Formation control for multiple nonholonomic wheeled mobile robots based on integrated sliding mode controller and force function

Abstract

In this paper, we present a novel method to achieve collision-free formation control for multiple nonholonomic wheeled mobile robots. By utilizing the leader–follower strategy, an integrated controller that combines a sliding mode controller for formation control and a force function for obstacle avoidance is proposed. To address the common chattering issue associated with sliding mode control, a novel depletion model is developed, which constrains the controller input to vary smoothly within a small range and effectively eliminates chattering. Additionally, a dynamic-gain structure is introduced to reconcile the conflicting demands of formation control and obstacle avoidance, providing enhanced coordination between the two tasks. Simulation and experiments’ results are provided to illustrate the effectiveness of the proposed method.

Introduction

With the improvement of the embedded computing and communication capabilities, the multi-agent systems (MASs) that possess good robustness, high fault tolerance, and strong scalability, could carry out complex tasks more flexibly and effectively (Sun et al., 2022; Wang et al., 2023a, 2023b). The formation control of multiple mobile robot systems has wide applications in both military and civilian fields (Hosseinzadeh et al., 2018; Li et al., 2022; Lin et al., 2018), such as target tracking, search and rescue, and patrol. In the formation control, multiple agents move toward a specific target or direction, while maintaining a predetermined geometric shape among each other and adapting to environmental constraints (Oh et al., 2015). To achieve this objective, various formation control strategies have been developed, including virtual-structure approach (Zhang et al., 2012), behavior-based control (Antonelli et al., 2009), and leader–follower strategy (Consolini et al., 2008; Dai et al., 2019; Zhao et al., 2022). Due to its simplicity and ease of implementation, the leader–follower strategy has received much attention (Liu et al., 2024b).

Formation control imposes high demands on the controller. The sliding mode control is a special type of variable structure control (Utkin et al., 1978), which has strong robustness against the external disturbances and the internal uncertainties of the system (Sabiha et al., 2022; Sun et al., 2021; Zheng et al., 2015). The trajectory tracking performance of nonholonomic wheeled mobile robots (NWMRs) may be affected by the disturbances and the internal uncertainties (Xie et al., 2022). Therefore, the sliding mode control is introduced into formation control (Chang et al., 2011; Damani et al., 2023; Elayaperumal et al., 2023; Fei et al., 2020; Li et al., 2018; Yang et al., 2023). For extremely complex environments, the formation tracking control of autonomous underwater vehicles with model parameter uncertainties and external disturbances was investigated, and the followers could track and guide the leader robot within a given stability time based on sliding mode control (Gao et al., 2020). In Yu et al. (2023), the neural adaptive sliding mode controllers were proposed for multiple unmanned aerial vehicles (UAVs), which could compensate for composite uncertainties to ensure good tracking performance. Despite its excellent characteristics, chattering is commonly present in sliding mode control (Norsahperi et al., 2020). In this paper, a sliding mode tracking controller for nonholonomic wheeled robots is designed, and a depletion model is constructed to solve the chattering problem commonly existing in the sliding mode controller.

Furthermore, the application of robot formation control in the obstacle environment remains challenging (Liu et al., 2024a). The artificial potential field (APF) method (Khatib, 1986) is a classic obstacle avoidance method for robots, which has good real-time performance and smooth obstacle avoidance trajectory and is widely used in path planning for mobile robots (Chen et al., 2016; Hao et al., 2022; Peng et al., 2023; Wu et al., 2023). The APF method was adopted in formation control of the multi-robot by designing the appropriate controller (Mu et al., 2020; Pan et al., 2021; Shou et al., 2022). By adopting an improved APF method, the formation obstacle avoidance method was proposed for multi-robot path planning and preventing collisions between machines based on the adaptive proportional–integral–derivative (PID) controller (Pan et al., 2019). In order to ensure collision-free formation of MASs, the APFs and time-varying topology were introduced to the original controller (Fei et al., 2023). Path-planning methods based on APF are also commonly used to address obstacle avoidance issues in robot formation (Tong et al., 2022). Different from most obstacle avoidance methods by adopting the pre-path-planning (Wang et al., 2017), this paper introduces a force function and a dynamic-gain structure to achieve obstacle avoidance in formation, which could effectively save computing resources and improve the formation efficiency of multi-robot systems.

Motivated by the aforementioned research, we propose an integrated collision-free formation control method based on sliding mode control and force function for NWMRs. The primary contributions are summarized as follows:

1. Based on the leader–follower strategy, a sliding mode tracking controller is designed for NWMRs. In order to address the chattering problem caused by abrupt changes in controller output due to large or rapidly changing inputs, a novel depletion model is constructed, which smoothly constrains the controller input within a small range and effectively eliminates the chattering issue.

2. In Wang et al. (2017), the obstacle avoidance is achieved through path -planning using the APF method, which requires each robot to complete its path planning independently. To relax this limitation, attractive force and repulsive force functions are constructed to address the obstacle avoidance problem, where the attractive and repulsive forces act as control inputs on the robot rather than being used for path planning.

3. In most previous results, the integrated controller adopts the sum of the formation control input and the obstacle avoidance control input. However, robots may deviate from the formation in order to avoid obstacles. Hence, a dynamic-gain structure is developed to adjust the formation control input and obstacle avoidance control input, which enables robots to have a certain autonomous -decision-making capability between maintaining formation and avoiding obstacles.

The structure of this article is as follows. In Section “Problem formulation,” the system models and the control objective are introduced. The main results are presented in Section “Main results.” The simulation and experiment results are provided in Section “Evaluation and experimental results.” The conclusion is drawn in Section “Conclusion.”

Problem formulation

Based on the leader–follower strategy, the formation control problem of the nonholonomic wheeled robots is described in this section.

System model

Consider a multi-mobile robots system with $n$ follower robots, and one leader robot in the global coordinate system $\{XOY\}$ . As illustrated in Figure 1, we denote the leader as the $0$ th robot and the follower as the $i$ th robot $(i=1,2,\dots ,n)$ . $(x_i,y_i)$ is the $i$ th robot’s position, where $i=0,1,2,\dots ,n$ . ${\theta }_i$ is the heading angle of the $i$ th robot. $v_i$ and ${\omega }_i$ are velocity and angular velocity of the $i$ th robot. The kinematic model of the $i$ th nonholonomic mobile robot can be described as follows:

$${\stackrel{·}{R}}_i=\left[\begin{array}{c}{\stackrel{·}{x}}_i\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{y}}_i}\\ \multicolumn{1}{c}{{\stackrel{·}{\theta }}_i}\end{array}\right]=\left[\begin{array}{cc}\cos {\theta }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin {\theta }_i}& 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]\left[\begin{array}{c}{v}_i\hfill \\ \multicolumn{1}{c}{{\omega }_i}\end{array}\right],$$

(1)

where $R_i$ denotes the states of the $i$ th robot.

Figure 1.

The leader–follower formation.

In order to achieve multi-robot formation, we adopt the leader–follower strategy. The trajectory of the leader is planned in advance by the upper computer, and followers are driven to follow the leader with a desired formation. The $i$ th virtual robot is introduced to always maintain the desired states of the $i$ th robot, that is, desired position and heading angle. Therefore, the $i$ th virtual robot’s states $R_{\mathrm{vi}}$ are described as follows:

$$R_{\mathrm{vi}}=\left[\begin{array}{c}{x}_{\mathrm{vi}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{vi}}}\\ \multicolumn{1}{c}{{\theta }_{\mathrm{vi}}}\end{array}\right]=R_0+\left[\begin{array}{c}{l}_i\cos {\phi }_i\hfill \\ \multicolumn{1}{c}{l_i\sin {\phi }_i}\\ \multicolumn{1}{c}{0}\end{array}\right]=\left[\begin{array}{c}{x}_0+l_i\cos {\phi }_i\hfill \\ \multicolumn{1}{c}{y_0+l_i\cos {\phi }_i}\\ \multicolumn{1}{c}{{\theta }_0}\end{array}\right],$$

(2)

where $[x_{\mathrm{vi}},y_{\mathrm{vi}}{]}^T$ and ${\theta }_{\mathrm{vi}}$ are the $i$ th virtual robot’s position and heading angle, respectively. $l_i$ and ${\phi }_i$ are the distance and bearing angle between the leader and the $i$ th virtual robot, respectively.To achieve formation, it is desirable for the $i$ th follower to track the trajectory of the $i$ th virtual robot. Furthermore, the tracking error $[x_{\mathrm{ei}},y_{\mathrm{ei}},{\theta }_{\mathrm{ei}}{]}^T$ between the $i$ th virtual robot and the $i$ th follower is obtained as follows:

$$\left[\begin{array}{c}{x}_{\mathrm{ei}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{ei}}}\\ \multicolumn{1}{c}{{\theta }_{\mathrm{ei}}}\end{array}\right]=R_{\mathrm{vi}}-R_i=\left[\begin{array}{c}{x}_{\mathrm{vi}}-x_i\hfill \\ \multicolumn{1}{c}{y_{\mathrm{vi}}-x_i}\\ \multicolumn{1}{c}{{\theta }_{\mathrm{vi}}-{\theta }_i}\end{array}\right].$$

(3)

Then, the tracking error $[e_{\mathrm{xi}},e_{\mathrm{yi}},e_{\theta i}{]}^T$ in the local coordinates can be deduced as follows:

$$\left[\begin{array}{c}{e}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{e_{\mathrm{yi}}}\\ \multicolumn{1}{c}{e_{\theta i}}\end{array}\right]=\left[\begin{array}{ccc}\cos {\theta }_i\hfill & \sin {\theta }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin {\theta }_i}& \cos {\theta }_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right]\left[\begin{array}{c}{x}_{\mathrm{ei}}\hfill \\ \multicolumn{1}{c}{y_{\mathrm{ei}}}\\ \multicolumn{1}{c}{{\theta }_{\mathrm{ei}}}\end{array}\right].$$

(4)

Combining (1) and (3), the time derivative of $[e_{\mathrm{xi}},e_{\mathrm{yi}},e_{\theta i}{]}^T$ is calculated as

$$\left[\begin{array}{c}{\stackrel{·}{e}}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{\mathrm{yi}}}\\ \multicolumn{1}{c}{{\stackrel{·}{e}}_{\theta i}}\end{array}\right]=\left[\begin{array}{c}{v}_0\cos e_{\theta i}-v_i+{\omega }_i{e}_{\mathrm{yi}}\hfill \\ \multicolumn{1}{c}{v_0\sin e_{\theta i}-{\omega }_i{e}_{\mathrm{xi}}}\\ \multicolumn{1}{c}{{\omega }_0-{\omega }_i}\end{array}\right].$$

(5)

Therefore, the tracking control problem is transformed into an origin stability problem in the error system (5).

Objective

The objective of our work is to devise a distributed controller for the error system (5) in an obstacle environment, such that

• all the tracking errors $[e_{\mathrm{xi}},e_{\mathrm{yi}},e_{\theta i}{]}^T$ globally asymptotically converge to zero;

• follower robots could autonomously avoid obstacles upon encounter and resume formation after obstacle avoidance, that is, the formation performance is resilient.

Main results

By combining the sliding mode controller and force function, the integrated controller is constructed in this paper, and the entire controller structure is shown in the Figure 2.

Figure 2.

The integrated controller structure.

Design of sliding mode controller

In order to make the error system (5) converge to the origin, the sliding mode controller is designed in this section.

We adopt the backstepping method to construct the sliding mode surface as follows:

$$s_i=\left[\begin{array}{c}{s}_{1i}\hfill \\ \multicolumn{1}{c}{s_{2i}}\end{array}\right]=\left[\begin{array}{c}{e}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{\mu e_{\theta i}+\arctan (v_0{e}_{\mathrm{yi}})}\end{array}\right],$$

(6)

where $\mu >1/2$ is a constant. Then, ${\stackrel{·}{s}}_i$ is computed as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{s}}_i=\left[\begin{array}{c}{\stackrel{·}{s}}_{1i}\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{s}}_{2i}}\end{array}\right]=\left[\begin{array}{c}{\stackrel{·}{e}}_{\mathrm{xi}}\hfill \\ \multicolumn{1}{c}{\mu {\stackrel{·}{e}}_{\theta i}+\frac{{\stackrel{·}{v}}_0{e}_{\mathrm{yi}}+v_0{\stackrel{·}{e}}_{\mathrm{yi}}}{1+{(v_0{e}_{\mathrm{yi}})}^2}}\end{array}\right]}\\ \multicolumn{1}{c}{=\left[\begin{array}{c}{\omega }_0{e}_{\mathrm{yi}}-v_i+v_0\cos e_{\theta i}\hfill \\ \multicolumn{1}{c}{\mu {\omega }_0-\mu {\omega }_i+\frac{v_0(-{\omega }_i{e}_{\mathrm{xi}}+v_0\sin e_{\theta i})+{\stackrel{·}{v}}_0{e}_{\mathrm{yi}}}{1+{(v_0{e}_{\mathrm{yi}})}^2}}\end{array}\right].}\end{array}$$

(7)

Then, in order to converge system states to the designed manifold $s$ from arbitrary initial conditions within a finite time, the reaching law (Elayaperumal et al., 2023) is selected as follows:

$${\stackrel{·}{s}}_{1i}^{\prime }=-{\epsilon }_1|s_{1i}{|}^{{\alpha }_1}\mathrm{sgn}\left(s_{1i}\right)-{\epsilon }_{1^{\prime }}{s}_{1i},$$

(8)

$${\stackrel{·}{s}}_{2i}^{\prime }=-{\epsilon }_2|s_{2i}{|}^{{\alpha }_2}\mathrm{sgn}\left(s_{2i}\right)-{\epsilon }_{2^{\prime }}{s}_{2i},$$

(9)

where ${\alpha }_1,{\alpha }_2,{\epsilon }_1,{\epsilon }_2,{\epsilon }_{1^{\prime }},{\epsilon }_2^{\prime }$ are positive constants, and $0<{\alpha }_1,{\alpha }_2<1$ .

By setting ${\stackrel{·}{s}}_{1i}={\stackrel{·}{s}}_{1i}^{\prime }$ and ${\stackrel{·}{s}}_{2i}^{\prime }={\stackrel{·}{s}}_{2i}^{\prime }$ , the sliding mode control law is obtained as

$$\left[\begin{array}{c}{v}_i\hfill \\ \multicolumn{1}{c}{{\omega }_i}\end{array}\right]=\left[\begin{array}{c}{v}_0\cos e_{\theta i}+{\omega }_i{e}_{\mathrm{yi}}+{\epsilon }_1|s_{1i}{|}^{{\alpha }_1}\mathrm{sgn}\left(s_{1i}\right)+{\epsilon }_{1^{\prime }}{s}_{1i}\hfill \\ \multicolumn{1}{c}{\mu {\omega }_0+((v_0^2\sin e_{\theta i}+{\stackrel{·}{v}}_0{e}_{\mathrm{yi}})/(1+{(v_0{\omega }_0)}^2))}\\ \multicolumn{1}{c}{\frac{+{\epsilon }_2|s_{2i}{|}^{{\alpha }_2}\mathrm{sgn}\left(s_{2i}\right)+{\epsilon }_{2^{\prime }}{s}_{2i}}{\mu +(v_0{e}_{\mathrm{xi}}/(1+{(v_0{e}_{\mathrm{yi}})}^2))}}\end{array}\right].$$

(10)

Theorem 1. Under formation control law (10), the closed-loop error system (5) is globally asymptotically stable.

Proof. First, we prove that the reaching phase is completed within a finite time. Then, we prove that $e_{\mathrm{xi}}$ , $e_{\mathrm{yi}},$ and $e_{\theta i}$ converge to zero on the sliding mode surface.

Step 1. Consider the following Lyapunov function:

$$W_{1i}(t)=\left|s_{1i}\right|.$$

(11)

Combining (8), the derivative of $W_{1i}$ is calculated as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{W}}_{1i}(t)=-{\epsilon }_1|s_{1i}{|}^{{\alpha }_1}-{\epsilon }_{1^{\prime }}\left|s_{1i}\right|}\\ \multicolumn{1}{c}{=-{\epsilon }_1{W}_{1i}(t{)}^{{\alpha }_1}-{\epsilon }_{1^{\prime }}{W}_{1i}(t)}\\ \multicolumn{1}{c}{\le 0.}\end{array}$$

(12)

From Yu et al. (2005), it can be deduced that $s_{1i}$ will converge to zero within a finite time

$$T_1\le \frac{1}{{{\epsilon }_1}^{\prime }(1-{\alpha }_1)}\ln \frac{{{\epsilon }_1}^{\prime }{W}^{1-{\alpha }_1}(0)+{\epsilon }_1}{{\epsilon }_1}.$$

The convergence of $s_{2i}$ can be proved according to a similar procedure.

Step 2. When $s_i=0$ , we have $e_{\mathrm{xi}}=0$ . Consider the Lyapunov function as follows:

$$V_{\mathrm{yi}}=\frac{1}{2}{e}_{\mathrm{yi}}^2.$$

(13)

From $s_{2i}=0$ , that is,i.e. $\mu e_{\theta i}=-\arctan (v_0{e}_{\mathrm{yi}}),\mu >\frac{1}{2}$ , the time derivative of $V_{\mathrm{yi}}$ is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\mathrm{yi}}=e_{\mathrm{yi}}{\stackrel{·}{e}}_{\mathrm{yi}}}\\ \multicolumn{1}{c}{=v_0{e}_{\mathrm{yi}}\sin (e_{\theta i})}\\ \multicolumn{1}{c}{=-v_0{e}_{\mathrm{yi}}\sin (\frac{1}{\mu }\arctan (v_0{e}_{\mathrm{yi}}))}\\ \multicolumn{1}{c}{\le 0.}\end{array}$$

(14)

Thus, $e_{\mathrm{yi}}$ will converge to zero.□

Remark 1. Note that $s_i$ will converge to zero within a finite time, and $e_{\mathrm{yi}}$ will converge to zero. According to the LaSalle’s invariance principle, $s_{2i}$ will converge within the set $\{[e_{\mathrm{xi}},e_{\mathrm{yi}},e_{\theta i}{]}^T|e_{\mathrm{xi}}=0\mathrm{and}{e}_{\mathrm{yi}}=0\mathrm{and}\mu e_{\theta i}+\arctan (v_0{e}_{\mathrm{yi}})=0\}$ . Therefore, $e_{\theta i}$ will converge to zero. Furthermore, the closed-loop error system is globally asymptotically stable.

Depletion model

In order to solve the chattering, the following depletion model (15) is introduced to process the system input:

$$h(x)=\frac{c_{2}x}{c_1+|x|}(1-e^{-(c_1+|x|)\delta }),$$

(15)

where $c_1$ , $c_2$ , and $\delta$ are positive constants. This model is a continuous bounded function, and its output only varies smoothly within a small range. Furthermore, the tracking errors $[e_{\mathrm{xi}}^{\prime },e_{\mathrm{yi}}^{\prime },e_{\theta i}^{\prime }{]}^T$ processed by the depletion model can be obtained as follows:

$$e_{\mathrm{xi}}^{\prime }=h(e_{\mathrm{xi}})$$

(16)

$$e_{\mathrm{yi}}^{\prime }=h(e_{\mathrm{yi}}),$$

(17)

$$e_{\theta i}^{\prime }=h(e_{\theta i}).$$

(18)

Remark 2. In the sliding mode control law (10), $\mu$ is an important parameter, which has an impact on the response speed and stability of the system. Increasing the value $\mu$ appropriately can improve the system’s response speed, but it may also lead to chattering. The reason for chattering is due to the controller’s input changing too quickly, resulting in velocit y a velocity jump. Therefore, we utilize (15) to process the controller inputs, ensuring they vary smoothly while remaining within a relatively small range.

From (16) to (18), the modified sliding mode control law is proposed as

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{c}{v}_i\hfill \\ \multicolumn{1}{c}{{\omega }_i}\end{array}\right]=\left[\begin{array}{c}{v}_0\cos {e^{\prime }}_{\theta i}+{\omega }_i{e^{\prime }}_{\mathrm{yi}}+{\epsilon }_1|{s_{1i}}^{\prime }{|}^{{\alpha }_1}\mathrm{sgn}\left({s^{\prime }}_{1i}\right)+{\epsilon }_{1^{\prime }}{s^{\prime }}_{1i}\hfill \\ \multicolumn{1}{c}{\mu {\omega }_0+((v_0^2\sin {e^{\prime }}_{\theta i}+{\stackrel{·}{v}}_0{e}_{\mathrm{yi}})/(1+{(v_0{\omega }_0)}^2))}\\ \multicolumn{1}{c}{\frac{+{\epsilon }_2|{s^{\prime }}_{2i}{|}^{{\alpha }_2}\mathrm{sgn}\left({s^{\prime }}_{2i}\right)+{\epsilon }_{2^{\prime }}{s^{\prime }}_{2i}}{\mu +(v_0{e^{\prime }}_{\mathrm{xi}}/(1+{(v_0{e^{\prime }}_{\mathrm{yi}})}^2))}}\end{array}\right]}\\ \multicolumn{1}{c}{+\left[\begin{array}{c}{\phi }_{1i}\hfill \\ \multicolumn{1}{c}{{\phi }_{2i}}\end{array}\right],}\end{array}$$

(19)

where $s_{1i}^{\prime }=e_{\mathrm{xi}}^{\prime }$ , $s_{2i}^{\prime }=\mu e_{\theta i}^{\prime }+\arctan (v_0{e}_{\mathrm{yi}}^{\prime })$ . Two first-order low-pass filters ${\phi }_1$ and ${\phi }_2$ for increasing system stability are described as follows:

$$\left[\begin{array}{c}{\phi }_{1i}(n)\hfill \\ \multicolumn{1}{c}{{\phi }_{2i}(n)}\end{array}\right]=\left[\begin{array}{c}(1-{\gamma }_1){\phi }_{1i}(n-1)+{\gamma }_1\frac{e_{\mathrm{xi}}}{\sqrt{1+e_{\mathrm{xi}}^2+e_{\mathrm{yi}}^2}}\hfill \\ \multicolumn{1}{c}{(1-{\gamma }_2){\phi }_{2i}(n-1)+{\gamma }_2\sin e_{\theta i}}\end{array}\right],$$

(20)

where $0<{\gamma }_1,{\gamma }_2<1$ are constants. From the modified control law (19), the system chattering could be effectively eliminated in this paper.

Formation obstacle avoidance

In order to achieve the formation task of realistic scenarios, the obstacle avoidance method combined with the sliding mode control law is given to form an integrated formation controller.

Inspired by the APF method,the obstacles exert a repulsive force on the follower, while the virtual robot exerts an attractive force on the corresponding follower. Under the resultant force, the robots could avoid the obstacles. The attractive force function (21) and repulsive force function (22) are constructed as follows:

$$F_{\mathrm{att}}^i(d_{l-f}^i)=\frac{d_{l-f}^i-l_i}{l_i-2r_i}-\ln \frac{d_{l-f}^i-2r_i}{l_i-2r_i},$$

(21)

$$F_{\mathrm{rep}}^{i,l}(d_{i,l})=\{\begin{array}{cc}(\frac{1}{d_{i,l}}-\frac{1}{\rho })\frac{1}{d_{i,l}^2}\hfill & d_{i,l}\le \rho ,\hfill \\ \multicolumn{1}{c}{0}& d_{i,l}>\rho ,\hfill \end{array}$$

(22)

where $F_{\mathrm{att}}^i$ denotes the attractive force exerted on the $i$ th follower, and $F_{\mathrm{rep}}^{i,l}$ represents the repulsive force exerted by the $l$ th $(l=1,2,\dots ,m)$ obstacle on the $i$ th follower. $d_{l-f}^i$ denotes the distance between the $i$ th followers and the leader. $r_i$ is the collision radius of the $i$ th followers, and $d_{i,l}$ is the distance between the $i$ th followers and the $l$ th obstacle. $\rho$ represents the maximum impact range of the obstacles.

As shown in Figure 3, ${\varphi }_{\mathrm{att}}^i$ and ${\varphi }_{\mathrm{rep}}^{i,l}$ represent the $i$ th robot’s line of sight angle relative to the $i$ th virtual robot and the $l$ th obstacle in the global coordinate system, respectively. The force acting on the $i$ th robot in the local coordinate system $\{x_{\mathrm{ri}}o{y}_{\mathrm{ri}}\}$ can be represented as

$$F_{\mathrm{xi}}={\lambda }_{\mathrm{att},x}{F}_{\mathrm{att}}^i\cos ({\varphi }_{\mathrm{att}}^i-{\theta }_i)-{\lambda }_{\mathrm{rep},x}\sum _{l=1}^m{F}_{\mathrm{rep}}^{i,l}\cos ({\varphi }_{\mathrm{rep}}^{i,l}-{\theta }_i),$$

(23)

$$F_{\mathrm{yi}}={\lambda }_{\mathrm{att},y}{F}_{\mathrm{att}}^i\sin ({\varphi }_{\mathrm{att}}^i-{\theta }_i)-{\lambda }_{\mathrm{rep},y}\sum _{l=1}^m{F}_{\mathrm{rep}}^{i,l}\sin ({\varphi }_{\mathrm{rep}}^{i,l}-{\theta }_i),$$

(24)

where $F_{\mathrm{xi}}$ and $F_{\mathrm{yi}}$ denote the resultant force in the $x_{\mathrm{ri}}$ -axis and $y_{\mathrm{ri}}$ -axis, respectively. ${\lambda }_{\mathrm{att},x}$ , ${\lambda }_{\mathrm{att},y}$ , ${\lambda }_{\mathrm{rep},x},$ and ${\lambda }_{\mathrm{rep},y}$ are positive constants. Furthermore, the velocity increment $\mathrm{\Delta }{v}_i$ and angular velocity increment $\mathrm{\Delta }{\omega }_i$ can be obtained as follows:

$$\left[\begin{array}{c}\mathrm{\Delta }{v}_i\hfill \\ \multicolumn{1}{c}{\mathrm{\Delta }{\omega }_i}\end{array}\right]=\left[\begin{array}{c}{F}_x\hfill \\ \multicolumn{1}{c}{F_y}\end{array}\right]\mathrm{\Delta }t,$$

(25)

where $\mathrm{\Delta }t$ denotes the sampling time.

Figure 3.

The decomposition of forces acting on the ith follower robot.

In order to coordinate formation and obstacle avoidance capabilities, we introduce a dynamic-gain structure as follows:

$$\left[\begin{array}{c}{k}_{\mathrm{smc}}^i\hfill \\ \multicolumn{1}{c}{k_{\mathrm{att}}^i}\\ \multicolumn{1}{c}{k_{\mathrm{rep}}^i}\end{array}\right]=\{\begin{array}{cc}{\left[\begin{array}{ccc}1-e^{-d_{i,min}}\hfill & \beta (1-e^{-d_{i,min}})\hfill & 1\hfill \end{array}\right]}^T\hfill & d_{i,min}\le \rho ,\hfill \\ \multicolumn{1}{c}{{\left[\begin{array}{ccc}1\hfill & 1\hfill & 0\hfill \end{array}\right]}^T}& d_{i,min}>\rho ,\hfill \end{array}$$

(26)

where $k_{\mathrm{smc}}^i,k_{\mathrm{att}}^i,$ and $k_{\mathrm{rep}}^i$ denotehe sliding mode controller (19) output gain, attractive force output gain (21), and repulsive force (22) output gain for the $i$ th robot, respectively. $d_{i,min}$ represents the distance between the $i$ th robot and the nearest obstacle. $\beta$ is a positive constant. By utilizing the output gains in (26), the output of (19), (23), and (24) can be adjusted according to the environment in real-time. Thus, the integrated formation controller can be obtained as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{c}{v}_i\hfill \\ \multicolumn{1}{c}{{\omega }_i}\end{array}\right]=k_{\mathrm{smc}}^i\left[\begin{array}{c}{v}_0\cos {e^{\prime }}_{\theta i}+{\omega }_i{e^{\prime }}_{\mathrm{yi}}+{\epsilon }_1|{s^{\prime }}_{1i}{|}^{{\alpha }_1}\mathrm{sgn}\left({s^{\prime }}_{1i}\right)+{\epsilon }_{1^{\prime }}{s^{\prime }}_{1i}\hfill \\ \multicolumn{1}{c}{\frac{\mu {\omega }_0+((v_0^2\sin {e^{\prime }}_{\theta i}+{\stackrel{·}{v}}_0{e}_{\mathrm{yi}})/(1+{(v_0{\omega }_0)}^2))+{\epsilon }_2|{s^{\prime }}_{2i}{|}^{{\alpha }_2}\mathrm{sgn}\left({s^{\prime }}_{2i}\right)+{\epsilon }_{2^{\prime }}{s^{\prime }}_{2i}}{\mu +(v_0{e^{\prime }}_{\mathrm{xi}}/(1+{(v_0{e^{\prime }}_{\mathrm{yi}})}^2))}}\end{array}\right]+k_{\mathrm{smc}}^i\left[\begin{array}{c}{\phi }_{1i}\hfill \\ \multicolumn{1}{c}{{\phi }_{2i}}\end{array}\right]}\\ \multicolumn{1}{c}{+\left[\begin{array}{c}{k}_{\mathrm{att}}^i{\lambda }_{\mathrm{att},x}{F}_{\mathrm{att}}^i\cos ({\varphi }_{\mathrm{att}}^i-{\theta }_i)-k_{\mathrm{rep}}^i{\lambda }_{\mathrm{rep},x}\sum _{l=1}^m{F}_{\mathrm{rep}}^{i,l}\cos ({\varphi }_{\mathrm{rep}}^{i,l}-{\theta }_i)\hfill \\ \multicolumn{1}{c}{k_{\mathrm{att}}^i{\lambda }_{\mathrm{att},y}{F}_{\mathrm{att}}^i\sin ({\varphi }_{\mathrm{att}}^i-{\theta }_i)-k_{\mathrm{rep}}^i{\lambda }_{\mathrm{rep},y}\sum _{l=1}^m{F}_{\mathrm{rep}}^{i,l}\sin ({\varphi }_{\mathrm{rep}}^{i,l}-{\theta }_i)}\end{array}\right]\mathrm{\Delta }t.}\end{array}$$

(27)

Remark 3. From (26) and (27), the composition of the integrated controller outputs varies in different environments. When $d_{i,min}>\rho$ , there is no obstacle in the vicinity of the robot, thus, repulsive force is not needed, and thus $k_{\mathrm{rep}}^{i,l}=0$ . Besides, $k_{\mathrm{smc}}^i=1$ and $k_{\mathrm{att}}^i=1$ imply that sliding mode controller and attractive force will output to maintain formation.

Otherwise, when $d_{i,min}\le \rho$ , it is indicated that the robot is in an obstacles environment, thus $k_{\mathrm{rep}}^{i,l}=1$ such that repulsive force output to avoid obstacles. It should be pointed out that the excessive output of the sliding mode controller and attractive forces is detrimental to the robot’s obstacle avoidance capabilities. Moreover, we aim for the robot to prioritize obstacle avoidance over maintaining formation in the presence of obstacles. Therefore, we design $k_{\mathrm{smc}}^i=1-e^{-d_{i,min}}$ and $k_{\mathrm{att}}^i=\beta (1-e^{-d_{i,min}})$ when $d_{i,min}\le \rho$ , which means that the sliding mode controller and attractive force will gradually weaken as $d_{i,min}$ decreases. Furthermore, the robot will rejoin the formation after successfully avoiding the obstacles.

Hence, the dynamic-gain structure (26) enables the robots to make better autonomous decisions between maintaining formation and avoiding obstacles. Therefore, the integrated controller (27) in this paper is more reasonable than the controller in Pan et al. (2021), that is,i.e. the sum of the formation control input and the obstacle avoidance control input.

Remark 4. It should be pointed out that when $d_{i,min}>\rho$ , the output of the attractive force is necessary. From (13) and (14), it is deduced that ${\stackrel{·}{V}}_{\mathrm{yi}}=0$ when $v_0=0$ , which means the tracking error $e_{\mathrm{yi}}$ will not converge to zero if the leader’s velocity $v_0=0$ equal to zero. Thus, if $v_0=0$ , the multi-robot system may not be able to form a formation. The attractive force $F_{\mathrm{att}}^i$ can move the follower to the desired formation position, even when $v_0=0$ .

Remark 5. In Wang et al. (2017), the formation obstacle avoidance is achieved by pre-planning the paths for each robot, which implies that each path needs to be independently planned before robot’s movement. It could not ensure good real-time performance because the system can only start working after all robots have completed their path planning. In fact, when the trajectory of the leader is determined, the trajectories of other followers are largely determined as well, which makes independent redundant planning unnecessary. In this paper, followers only need to receive the leader’s state information and detect obstacles in their immediate vicinity to achieve formation obstacle avoidance, which eliminates the need for path planning of the followers. Therefore, the method proposed in this paper is more applicable.

Evaluation and experimental results

The performance evaluation and experimental results of the proposed method are presented in this section to demonstrate its effectiveness. The MASs consisting of three robots are considered, and the communication topologies are shown in Figure 4.

Figure 4.

The communication topology.

Simulation

Firstly, the effectiveness of the proposed depletion model (15) will be demonstrated through simulation. The sliding mode controller parameters are set as follows: $\mu =12,{\epsilon }_1=0.001,{\epsilon }_{1^{\prime }}=3,{\epsilon }_2=0.0015,{\epsilon }_{2^{\prime }}=4.5,{\alpha }_1={\alpha }_2=0.2,c_2=3.0,c_1=0.8,\delta =0.1$ , and ${\gamma }_1={\gamma }_2=0.8$ . The initial values of the robots are set as follows: $[x_0(0),y_0(0),{\theta }_0(0){]}^T=[0,0,\pi /8{]}^T$ , $[x_1(0),y_1(0),{\theta }_1(0){]}^T=[5,0,0{]}^T,[x_2(0),y_2(0),{\theta }_2(0){]}^T,=[0,-5,0{]}^T$ . The formation is configured as a linear formation with $l_i=5\mathrm{m}$ . The leader’s velocity and angle velocity are given as $v_0=2\mathrm{m}/s$ and ${\omega }_0=0.6\sin (t)\mathrm{rad}/s$ , respectively. Besides, the maximum velocity and angular velocity of the followers are limited to $4\mathrm{m}/s$ and $5\mathrm{rad}/s$ , respectively.

The trajectories, tracking errors, and controller outputs of robots with (10) and (19) are shown in the left and right sides of Figure 5, respectively. By adopting (10), the controller output $v_i$ exhibits chattering, which prevents the tracking errors from quickly converging to zero. In contrast, through adopting (19), no chattering occurs, and the tracking errors could quickly converge. Therefore, the depletion model (15) is effective in eliminating chattering.

Figure 5.

The simulation results with control law (10) (left) and with modified control law (19) (right): (a) the trajectories of robots with control law (10); (b) the trajectories of robots with modified control law (19); (c) the formation tracking error for each follower with control law (10); (d) the formation tracking error for each follower with control law (19); (e) the control input for each follower with control law (10); and (f) the control input for each follower with control law (19).

Experiment

To verify the proposed method in actual implementation, we will examine integrated controller (27) performance based on SCOUT MINI unmanned ground vehicles (UGVs) in obstacle environments.

The real formation experiments are carried out in an indoor obstacle environment. The diagram of the experimental system is illustrated in Figure 6. The four-wheeled UGVs SCOUT MINI (Zhang et al., 2023) is adopted as the experiment platforms. The leader’s trajectory and formation shape are preseted on a laptop (Intel Core i5, CPU@ 2.4 Hz equipped). The directed communication topology is imitated with a robot operating system (ROS Noteic). For the $i$ th UGV, thread $i$ is adopted to receive the UGV’s state, compute the controller output, and send the control command to the onboard computing device (NVIDIA Jeston Nano and NVIDIA Jeston AGX Xavier). The onboard computing device will send control commands to the UGV and transmit the UGV’s state back to the laptop. Due to the limited size of the environment, UGVs rely on odometry for localization. In the majority of practical application scenarios, UGVs can be located using relative methods such as Global Positioning System (outdoor environment) and visual-inertial odometry (indoor environment). The obstacles position is prior knowledge in experiments. Note that the obstacles position and environment map can also be obtained online with sensors such as depth cameras and laser radar.

Figure 6.

The diagram of experimental system.

The integrated controller parameters are set as follows: $\mu =60$ , $c_1=0.8$ , $c_2=0.5$ , $\delta =0.405$ , $l_i=2$ , $2r_i=0.845$ , $\rho =1.5$ , ${\lambda }_{\mathrm{att},x}=2.25$ , ${\lambda }_{\mathrm{att},y}=3.25$ , ${\lambda }_{\mathrm{rep},x}=2.75$ , ${\lambda }_{\mathrm{rep},y}=5.75$ , $\beta =0.9$ , $\mathrm{\Delta }t=0.1$ , and the other parameters are the same as those in the simulation. To achieve better system performance, $\mu =60$ is chosen as a relatively large parameter. According to previous analysis, the large $\mu$ can lead to chattering in the system. Therefore, it is necessary to employ the proposed depletion model (14) in the actual system. The initial values of UGVs are set as follows: $[x_0(0),y_0(0),{\theta }_0(0){]}^T=[-1,0,0{]}^T$ , $[x_1(0),y_1(0),{\theta }_1(0){]}^T=[-2.732,1,0{]}^T$ , and $[x_2(0),y_2(0),$ ${\theta }_2(0){]}^T=[-2.732,-1,0{]}^T$ . The leader’s velocity and angle velocity are given as $v_0=0.24$ m/s and ${\omega }_0=0$ rad/s. The maximum velocity and angular velocity of the UGVs are limited to 0.6 m/s and 0.4 rad/s, respectively.

The experiment results are displayed in Figure 7. It can be observed that UGVs can leave the formation to avoid obstacles and quickly return to the desired formation after obstacle avoidance is completed. Therefore, the integrated controller (27) is effective. The formation trajectories of obstacle avoidance experiments in different environments are shown in Figure 8, which further demonstrates the effectiveness and reliability of the proposed method.

Figure 7.

The experiment results in the obstacle environment with (27): (a) the trajectories of unmanned ground vehicles (UGVs) in the obstacle environment; (b) the formation tracking error for each follower; and (c) the velocities and angle velocities of UGVs.

Figure 8.

The formation trajectories in different obstacle environments: (a) Case I; (b) Case II; (c) Case III; and (d) Case IV.

Conclusion

In this paper, the formation obstacle avoidance problem has been investigated for multiple NWMRs. A novel integrated controller that combines a sliding mode controller and a force function is proposed, and a dynamic-gain structure is introduced to coordinate the outputs of the sliding mode controller and the force function. Then, the formation obstacle avoidance task is realized by the proposed integrated controller using minimal computational resources, which ensure good real-time performance. Simulation and experiment results demonstrate the effectiveness of the proposed collision-free formation control scheme. However, the formation approach in this paper only tracks the virtual robot’s trajectory, without employing distributed consensus theory in the controller design. Additionally, the designed controller does not account for uncertainties, such as unknown model parameters and external disturbances. The possible future work includes formation control for multiple NWMRs with unknown disturbances and cyber attacks.