## 1. Introduction

With the emergence of artificial intelligence technologies, multirobot systems have drawn great attention [

1]. Such systems not only strengthen and refine the ability of individual robots, but they also provide a platform to display collective behaviors [

2,

3]. Compared with a complex robot, multirobot systems rooted in the real world have broad applications, for example, collaborative projects, military reconnaissance, and search and rescue [

4,

5].

In many industrial, agricultural, and maritime situations, multiple robots have to form up into some given patterns in order to fulfill a task [

6]. In order to manage and coordinate the robots, the formation problem must be addressed. This problem originated in biological phenomena in nature, such as schools of fish swimming or a team of ants moving [

7,

8]. Concerning these biological systems, their formation behaviors exhibit high robustness and hierarchy in that a certain formation mechanism inherently exists. Similarly, the multiple robots call for such a mechanism. Some typical mechanisms have been developed for the robots, that is, the behavior-based algorithm, the virtual structure technique, the leader–follower framework, and the artificial potential field approach [

9]. From the aspect of control design, the leader–follower framework has blossomed notably, although the mechanism is criticized for its drawback of a “single point of failure” [

10]. This paper does not focus on how to design a novel mechanism, but it provides a formation control design. Consequently, the leader–follower framework was directly adopted because the mechanism is of merit for small- and medium-scale formation problems.

From the viewpoint of reality, the individual robots of a multirobot system are inevitably subject to uncertainties and disturbances which will make the formation dynamics of this multirobot system uncertain [

11,

12]. Affected by these adverse factors, the formation control problem of multiple robots becomes challenging. Many control strategies have been reported, i.e., iterative learning control [

13,

14], model predictive control [

15], interval type-2 fuzzy control [

16,

17], and so on.

As a synthetic tool, sliding mode control is an alternative for the formation problem of uncertain multirobot systems. So far, some sliding-mode-based formation methods have been presented, that is, first-order sliding mode control [

18,

19], integral sliding mode control [

8], derivative and integral terminal sliding mode control [

4], and terminal sliding mode control [

20]. Sliding mode control’s most attractive property is its invariance, which can guarantee that a sliding mode control system is completely robust despite the matched uncertainties and disturbances [

21].

On the other side, sliding mode control is also confronted with the dilemma of chattering. As a result, many ideas have been devoted to the decrease and elimination of chattering. Among these ideas, the super-twisting-based sliding mode control technique is advocated because it only needs the information of a sliding mode variable and gets rid of the dependence on the time derivative of this sliding mode variable [

22].

On the assumption that the bounds of uncertainties and disturbances are known, this technique is able to effectively force the sliding mode variable and its time derivative to the origin in finite time [

23,

24]. Unfortunately, this assumption is not mild in uncertain multirobot systems. In reality, one has to overestimate the bounds from the aspect of the closed-loop formation stability [

25,

26]. However, the overestimate definitely enlarges the gain of the super-twisting sliding mode control technique. A potential solution is to design a module that can adaptively estimate the bounds.

Motivated by this solution idea, some technical methods have been explored, i.e., disturbance observers, adaptive law design, fuzzy or neural network compensators, and so on. In this paper, the extreme learning machine (ELM) [

27] is taken into consideration. The ELM is a kind of feed-forward neural network with a single hidden layer. The parameters in its hidden layer need no tuning, as they are generated randomly and independent of the training data. Compared with the back-propagation algorithm, the training and learning speed of the ELM is much faster. So far, the ELM has been successfully applied to microwave filters [

28], traffic accident detection [

29], air–fuel ratio control [

30], and so on. However, application of the ELM technique to the formation problem of multi-agent mobile robots has not been reported. In this paper, we adopted the ELM for the super-twisting sliding mode formation maneuvers of uncertain multirobot systems. The purpose of this was to refine the formation performance when the bounds of the uncertainties and disturbances are unknown.

The highlights of the paper are summarized as follows.

An architecture that combines second-order sliding mode control and the extreme learning machine technique is investigated.

The closed-loop stability of this combination is presented in the sense of Lyapunov.

Some numerical results for different formation patterns are demonstrated to support the combination.

The remainder of this paper is organized as follows.

Section 2 models both a single mobile robot and a leader–follower pair.

Section 3 addresses super-twisting sliding mode control, adopts an ELM to estimate the bounds of the uncertainties and disturbances, and analyzes the closed-loop formation stability in the sense of Lyapunov. In

Section 4, we implement the presented control method in a multirobot system platform. Some numerical results and comparisons are illustrated in

Section 4. Finally, conclusions are drawn in

Section 5.

## 3. Formation Control Design

#### 3.1. Sliding Surfaces and Input–Output Dynamics

The super-twisting law is a powerful and effective technique that can realize a second-order sliding mode control design. The technique can effectively deal with a controlled plant with a relative degree equal to 1 with respect to the control input. With regard to the matched uncertainties and disturbances, it can make the sliding mode variable and its time derivative converge to the origin in finite time. Consequently, we considered this technique as a solution for formation maneuvers of the leader–follower pair in

Figure 2. In order to implement the control design, the sliding surfaces, that is, the sliding-mode vector, have to be predefined.

Here,

${l}_{ik}^{d}$ and

${\psi}_{ik}^{d}$ are the desired relative distance and the desired relative bearing angle, respectively, of the leader–follower pair.

${C}_{1}$ and

${C}_{2}$ are 2 × 2 constant diagonal matrices, given by

where both

${c}_{1}$ and

${c}_{2}$ are positive and predefined constants.

We differentiate the sliding-mode vector

${s}_{ik}$ in Equation (11) with respect to time and substitute the formation dynamics Equation (8) into the derivative of

${s}_{ik}$. Then, the input–output dynamics are determined by

In order to achieve a super-twisting sliding mode control design, the first step is to calculate the relative degree of the dynamics via Equation (11) with respect to the control input. From Equations (11) and (12), we have

From Equation (13), it is apparent that the relative degree of ${s}_{ik}$ with respect to ${u}_{k}$ is equal to 1. In other words, a super-twisting sliding mode control design is available for the formation maneuvers of multiple robots under the leader–follower scheme.

**Assumption** **1.** The matrix$b({x}_{ik},{\Delta}_{k},t)\in {\Re}^{2\times 2}$in Equation (14) contains both known and unknown parts, written as Here,

${b}_{0}({x}_{ik},t)$ is a known positively definite matrix,

${b}_{1}({x}_{ik},{\Delta}_{k},t)$ is bounded but unknown, and the two parts of

$b({x}_{ik},{\Delta}_{k},t)$ satisfy

where

${\gamma}_{1}$ is a unknown constant.

**Assumption** **2.** The vector$a({x}_{ik},{d}_{ik},t)\in {\Re}^{2\times 1}$contains both known and unknown parts, depicted by Here,

${a}_{0}({x}_{ik},t)$ is a known and bounded vector and

${a}_{1}({x}_{ik},{d}_{ik},t)$ is bounded but unknown. Their ∞-norms satisfy

where

${\delta}_{1}$ and

${\delta}_{2}$ are positive but unknown.

Concerning the two assumptions, the input–output dynamics of the sliding-mode vector

${s}_{ik}$ in Equations (12) can have the form

Here, ${a}_{0}({x}_{ik},t)$, ${a}_{1}({x}_{ik},{d}_{ik},t)$, ${b}_{0}({x}_{ik},t)$, and ${b}_{1}({x}_{ik},{\Delta}_{k},t)$ are abbreviated to ${a}_{0}$, ${a}_{1}$, ${b}_{0}$, and ${b}_{1}$ for brevity.

#### 3.2. Super-Twisting Sliding Mode Control Design

According to the nominal system in Equation (19), the super-twisting sliding mode control can be designed as

where

In (21),

${\alpha}_{k}$ and

${\chi}_{k}$ are positive and they need to be predefined. The signum function

$\mathrm{sgn}({s}_{ik})$ in Equation (22) is defined by

$\mathrm{sgn}({s}_{ik})={\left[\begin{array}{cc}\mathrm{sgn}({s}_{ik,1})& \mathrm{sgn}({s}_{ik,2})\end{array}\right]}^{T}$. We select a Lyapunov function candidate

We consider the input–output dynamics in Equation (19) and substitute Equations (20) and (21) into Equation (19). Given Assumptions 1 and 2, the time derivative of Equation (22) has the following form:

Note that the following equalities exist.

Considering (18), (23) can be written as

Concerning Equation (16), $0<{\gamma}_{1}<1$. Consequently, one can have ${\dot{V}}_{0}<0$ by picking up ${\alpha}_{k}$ and ${\chi}_{k}$ if ${\gamma}_{1}$, ${\delta}_{1}$, and ${\delta}_{2}$ are known. Unfortunately, these constants are hardly known in advance, that is, Equation (25) theoretically holds true but it is not available in reality. In order to make Equation (25) hold true, one possible approach is to overestimate ${\alpha}_{k}$ and ${\chi}_{k}$ so that ${\dot{V}}_{0}<0$ can be guaranteed and the closed-loop formation system can have stability in the sense of Lyapunov. However, the approach inevitably enlarges the gain of the super-twisting sliding mode control technique, which can definitely have adverse effects on the formation performance. To address this issue, in this paper we selected the extreme learning machine and fused it with super-twisting sliding mode control. Their integration can guarantee the formation stability while the super-twisting sliding mode control technique can have a suitable gain.

#### 3.3. Super-Twisting Sliding Mode Control Design via ELM

The ELM is a learning algorithm for single-hidden-layer feed-forward networks. By the algorithm, the input weights are randomly chosen, the hidden layer biases are randomly assigned, and the output weights are analytically determined. The reason why the gain of the super-twisting sliding mode is overestimated is that some bounds are unknown. With the help of the ELM, one possible approach is to estimate these uncertainties and disturbance online, which can avoid the drawback of overestimating the bounds.

Considering the formation dynamics in the form of the second-order differential equations in Equation (10), the sliding surfaces in Equation (11) can be written as

Then, the time derivative of Equation (26) is determined by

For the formation maneuvers with constants

${l}_{ik}^{d}$ and

${\psi}_{ik}^{d}$, we have

Consequently, Equation (27) has the form

Substituting Equation (10) into Equation (29) yields

Let

where

${\varpi}_{k}$ is defined in Equation (21). Then,

${u}_{k}$ can be obtained by

From Equation (10),

${\mathbb{D}}_{ik}$ contains all the uncertainties and disturbances. Here the ELM is designed to estimate

${\mathbb{D}}_{ik}$ in real time. Replacing

${\mathbb{D}}_{ik}$ by its estimate

${\widehat{\mathbb{D}}}_{ik}$ in Equation (32) yields

In Equation (9), ${\mathbb{D}}_{ik}$ is the function of some variables, where ${\phi}_{ik}$, ${l}_{ik}$, and ${\psi}_{ik}$ play an important role. Here, the three variables were chosen as the input nodes of the ELM, that is, $z\in {\Re}^{3\times 1}$ is the input vector and $z={\left[\begin{array}{ccc}{l}_{ik}& {\psi}_{ik}& {\phi}_{ik}\end{array}\right]}^{T}$. Without doubt, the output vector is just ${\widehat{\mathbb{D}}}_{ik}\in {\Re}^{2\times 1}$, that is, there are two output nodes located at the output layer.

We assigned

M hidden nodes as the hidden layer. Then, the weights between the input and hidden layers can be defined by

$w\in {\Re}^{M\times 3}$. The input bias vector of the hidden nodes was defined as

$c\in {\Re}^{M\times 1}$. A sigmoidal function was selected as the activation function of the hidden layer. Then, the output of the

lth

$(l=1,2,\cdots ,M)$ hidden layer node can be calculated by

Here, ${w}_{l}\in {\Re}^{1\times 3}$ is the lth row of $w$ and ${c}_{l}$ is the lth element of ${c}_{l}$.

The output weights between the hidden layer nodes and the output layer nodes were defined as

$\Theta =\left[\begin{array}{cc}{\Theta}_{1}& {\Theta}_{2}\end{array}\right]\in {\Re}^{M\times 2}$,

${\Theta}_{1}\in {\Re}^{M\times 1}$, and

${\Theta}_{2}\in {\Re}^{M\times 1}$. Finally, the output vector of the ELM can be calculated by

Here, $\mathbb{H}={[\begin{array}{cccccc}{h}_{1}& {h}_{2}& \cdots & {h}_{l}& \cdots & {h}_{M}\end{array}]}^{T}$.

According to the universal approximation theorem of the single-hidden-layer feed-forward networks in [

27], there exist optimal output weights

${\Theta}^{*}\in {\Re}^{M\times 2}$ to approximate

${\mathbb{D}}_{ik}$ so that

where

$\epsilon (z)\in {\Re}^{2\times 1}$ is an approximation error vector, and it can be arbitrarily reduced by increasing the number of hidden layer nodes. Therefore, it is assumed that

where

$\overline{\epsilon}$ is an arbitrary small constant.

Finally, a schematic diagram of the super-twisting second-order sliding mode formation control by the extreme learning machine is presented in

Figure 3.

**Theorem** **1.** Consider the formation dynamics in Equations (8) and (10) given Assumptions 1 and 2, utilizing the super-twisting sliding mode control in Equation (33). Suppose the ELM is designed to estimate${\widehat{\mathbb{D}}}_{ik}$online by Equation (35). If the weights between the hidden and output layers of the ELM are adjusted by Equation (38), then the formation control system is asymptotically stable. Here, $\eta $ is a positive constant, and both ${c}_{2}$ and ${s}_{ik,q}$ are defined by Equation (11).

**Proof.** Take the following Lyapunov function candidate into consideration.

Here,

$\tilde{\Theta}=\left[\begin{array}{cc}{\tilde{\Theta}}_{1}& {\tilde{\Theta}}_{2}\end{array}\right]$
is defined by

The time derivative of

$V$ can have the form of

Substituting Equation (30) into Equation (41) yields

Then, according to the designed formation control Equation (33), we have

From Equation (21), we can obtain

Since

${C}_{2}$ is a diagonal matrix, (44) can be written as

Consider the condition in Equation (38) and the definition in Equation (40). We can have

Then, Equation (45) becomes

From Equation (47), we can select suitable ${\alpha}_{k}$ and ${\chi}_{k}$ to make $\dot{V}<0$ so that the formation system becomes asymptotically stable.

Although

$\overline{\epsilon}$ in Equation (47) is still unknown in advance, the universal approximation theorem of the single-hidden-layer feed-forward networks (Equation (37)) in [

27] indicates that

$\overline{\epsilon}$ can be an arbitrary small constant. This fact avoids the drawback of overestimating

${\alpha}_{k}$ and

${\chi}_{k}$ in Equation (20) from the point of view of the formation stability. Thus, the super-twisting sliding mode formation maneuvers via ELM can contribute to the improvement of the formation performance. □