − p oi (cid:107) > d and (cid:107) p i − ˆ p i − p i (cid:107) = ∀ t ≥ T i , o . Case B : If | ˜ ρ io ( t ) | = | ˜ ρ i f ( t ) | >
0, then accordingto Lemma 2, there exists a settling time T i , f > (cid:107) p i − p oi (cid:107) > d and (cid:107) p i − ˆ p i − p i (cid:107) = ∀ t ≥ T i , f . Case C : If | ˜ ρ io ( t ) | > | ˜ ρ i f ( t ) | >
0, then ˜ ρ io ( t ) < ρ i f ( t ) <
0. The rest of the proof is focused on the analysisof this case. For each robot i ∈ V , consider the two differentsubtask errors and define the Lyapunov function as follows: V i , M ( ˜ ρ io , ˜ ρ i f ) : = γ i , o ˜ ρ io + γ f ˜ ρ i f , (17)where γ i , o , γ f ∈ R > satisfies γ i , o ≥ γ f J i f ( ) J † io ( ) + γ i , ε with γ i , ε ∈ R > . Moreover, we define L such that0 < max i ∈ V {| ˜ ρ io ( ) | , | ˜ ρ i f ( ) |} ≤ L , (18)where ˜ ρ io ( ) , ˜ ρ i f ( ) are the initial values of ˜ ρ io ( t ) and ˜ ρ i f ( t ) ,respectively. Taking the time derivative of V i , M along the desired velocity (16) yields ˙ V i , M = − γ i , o ˜ ρ io J io [ v io + ( I − J † io J io ) v i f ] − γ i , o ˜ ρ io J oi v oi − γ f ˜ ρ i f J i f [ v io + ( I − J † io J io ) v i f ] − γ f ˜ ρ i f ( J i ˆ f ˙ˆ p i + J i ˙ p i )= − γ i , o ˜ ρ io J io J † io λ io ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] − γ f ˜ ρ i f J i f J † io [ λ io ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] − J oi v oi ]+ γ f ˜ ρ i f ˜ J [ λ f ( β ˜ ρ [ r ] i f + β ˜ ρ [ r ] i f ) [ r ] − J i ˆ f ˙ˆ p i − J i ˙ p i ] − γ f ˜ ρ i f ( J i ˆ f ˙ˆ p i + J i ˙ p i ) , (19) where ˜ J : = J i f ( I − J † io J io ) J † i f , and J io ( I − J † io J io ) =
0. Wefurther scale (19) as ˙ V i , M ≤ − γ i , o λ io ˜ ρ io ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] + γ f λ io | J i f J † io || ˜ ρ i f || ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] |− γ f λ f ˜ ρ i f ˜ J ( β ˜ ρ [ r ] i f + β ˜ ρ [ r ] i f ) [ r ] + γ f | ˜ ρ i f |(cid:107) J i f (cid:107)(cid:107) v oi (cid:107) + γ f | ˜ ρ i f |(cid:107) J i ˆ f (cid:107)(cid:107) ˙ˆ p i (cid:107) + γ f | ˜ ρ i f |(cid:107) J i (cid:107)(cid:107) ˙ p i (cid:107) , (20) From this point, the proof goes into the following twodirections. (a) | ˜ ρ io | ≥ | ˜ ρ i f | > | ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] | ≥ | ( β ˜ ρ [ r ] i f + β ˜ ρ [ r ] i f ) [ r ] | >
0. First, we prove that ˜ ρ io ( t ) is bounded ifinitial value ˜ ρ i f ( ) is bounded. From the relations γ i , o ≥ γ f J i f ( ) J † io ( ) + γ i , ε and | ˜ ρ io ( t ) || ˜ ρ if ( t ) | ≥
1, we obtain from (20) that ˙ V i , M ( ) ≤ − γ i , o λ io ˜ ρ io ( )( β ˜ ρ [ r ] io ( ) + β ˜ ρ [ r ] io ( )) [ r ] + γ f λ io | J i f ( ) J † io ( ) || ˜ ρ i f ( ) || ( β ˜ ρ [ r ] io ( ) + β ˜ ρ [ r ] io ( )) [ r ] |− γ f λ f ˜ ρ i f ( ) J i f ( )( I − J † io ( ) J io ( )) J † i f ( ) × ( β ˜ ρ [ r ] i f ( ) + β ˜ ρ [ r ] i f ( )) [ r ] + γ f | ˜ ρ i f ( ) |(cid:107) J i f ( ) (cid:107)(cid:107) v oi ( ) (cid:107) + γ f | ˜ ρ i f ( ) |(cid:107) J i ˆ f ( ) (cid:107)(cid:107) ˙ˆ p i ( ) (cid:107) + γ f | ˜ ρ i f ( ) |(cid:107) J i ( ) (cid:107)(cid:107) ˙ p i ( ) (cid:107)≤ − γ i , o λ iv | ˜ ρ io ( ) || ( β ˜ ρ [ r ] io ( ) + β ˜ ρ [ r ] io ( )) [ r ] | ≤ , (21) where λ i , o is state-dependent and designed to satisfy λ i , o ≥ γ f (cid:107) J i f ( ) (cid:107) sup t ≥ ( (cid:107) v oi (cid:107) + (cid:107) ˙ˆ p i (cid:107) + (cid:107) ˙ p i (cid:107) ) γ i , ε | ( β ˜ ρ [ r ] i f ( ) + β ˜ ρ [ r ] i f ( )) [ r ] | + γ i , o λ iv γ i , ε , (22) with λ iv > V i , M ( ) ≤
0. It follows from (18)and (21) that | ˜ ρ io ( ∆ t ) | ≤ | ˜ ρ io ( ) | ≤ L , for any ∆ t >
0. Wecan further show that ˙ V i , M ( ∆ t ) ≤ V i , M ( t ) ≤ t , which implies that | ˜ ρ i f ( t ) | ≤ | ˜ ρ i f ( ) | ≤ L .Next, with the relations −| ˜ ρ io | ≤ −| ˜ ρ i f | and −| ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] | ≤ −| ( β ˜ ρ [ r ] i f + β ˜ ρ [ r ] i f ) [ r ] | , we rewrite (21) as ˙ V i , M ≤ − γ i , o λ iv | ˜ ρ io || ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] |− γ i , o λ iv | ˜ ρ i f || ( β ˜ ρ [ r ] i f + β ˜ ρ [ r ] i f ) [ r ] | , ≤ − η iM V r r + i , M − η iM V r r + i , M , (23) where η iM : = min (cid:40) γ i , o λ iv β − r r (cid:18) γ i , o (cid:19) r r + , γ i , o λ iv β − r r (cid:18) γ f (cid:19) r r + (cid:41) , η iM : = min (cid:40) γ i , o λ iv β (cid:18) γ i , o (cid:19) r r + , γ i , o λ iv β (cid:18) γ f (cid:19) r r + (cid:41) . It then obtain from Lemma 1 that for any initial values ( ˜ ρ io ( ) , ˜ ρ i f ( )) , there exists a settling time T i , = η iM ( r r − ) + η iM ( − r r ) (24)such that (cid:107) p i − p oi (cid:107) ≥ (cid:107) p i − ˆ p i − p i (cid:107) = ∀ t ≥ T i , . (b) < | ˜ ρ io | < | ˜ ρ i f | ≤ L Now, we have 0 < | ( β ˜ ρ [ r ] io + β ˜ ρ [ r ] io ) [ r ] | < | ( β ˜ ρ [ r ] i f + β ˜ ρ [ r ] i f ) [ r ] | ≤ L ∗ , where L ∗ : = ( β L r + β L r ) r . We firstshow the boundedness of ˜ ρ i f ( t ) for a bounded initial value˜ ρ i f ( ) . It follows from γ i , o ≥ γ f J i f ( ) J † io ( ) sup t ≥ | ˜ ρ if ( t ) || ˜ ρ io ( t ) | + γ i , ε and | ˜ ρ if ( t ) || ˜ ρ io ( t ) | ≥ ˙ V i , M ( ) ≤ − γ i , o λ io ˜ ρ io ( )( β ˜ ρ [ r ] io ( ) + β ˜ ρ [ r ] io ( )) [ r ] + γ f λ io | J i f ( ) J † io ( ) || ˜ ρ i f ( ) || ( β ˜ ρ [ r ] io ( ) + β ˜ ρ [ r ] io ( )) [ r ] |− γ f λ f ˜ ρ i f ( ) J i f ( )( I − J † io ( ) J io ( )) J † i f ( ) × ( β ˜ ρ [ r ] i f ( ) + β ˜ ρ [ r ] i f ( )) [ r ] + γ f | ˜ ρ i f ( ) |(cid:107) J i f ( ) (cid:107)(cid:107) v oi ( ) (cid:107) + γ f | ˜ ρ i f ( ) |(cid:107) J i ˆ f ( ) (cid:107)(cid:107) ˙ˆ p i ( ) (cid:107) + γ f | ˜ ρ i f ( ) |(cid:107) J i ( ) (cid:107)(cid:107) ˙ p i ( ) (cid:107)≤ − γ i , ε λ i , o | ˜ ρ io ( ) || ( β ˜ ρ [ r ] io ( ) + β ˜ ρ [ r ] io ( )) [ r ] |− γ f λ iv | ˜ ρ i f ( ) || ( β ˜ ρ [ r ] i f ( ) + β ˜ ρ [ r ] i f ( )) [ r ] | ≤ , (25) where λ f is state-dependent and designed to satisfy λ f ≥ γ i , f λ iv + (26) γ f (cid:107) J i f ( ) (cid:107) sup t ≥ ( (cid:107) v oi (cid:107) + (cid:107) ˙ˆ p i (cid:107) + (cid:107) ˙ p i (cid:107) ) J i f ( )( I − J † io ( ) J io ( )) J † i f ( ) | ( β ˜ ρ [ r ] i f ( ) + β ˜ ρ [ r ] i f ( )) [ r ] | . With (26), ˙ V i , M ( t ) ≤ t , and thus | ˜ ρ i f ( t ) | ≤| ˜ ρ i f ( ) | ≤ L is guaranteed. Furthermore, according to (25),we can be rewrite (20) as ˙ V i , M ≤ − γ i , ε λ i , o | ˜ ρ io ( ) || ( β ˜ ρ [ r ] io ( ) + β ˜ ρ [ r ] io ( )) [ r ] |− γ f λ iv | ˜ ρ i f ( ) || ( β ˜ ρ [ r ] i f ( ) + β ˜ ρ [ r ] i f ( )) [ r ] | , ≤ − η iM V r r + i , M − η iM V r r + i , M , (27) where η iM : = min (cid:40) γ i , ε λ iv β − r r (cid:18) γ i , o (cid:19) r r + , γ f λ iv β − r r (cid:18) γ f (cid:19) r r + (cid:41) , η iM : = min (cid:40) γ i , ε λ iv β (cid:18) γ i , o (cid:19) r r + , γ f λ iv β (cid:18) γ f (cid:19) r r + (cid:41) . Then from Lemma 1, we obtain (cid:107) p i − p oi (cid:107) ≥ d and (cid:107) p i − ˆ p i − p i (cid:107) = t ≥ T i , : = η iM ( r r − ) + η iM ( − r r ) . (28)From the above discussions, we conclude that if themerged driving velocity in (16) is applied to each robot i ∈ V , then for any initial values ( ˜ ρ io ( ) , ˜ ρ i f ( )) , there existsa settling time T i : = max { T i , o , T i , f , T i , , T i , } with T i , o , T i , f defined in Lemma 2 and Lemma 4, respectively, such that (cid:107) p i − p oi (cid:107) ≥ d and (cid:107) p i − ˆ p i − p i (cid:107) = ∀ t ≥ T i .R EFERENCES[1] Y. Zou, X. Su, S. Li, Y. Niu, and D. Li, “Event-triggered distributedpredictive control for asynchronous coordination of multi-agent sys-tems,”
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