Distributed Control of Nonholonomic Robots Without Global Position Measurements Subject to Unknown Slippage Constraints

I. Introduction

IN recent years, there has been great development in cooperative control of multi-agent systems [1]-[5]. Compared with single agent, multi-agent systems have many advantages, such as better work efficiency, wider detection range, stronger load capacity, robustness, etc. Among these studies, more and more researchers have shown greater interest in formation control of multi-robot systems due to their potential applications, such as exploration, cooperative transportation, in search and rescue operations[6]-[8], etc. The objective of formation control for multi-robot systems is to make a group of robots move towards maintaining a desired geometric structure and overcoming disturbances from the environment. There are many methods to achieve formation control of multi-robot systems, such as leader-follower approach [9]-[12], artificial potential approach [13], virtual structure approach [14]-[15], behavior-based approach [16]-[17], etc.

The most widely adopted leader-follower formation structure is usually made up of one leader robot and multiple follower robots, in which all follower robots need to follow the leader robot while maintaining an expected geometric structure relative to the leader robot and overcoming any disturbances in the environment. In [18], a nonlinear model predictive control method based on neural-dynamic optimization was proposed for formation control of multi-robot systems, where a separation-bearing-orientation scheme was designed for regular leader-follower formation and a separation-distance scheme was designed for obstacle avoidance. In [19], an adaptive PID formation control strategy was proposed to solve the formation control problem of wheeled mobile robots without knowing the leader robot’s velocity information in the leader-follower framework, which improves the robustness of formation system. In [20], a robust controller was designed for leader-follower formation by using the Lyapunov redesign technique, where the proposed controller can make the closed-loop system robust to the uncertainty associated with the state of leader robot. Nevertheless, in these papers, it is assumed that the state information of leader robot is accessible to all follower robots. When the number of robots in the system becomes large, the leader needs to communicate with all the followers; thus, the communication burden of the leader will be heavy. With the limitation of the measurement range and communication conditions in some environments, only a part of the followers can obtain the states of the leader while others can not.

Recently, many works have studied distributed formation control of multi-robot systems in order to solve the problem mentioned above. In [21], a distributed formation controller subject to velocity constraints was proposed for multiple vehicles, and the leader-follower formation was achieved without using global position measurements. In [22], a novel distributed event-triggered control strategy was proposed for multi-robot systems under fixed topology and switching topology. By using the event triggering mechanism, communication requirements and energy consumption of the whole system can be effectively reduced. In [23], the distributed event-triggered control scheme and self-triggered control scheme were developed. In [24], a distributed formation method was also proposed for switching topology. In [25], a distributed finite-time observer was designed for each follower robot such that they can get the leader robot’s states in finite time. Based on the finite-time observer, a distributed finite-time formation controller was further proposed. In [26], a distributed fixed-time controller was designed to solve the consensus tracking problem of multiple agents in the form of second-order dynamics, and the result was further extended to formation control of multi-robot systems by using linear feedback technique. The authors in [27] studied the fixed-time fuzzy adaptive bipartite containment quantized control problem for nonlinear multi-agent systems subject to unknown external disturbances and unknown Bouc-Wen hysteresis. In [28], a reinforcement learning algorithm was adopted to investigate the adaptive fault-tolerant tracking control problem for a class of discrete-time multi-agent systems. These works have made great contributions to distributed formation control of wheeled mobile robots. However, the proposed controllers in these papers can not be directly applied to the environments with slippage constraints.

As stated in [29], slippage is hard to avoid due to the possible existence of ice, sand, or muddy roads, which may lead to insufficient friction in the actual working environment of wheeled mobile robots. The existence of unknown slippage not only affects the control performance of conventional controllers, but also limits the application scene of wheeled mobile robots, bringing huge losses to users. So it is necessary to consider the effect of slippage when studying formation control of multi-robot systems. Till now, few works have considered the slippage constraints in formation control of multi-robot systems. In [30], formation control of wheeled robots with slippage constraints were studied for the first time, and a slip-sensitive controller was proposed such that the formation can be stabilized when slippage exists. In [31], an adaptive formation control law was proposed for wheeled mobile robots in the presence of lateral slip and parametric uncertainties. In [32], dynamic surface design method and virtual structure approach were combined to realized formation maintenance, and an adaptive technique was used to compensate for the unknown slippage. However, these papers are all based on a centralized framework, which means the states of leader robot are accessible to all follower robots. Therefore, as mentioned in the second paragraph, it is necessary to study the distributed method to overcome the effect of unknown slippage in formation control.

In addition, in most of the research results about formation control, global position information $ (x,y) $ is required for each robot in the system. However, it is difficult, and sometimes even impossible, for robots to obtain accurate and stable global position information. Therefore, it is better to use relative position and one can obtain accurate and stable relative position between different robots by using equipment such as laser radars and gyroscopes.

Followers can not access the leader’s states directly in the distributed control of leader-follower formation structure in the most of the current research. It is difficult to use the consensus-based distributed method to achieve adaptive formation control under the constrains of the existence of unknown slippage and the unavailability of global position information. Therefore, this paper adopts observer-based distributed formation control method to overcome the two constraints simultaneously. In [21], [25], [29], and [33], the observer-based method was adopted without considering the convergence rate. Compared with asymptotic convergence, finite time convergence is more conducive to formation control. However, none of the observers in [21], [29], and [33] are convergent in finite time. The observer in [25] was convergent in finite time, however, the global position information was needed for each robot.

In this paper, observer-based fully distributed adaptive control of multi-robot systems with only relative position information subject to unknown slippage constraint are studied. Firstly, a novel distributed finite time state observer without using any global position information is proposed. Compared with existing finite time observers, such as [25] and [29], this paper only needs to use the relative position information between robot $ R_i $ and its neighbors to estimate the position errors relative to the leader in finite time. Secondly, based on the finite time observer, novel distributed adaptive formation controllers are further proposed without using global coordinate information and the effect of unknown slippage can be overcome. In order to deal with larger slippage, just as [34], [35] did, “slipping ratio” is adopted to model the unknown sliding disturbance, and a novel adaptive law is further designed to estimate the unknown slipping ratios. Note that a gyroscope is used for each robot to establish a common reference direction.

The major contributions of this paper can be summarized as: The fully distributed formation control of multi-robot systems without global position information while overcoming the effect of the unknown slippage constraint is proposed. Novel adaptive formation controllers and adaptive laws are proposed in the paper to achieve distributed control under the unknown slippage constraint. At the same time, in order to overcome the dependence on global position information and simplify the adaptive controllers, a novel distributed finite time state observer without using any global position information is proposed to obtain the related states of the leader.

This paper is organized as follows. Section II presents the problem formulation. In Section III, a novel distributed finite time observer and distributed formation controllers are proposed, and a strict stability analysis is given. Section IV shows simulation results of the proposed observers and controllers, which is followed by the conclusion in Section V.

II. Problem Statement

Consider a group of $ N+1 $ wheeled mobile robots, labelled as $R_0,R_1,\ldots,R_N$. The kinematic model of robot $ R_i $ is shown as Fig. 1, where $ [x_i, y_i]^T $ is the position of robot $ R_i $ ($i \in\{0, 1,\ldots, N\}$), $ \theta_i $ is the heading angle of robot $ R_i $ $ (0\leq\theta_i<2\pi) $, $ \nu_i $ and $ \omega_i $ are the linear velocity and angular velocity. $ u_{1i} $ and $ u_{2i} $ are actual control inputs of robot $ R_i $, which denote the control inputs of the left and right wheels, respectively. The radius of each driving wheel of robot $ R_i $ is $ r $, and the distance between the centers of two driving wheels of robot $ R_i $ is $ 2R $. The kinematic model of robot $ R_i $ is described by

Figure  1.  Kinematic model of wheeled mobile robot.

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For the group of $ N+1 $ robots, robot $ R_0 $ is the leader and the other robots are the followers, and all follower robots must follow the leader robot’s motion while surrounding the leader with a desired geometric structure. In this paper, it is considered that self-localization information is not available for each robot, so it is assumed that the leader robot can carry out local path planning autonomously by SLAM (simultaneous localization and mapping) or other technology, and the leader robot has the ability to overcome the unknown slippage from the ground. The design of the leader’s controller can refer to the follower robots’ controllers in the following chapters. In addition, consider the actual situation, the input states of the leader robot $ R_0 $ should satisfy the following assumptions.

Assumption 1: $ \nu_0(t) $, $ \omega_0(t) $, $ \dot{\nu}_0(t) $, $ \dot{\omega}_0(t) $, $ x_0(t) $, $ y_0(t) $ and $ \theta_0(t) $ of the leader robot $ R_0 $ are all continuous and bounded, and $ \nu_0(t)>0 $.

The communication network among the multi-robot systems is set up by onboard communication devices and sensors of each robot. To describe the communication network among the $ N $ follower robots, a directed graph $ \mathcal {G} = \{\mathcal {V}, \mathcal {E}\} $ is adopted, where $\mathcal {V} = \{1,\ldots,N\}$ represents $ N $ follower robots and $ \mathcal {E} = \{(j,i):j\neq i,\,i,j\in\mathcal {V}\} $ is an edge set with directed edge $ (j,i) $ representing the communication from robot $ j $ to robot $ i $. If $ (j,i)\in\mathcal{E} $, then the state of robot $ j $ is available to robot $ i $, and robot $ j $ is a neighbor of robot $ i $; each robot has access to its own state information and its neighbors’ directly. The set of neighbors of robot $ i $ is denoted as $ \mathcal {N}_i = \{j\in\mathcal {V}:(j,i)\in\mathcal {E}\} $. The adjacency matrix of $ \mathcal {G} $ can be represented as $\mathcal {A} = [a_{ij}]\in \mathbb{R}^{N\times N}$, where $ a_{ii} = 0 $, and $ a_{ij} = 1 $ when $ (j,i)\in\mathcal {E} $, otherwise $ a_{ij} = 0 $. Let diagonal matrix $ \mathcal {D} = $ diag$\{d_{1},d_{2},\ldots,d_{N}\}\in \mathbb{R}^{N\times N}$ be the degree matrix of a directed graph, where $ d_{i} = \sum_{j\in \mathcal {N}_i}a_{ij} $, then the Laplacian matrix of $ \mathcal {G} $ is defined as $ \mathcal {L} = \mathcal {D}-\mathcal {A} $. In addition, to describe the communication between the leader robot and followers, diagonal matrix $ B = $ diag$\{b_1,b_2,\ldots,b_N\}$ is defined, where $ b_i = 1 $ if follower robot $ i $ can get the information from the leader, and $ b_i = 0 $ otherwise. Also, the following assumption is made on the multi-robot systems.

Assumption 2: The directed graph among the multi-robot systems contains a directed spanning tree with the leader robot $ R_0 $ being the root.

Now, the destination of leader-follower formation control of multi-robot systems is to design distributed controller for each follower robot $ R_i $ such that the following objectives can be achieved.

where $ [d_{i0}^x, d_{i0}^y]^T $ are the desired relative position between robot $ R_i $ and $ R_0 $ on the X-axis and Y-axis, respectively.

Remark 1: Equation (2) ensures that the positions of all follower robots converge to the desired geometric structure gradually. That is to say, if (2) is satisfied, the desired formation goal can be reached.

Definition 1: To facilitate the design of controller, the following formation tracking errors are defined.

Before presenting the main results, two lemmas must be introduced, which are as follows.

Lemma 1 [25]: Consider the following system: $ \dot{x} = f(x), \ f(0) = 0,\ x\in\mathbb{R}^n $. Suppose there exists a continuous function $ V(x):U\rightarrow \mathbb{R} $ such that the following conditions hold:

i) $ V(x) $ is positive definite.

ii) There exist real numbers $ c> 0 $ and $ \alpha \in (0,1) $ and an open neighborhood $ U_0 \in U $ of the origin such that $\dot{V}(x)+ cV^\alpha(x)\leq 0, x\in U_0 \setminus \{0\}$.

Then, the origin is a finite-time stable equilibrium of the system. If $ U = U_0 = \mathbb{R}^n $, then the origin is a globally finite-time stable equilibrium of the system.

Lemma 2 [36]: Consider a first-order system $ \dot{\eta} = u $, where $ \eta $ and $ u $ denote its state and input, respectively. Take a continuous control law

where $ {k}> 0 $ and $ {\alpha'} \in (0,1) $, $ sgn(\cdot) $ is the standard signum function. And the closed-loop system can be described as

Then, there exists a finite time $T_f = {V^\beta(0)}/{k\beta2^{1-\beta}}$, where $V(0) = {\eta^2(0)}/{2}$, $\beta = ({1-{\alpha'}})/{2}$, such that the solution of the system converges to 0 as $ t\rightarrow T_f $, and $ \eta(t)\equiv 0 $ for any $ t\geq T_f $.

III. Main Results

In this section, the main results will be presented in two parts. Firstly, a distributed finite time state observer for each follower robot is proposed such that the state information associated with the leader robot can be obtained in finite time. It is noted that the finite time state observer does not need the robots’ global position information. Secondly, distributed adaptive formation controllers are designed for each follower robot such that the desired leader-follower formation can be achieved and the effect of unknown slippage can be overcome. Strict stability analysis is also given.

A Distributed Finite Time State Observer Design

Since only part of the followers can get the information associated with the leader robot, while other robots can not obtain the information from the leader directly, including the formation errors $ [e_i^x, e_i^y, e_i^\theta]^T $ and the velocities of the leader robot $ [\nu_0,\omega_0]^T $, a distributed finite time state observer was designed as follows such that each follower robot can achieved leader’s information in finite time.

Firstly, assign a virtual leader robot with states $ [\hat{x}_i,\hat{y}_i,\hat{\theta}_i] $ and velocities $ [\hat{\nu}_i,\dot{\hat{\theta}}_i] $ for each follower robot $ R_i $ ($i \in\{1,2,\ldots,N\}$), which can be seen as the estimations of the actual states of leader robot $ R_0 $. Then, the estimated formation tracking errors of each follower robot are defined as followers:

Then, the distributed finite time observer is designed as

where $ [d_{ij}^x,\, d_{ij}^y]^T $ are the desired relative distance between robot $ R_i $ and $ R_j $ on the X-axis and Y-axis, respectively, $ \eta_1, \eta_2, \eta_3, \eta_4>0 $, $ 0<a_1, a_2, a_3, a_4<1 $, and $sig^\alpha(y) = |y|^\alpha sgn(y), \alpha > 0$, where $ sgn(\cdot) $ denotes the standard signum function. If follower robot $ R_i $ can get information from the leader, then $ b_i = 1 $; otherwise, $ b_i = 0 $.

Remark 2: It is noted that the global coordinate $ [x_i, y_i]^T $ of robot $ R_i $ can not be used, while the relative distance $ (x_j-x_i) $ or $ (y_j-y_i) $ can be obtained by onboard device. The observed information $ [\hat{e}_i^x,\, \dot{\hat{e}}_i^x,\, \hat{e}_i^y,\, \dot{\hat{e}}_i^y,\, \hat{\theta}_i,\, \dot{\hat{\theta}}_i,\, \hat{v}_i,\, \dot{\hat{v}}_i]^T $ of robot $ R_i $ can be seen as internal states. For each follower robot, the designed observer only uses relative position information between the robots, while the self-localization information is not needed.

Remark 3: When the proposed observers (5)–(8) are used in simulations or experiments, it is not easy for each robot $ R_i $ to acquire $ [\dot{\hat{e}}_i^x,\, \dot{\hat{e}}_i^y,\, \dot{\hat{\theta}}_i,\, \dot{\hat{v}}_i]^T $ directly. However, one can get them by taking the derivative of the difference values of $ [\hat{e}_i^x,\, \hat{e}_i^y,\, \hat{\theta}_i,\, \hat{v}_i]^T $ between two consecutive sampling along the sampling time.

Theorem 1: Consider the distributed finite time observer (5)–(8). Under Assumptions 1 and 2, for any initial states $ [\hat{e}_i^x (t_0),\, \hat{e}_i^y (t_0),\, \hat{\theta}_i (t_0),\, \hat{v}_i (t_0),\, \dot{\hat{\theta}}_i (t_0)]^T $, one can obtain $[\hat{e}_i^x (t),\, \hat{e}_i^y (t),\, \hat{\theta}_i (t),\, \hat{v}_i (t),\, \dot{\hat{\theta}}_i (t)]^T$ converge to $ [e_i^x (t),\, e_i^y (t),\, \theta_0 (t),\, v_0 (t),\, \omega_0 (t)]^T $ in finite time.

Proof: Firstly, it will show $ \hat{e}_i^x (t) $ converge to $ e_i^x (t) $ in finite time.

Multiply both sides of (5) by ${\sum\nolimits_{j\in\mathcal {N}_i}a_{ij}+b_i}$ and since $ \dot{x}_i = \nu_i\cos\theta_i $, (5) can be rewritten as

By using (4), one has

Then, it follows from (9) and (10), one can obtain

Define $e_i = \sum\nolimits_{j\in\mathcal {N}_i}a_{ij}(\hat{x}_i-\hat{x}_j)+b_i(\hat{x}_i-x_0)$, and (11) can be rewritten as

Consider a Lyapunov function $V_i^* = {1}/{2}e_i^2,\ i = 1,2,\ldots,n$, and take the derivative of $ V_i^* $ along system (12), one can obtain

where $ \eta_1>0 $, $ 0<\frac{1+a_1}{2}<1 $.

It follows from Lemmas 1 and 2 that $ V_i^* $ converges to zero in finite time, which means there exists a finite time $T_i^* = {2V_i^*(e_i(0))^{({1-a_1})/{2}}}/{(1-a_1)\eta_12^{({1+a_1})/{2}}}$ such that $ V_i^* = 0 $ for $ \forall t\geq T_i^* $. Namely

when $ t \geq T_i^* $.

Define $T^* = {\rm{max}}\{T_1^*,T_2^*,\ldots ,T_n^*\}$, then according to (14), one can obtain

when $ t \geq T_i^* $.

According to Assumption 2, $ (\mathcal {L}+B) $ is a non-singular matrix, therefore the unique solution of (15) is zero and one has $ \hat{x}_i = x_0 $ when $ t \geq T^* $ ($i \in\{1,2,\ldots,N\}$). Then it follows from (4) that one can obtain $\hat{e}_i^x = \hat{x}_i-x_i+d_{i0}^x = x_0-x_i+d_{i0}^x = e_i^x$, $ \forall t\geq T^* $.

Similarly, we can use the same method to prove that there exists finite times $ T^{1*},T^{2*},T^{3*} $ such that $ \hat{e}_i^y = e_i^y $ when $ t\geq T^{1*} $, $ \hat{\theta}_i = \theta_0 $ when $ t\geq T^{2*} $, and $ \hat{\nu}_i = \nu_0 $ when $ t\geq T^{3*} $. Select $T_0 = {\rm{max}}\{T^*,T^{1*},T^{2*},T^{3*}\}$, one has $[\hat{e}_i^x (t), \hat{e}_i^y (t), \hat{\theta}_i (t), \hat{v}_i (t), \dot{\hat{\theta}}_i (t)]^T = [e_i^x (t), e_i^y (t), \theta_0 (t), v_0 (t), \omega_0 (t)]^T$, $ \forall t\geq T_0 $.

In addition, since the initial value of $ e_i $ is bounded, then one can obtain that $ e_i(t) $ is always bounded when $ 0<t \leq T_0 $ by using (12). According to (5), $ \hat{e}_i^x $ is also bounded when $ 0<t \leq T_0 $. Similarly, $ [\hat{e}_i^y (t), \hat{\theta}_i (t), \hat{v}_i (t), \dot{\hat{\theta}}_i (t)]^T $ are also bounded when $ 0<t \leq T_0 $.■

Remark 4: Compared with the distributed finite time state observer in [25], the main improvements are the proposition of (5) and (6), the state observer in [25] estimated the absolute position of leader robot directly in global coordinate system, while in this paper, no global coordinate system can be referred to, so we use (5) and (6) to estimate the relative position errors between leader $ R_0 $ and follower $ R_i $ instead.

B Distributed Adaptive Formation Controller Design

In order to solve the effect of unknown slippage on distributed formation control of multi-robot systems, distributed adaptive formation controllers $ [u_{1i},u_{2i}]^T $ need to be designed for each follower robot such that the desired formation objective can be realized.

When considering the unknown slippage, there is such a relationship between the actual control input $ [u_{1i},u_{2i}]^T $ and the actual output $ [\nu_i, \omega_i]^T $ for robot $ R_i $, which is as follows:

where $ [u_{1i},u_{2i}]^T $ are the control inputs for the left and right wheels of robot $ R_i $, respectively, $ [\eta_{1i},\eta_{2i}]^T $ are the actual slipping ratios corresponding to the left wheel and right wheel of wheeled robot $ R_i $, which are defined as $\eta_{1i} = {u_{1i}}/ {u_{s1i}}, \ \eta_{2i} = {u_{2i}}/{u_{s2i}}$, where $ u_{s1i} $ and $ u_{s2i} $ are actual forward velocity of left wheel and right wheel of $ R_i $ due to the existence of slippage. When $ \eta_{1i} = 1 $ and $ \eta_{2i} = 1 $, then there is no slippage. In addition, the slipping ratios should satisfy the following assumption.

Assumption 3: The slipping ratios $ \eta_{1i} $ and $ \eta_{2i} $ are bounded but unknown constant.

Before designing the distributed formation controllers, the following coordinate transformation (17) and (18) need to be defined first.

where $ [\hat{x}_i^e, \hat{y}_i^e, \hat{\theta}_i^e]^T $ and $ [x_i^e, y_i^e, \theta_i^e]^T $ are the estimated formation tracking errors and the true formation tracking errors in the coordinate system of robot $ R_i $, respectively.

According to Theorem 1, one can obtain $[\hat{e}_i^x, \hat{e}_i^y, \hat{e}_i^\theta]^T = [e_i^x, e_i^y, e_i^\theta]^T$ when $ t \geq T_0 $. It follows from (17) and (18), one has

To satisfy the formation target (2), the distributed adaptive formation controllers $ [u_{1i},u_{2i}]^T $ are designed as follows:

where

$ 0<k_0<1 $, $ k_{1i} $, $ k_{2i} $, $ \beta_{1i} $ and $ \beta_{2i} $ are positive constants, $ [\hat{\eta}_{1i}^*,\,\hat{\eta}_{2i}^*]^T $ are the estimated values of unknown slipping ratios corresponding to the left and right wheel of robot $ R_i $, respectively, according to the estimated formation tracking errors when $ t \geq 0 $. $ [\nu_{di}^*,\omega_{di}^*]^T $ are the desired velocity and angular velocity of robot $ R_i $ that can achieve the desired formation based on the observer when $ t \geq 0 $. Variables in (22) and (23) are intermediate variables to simplify later proof.

Remark 5: When designing the distributed adaptive formation controller, each follower robot can not use any global position information. In addition, it is necessary to ensure that the output of the designed controller is bounded.

Theorem 2: The distributed formation control problem of multi-robot systems without global position measurements and which are subject to slippage constraints can be solved by the distributed finite-time observer (5)–(8) and the distributed adaptive formation controller (20)–(25). The estimated values $ [\hat{\eta}_{1i}^*,\hat{\eta}_{2i}^*]^T $ of unknown slipping ratios will converge to its true value $ [\eta_{1i},\eta_{2i}]^T $ gradually by the adaptive laws (24) and (25).

Proof: First of all, it will be proved that the distributed adaptive formation controller (20) is always bounded when $ 0<t \leq T_0 $. According to Theorem 1, $[\hat{e}_i^x (t), \hat{e}_i^y (t), \hat{\theta}_i (t), \hat{v}_i (t), \dot{\hat{\theta}}_i (t)]^T$ are always bounded when $ 0<t \leq T_0 $. In addition, since $ 0<k_0<1 $, $ |\hat{y}_i^e|<|\hat{\gamma}_{1i}| $, $ 1\leq|\hat{\gamma}_{2i}| $, $ |\hat{\bar{\theta}}_i^e|<|\hat{\gamma}_{3i}| $, $ |\hat{x}_i^e|<|\hat{\gamma}_{2i}| $, $ |\hat{\gamma}_{1i}|<|\hat{\gamma}_{2i}| $, then $| {k_0\hat{y}_i^e\hat{\bar{\theta}}_i^e}/{\hat{\gamma}_{1i}\hat{\gamma}_{2i}^2\hat{\gamma}_{3i}}| < 1$ and $\left| {k_0\gamma_{1i}x_i^e}/{\gamma_{2i}^2}\right| < 1$. It follows from (17) and (20)–(25) that one can obtain the distributed adaptive formation controller (20) is always bounded when $ 0<t \leq T_0 $.

Then, according to Theorem 1 and (19), when $ t \geq T_0 $, the distributed formation controller (20) is equivalent to the following control law:

where

where $ [\hat{\eta}_{1i},\,\hat{\eta}_{2i}]^T $ are the estimated values of unknown slipping ratios corresponding to the left and right wheel of robot $ R_i $, respectively, according to the true formation tracking errors when $ t \geq T_0 $. $ [\nu_{di},\omega_{di}]^T $ are the desired velocity and angular velocity of robot $ R_i $ according to the true formation tracking errors when $ t \geq T_0 $.

Taking the derivative of (18), one can obtain the tracking errors space model of wheeled mobile robot $ R_i $ as follows:

In order to prove the stability of the formation controller (26), a new state variable $ \bar{\theta}_i^e $ needs to be defined as follow:

where $ \gamma_{1i} = \sqrt{1+(x_i^e)^2+(y_i^e)^2} $.

Taking the derivative of (32), one has

where $ \gamma_{2i} = \sqrt{1+(x_i^e)^2+(1+k_0^2)(y_i^e)^2} $.

It follows from (23) and (32), one has $\sin\theta_i^e = \alpha_{0i}\bar{\theta}_i^e- \sin\times (\arctan({k_0y_i^e}/{\gamma_{1i}}))$. Take $ [x_i^e,y_i^e,\bar{\theta}_i^e]^T $ as new state variables of wheeled robot $ R_i $, then the errors space model (31) can be rewritten as

where $M_i = {k_0v_0\sin\theta_i^e(1+(x_i^e)^2)-k_0x_i^ey_i^e\nu_0\cos\theta_i^e}/{\gamma_{1i}\gamma_{2i}^2}$.

The following proof process will be divided into two steps:

Step 1: In this step, we show that the ideal control output $ [\nu_{di},\omega_{di}]^T $ in (27) can realize formation objective with the assumption that there is no slippage.

Consider a Lyapunov function candidate $ V_{1i} $ as

where $ \gamma_{3i} = \sqrt{1+(\bar{\theta}_i^e)^2} $.

Taking the derivative of (35) and using (27), (34), one has

which means all the tracking errors $ [x_i^e,y_i^e,\bar{\theta}_i^e]^T $ converge to zero gradually.

Remark 6: It is noted that when $ [x_i^e,y_i^e,\bar{\theta}_i^e]^T $ converge to zero asymptotically, $ [x_i^e,y_i^e,\theta_i^e]^T $ also converge to zero according to (32). In addition, since $ 0<k_0<1 $, $ |y_i^e|<|\gamma_{1i}| $, $ 1\leq|\gamma_{2i}| $, $ |\bar{\theta}_i^e|<|\gamma_{3i}| $, $ |x_i^e|<|\gamma_{2i}| $, $ |\gamma_{1i}|<|\gamma_{2i}| $, then $\left| {k_0y_i^e\bar{\theta}_i^e}/{\gamma_{1i}\gamma_{2i}^2\gamma_{3i}}\right| < 1$ and $\left| {k_0\gamma_{1i}x_i^e}/{\gamma_{2i}^2}\right| < 1$, so the denominator of (27) can not be 0.

Step 2: In this step, slippage is further taken into consideration. It will show that the distributed adaptive formation controllers $ [u_{1i},u_{2i}]^T $ in (26) can realize formation objective of multi-robot systems with slippage constraints, and the estimated values $ [\hat{\eta}_{1i},\hat{\eta}_{2i}]^T $ of unknown slipping ratios will converge to their true values $ [\eta_{1i},\eta_{2i}]^T $ gradually by the adaptive laws (29) and (30).

Take $ \hat{\eta}_{1i} $ and $ \hat{\eta}_{2i} $ as estimated values of slipping ratios $ \eta_{1i} $ and $ \eta_{2i} $, respectively, and the estimation errors of $ \eta_{1i} $ and $ \eta_{2i} $ and their derivatives are as follows.

Due to the existence of unknown slippage, the actual outputs $ [\nu_i, \omega_i] $ of wheeled mobile robot $ R_i $ are no longer the same as the desired values $ [\nu_{di}, \omega_{di}] $. It follows from (16) and the proposed adaptive formation controllers $ [u_{1i},u_{2i}]^T $ (26), one can obtain the actual outputs $ [\nu_{i},\omega_{i}]^T $ of $ R_i $ as

Firstly, it will show that the formation tracking errors $ [x_i^e, y_i^e, \theta_i^e]^T $ of each follower robot $ R_i $ converge to 0 asymptotically.

Consider a Lyapunov function $ V_{2i} $ for robot $ R_i $ as

where $ \beta_{1i} $ and $ \beta_{2i} $ are positive constants.

Taking the derivative of $ V_{2i} $ and use (29), (30) and (38), one has

Since $ V_{2i} \geq0 $ and $ \dot{V}_{2i} \leq 0 $, thus $ V_{2i} $ is bounded. Then one can obtain that $ x_i^e,y_i^e,\theta_i^e,\dot{\theta}_i^e,\gamma_{1i},\gamma_{2i},\gamma_{3i},\tilde{\eta}_{1i},\tilde{\eta}_{2i} $ are all bounded. According to Assumption 3, $ \eta_{1i} $ and $ \eta_{2i} $ are bounded, thus one can obtain that $ \hat{\eta}_{1i} $ and $ \hat{\eta}_{2i} $ are also bounded by using (37).

In addition, it follows from (27), (29), (31), (34) and (38) that $ \dot{x}_i^e, \dot{y}_i^e, $ $ \dot{\theta}_i^e, \dot{\bar{\theta}}_i^e $, $ \nu_{di}, \omega_{di}, \nu_{i}, \omega_{i} $, $ \dot{\hat{\eta}}_{1i}, \dot{\tilde{\eta}}_{1i}, \dot{\hat{\eta}}_{2i} $, and $ \dot{\tilde{\eta}}_{2i} $ are all bounded. Taking the derivative of $ \gamma_{1i}, \gamma_{2i} $ and $ \gamma_{3i} $, it can be inferred that $ \dot{\gamma}_{1i}, \dot{\gamma}_{2i} $ and $ \dot{\gamma}_{3i} $ are also bounded.

Then taking the derivative of $ \dot{V}_{2i} $, one can obtain

Using (41), one can obtain that $ \ddot{V}_{2i} $ is bounded. According to Barbalat’s lemma, one has $\lim_{t\rightarrow \infty}\dot{V}_{2i} = 0$, which means $ [x_i^e, y_i^e, \bar{\theta}_i^e]^T $ converge to zero asymptotically. Then it follows from (32) that all the formation errors $ [x_i^e, y_i^e, \theta_i^e]^T $ of each follower robot $ R_i $ converge to 0 gradually.

Secondly, it will prove that the estimated values $ [\hat{\eta}_{1i},\hat{\eta}_{2i}]^T $ of slipping ratios will converge to their true values $ [\eta_{1i},\eta_{2i}]^T $ asymptotically.

Since $ [x_i^e, y_i^e, \theta_i^e]^T $ converge to zero gradually, it follows from (27) and (31) that $\lim_{t\rightarrow \infty}\nu_{di} = \nu_{0}$, $\lim_{t\rightarrow \infty}\omega_{di} = \omega_{0}$, $\lim_{t\rightarrow \infty}\nu_{i} = \nu_{0}$, and $\lim_{t\rightarrow \infty}\omega_{i} = \omega_{0}$. According to (38), one has

It can be inferred from (42) that

For arbitrary $ \nu_{di} $ and $ \omega_{di} $, (42) is satisfied. By using (42) and (43), one has $\lim_{t\rightarrow \infty}\tilde{\eta}_{1i} = 0,\ \lim_{t\rightarrow \infty}\tilde{\eta}_{2i} = 0$, which means $\lim_{t\rightarrow \infty}\hat{\eta}_{1i} = \eta_{1i}$ and $\lim_{t\rightarrow \infty}\hat{\eta}_{2i} = \eta_{2i}$.■

Remark 7: By using the distributed finite time state observer (5)–(8), the estimated formation errors $ [\hat{x}_i^e, \hat{y}_i^e, \hat{\theta}_i^e]^T $ equal the actual formation errors $ [x_i^e, y_i^e, \theta_i^e]^T $ when $ t \geq T_0 $, then the proposed controllers (20)–(25) can realize the desired formation objective and overcome the effect of the unknown slippage adaptively. Also, the estimated values $ [\hat{\eta}_{1i}^*,\hat{\eta}_{2i}^*]^T $ of unknown slipping ratios converge to their true values $ [\eta_{1i},\eta_{2i}]^T $ gradually with the adaptive laws (24) and (25). It is noted that the proposed controllers (20)–(25) do not contain any global position information.

IV. Simulation Results

In this section, the effectiveness of the proposed finite time state observer (5)−(8) and distributed adaptive formation controllers (20)−(25) in Section III are verified by simulation. The communication topology among the multi-robot systems adopts a directed graph satisfying Assumption 2, which is shown as Fig. 2. By default, all variables are in SI units. The desired relative positions between leader and followers are selected as $[d_{10}^x\;\;d_{20}^x\;\;d_{30}^x\;\;d_{40}^x\;\;d_{50}^x\;\;d_{60}^x\;\;d_{70}^x\;\;d_{80}^x\;\;d_{90}^x\;\;]^T = [-30\;\; -30\; \; 0\;\;30\;\;30\;\; -60\;\;30\;\;90\;\;60]^T$ and $[d_{10}^y\;\;d_{20}^y\;\;d_{30}^y\;\;d_{40}^y \;\;d_{50}^y $ $d_{60}^y\;\;\,d_{70}^y\,\;\;d_{80}^y\,\;\; d_{90}^y\;\;]^T = [0\;\;-30\;\;-30\;\;-30\;\;0\;\;-60\;\;-60 \;\;-60$ $-90]^T $, as shown in Fig. 3. The leader robot $ R_0 $ starts from $[x_0(0),y_0(0),\theta_0(0)]^T = [0.5\;({\rm{m}}),1\;({\rm{m}}),\pi/6\;({\rm{rad}})]^T$, and the velocities of $ R_0 $ are chosen as $\nu_0(t) = 3.25-0.25\cos0.24t\;({\rm{m/s}})$ and $\omega_0(t) = 0.1\cos 0.2t \; ({\rm{rad/s}})$. The actual slipping ratios are set as $ \eta_{1i} = 1.5 $ and $\eta_{2i} = 1.25,\;i = 1,2,\ldots,N$, which are unknown for each follower robot. Note that the friction coefficient of different wheels can be different. The control parameters of follower robots are chosen as $k_0 = 0.9,\; k_{1i} = 10,\; k_{2i} = 15,\; \lambda_{1i} = 12,\; \lambda_{2i} = 4,\;i = 1, 2,\ldots,N$.

Figure  2.  Communication topology among the multi-robot systems.

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Figure  3.  Desired formation of the multi-robot systems.

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Fig. 4 presents the trajectories of all robots during $0-100\;{\rm{s}}$. It can be seen from the figure that the multi-robot systems converges to the desired formation pattern (as shown in Fig. 3) gradually from the initial pattern (represented by circles in Fig. 4), which shows the proposed algorithm achieves the desired effect.

Figure  4.  Trajectories of all robots.

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Fig. 5 shows the estimated formation tracking errors $ [\hat{e}_{i}^x\;\hat{e}_{i}^y\;\hat{e}_{i}^\theta]^T $ converge to the true values $ [e_{i}^x\;e_{i}^y\;e_{i}^\theta]^T $. Fig. 6 shows that the observation errors between observed and true values of the leader’s velocity and angular velocity converge to zero. The two figures verify the effectiveness of the finite time state observer.

Figure  5.  Formation tracking observation errors of follower robots.

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Figure  6.  Velocity and angular velocity observation errors of follower robots.

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Figs. 7 and 8 show the estimated values $ [\hat{\eta}_{1i}^*,\hat{\eta}_{2i}^*]^T $ of unknown slipping ratios converge to the true values $ [\eta_{1i},\eta_{2i}]^T $ gradually. In addition, it can be seen from Fig. 9 that all the tracking errors $ [x_i^e, y_i^e, \theta_i^e]^T $ of each follower robot converges to zero gradually. That is the proposed adatptive formation control algorithm can achieve the desired formation while overcoming the unknown slippage.

Figure  7.  Estimation errors of slipping ratio (left wheel of each robot).

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Figure  8.  Estimation errors of slipping ratio (right wheel of each robot).

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Figure  9.  Formation tracking errors of all follower robots.

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In order to test the robustness of the designed controllers to the change of slipping ratios, slipping ratios are chosen as $ \eta_{1i} = 1.5 $ and $ \eta_{2i} = 1.25 $ when $0\leq t < 30\;{\rm{s}}$, $ \eta_{1i} = 24 $ and $ \eta_{2i} = 20 $ when $30\leq t < 60\;{\rm{s}}$, $ \eta_{1i} = 3 $ and $ \eta_{2i} = 2.5 $ when $t\geq 60\;{\rm{s}}$, $i = 1,2,\ldots,N$. Then running the simulation again, one can get the formation tracking errors of all followers as Fig. 10. It can be seen from the figure that when changing the slipping ratios a lot, it has little influence on formation errors, which means the proposed controllers have good robustness to the change of slipping ratios and are more applicable. According to the simulation results above, the effectiveness of the proposed controllers is verified.

Figure  10.  Formation tracking errors of all follower robots when slipping ratios change.

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V. Conclusion

In this paper, distributed formation control problem of multi-robot systems without global position information and subject to slippage constraints is studied. The formation pattern of the system adopted is leader-follower structure and the design process can be divided into two parts. Firstly, a distributed finite time state observer without using global position information is designed such that each follower robot can estimate leader’s states in finite time. Secondly, a distributed adaptive formation controller is designed for each follower robot such that the desired formation objective can be achieved while overcoming the effect of unknown slippage. Finally, the effectiveness of the proposed observers and controllers are verified by simulation. In this paper, the communication topology among multi-robot systems is assumed to be fixed and communication delay is not considered. Hence, future works will take these factors into account.