Distributed Consensus-Based Robust Adaptive Formation Control for Nonholonomic Mobile Robots with Partial Known Dynamics

This paper investigates the distributed consensus-based robust adaptive formation control for nonholonomic mobile robots with partially known dynamics. Firstly, multirobot formation control problem has been converted into a state consensus problem. Secondly, the practical control strategies, which incorporate the distributed kinematic controllers and the robust adaptive torque controllers, are designed for solving the formation control problem. Thirdly, the specified reference trajectory for the geometric centroid of the formation is assumed as the trajectory of a virtual leader, whose information is available to only a subset of the followers. Finally, numerical results are provided to illustrate the effectiveness of the proposed control approaches.


Introduction
In the past decades, cooperative control of multiple mobile robots has been receiving significant attention owing to many potential advantages of such systems over single robot.In fact, multirobot cooperative control means a group of mobile robots working cooperatively that can achieve great benefits including low cost, greater flexibility, adaptability to unknown environments, and robustness [1][2][3][4].In the field of cooperative control, formation control has received a lot of attention from the researchers for its potential applications such as surveillance-and-security, object transportation, object manipulation, search-and-rescue, intelligent transportation systems, and exploration.The formation control means the problem of controlling the relative position and orientation of mobile robots in a group according to some desired pattern for executing a given task.
Various control approaches have been proposed in the literature for mobile robot formations, including leaderfollower approach [5][6][7][8][9][10], behavior-based approach [11][12][13], virtual-structure approach [14][15][16][17][18], artificial potential approach [19][20][21][22], and graph theory [23,24].The main idea behind these approaches is to find suitable velocity control inputs to stabilize the closed-loop system.In the literature, formation control for multiple nonholonomic mobile robots, just simply consider the kinematic model by ignoring the robot dynamics.To design the control inputs to guarantee the stability of the closed-loop system, it is assumed that there is "perfect velocity tracking." Reference [25] proposed an errorbased tracking model and designed a stable kinematic tracking controller for the nonholonomic mobile robot.Reference [26] presents a kinematic controller based on the recedinghorizon leader-follower (RH-LF) control framework to solve the formation problem of multiple nonholonomic mobile robots.Reference [27] studied the tracking control problem for nonholonomic mobile robots with limited information of a desired of trajectory.Reference [28] proposed a kinematic controller for the distributed consensus-based formation control.However, the perfect velocity tracking assumption does not hold in practice, and the dynamics of robot should not be ignored and practical control strategies accounting for both the kinematic and dynamic affect should be implemented [29][30][31].In [32], the decentralized cooperative robust controllers are proposed for the formation control of a group of wheeled mobile robots with dynamics.In [31], an adaptive tracking controller for the dynamic model with unknown parameters was designed for a nonholonomic mobile robot by using an adaptive backstepping approach.Though these works consider the dynamics of the mobile robot, the dynamics of the mobile robot do not have the friction and bounded disturbance.It is well known that friction plays a central, controlling role in a rich variety of physical systems.Therefore, the friction term and bounded disturbance term should not be ignored and practical control strategies accounting for the friction term and bounded disturbance term should be implemented in practice.
Motivated by the above discussions, this paper investigates the distributed consensus-based robust adaptive formation control for nonholonomic mobile robots with partial known dynamics.The contribution of this paper is given as follows.Firstly, a variable transformation is given to convert the formation control problem into a state consensus problem.Then, the distributed consensus-based kinematic controllers are developed to make a group of robots asymptotically converge to a desired geometric pattern.In this paper, the specified reference trajectory for the geometric centroid of the formation is assumed as the trajectory of a virtual leader whose information is available to only a subset of the followers.Also the followers are assumed to have only local interaction with their neighbors.It is well known in practice that the perfect knowledge of dynamic model of the wheeled mobile robot is unattainable, and it is almost impossible to obtain exact values of the parameters of the mobile robot.Therefore, this paper considers that the dynamics of the mobile robot is partial known, in which there exist some unknown factors that will affect the robust trajectory tracking of the system.Then the corresponding robust adaptive torque controllers for mobile robots are developed for guaranteeing the robust velocity tracking, and the corresponding sufficient conditions are obtained for a group of nonholonomic mobile robots asymptotically converge to a desired geometric pattern with its centroid moving along the specified reference trajectory.The rigorous proofs are given by using graph theory, matrix theory, and Lyapunov theory.Finally, simulation examples illustrate the effectiveness of the proposed controllers.Compared with existing works in the literature, the current paper has the following advantages.Firstly, the relative distance and angular for each robot with its leader are not required to be known that is different from the traditional leader-follower approach [5,6,8,9,33].Secondly, in contrast to that only kinematic control models considered in [26,27,[34][35][36], the controllers designed in this paper are based on both the kinematic and dynamic models of robots.Moreover, the dynamics of wheeled mobile robots with possible uncertainty are considered.Thirdly, in contrast to that complete knowledge of the dynamics needed in [32,37], only partial knowledge of the dynamics is needed.Fourthly, the control laws proposed in this paper are distributed.It is not necessary to know the global information for each robot.In fact, each robot can obtain information only from its neighbors.The remainder of this paper is organized as follows.Section 2 introduces some preliminaries and gives the problem formulation.Section 3 and Section 4 present some new results on distributed formation control problem for multiple nonholonomic mobile robots.Simulations results are provided to verify the theoretical analysis in Section 5. Section 6 concludes this article.

Background
In this section, the model of nonholonomic wheeled mobile robot is first briefly presented.Then some notations for graph theory and nonsmooth analysis are introduced.Finally, the problem description is given.

Dynamics of Nonholonomic Wheeled Mobile Robot.
Consider a multirobot system consisting of  nonholonomic wheeled mobile robots indexed by 1, 2, . . ., .The nonholonomic mobile robot is shown in Figure 1.The kinematic model and dynamic model of the mobile robot  can be described as follows [38]: where ].   (  ) ∈ R 3×3 is a symmetric positive definite inertia matrix,   (  , q  ) ∈ R 3×3 is the bounded centripetal and coriolis matrix,   ( q  ) ∈ R 3×1 denotes surface friction,   (  ) ∈ R 3×1 is the gravitational vector, and   denotes bounded unknown disturbances including unstructured unmodeled dynamics.  (  ) ∈ R 3×2 is the input transformation matrix,   ∈ R 2×1 is the control torque vector,   () ∈ R 1×3 is the matrix associated with the constraints, and   ∈ R 1×1 is the vector of constraint forces.
The dynamic model (2) has the following properties [38].
Property 1.The inertia matrix   (  ) is symmetric positive definite and satisfies the following inequality: where  1 ,  2 are positive constants, and ‖ ⋅ ‖ is the standard Euclidean norm.
Property 2. Ṁ (  ) − 2  (  , q  ) is skew symmetric; that is to say, A path in the undirected graph G is a sequence of edges in the form (V  1 , V  2 ), (V  2 , V  3 ), . .., where V   ∈ V. We call an undirected graph G connected if for any different nodes V  and V  in V there exists an undirected path.The Laplacian matrix  = (  ) × associated with A for graph G is defined as   = −  for  ̸ = , and   = ∑  =1, ̸ =    , ,  ∈ {1, . . ., }.For an undirected graph,  is symmetric positive semidefinite.
Lemma 1 (Chung [39]).Assume that G is a weighted undirected graph with Laplacian matrix ; then G is connected if and only if the matrix  has an eigenvalue zero with multiplicity 1 and corresponding eigenvector 1, and all other eigenvalues are positive.

Nonsmooth Analysis.
In what follows, some elements from nonsmooth analysis will be presented.Consider a vector differential equation with a discontinuous right-hand side as where (, ) is measurable and essentially locally bounded.The vector function (⋅) is called a Filippov solution [40] of (6) if (⋅) is absolutely continuous and satisfies almost everywhere where where Ω V denotes the set of measure zero that contains the set of points where  is not differentiable and co denotes the convex closure.
Lyapunov theorems have been extended to nonsmooth systems in [41].The following chain rule provides a calculus for the time derivative of the energy function in the nonsmooth case.
Definition 3 (see [42]).Let () be a locally Lipschitz continuous function.The generalized gradient of () is given by (11) where co denotes the convex hull, Ω V is the set of Lebesgue measure zero, where ∇ does not exist, and  is an arbitrary set of zero measure.
In this paper, the candidate Lyapunov function  we use is smooth and hence regular, while its generalized gradient is a singleton which is equal to its usual gradient everywhere in the state space: () = {∇()}.
Definition 4 (see [40]).Consider the vector differential equation ( 6), a set-valued map K : R  → B(R), the set-valued Lie derivative of  with respect to ( 6) is defined as In what follows, we introduce a Lyapunov stability theorem in terms of the set-valued map V.
Lemma 5 (see [41]).For (6), let (, ) be locally essentially bounded and 0 where  > 0. Also, let  : R  → R be a regular function satisfying Mathematical Problems in Engineering in  for some  1 and  2 belonging to class K.If there exists a class K function (⋅) in  such that the set-valued Lie derivative of () satisfies max V (, ) ≤ − () < 0, for  ̸ = 0, (14) then the solution  ≡ 0 is asymptotically stable.
2.4.Problem Formulation.In this paper, the desired geometric pattern F of  mobile robots is described by the orthogonal coordinates (  ,   ) as follows: where ( 0 ,  0 ) denotes the center of F. Without loss of generality, assume that  0 = 0,  0 = 0.The objective of this paper is to design the control inputs V  and   for the wheeled nonholonomic mobile robot  using its states (  , q  ) and (  ,   ) as well as its neighbors' states (  , q  ) and (  ,   ) for  ∈ N  , such that (a) the group of mobile robots converges to the desired formation F; (b) each robot in group converges to the desired orientation  0 ; (c) the geometric centroid of the formation converges to the desired reference trajectory ( 0 ,  0 ); lim lim where ( 0 ,  0 ,  0 ) can be considered as the posture of a virtual leader 0, which does not have to be an actual robot but is specified by Hereafter, the  robots in system (1) are called followers.The connection weight between robot  and the virtual leader 0 is described by  = diag{ 1 ,  2 , . . .,   }, in which   > 0 if robot  can obtain information from the virtual leader 0,   = 0 otherwise.Note that if the undirected graph G is connected, it then follows that the matrix  +  =  + diag{ 1 , . . .,   } and the matrix M = diag{ + ,  + } are symmetric positive definite.
In this paper, the following assumptions are needed for achieving our control objective.Assumption 6.The   for (0 ≤  ≤ ) is bounded,   for (0 ≤  ≤ ) is persistently exciting, and |  | ≤  max .Remark 7.   is persistently exciting, which means that   does not converge to 0. The assumption is because of the fact that the wheeled mobile robot system is nonholonomic.Assumption 8.There exists at least one follower which can obtain information from the virtual leader.Remark 9. Note from Assumption 8 that all follower robots do not need to obtain the information from the virtual leader; that is to say, the desired reference trajectory is not required to available for each robot, which is different from the existing works in [35,43].
The following notations will be used throughout this paper.Let   denote the  ×  identity matrix, 0 × denote the  ×  zero matrix, and 1  = [1, 1, . . ., 1]  ∈   (1 for short, when there is no confusion). min (M) and  max (M) are the smallest and the largest eigenvalues of the matrix M, respectively.

Distributed Control Algorithm
To achieve the control objective ( 16)-( 18), the following transformation is defined to convert the formation control problem for multiple nonholonomic mobile robots into a state consensus problem: where  1 and  2 are control inputs, 0 ≤  ≤ ,  0 > 0, and sign(⋅) is the signum function.The definitions in (20) yield the following dynamic system as Then the control objective is changed to design  1 and  2 such that the following equations are satisfied: lim lim lim Lemma 10.If ( 22)-( 25) hold for 0 ≤  ≤ , then the  mobile robots can converge to the formation pattern F; that is, (16)-( 18) can be satisfied.
Proof.Due to the fact that it is similar to the proof of Lemma 3.1 in [28], it is therefore omitted.
In practice, it is well known that the dynamics model of the wheel mobile robot may have unknown dynamical parameters and bounded unknown disturbances, which will affect the robust trajectory tracking of the system; that is to say, the "perfect velocity tracking" for robot may not hold.Hence, the following desired control inputs for the mobile robot  are proposed in this paper as where  = 1, . . ., ,   is a positive constant if the virtual leader's position is available to the follower , and   = 0 otherwise, | ż 20 | ≤ ,  is a positive constant,  is a nonnegative constant, and  is a positive constant and satisfies  > .

Robot Model and Its
Properties.According to (20) and the definition of V  in Section 2.2, it is easy to obtain that where  = (1 +  2 0 ) −  0 sign( 1 ) 2 .Then, it follows from (1) that we have where Hence, the dynamics (2) of the mobile robot can be rewritten as follows: that is, where   = Ŝ   Ŝ is a symmetric positive definite inertia matrix.  = Ŝ (  Ṡ +   Ŝ) is the centripetal and coriolis matrix,   = Ŝ   is the gravitation vector,   = 0.   = Ŝ   is the surface friction,   = Ŝ   denotes the bounded unknown disturbances including unstructured unmodeled dynamics, and   = Ŝ   is the input vector.Similar to the Properties 1 and 2 in Section 2.2, (39) has the following properties.Proof.It is easy to verify the result, and it is therefore omitted here.
Proof.The derivative of the inertia matrix and the centripetal and coriolis matrix are given by Since Ṁ − 2  is skew symmetric and   is symmetric positive definite, it follows that Hence, the matrix Ṁ − 2  is skew symmetric.

Controller Design.
Taking the derivative of ( 28) and multiplying by the inertia matrix   to both sides of (28) give where   (  , u  ) =   u  +     is composed of known quantities and the disturbance term is with Δ  representing any model uncertainties and unmodeled dynamics and   being the unknown bounded disturbance which could represent any inaccurately modeled dynamics.
with  0 ,  1 ,  2 depending on the terms like the disturbance bound, the changes in the mass of the robot due to payload, and friction coefficients with   being a known regression vector.
When the robot dynamics are partially known, the torque control algorithm for the dynamics system ( 42) is designed to be where   is a symmetric positive-definite matrix defined by   =    2 with   being a positive gain constant and  2 ∈ R 2×2 being the identity matrix.The nonlinear term   is an adaptive robustifying term and is defined as [44]   = ũ (  Θ ) where   and   are positive design constants,   Θ is the adaptive estimate of the known function   Θ  , Θ is the estimate of Θ  , and the parameter turning law for the estimate Θ is defined as with Γ  being a symmetric and positive definite matrix.Let Θ be the estimation error of the parameter turning law, and Θ = Θ  − Θ .It then follows that Θ = − Θ .Substituting (45) into (42) and writing it in a vector form give where (), (, q ), and  are the block diagonal matrices of   (  ),   (  , q  ), and   , respectively,   = [ 1 , . . .,   ]  , () = [ 1 , . . .,   ]  with   = [ 1 , . . .,   ]  and Δ = [Δ 1 , . . ., Δ  ]  under (43).

Proof. Choose the Lyapunov candidate as
where  1 and  2 are chosen as with Θ = [ Θ1 , . . ., Θ ]  and Γ being the block diagonal matrices of Γ  .Using the properties of K[⋅], the set-valued Lie derivative of  can be obtained as follows: Since  2 is continuous, it follows from Lemma 2 that the equality (51) holds.
According to Definition 4, the set-valued Lie derivative of  1 is given as where ( Z) is the generalized gradient of  at Z.Because  is continuously differentiable with respect to Z, ( Z) = { Z}, which is a singleton.Therefore, it follows that where the fact that   sign() = ‖‖ 1 has been used.By Lemma 2 and [42], if  is continuous, then K[] = {}.Note that the set-valued Lie derivative V is a singleton, whose only element is actually V. Therefore, it follows that max where  ≥  and  is positive.It is easy to verify that M 2 is symmetric positive definite.Since  2 is continuous, it follows that the set-valued Lie derivative of  2 satisfies max V2 = V 2 .Hence, we have Since the matrix ( Ṁ − 2  ) is skew symmetric, we have Let Hence, it can be obtained that Substituting ( 58) into (55) gives the following inequality: Now, substituting (54) and ( 59) into the set-valued Lie derivative V reveals max Therefore, max V ≤ 0 as ż 20 ≤ ,  > 1/2 max (M),  >  and  max >  max (M)/2.It then follows from Lemma 5 that ũ → 0 and Z → 0 as  → ∞; that is, z1 → 0, z2 → 0, ũ → 0, and ũV → 0 as  → ∞.Therefore, the errors z1 = 0, z2 = 0, ũ = 0, and ũV = 0 are globally asymptotically stable.This proof is completed.61) is given as follows: According to Theorem 12, z2 asymptotically converges to zero, and  1 asymptotically converges to  10 .It then follows from the definition of  2 () that  2 () also asymptotically converges to zero.Hence, according to the definition of asymptotic stable, for an arbitrary positive value  > 0,  > 0 exists; when the | 2 (0)| < , it has | 2 ()| < .
From Assumption 6, the  1 is bounded, and The solution of the differential equation (62) satisfies the inequality Hence, when  → +∞, |z 3 ()| ≤ .Since  is an arbitrary positive value, from the definition of asymptotic stable, the z3 () is asymptotic stable at the neighborhood of origin.This proof is completed.Remark 15.From Theorems 12 and 14, our control objectives ( 22)-( 25) hold under the distributed kinematic controller (26) and the torque controller (45).Therefore, from Lemma 10, the  mobile robots converge to the formation pattern F; that is, ( 16)-( 18) are satisfied.

Verification of Formation Control Based on Robust Adaptive Techniques.
In this simulation, Figure 3 shows the trajectory of virtual leader (black line), the trajectory of the six followers' centroid (blue line), and the formation positions and pattern of the six followers at several moments.We could see from Figure 3 that the six robots converge to the desired geometry pattern under the proposed controllers ( 27), (26), and (45); that is to say, (16) has been verified.shows the position error between  0 and the centroid of   .We could see, from Figures 4 and 5, the trajectory of the formation geometric centroid converges to the trajectory of virtual leader; that is to say, (18) has been verified.Figure 6 shows the tracking error ũ for (1 ≤  ≤ 6) and ũV for (1 ≤  ≤ 6) under the torque controller (45).From Figure 6, ũ and ũV , respectively, converge to zeros.The perfect tracking of velocity and angular velocity has been guaranteed.Figure 7, respectively, shows the angular velocity tracking errors   − 0 and the orientation tracking errors   −  0 between follower   (1 ≤  ≤ 6) and virtual leader.It can be seen from Figure 7 that   −  0 and   −  0 converge to zero over time; that is, (17) and ( 25) have been verified.

Conclusion
In this paper, the distributed consensus-based robust adaptive formation control problem for nonholonomic mobile robots with partial known dynamics has been investigated, in which the dynamics model of the wheeled mobile robot has the friction term and bounded disturbance term in the dynamic model.The partial knowledge of the mobile robot dynamics has been assumed to be available.Then an asymptotically stable torque controller has been proposed by using robust adaptive control techniques to account for unmolded dynamics and bounded disturbances.
As further extensions of this study, there still exist a number of topics for future works.In practice, formations have to avoid obstacles and need to admit changes in the formation speed and strong deformations in formation shape.The obstacles avoidance problem for formation control of nonholonomic mobile robots will be considered.In addition, the formation control is assumed to be noiseless in this paper.However, it is inevitable in reality.Hence, in the future it is necessary to investigate the formation control problem with measurement noise.Finally, it is well known that most operations in mobile robots systems are naturally delayed.Moreover, it has been observed from numerical experiments that formation control algorithms without considering time delays may lead to unexpected instability.Hence, in future, we may consider the formation control with time-varying delays.

Figure 1 :
Figure 1: Model of a differential wheeled mobile robot.

Property 3 .
The inertia matrix   (  ) is symmetric positive definite.