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Abstract

In this article, an adaptive coordinated tracking control problem for a group of nonholonomic chained systems has been discussed under the assumption that the desired trajectory is available only to a part of neighboring agents. Firstly, coordinated tracking controllers under the directed communication topology graph containing a spanning tree are designed for two subsystems, adaptive control gains are employed in the linear subsystem based on the state information of neighboring agents, then the global information of the topology graph is not required to be known in the control laws. Furthermore, the backstepping strategy is applied in the rest of chained-form subsystem such that the state of all robots converge to the desired reference trajectory. And then, the results are further evolved with the case of switching topology. Finally, an application is introduced and the simulation results are given to show the validity of the proposed theoretical results.

Keywords

Multi-robot system distributed control adaptive control coordinated tracking

Introduction

With the rapid development of distributed computing technology and modern control theory, distributed cooperative control of multi-agent systems has caused researchers’ tremendous attentions in the last few years. This research direction includes consensus, formation, rendezvous, containment, and tracking control.¹ These problems have been discussed in pioneering works via different control strategies, such as leader–follower,² virtual structure,³ behavior-based,⁴ and graph theory methods.⁵ Obviously, the topic of coordinated tracking has a wide range of applications in engineering and physics, such as tracking control of mobile robots, satellites clustering, unmanned helicopter, and autonomous underwater vehicles.^6
–8

The main objective of coordinated tracking control is to make a group of autonomous vehicles to track a target in a cooperative manner via distributed control protocols such that some challenging tasks can be completed and many inevitable physical constraints including sensor ranging are removed. Hence, this distributed control behavior not only reduces the operational costs but also improves the robustness and adaptivity of the systems. In most of the previous technical notes concerned with decentralized control, the coordinated control problem of multi-agent systems which are usually considered as linear dynamic systems has been studied by many researchers.⁹ In the study by Qu et al.,¹⁰ by the use of matrix theory, a sufficient and necessary condition is given to guarantee the convergence of the closed-loop system. Two different distributed tracking controllers for linear dynamic systems in the study by Li et al.¹¹ are proposed to guarantee that the states reach an agreement. Ni and Cheng¹² considered leader-following consensus problem of high-order linear systems, the Lyapunov inequality and Riccati inequality are introduced to solve the consensus problem.

In practical applications, the practical physical model cannot be described by the linear model, and the most practical model may be nonlinear model. Moreover, most of the mobile models have to satisfy nonholonomic constraints. The design of control laws for nonholonomic systems is involved, mainly due to Brockett’s condition which indicates that continuous pure state feedback law is not applied. Therefore, when agents with nonholonomic constraint are considered, the coordinated tracking control problem becomes more challenging. Some proposed effective control strategies, such as time varying,¹³ discontinuous feedback,¹⁴ and hybrid control method,¹⁵ establish the foundation for our further study. The distributed tracking controller is designed in the study by Yang et al.¹⁶ with the aid of the dynamic oscillator strategy. In the study by Dong,¹⁷ the state of each follower converges to the leader by utilizing the adaptive control technique. Some other methods including small gain and dynamic feedback linearization are employed in the study by Liu and Jiang¹⁸ to deal with the formation problem of nonholonomic mobile robots. Cao et al.¹⁹ also considered the consensus problem of high-order chained-form systems, the cascading theory is applied to construct hybrid distributed controllers. In the studies by Peng et al.²⁰ and Qiu et al.,²¹ distributed adaptive controller is designed to steer a class of mobile robots to achieve a prespecified geometric pattern and track a reference trajectory. Xu et al.²² consider the output consensus of nonholonomic system, a time-varying control law is given to demonstrate the stability of the overall closed-loop system. However, for the tracking control problem in the study by Zhang et al.⁹, and Ni and Cheng,¹² the input information of a reference trajectory is either available to all the other neighboring agents or equal to zero, which is impractical and restrictive due to the constraints of large scale of network. All the above studies^{13

–22} consider the nonholonomic chained-form system, then the global information of the communication topology graph is required to be known.

Motivated by the previous studies and investigations on coordinated tracking control of multiple mobile robots, the distributed coordinated tracking problem for a group of nonholonomic chained systems is discussed in this article. Compared to the result in the study by Xu et al.,²² the contributions of this work are stated as follows. First, the time-varying control strategy is applied in the cooperative control law instead of strict time-invariant continuous feedback linear system. Second, the controllers are designed on the basis of the neighboring state information of the agent with the coupling adaptive gains. The global information of communication topology is not required to be known in the controllers. Furthermore, it is worth mentioning that the tracking problem is solved under a practical assumption that the information of the reference trajectory is available to a part of neighboring systems rather than all neighboring systems.

Preliminaries and problem statement

Graph theory and notations

In this article, the interconnection topology of multi-agent systems is modeled as a directed graph. The communication topology of the N agents can be described as the weighted graph $G = (V, E, A), where V = {v_{i}, v_{2}, \dots, v_{N}}$ is a finite vertex set which represents each individual agent, and $E \subseteq V \times V$ is a set of edge with an ordered pair $(v_{i}, v_{j})$ which describes the information exchange between nodes i and j. $(j, i) \in E$ is a directed edge which means the state information of agent j is available to agent i. If the node j can receive the information from the node i, then the set of all neighboring nodes j is defined by $N_{i} = {j \in V : (j, i) \in E} . The N \times N$ adjacent matrix $A (G) = a_{i j}$ associated with edges of the directed graph is denoted such that $a_{i j} > 0 if (v_{j}, v_{i}) \in E (G); a_{i i} = 0 for i = 1, 2, \dots, N$ , otherwise. The Laplacian matrix $L = [l_{i j}] \in R^{N \times N}$ is defined as $L = D - A, where D = diag {d_{1}, d_{2}, \dots, d_{N}}, l_{i i} = d_{i} = \sum_{j = 1, j \neq i}^{N} a_{i j} and l_{i j} = - a_{i j}, i \neq j$ which satisfies that $\sum_{j = 1}^{N} l_{i j} = 0$ . A directed path from $v_{i_{1}} to v_{i_{l}}$ is a sequence of ordered edges in the form $(i_{k}, i_{k + 1}), k = 1, \dots, l - 1$ with distinct vertices $i_{k} \in V, k = 1, \dots, l$ . A directed graph has a spanning tree if there exists at least one node, which has directed paths to all other nodes, and such node is called a root node.

Throughout this article, the following notations are used. 1_N represents a column vector with all elements equal to one. For a vector $x = (x_{1}, x_{2}, \dots, x_{N})^{T} \in R^{N}, let {‖ x ‖}_{1}$ denote its 1-norm, where ${‖ x ‖}_{1} = \sum_{i = 1}^{N} | x_{i} |$ . The sign function is denoted by $sign (x) = - 1$ if the variable x is negative; $sign (x) = 1$ if the variable x is positive; and otherwise, $sign (x) = 0$ . For the matrix $L \in R^{N \times N}, λ_{max} (L) and λ_{min} (L)$ denote the maximum and minimum eigenvalue of the matrix L, respectively.

Assumption 1

Graph $G$ contains a directed spanning tree, and $C = diag (c_{1}, \dots, c_{N})$ is the connection matrix, where the vector $c = [c_{1}, \dots, c_{N}]^{T} \in R^{N}, c_{i} > 0, i = 1, \dots, N$ , and at least one of the elements of c is nonzero (equivalently, at least one of the N systems can connect with system 0).

Problem statement

Many kinematic systems in our life such as nonholonomic wheeled mobile robots can be transformed into chained form by coordinate transformation.²² In this article, we consider a group of chained-form nonholonomic systems as follows

{\begin{array}{l} {\dot{x}}_{1 i} = u_{1 i} \\ {\dot{x}}_{p i} = x_{p + 1, i} u_{1 i}, p = 2, 3, ..., n - 1 \\ {\dot{x}}_{n i} = u_{2 i} \end{array}

where $x = (x_{1 i}, x_{2 i}, \dots, x_{n i})^{T} and u = (u_{1 i}, u_{2 i})^{T}$ are the state and control input of agent i, $i = 1, 2, \dots, N$ , respectively.

In chained-form systems, the formation tracking control problem is taken into consideration in this article. Assume that the desired reference trajectory $x_{0} = (x_{10}, x_{20}, \dots, x_{n 0})^{T}$ is described by the following chained form

{\begin{array}{l} {\dot{x}}_{10} = u_{10} \\ {\dot{x}}_{p 0} = x_{p + 1, 0} u_{10}, p = 2, 3, ..., n - 1 \\ {\dot{x}}_{n 0} = u_{20} \end{array}

where $u_{0} = (u_{10}, u_{20})^{T}$ is the known time-varying reference input.

Assumption 2

The reference inputs $u_{0} = (u_{10}, u_{20})^{T}$ are bounded, and the states $x_{10}, x_{n 0}, x_{p 0}$ are also bounded.

Let us define the tracking error as ${\hat{x}}_{k} = x_{k i} - x_{k 0}, k = 1, 2, ..., n$ . It is easy to derive the following differential equations by utilizing equations (1) and (2)

\begin{matrix} {\dot{\hat{x}}}_{1} = u_{1 i} - u_{10} \\ {\dot{\hat{x}}}_{2} = {\dot{x}}_{2 i} - {\dot{x}}_{20} = x_{3 i} u_{1 i} - x_{30} u_{10} = {\hat{x}}_{3} u_{1 i} + x_{30} (u_{1 i} - u_{10}) \\ ⋮ \\ {\dot{\hat{x}}}_{n - 1} = {\dot{x}}_{n i} - {\dot{x}}_{n 0} = {\hat{x}}_{n} u_{1 i} + x_{n 0} (u_{1 i} - u_{10}) \\ {\dot{\hat{x}}}_{n} = u_{2 i} - u_{20} \end{matrix}

The Coordinated Tracking Control Problem is to design the control laws u_1i and u_2i for each agent i, $i = 1, 2, ..., N$ based on x_ki and its neighboring state x_kj, $k = 1, 2, ..., n, j \in N_{i}$ , such that, for an initial coordinated tracking state error ${\hat{x}}_{k} (0) = x_{k i} (0) - x_{k 0} (0)$ , the state of each agent can converge to the desired reference trajectory, that is

lim_{t \to \infty} (x_{k i} (t) - x_{k j} (t)) = 0

lim_{t \to \infty} {\hat{x}}_{k} (t) = lim_{t \to \infty} (x_{k i} (t) - x_{k 0} (t)) = 0

Lemma 1

If graph $G$ has a directed spanning tree, zero is an eigenvalue of L with associated right eigenvector 1_N, the other nonzero eigenvalues have positive real parts, and the matrix [L + C] is positive definite.¹¹

Controller design and convergence analysis

In this section, we will design a coordinated tracking controller under a directed communication topology containing directed spanning tree by two parts. In the first part, we add a new auxiliary variable in the traditional cooperative tracking control protocol instead of linear control strategy. An adaptive control law is also proposed for updating the linear subsystem gain. In the second part, the controller for the rest of system in chained form is constructed by the backstepping design method.

Control law design for the linear subsystem

Consider the linear subsystem in equation (1)

{\dot{x}}_{1 i} = u_{1 i}

The control law u_1i is designed by adopting the adaptive scheme for updating coupling gain and adding a new auxiliary variable in the traditional cooperative tracking protocol as follows

u_{1 i} = - \sum_{j \in N_{i}} a_{i j} (x_{1 i} - x_{1 j}) - c_{i} (x_{1 i} - x_{10}) + z_{1 i}

{\dot{z}}_{1 i} = - β_{1 i} [\sum_{j \in N_{i}} a_{i j} (z_{1 i} - z_{1 j}) + c_{i} (z_{1 i} - z_{10})] - β_{1 i} sign [\sum_{j \in N_{i}} a_{i j} (z_{1 i} - z_{1 j}) + c_{i} (z_{1 i} - z_{10})]

{\dot{β}}_{1 i} = {[\sum_{j \in N_{i}} a_{i j} (z_{1 i} - z_{1 j}) + c_{i} (z_{1 i} - z_{10})]}^{2} + | \sum_{j \in N_{i}} a_{i j} (z_{1 i} - z_{1 j}) + c_{i} (z_{1 i} - z_{10}) |

where $u_{10} = z_{10}$ , and a_ij is the element of the adjacency matrix of $G$ . The parameter c_i > 0 if the system 0 is available to system i, 1 ≤ i ≤ N and c_i = 0, otherwise. β_1i (t) denotes the time-varying adaptive gain.

Theorem 1

For the system (6) with controller (7) to (9), if assumptions 1 and 2 hold, then we have ${lim}_{t \to \infty} (x_{1 i} (t) - x_{10} (t)) = 0 and {lim}_{t \to \infty} (z_{1 i} (t) - z_{10} (t)) = 0$ , for all i, 1 ≤ i ≤ N.

Proof

The closed-loop system can be written as in the following form

{\dot{z}}_{1 *} = - β_{1} L z_{1 *} - β_{1} sign [L z_{1 *} + C (z_{1 *} - 1_{N} z_{10})] - β_{1} C (z_{1 *} - 1_{N} z_{10})

{\dot{x}}_{1 *} = z_{1 *} - L x_{1 *} - C (x_{1 *} - 1_{N} x_{10})

where $β_{1} = diag (β_{11}, β_{12}, \dots, β_{1 N}), x_{1 *} = (x_{11}, x_{12}, \dots, x_{1 N})^{T}, C = diag (c_{1}, c_{2}, \dots, c_{N}),$ . $z_{1 *} = (z_{11}, z_{12}, \dots, z_{1 N})^{T} . Let {\hat{z}}_{1 *} = z_{1 *} - 1_{N} z_{10}$ . It follows that

{\dot{\hat{z}}}_{1 *} = - β_{1} (L + C) {\hat{z}}_{1 *} - β_{1} sign [(L + C) {\hat{z}}_{1 *}] - 1_{N} {\dot{z}}_{10}

{\dot{\hat{x}}}_{1 *} = - (L + C) {\hat{x}}_{1 *} + {\hat{z}}_{1 *}

where L1_N = 0 is applied. Choosing the following Lyapunov candidate function

V_{1} = \frac{1}{2} {\hat{z}}_{1 *}^{T} (L + C) {\hat{z}}_{1 *} + \frac{1}{2} \sum_{i = 1}^{N} {(β_{1 i} - α)}^{2}

where α is a positive constant. It is directly checked that V₁ ≥ 0. Taking the time derivative of V₁ along the solution of equation (14) as follows

\begin{matrix} {\dot{V}}_{1} = {\hat{z}}_{1 *}^{T} (L + C) {\dot{\hat{z}}}_{1 *} + \sum_{i = 1}^{N} (β_{1 i} - α) {\dot{β}}_{1 i} \\ = - β_{1} {\hat{z}}_{1 *}^{T} {(L + C)}^{2} {\hat{z}}_{1 *} - {\hat{z}}_{1 *}^{T} (L + C) β_{1} sign [(L + C) {\hat{z}}_{1 *}] \\ - {\hat{z}}_{1 *}^{T} (L + C) 1_{N} {\dot{z}}_{10} + \sum_{i = 1}^{N} (β_{1 i} - α) {\dot{β}}_{1 i} \\ \leq - α {\hat{z}}_{1 *}^{T} {(L + C)}^{2} {\hat{z}}_{1 *} - α {‖ {\hat{z}}_{1 *}^{T} (L + C) ‖}_{1} + sup | {\dot{z}}_{10} | {‖ {\hat{z}}_{1 *}^{T} (L + C) ‖}_{1} \\ = - α {\hat{z}}_{1 *}^{T} {(L + C)}^{2} {\hat{z}}_{1 *} - (α - sup | u_{10} |) {‖ {\hat{z}}_{1 *}^{T} (L + C) ‖}_{1} \end{matrix}

In light of Lemma 1 in the study by Li et al.,¹¹ it yields that the matrix (L + C)² is positive definite. Choosing α large enough such that $α \geq sup | u_{10} |$ , it can obtain that

{\dot{V}}_{1} \leq - α {\hat{z}}_{1 *}^{T} {(L + C)}^{2} {\hat{z}}_{1 *} = - W (\hat{x})

By noting that $W (\hat{x})$ is positive definite, then it follows that ${\dot{V}}_{1} \leq 0$ , implying that V₁(t) is not increasing. Therefore, in view of equation (14), we know that $β_{1 i}, {\hat{z}}_{1 *}$ are bounded. Since u₁₀ is bounded, it implies that ${\dot{\hat{z}}}_{1 *}$ is bounded from equation (12). As V₁(t) is nonincreasing, it indicates that V₁(+∞) exists as t → +∞. Integrating equation (15), it has $\int_{0}^{+ \infty} W (\hat{z} (τ)) d τ \leq V_{1} (\hat{z} (0)) - V_{1} (+ \infty)$ . Hence, $\int_{0}^{+ \infty} W (\hat{z} (τ)) d τ$ exists and is finite. Because ${\hat{z}}_{1 *} and {\dot{\hat{z}}}_{1 *}$ are bounded, it is easy to be found that $\dot{W} (\hat{z})$ is also bounded, which accordingly guarantees the uniform continuity of $W (\hat{z})$ . Therefore, by Barbalat’s Lemma, it can obtain that $W (\hat{z}) \to 0$ as t → + ∞, it implies that ${\hat{z}}_{1 *} \to 0 as t \to + \infty$ . Equation (13) can be treated as a stable system, hence ${\hat{x}}_{1 *}$ exponentially converges to zero as ${\hat{z}}_{1 *}$ converges to zero. Consequently, it shows that ${lim}_{t \to \infty} {\hat{x}}_{1} (t) = {lim}_{t \to \infty} (x_{1 i} (t) - x_{0} (t)) = 0 and {lim}_{t \to \infty} (z_{1 i} (t) - z_{10} (t)) = 0$ .

Control law design for the remainder of chained-form subsystem

Noting that the remainder of chained-form subsystem in equation (1), the control law u_2i is designed by utilizing the backstepping design method.

Theorem 2

Consider the rest of chained-form subsystem in equation (1), if assumptions 1 and 2 hold, then the coordinated tracking problem of chained-form subsystems is solved with the following controllers

u_{2 i} = u_{20} + {\dot{\bar{x}}}_{n} - {\tilde{x}}_{n - 1} u_{1 i} + k_{2} [\sum_{j \in N_{i}} a_{i j} ({\tilde{x}}_{n j} - {\tilde{x}}_{n i}) - c_{i} {\tilde{x}}_{n i}]

{\bar{x}}_{3 i} = - k_{1} \frac{{\hat{x}}_{2 i}}{u_{1 i}} - \frac{x_{30} (u_{1 i} - u_{10})}{u_{1 i}}, {\tilde{x}}_{3 i} = {\hat{x}}_{3 i} - {\bar{x}}_{3 i}

{\bar{x}}_{4 i} = - \frac{k_{1} {\tilde{x}}_{3 i}}{u_{1 i}} - \frac{x_{40} (u_{1 i} - u_{10})}{u_{1 i}} + \frac{{\dot{\bar{x}}}_{3 i}}{u_{1 i}} - \frac{{\hat{x}}_{2 i} u_{1 i}}{u_{1 i}}, {\tilde{x}}_{4 i} = {\hat{x}}_{4 i} - {\bar{x}}_{4 i}

{\bar{x}}_{p i} = - \frac{k_{1} {\tilde{x}}_{p - 1, i}}{u_{1 i}} - \frac{x_{p 0} (u_{1 i} - u_{10})}{u_{1 i}} + \frac{{\dot{\bar{x}}}_{p - 1, i}}{u_{1 i}} - \frac{{\tilde{x}}_{p - 2, i} u_{1 i}}{u_{1 i}}, {\tilde{x}}_{p i} = {\hat{x}}_{p i} - {\bar{x}}_{p i}

where $p = 5, 6, ..., n, i = 1, 2, ..., N, k_{1} > (n - 2) λ_{max} (L + C), k_{2} > 0$ .

Proof

Firstly, consider the error dynamic chained-form subsystem in equation (3)

{\dot{\hat{x}}}_{2} = {\hat{x}}_{3} u_{1 i} + x_{30} (u_{1 i} - u_{10})

In order to simplify the variables, we will omit partial second variable i of right subscript. Take ${\bar{x}}_{3}$ as a virtue control input to make this subsystem (20) stable, we have

{\dot{\hat{x}}}_{2} = {\bar{x}}_{3} u_{1 i} + x_{30} (u_{1 i} - u_{10})

There exists a function to satisfy $V_{2} (\hat{x}) \geq 0$ , then we choose $V_{2} = {\hat{x}}^{2} / 2$ as a Lyapunov candidate function to verify the stability of equation (20). Since u_1i ≠ 0, setting the virtue control in equation (17), we get

{\dot{V}}_{2} (x) = {\hat{x}}_{2} {\bar{x}}_{3} u_{1 i} + {\hat{x}}_{2} x_{30} (u_{1 i} - u_{10}) = - k_{1} {\hat{x}}_{2}^{2} = - 2 k_{1} V_{2} < 0

which ensures the exponential stability of the subsystem (20). Obviously, as V₂ decays with 2_k1, ${\hat{x}}_{2}$ decays with k₁. The attenuation index of u_1i is $λ_{max} (L + C), where λ_{max} (L + C)$ is the maximum eigenvalue of matrix (L + C). Based on equation (17), we require that ${\hat{x}}_{2}$ converges faster than u_1i, hence choosing the parameter $k_{1} > λ_{max} (L + C)$ , which guarantees the boundedness of ${\bar{x}}_{3}$ .

Secondly, we introduce the error variable ${\tilde{x}}_{3} = {\hat{x}}_{3} - {\bar{x}}_{3}$ and extend equation (21), applying this error variable

{\dot{\hat{x}}}_{2} = ({\tilde{x}}_{3} + {\bar{x}}_{3}) u_{1 i} + x_{30} (u_{1 i} - u_{10})

{\dot{\tilde{x}}}_{3} = {\hat{x}}_{4} u_{1 i} + x_{40} (u_{1 i} - u_{10}) - {\dot{\bar{x}}}_{3}

Similarly, regard ${\bar{x}}_{4}$ as a virtue control input

{\dot{\tilde{x}}}_{3} = {\bar{x}}_{4} u_{1 i} + x_{40} (u_{1 i} - u_{10}) - {\dot{\bar{x}}}_{3}

Using Lyapunov candidate function $V_{3} = V_{2} + {\tilde{x}}_{3}^{2} / 2$ and setting the virtue control in equation (18), then take the time derivative, we have

\begin{matrix} {\dot{V}}_{3} = {\dot{V}}_{2} + {\tilde{x}}_{3} {\dot{\tilde{x}}}_{3} \\ = {\hat{x}}_{2} ({\tilde{x}}_{3} + {\bar{x}}_{3}) u_{1 i} + {\hat{x}}_{2} x_{30} (u_{1 i} - u_{10}) \\ + {\tilde{x}}_{3} [{\bar{x}}_{4} u_{1 i} + x_{40} (u_{1 i} - u_{10}) - {\dot{\bar{x}}}_{3}] \\ = - k_{1} {\hat{x}}_{2}^{2} + {\hat{x}}_{2} {\tilde{x}}_{3} u_{1 i} + {\tilde{x}}_{3} [{\bar{x}}_{4} u_{1 i} + x_{40} (u_{1 i} - u_{10}) - {\dot{\bar{x}}}_{3}] \\ = - k_{1} ({\hat{x}}_{2}^{2} + {\tilde{x}}_{3}^{2}) = - 2 k_{1} V_{3} < 0 \end{matrix}

In this step, ${\bar{x}}_{4}$ is a function of ${\dot{\bar{x}}}_{3} / u_{1 i}$ , that also is the function of ${\hat{x}}_{2} / u_{1 i}^{2}$ , based on equation (18). Using the same description method as before, we choose the parameter $k_{1} > 2 λ_{max} (L + C)$ that guarantees the boundedness of ${\bar{x}}_{4}$ .

For all further steps, similarly introduce an error variable ${\tilde{x}}_{n - 1} = {\hat{x}}_{n - 1} - {\bar{x}}_{n - 1}$ , and we have

{\dot{\hat{x}}}_{n - 1} = ({\tilde{x}}_{n - 1} + {\bar{x}}_{n - 1}) u_{1 i} + x_{n - 1, 0} (u_{1 i} - u_{10})

{\dot{\tilde{x}}}_{n - 1} = {\hat{x}}_{n} u_{1 i} + x_{n 0} (u_{1 i} - u_{10}) - {\dot{\bar{x}}}_{n - 1}

Regard ${\bar{x}}_{n}$ as a virtue control input

{\dot{\tilde{x}}}_{n - 1} = {\bar{x}}_{n} u_{1 i} + x_{n 0} (u_{1 i} - u_{10}) - {\dot{\bar{x}}}_{n - 1}

Using Lyapunov candidate function $V_{n - 1} = V_{n - 2} + {\tilde{x}}_{n - 1}^{2} / 2$ , then the time derivative of V_{n − 1} is given as follows

\begin{matrix} {\dot{V}}_{n - 1} = {\dot{V}}_{n - 2} + {\tilde{x}}_{n - 1} {\dot{\tilde{x}}}_{n - 1} \\ = - 2 k_{1} V_{n - 2} + {\tilde{x}}_{n - 2} {\tilde{x}}_{n - 1} u_{1 i} \\ + {\tilde{x}}_{n - 1} [{\bar{x}}_{n} u_{1 i} + x_{n 0} (u_{1 i} - u_{10}) - {\dot{\bar{x}}}_{n - 1}] \\ = - 2 k_{1} V_{n - 2} - k_{1} {\tilde{x}}_{n - 1}^{2} = - 2 k_{1} V_{n - 1} < 0 \end{matrix}

Using a similar method as before, choosing the parameter $k_{1} > (n - 2) λ_{max} (L + C)$ that assure the convergence and boundedness of the virtue control ${\bar{x}}_{n}$ .

Finally, in the last step, introduce an error variable ${\tilde{x}}_{n} = {\hat{x}}_{n} - {\bar{x}}_{n}$ , and we have

{\dot{\hat{x}}}_{n - 1} = (u_{2 i} - u_{20}) u_{1 i} + x_{n,0} (u_{1 i} - u_{10})

{\dot{\tilde{x}}}_{n} = u_{2 i} - u_{20} - {\dot{\bar{x}}}_{n}

Choosing the Lyapunov candidate function $V_{n} = V_{n - 1} + {\tilde{x}}_{n}^{2} / 2$ and using equation (16), we have

\begin{matrix} {\dot{V}}_{n} = - 2 k_{1} V_{n - 1} + {\tilde{x}}_{n} {\tilde{x}}_{n - 1} u_{1 i} + {\tilde{x}}_{n} [u_{2 i} - u_{20} - {\dot{\bar{x}}}_{n}] \\ = - 2 k_{1} V_{n - 1} + k_{2} {\tilde{x}}_{n} [\sum_{j \in N_{i}} a_{i j} ({\tilde{x}}_{n j} - {\tilde{x}}_{n i}) - c_{i} {\tilde{x}}_{n i}] \end{matrix}

Let $V_{n} = \sum_{i = 1}^{N} V_{n, i}$ , and define ${\hat{x}}_{2 *} = [{\hat{x}}_{21}, \dots, {\hat{x}}_{2 N}]^{T}, {\hat{x}}_{p *} = [{\hat{x}}_{p 1}, \dots {\hat{x}}_{p N}]^{T}, p = 3, 4, ..., n$ , then we have

\begin{matrix} {\dot{V}}_{n} = \sum_{i = 1}^{N} {\dot{V}}_{n, i} \\ = - 2 k_{1} \sum_{i = 1}^{N} V_{n - 1, i} \\ + k_{2} \sum_{i = 1}^{N} ({\tilde{x}}_{n, i} \sum_{j = 1}^{N} a_{i j} ({\tilde{x}}_{n j} - {\tilde{x}}_{n i}) - c_{i} {\tilde{x}}_{n, i} {\tilde{x}}_{n, i}) \\ = - 2 k_{1} \sum_{i = 1}^{N} V_{n - 1, i} - k_{2} {\tilde{x}}_{n}^{T} (L + C) {\tilde{x}}_{n} \end{matrix}

Since L + C is positive definite matrix. Hence, we get ${\dot{V}}_{n} \leq 0 . As V_{n}$ is nonincreasing and bounded from below by zero, it indicates that ${lim}_{t \to \infty} V_{n}$ exists and ${\tilde{x}}_{p}$ is bounded, hence $\int_{0}^{+ \infty} {\dot{V}}_{n} (\hat{x} (τ)) d τ = V_{n} (+ \infty) - V_{n} (\hat{x} (0))$ exists and is finite. And by the recursive formula from equations (17) to (19), it can be obtained that ${\dot{\hat{x}}}_{2}, {\dot{\tilde{x}}}_{p}, p = 3, 4, ..., n,$ are bounded. The derivative of ${\dot{V}}_{n} is {\ddot{V}}_{n} = - 2 k_{1} \sum_{i = 1}^{N} {\dot{V}}_{n - 1, i} - 2 k_{2} {\tilde{x}}_{n}^{T} (L + C) {\dot{\tilde{x}}}_{n}$ which is bounded, it implies that ${\dot{V}}_{n}$ is uniformly continuous. Therefore, with the aid of Barbalat’s Lemma, it can be obtained that ${\dot{V}}_{n} \to 0, p = 3, 4, ..., n - 1$ as t → ∞. Moreover ${\hat{x}}_{2} \to 0, {\tilde{x}}_{p} \to 0, p = 3, 4, ..., n - 1, and {\tilde{x}}_{n} = 0$ , that is, ${\hat{x}}_{p} \to 0, p = 2, 3, ..., n - 1, and {\hat{x}}_{n} = 0$ . Therefore, we can get ${lim}_{t \to \infty} (x_{p i} - x_{p 0}) = 0, p = 2, 3, ..., n, i = 1, 2, ..., N$ .

Remark 1

Compared to the static controller in the study by Li et al.,¹¹ the proposed adaptive control law (7) to (9) does not need to calculate the minimal eigenvalue of L, then the global information of the topology graph is not required to be known. On the other hand, the coupling gains need to be dynamically updated in equations (7) to (9), implying that the adaptive control law (7) to (9) is more complex than the static controller.

Coordinated tracking for switching communication graph

In the preceding sections, the fix communication graph is considered, If the communication graph $G (t)$ is switching at time t, the previous theorem also holds and is extended in the following results.

Theorem 3

For the system (6) (1 ≤ i ≤ N) with controller (7) to (9), if the switching communication topology $G (t)$ contains a directed spanning tree at each instant t, assumption 1 holds, then we have ${lim}_{t \to \infty} {\hat{x}}_{1} (t) = {lim}_{t \to \infty} (x_{1 i} (t) - x_{10} (t)) = 0 and {lim}_{t \to \infty} (z_{1 i} (t) - z_{10} (t)) = 0$ , for all i, 1 ≤ i ≤ N.

Proof

In any time instant t, we have the following closed-loop systems
${\dot{\hat{z}}}_{1 } = - β_{1} {(L + C)}_{t} {\hat{z}}_{1 } - β_{1} sign [(L + C)_{t} {\hat{z}}_{1 }] - 1_{N} {\dot{z}}_{10}$
31

${\dot{\hat{x}}}_{1 } = - {(L + C)}_{t} {\hat{x}}_{1 } + {\hat{z}}_{1 }$
32

Choosing Lyapunov candidate function
$V_{1 t} = \frac{1}{2} {\hat{z}}_{1 }^{T} {(L + C)}_{t} {\hat{z}}_{1 } + \frac{1}{2} \sum_{i = 1}^{N} {(β_{1 i} - α)}^{2}$
33

And taking the time derivative of V_1t along the solution of equation (31) with the aid of the equations (7) to (9). One gets
${\dot{V}}_{1 t} \leq - α {\hat{z}}_{1 }^{T} {(L + C)}_{t}^{2} {\hat{z}}_{1 } \leq - α λ_{min, t}^{2} (L + C) {\hat{z}}_{1 }^{T} {\hat{z}}_{1 }$
34

where α > 0, and $λ_{min, t} (L + C)$ is the smallest eigenvalue of symmetric matrix (L + C)_t at each time instant t. Because the symmetric matrix ${(L + C)}_{t}^{2}$ is positive definite, $λ_{min, t} (L + C)$ is greater than zero for the each time interval t. Therefore, by Barbalat’s Lemma, similar to the proof of theorem 1, it proves that ${\dot{V}}_{1 t} \leq 0$ and z_1i asymptotically converge to z₁₀ for 1 ≤ i ≤ N. Furthermore, on the basis of equation (32), it is easy to get that x_1i asymptotically converges to x₁₀ for 1 ≤ i ≤ N.

Theorem 4

Consider the remaining subsystem in equation (1) (1 ≤ i ≤ N), if the switching communication topology $G (t)$ contains a directed spanning tree at each instant t and at least one of the N systems can connect with system 0, assumption 1 holds, and the system is driven by the controller (16) and virtue control law (17) to (19), where the control parameter $a_{i j} > 0, k_{2} > 0, and k_{1} > (n - 2) λ_{max} (L + C)$ . Then, the coordinated tracking among chained-form subsystems is achieved, that is
$lim_{t \to \infty} (x_{k i} (t) - x_{k j} (t)) = 0, lim_{t \to \infty} {\hat{x}}_{k} (t) = lim_{t \to \infty} (x_{k i} (t) - x_{k 0} (t)) = 0$

where $k = 2, 3, ..., n, i = 1, 2, ..., N, j \in N_{i}$ .

Proof

Similar process of proof in Theorem 2, the variables ${\bar{x}}_{p i}, {\tilde{x}}_{p i}, p = 3, 4, ..., n, i = 1, 2, ..., N$ are estimated recursively by backstepping design method, the condition that the parameter $k_{1} > (n - 2) λ_{max, t} (L + C)$ is satisfied to ensure the virtue control bounded, where $λ_{max, t} (L + C)$ is the maximum eigenvalue of Laplacian matrix L + C at each time instant t. Hence, in the last step, since L + C is positive definite at each time instant, we get ${\dot{V}}_{n} \leq 0$ . Therefore, by Barbalat’s Lemma, it is easy to obtain that ${\hat{x}}_{p} \to 0, p = 2, 3, ..., n - 1 and {\hat{x}}_{n} = 0$ . Finally, we conclude that ${lim}_{t \to \infty} (x_{p i} - x_{p 0}) = 0, p = 2, 3, ..., n, i = 1, 2, ..., N$ .

Remark 2

In control laws (7) and (16), a_ij and c_i are control parameters. The parameter of the control law affects the convergence velocity of the system. Therefore, the convergence rate of $(x_{* i} - x_{* 0})$ relies on the switching communication topology graph at different time intervals.

Applications

Tracking control of wheeled mobile robots has extensive application in cooperative transportation and target tracking. In this section, an application example is showed in formation control with a desired trajectory by converting the wheeled mobile robots into nonholonomic chained form and integrating with the proposed control laws.

Consider a group of N wheeled mobile robots to track a presupposed target in a plane. The kinematic model of each mobile indexed by i ∈ N is given as follows
${\dot{x}}_{f i} = v_{i} cos θ_{i}, {\dot{y}}_{f i} = v_{i} sin θ_{i}, {\dot{θ}}_{i} = ω_{i}$
35

where $(x_{f i}, y_{f i})$ is the position of robot i in an inertial coordinate system. θ_i is the orientation of robot i and v_i, ω_i denote the linear velocity and angular velocity, respectively.

The trajectory of tracked target $(x_{l 0}, y_{l 0}, θ_{0})$ is described as
${\dot{x}}_{l 0} = v_{0} sin θ_{0}, {\dot{y}}_{l 0} = v_{0} sin θ_{0}, {\dot{θ}}_{0} = ω_{0}$
36

The position of desired formation is given by constant vectors $(p_{x i}, p_{y i})$ in local coordinate frame. Without loss of generality, if the origin of the local coordinate frame is the center of trajectory of desired formation, then we conclude that $\sum_{i = 1}^{N} p_{x i} = 0 and \sum_{i = 1}^{N} p_{y i} = 0$ .
Control goals: Design coordinated tracking controllers for each robot, on the basis of neighboring state information and tracked trajectory information, such that the multi-robot system can obtain desired formation and track the desired trajectory. Therefore, coordinated tracking control goals can be rewritten as follows

$\begin{array}{l} lim_{t \to \infty} [(x_{f j} - x_{f i}) - (p_{x j} - p_{x i})] = 0 \\ lim_{t \to \infty} [(y_{f j} - y_{f i}) - (p_{y j} - p_{y i})] = 0, 1 \leq i \neq j \leq N \end{array}$
37

$lim_{t \to \infty} (\sum_{i = 1}^{N} \frac{x_{f i}}{N} - x_{l 0}) = 0, lim_{t \to \infty} (\sum_{i = 1}^{N} \frac{y_{f i}}{N} - y_{l 0}) = 0$
38

For completing the above control goal, in this subsection, it is necessary to construct a new coordinate transformation as follows
${\begin{array}{l} x_{1 i} = - θ_{i} \\ x_{2 i} = - (x_{f i} - p_{x i}) sin θ_{i} + (y_{f i} - p_{y i}) cos θ_{i} \\ x_{3 i} = (x_{f i} - p_{x i}) cos θ_{i} + (y_{f i} - p_{y i}) sin θ_{i} \\ u_{1 i} = - ω_{i} \\ u_{2 i} = v_{i} + x_{2 i} ω_{i} \end{array}$
39

In the same way, letting
${\begin{array}{l} x_{10} = - θ_{0} \\ x_{20} = - x_{l 0} sin θ_{0} + y_{l 0} cos θ_{0} \\ x_{30} = x_{l 0} cos θ_{0} + y_{l 0} sin θ_{0} \\ u_{10} = - ω_{0} \\ u_{20} = v_{i} + x_{20} ω_{0} \end{array}$
40

The systems in equations (35) and (36) can transform into equations (1) and (2) with n = 3 through the coordinate transformation in equations (39) and (40). The following corollary can be easily derived through simple derivative and calculation to the above results, which the process of the proof is omitted here.

Corollary 1

By the changes of variables, the systems in equations (35) and (36) are converted to the nonholonomic chained system in equations (1) and (2) with n = 3, under theorems 1 and 2, the coordinated tracking can be achieved, that is ${lim}_{t \to \infty} (x_{p i} (t) - x_{p 0} (t)) = 0, p = 1, 2, 3, i = 1, 2, ..., N$ . Moreover, robots obtain desired formation and track the desired trajectory, that is, equations (37) and (38) are also reached.

Numerical illustrations

To verify the effectiveness of proposed theoretical results, we give the corresponding value to carry on the simple simulation. Consider the example described in the above section. There exist five robots, and their desired trajectory coordinates are given by $(p_{x 1}, p_{y 1}) = (- 6.5, 7.5), (p_{x 2}, p_{y 2}) = (- 9.0, - 4.0), (p_{x 3}, p_{y 3}) = (1 .0, - 10), (p_{x 4}, p_{y 4}) = (9.5, - 2), and (p_{x 5}, p_{y 5}) = (5.0, 8.5)$ . By choosing bounded reference input in equation (32) with $v_{0} = 5 and ω_{0} = 5$ , the tracked trajectory is generated by $(x_{l 0}, y_{l 0}, θ_{0}) = (sin (5 t), - cos (5 t),5 t)$ .

Let the communication topology graph be shown in Figure 1. The coordinated tracking control can be achieved by corollary 1 combining with previous results in the section “ Controller design and convergence analysis.” The control parameters are selected as $a_{i j} = 1, c_{i} = 1, k_{1} = 2, and k_{2} = 1$ . The formation trajectories of five robots with a tracked reference path are shown in Figure 2. Figures 3 and 4 show that the angular velocity and the original three different state tracking of five robots converge to the reference one, which confirm the result in theorems 1 and 2.

Figure 1.
The communication topology graph $G_{1}$ .

Figure 2.
The movement trajectories of five robots with a reference path.

Figure 3.
The angular velocity tracking error of five robots.

Figure 4.
The state tracking errors of five robots.

If the communication topology graph is different in each time interval, the control laws are valid in theorem 3. Assume that the switching communication topology graph shown in Figures 1 and 5 satisfies the following rules
$G = {\begin{array}{l} G_{1} & ift-round (t) \geq 0 \\ G_{2} & ift-round (t) < 0 \end{array}$

Figure 5.
The communication topology graph $G_{2}$ .

Figure 6 shows the formation trajectory of five robots with a tracked reference path for switching topology graph case. In the case of topology graph switching, Figures 7 and 8 show that the tracking error of angular velocity and three original states still converge to zero, which verifies the effectiveness of the proposed coordinated tacking control algorithm.

Figure 6.
The movement trajectories of five robots with a reference path.

Figure 7.
The angular velocity tracking error of five robots.

Figure 8.
The state tracking errors of five robots.

Conclusions

In this article, the distributed adaptive coordinated tracking control problem for a group of nonholonomic chained systems has been discussed under the condition that the communication topology graph has a spanning tree. At least one of the agents can receive the state information of the desired reference trajectory whose control input is bounded. The design of control laws is divided by two parts. The control law of the linear subsystem is proposed with the aid of adaptive control method, and the control protocol of the remaining subsystem in chained form is addressed on the basis of the backstepping techniques. The state of each agent exponentially converges to the reference trajectory by the virtue of Lyapunov techniques, Barbalat’s Lemma and graph theory. Finally, the coordinated tracking controllers design is also investigated under the switching topology. Simulation results show the validity of the proposed controllers.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Nature Science Foundation under grants 61503329, 61573307, 61473249, and 61773335; Jiangsu Planned Projects for Postdoctoral Research Funds 1601024B; and the Natural Science Foundation of Jiangsu Province BK20171289.

References

Ren

Beard

Atkins

. Information consensus in multivehicle cooperative control. IEEE Control Syst 2007; 27(2): 71–82.

Tanner

Pappas

Kumar

. Leader-to-formation stability. IEEE Trans Robot Autom 2004; 20(3): 443–445.

Ren

Beard

. Formation feedback control for multiple spacecraft via virtue structures. Control Theory Appl 2004; 151(3): 357–368.

Balch

Arkin

. Behavior-based formation control for multiagent robot teams. IEEE Trans Robot Autom 1998; 14(6): 926–936.

Lin

Francis

Maggiore

. State agreement for coupled nonlinear systems with time-varying interaction. Soc Ind Appl Mathem 2007; 46(1): 288–307.

Yoo

Kim

. Distributed formation tracking of networked mobile robots under unknown slippage effects. Automatica 2015; 54(C): 100–106.

Karimoddini

Lin

Chen

. Hybrid three-dimensional formation control for unmanned helicopters. Automatica 2013; 49(2): 424–433.

Wang

Yan

. Passivity-based formation control of autonomous underwater vehicles. IET Control Theory Appl 2012; 6(4): 518–525.

Zhang

Lewis

Das

. Optimal design for synchronization of cooperative systems: state feedback, observer, and output feed-back. IEEE Trans Autom Control 2011; 56(8): 1948–1952.

10.

Wang

Hull

. Cooperative control of dynamical systems with application to autonomous vehicles. IEEE Trans Autom Control 2008; 53(4): 894–911.

11.

Liu

Ren

. Distributed tracking control for linear multiagent systems with a leader of bounded unknown input. IEEE Trans Autom Control 2013; 58(2): 518–523.

12.

Cheng

. Leader-following consensus of multi-agent systems under fixed and switching topologies. Syst Control Lett 2010; 59(3): 209–217.

13.

Dong

Farrell

. Cooperative control of multiple nonholonomic mobile agents. IEEE Trans Autom Control 2008; 53(6): 1434–1448.

14.

Khoo

Xie

Man

. Leader–follower consensus control of a class of nonholonomic systems. In: Proceedings of the 11th international conference on control, automation, robotics and vision, Singapore, 7–10 December 2010, pp.1381–1386. IEEE.

15.

Kolmanovsky

McClamroch

. Hybrid feed-back laws for a class of cascade nonlinear control systems. IEEE Trans Autom Control 1996; 41(9): 1271–1282.

16.

Yang

Fang

Cao

. Distributed trajectory tracking control for multiple nonholonomic mobile robots. IFAC PapersOnLine 2016; 49(4): 31–36.

17.

Dong

. Tracking control of multiple-wheeled mobile robots with limited information of a desired trajectory. IEEE Trans Robot 2012; 28(1): 262–268.

18.

Liu

Jiang

. Distributed formation control of nonholonomic mobile robots without global position measurements. Automatica 2013; 49(2): 592–600.

19.

Cao

Jiang

Yue

. Consensus of multiple nonholonomic chained form systems. Syst Control Lett 2014; 72: 61–70.

20.

Peng

Yang

Wen

. Adaptive distributed formation control for multiple nonholonomic wheeled mobile robots. Neurocomputing 2016; 173(P3): 1485–1494.

21.

Qiu

Xiang

. Distributed adaptive coordinated tracking for coupled nonholonomic mobile robots. IET Control Theory Appl 2014; 8(18): 2336–2445.

22.

Tian

Chen

. Output consensus for multiple non-holonomic systems under directed communication topology. Int J Syst Sci 2015; 46(3): 451–463.

Adaptive coordinated tracking control for multi-robot system with directed communication topology

Abstract

Keywords

Introduction

Preliminaries and problem statement

Graph theory and notations

Assumption 1

Problem statement

Assumption 2

Lemma 1

Controller design and convergence analysis

Control law design for the linear subsystem

Theorem 1

Proof

Control law design for the remainder of chained-form subsystem

Theorem 2

Proof

Remark 1

Coordinated tracking for switching communication graph

Theorem 3

Proof

Theorem 4

Proof

Remark 2

Applications

Corollary 1

Numerical illustrations

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References