Keywords

1 Introduction

To date, the trajectory tracking and path following control of wheeled mobile robots have been widely studied. There are no continuous time-invariant controllers to achieve state stabilization of WMRs due to the limitation of Brockett necessary condition [1]. A trajectory tracking control law based on backstepping method is proposed in [2], within which the tracking errors converge to zero uniformly asymptotically. Using dynamic feedback linearization, a local asymptotical tracking control scheme is shown in [3]. Clearly, sliding-mode control method is also a good way to solve control problem and makes systems robust to uncertainties and disturbances. By describing system from cartesian coordinate to polar coordinate, a sliding-mode tracking control law proposed in [4] guaranties the tracking errors ultimately bounded, while a large control may appear near the origin. Considering a universal sliding-mode control scheme for a class of nonlinear systems and transform the model equations of WMRs into a special form, controller proposed in [5] makes the system globally asymptotically stable. By designing a PI-type sliding-mode surface and an adaptive algorithm, the trajectory tracking errors are steered to zero asymptotically [6].

Almost under all situations, the trajectory tracking or path following controllers can be directly used for the tracking control of two WMRs if the position/orientation and linear/angular velocity information of reference robot are completely known by the tracker robot. However, under real circumstance, not all the information of the reference WMR can be known or easily detected, and less communication burden in hardware-layer of controller helps to build a reliable apparatus and decreases error-code rate [7]. Based on above practical considerations, it is desired to solve the tracking control problem of WMRs using only position information of the reference robot, which can be easily obtained even in indoor environment by camera [8] or UWB [9].

In this work, we first refer results in [10] to design estimators of reference robot using only its position information. Later, we introduce a full-state feedback sliding-mode controller which drives the system states converging to the stable sliding surface in finite time despite the model parameter uncertainties. The combination of state feedback control law with estimators contributes to a tracking controller with only position information of reference robot.

The paper is organized as follows. Section 2 contains problem formation, controller design is included in Sect. 3, simulation results and conclusion are presented in Sects. 4 and 5 respectively.

2 Problem Formation

Consider the dynamic model of WMRs described by

$$\begin{aligned} \left\{ \begin{array}{l} \dot{x} = v\cos \theta ,\dot{y} = v\sin \theta ,\dot{\theta }= \omega \\ m\dot{v} = \displaystyle \frac{{{\tau _1}\mathrm{{ + }}{\tau _\mathrm{{2}}}}}{R},I\dot{\omega }\mathrm{{ = }}\displaystyle \frac{L}{R}\left( {{\tau _\mathrm{{1}}} - {\tau _\mathrm{{2}}}} \right) \end{array} \right. \end{aligned}$$
(1)

where \(\left( {x,y} \right) \) is the coordinate of mass center, \(\theta \) denotes the posture angle, v and \(\omega \) represent linear and angular velocity respectively. \(\left( {\dot{v},\dot{\omega }} \right) \) are linear and angular accelerations. \({\tau _1}\) and \({\tau _2}\) denote driving torques of the right and left wheels. \(\left( {m,I,R,L} \right) \) are mass, inertia around the mass center, wheel diameter, distance between right and left wheel respectively, which are unknown parameters bounded by known bounds, i.e.,

$$\begin{aligned} \begin{array}{l} 0< {m_m} \le m \le {m_M},0< {I_m} \le I \le {I_M}\\ 0< {R_m} \le R \le {R_M},0 < {L_m} \le L \le {L_M} \end{array} \end{aligned}$$
(2)

where \({m_m},{m_M},{I_m},{I_M},{R_m},{R_M},{L_m},{L_M}\) are known positive constants.

The kinematic equations of the reference robot are as follows:

$$\begin{aligned} {\dot{x}_r} = {v_r}\cos {\theta _r},{\dot{y}_r} = {v_r}\sin {\theta _r},{\dot{\theta }_r} = {\omega _r} \end{aligned}$$
(3)

Assumption 1

The reference speeds and their first- and second-order derivatives \((v_r,\omega _r,\dot{v}_r,\dot{\omega }_r,\ddot{v}_r,\ddot{\omega }_r)\) are bounded by

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{v_{rM}} \ge {v_r} \ge {v_{rm}} > 0,{{\dot{v}}_{rM}} \ge \left| {{{\dot{v}}_r}} \right| ,{{\ddot{v}}_{rM}} \ge \left| {{{\ddot{v}}_r}} \right| }\\ {{\omega _{rM}} \ge \left| {{\omega _r}} \right| ,{{\dot{\omega }}_{rM}} \ge \left| {{{\dot{\omega }}_r}} \right| ,{{\ddot{\omega }}_{rM}} \ge \left| {{{\ddot{\omega }}_r}} \right| } \end{array}} \right. \end{aligned}$$
(4)

where \({v_{rM}},{v_{rm}},{\omega _{rM}},{\dot{v}_{rM}},{\dot{\omega }_{rM}},{\ddot{v}_{rM}},{\ddot{\omega }_{rM}}\) are positive constants.

Assumption 2

The exact position \(({x_r},{y_r})\) of the reference robot is known.

Define the tracking errors as

$$\begin{aligned} {e_x} = x - {x_r},{e_y} = y - {y_r},{e_\theta } = \theta - {\theta _r} \end{aligned}$$
(5)

With Assumptions 1 and 2, the control task in this paper is to design control law

$$\begin{aligned} \left[ {\begin{array}{*{20}{l}} {{\tau _1}}\\ {{\tau _2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} {{U_1}\left( {x,y,\theta ,v,\omega ,{x_r},{y_r},\varOmega } \right) }\\ {{U_2}\left( {x,y,\theta ,v,\omega ,{x_r},{y_r},\varOmega } \right) } \end{array}} \right] \end{aligned}$$
(6)

such that

$$\begin{aligned} \mathop {\lim }\limits _{t \rightarrow \infty } {e_x} = 0,\mathop {\lim }\limits _{t \rightarrow \infty } {e_y} = 0,\mathop {\lim }\limits _{t \rightarrow \infty } {e_\theta } = 0 \end{aligned}$$
(7)

where \(\varOmega \) denotes the set of auxiliary variables.

3 Controller Design

In this section, we first give out some preliminary results that refer to [10] and estimate some values of reference robot that are not known exactly. Later, a robust state feedback controller will be introduced. Combining estimating algorithm and state feedback controller leads to the robust tracking controller with only position information of the target.

3.1 Target Observer Design

From Assumptions 1 and 2, we know that \((x_r,y_r)\) is measurable and their derivatives are bounded, so that we can estimate their first, second, and third derivatives by the exact differentiators proposed in [10] as follows:

$$\begin{aligned} \begin{array}{l} {{\dot{f}}_{0x}} = {w_{0x}},{w_{0x}} = - {\lambda _0}{\left| {{f_{0x}} - {x_r}} \right| ^{\frac{3}{4}}}\mathrm{{sign}}\left( {{f_0} - {x_r}} \right) + {f_{1x}}\\ {{\dot{f}}_{1x}} = {w_{1x}},{w_{1x}} = - {\lambda _1}{\left| {{f_{1x}} - {w_{0x}}} \right| ^{\frac{2}{3}}}\mathrm{{sign}}\left( {{f_{1x}} - {w_{0x}}} \right) + {f_{2x}}\\ {{\dot{f}}_{2x}} = {w_{2x}},{w_{2x}} = - {\lambda _2}{\left| {{f_{2x}} - {w_{1x}}} \right| ^{\frac{1}{2}}}\mathrm{{sign}}\left( {{f_{2x}} - {w_{1x}}} \right) + {f_{3x}}\\ {{\dot{f}}_{3x}} = - {\lambda _3}\mathrm{{sign}}\left( {{f_{3x}} - {w_{2x}}} \right) \end{array} \end{aligned}$$
(8)
$$\begin{aligned} \begin{array}{l} {{\dot{f}}_{0y}} = {w_{0y}},{w_{0y}} = - {\lambda _0}{\left| {{f_{0y}} - {y_r}} \right| ^{\frac{3}{4}}}\mathrm{{sign}}\left( {{f_{0y}} - {y_r}} \right) + {f_{1y}}\\ {{\dot{f}}_{1y}} = {w_{1y}},{w_{1y}} = - {\lambda _1}{\left| {{f_{1y}} - {w_{0y}}} \right| ^{\frac{2}{3}}}\mathrm{{sign}}\left( {{f_{1y}} - {w_{0y}}} \right) + {f_{2y}}\\ {{\dot{f}}_{2y}} = {w_{2y}},{w_{2y}} = - {\lambda _2}{\left| {{f_{2y}} - {w_{1y}}} \right| ^{\frac{1}{2}}}\mathrm{{sign}}\left( {{f_{2y}} - {w_{1y}}} \right) + {f_{3y}}\\ {{\dot{f}}_{3y}} = - {\lambda _3}\mathrm{{sign}}\left( {{f_{3y}} - {w_{2y}}} \right) \end{array} \end{aligned}$$
(9)

where \({\lambda _i} > {L_r}\left( {i = 0,1,2,3} \right) \) with \(L_r=\max \{|{\dot{x}_r}|,|{\ddot{x}_r}|,|{\dddot{x}_r}|,|{\dot{y}_r}|,|{\ddot{y}_r}|,|{\dddot{y}_r}|\}\). By using (8) and (9), the exact estimation of \(({\dot{x}_r},{\ddot{x}_r},{\dddot{x}_r},{\dot{y}_r},{\ddot{y}_r},{\dddot{y}_r})\) can be obtained by \(({w_{0x}},{w_{1x}},{w_{2x}},{w_{oy}},{w_{1y}},{w_{2y}})\) in finite time.

Taking (3) into account and calculating the first- to third-order derivatives of \((x_r,y_r)\), we get

$$\begin{aligned} \left\{ \begin{array}{lll} {{\dot{x}}_r} &{}=&{} {v_r}\cos {\theta _r},{{\ddot{x}}_r} = {{\dot{v}}_r}\cos {\theta _r} - {v_r}{\omega _r}\sin {\theta _r}\\ {{\dot{y}}_r} &{}=&{} {v_r}\sin {\theta _r},{{\ddot{y}}_r} = {{\dot{v}}_r}\sin {\theta _r} + {v_r}{\omega _r}\cos {\theta _r}\\ {{\dddot{x}}_r} &{}=&{} {{\ddot{v}}_r}\cos {\theta _r} - 2{{\dot{v}}_r}{\omega _r}\sin {\theta _r} - {v_r}{{\dot{\omega }}_r}\sin {\theta _r} - {v_r}\omega _r^2\cos {\theta _r}\\ {{\dddot{y}}_r} &{}=&{} {{\ddot{v}}_r}\sin {\theta _r} + 2{{\dot{v}}_r}{\omega _r}\cos {\theta _r} + {v_r}{{\dot{\omega }}_r}\cos {\theta _r} - {v_r}\omega _r^2\sin {\theta _r} \end{array} \right. \end{aligned}$$
(10)

which suggests

$$\begin{aligned} \left\{ \begin{array}{lll} {\theta _r} &{}=&{} \arctan 2\left( {{{\dot{y}}_r},{{\dot{x}}_r}} \right) ,{v_r} = \sqrt{\dot{x}_r^2 + \dot{y}_r^2} ,{{\dot{v}}_r} = {{\ddot{x}}_r}\cos {\theta _r} + {{\ddot{y}}_r}\sin {\theta _r}\\ {\omega _r} &{}=&{} \displaystyle \frac{{{{\ddot{y}}_r}\cos {\theta _r} - {{\ddot{x}}_r}\sin {\theta _r}}}{{{v_r}}},{{\dot{\omega }}_r} = \displaystyle \frac{{{{{\dddot{y}}_r}}\cos {\theta _r} - {{{\dddot{x}}_r}}\sin {\theta _r} - 2{{\dot{v}}_r}{\omega _r}}}{{{v_r}}}\\ \forall &{}{v_r}&{} > 0 \end{array} \right. \end{aligned}$$
(11)

Thus, the estimated values of \((\theta _r,v_r,\omega _r,\dot{v}_r,\dot{\omega }_r)\) can be obtained as

$$\begin{aligned} \left\{ \begin{array}{lll} \hat{\theta }_r&{}=&{}\arctan 2\left( w_{0y},w_{0x}\right) \\ \hat{v}_r &{}=&{} \sqrt{\left( w_{0x}\right) ^2+\left( w_{0y}\right) ^2}, \hat{\dot{v}}_r=w_{1x}\cos {{\hat{\theta }}_r}+w_{1y}\sin {{\hat{\theta }}_r}\\ \hat{\omega }_r&{}=&{}\displaystyle \frac{w_{1y}\cos {{\hat{\theta }}_r}-w_{1x}\sin {{\hat{\theta }}_r}}{\hat{v}_r},\hat{\dot{\omega }}_r=\displaystyle \frac{w_{2y}\cos {{\hat{\theta }}_r}-w_{2x}\sin {{\hat{\theta }}_r}-2{\hat{\dot{v}}}_r \hat{\omega }_r}{\hat{v}_r} \end{array} \right. \end{aligned}$$
(12)

Remark 1

As \(\hat{v}_r\) appears in denominators of \(\left( \hat{\omega }_r,\hat{\dot{\omega }}_r \right) \) and converges to real value in finite time, we adopt the following strategy in control to avoid possible singularity when \(\hat{v}_r\) cross zero during transient process.

$$\begin{aligned} {\hat{v}_r} = \left\{ \begin{array}{lll} {v_{rm}}, \sqrt{{{\left( {{w_{0x}}} \right) }^2} + {{\left( {{w_{0y}}} \right) }^2}} \le {v_{rm}}\\ \sqrt{{{\left( {{w_{0x}}} \right) }^2} + {{\left( {{w_{0y}}} \right) }^2}} , \sqrt{{{\left( {{w_{0x}}} \right) }^2} + {{\left( {{w_{0y}}} \right) }^2}} > {v_{rm}} \end{array} \right. \end{aligned}$$
(13)

3.2 Sliding-Mode Controller

Define the auxiliary position tracking errors

$$\begin{aligned} {e_1} = {e_x} + l\left( {\cos \theta - \cos {\theta _r}} \right) ,{e_2} = {e_y} + l\left( {\sin \theta - \sin {\theta _r}} \right) \end{aligned}$$
(14)

where constant \(l > 0\). Differentiating (14) along state trajectory of (5) results

$$\begin{aligned} \left[ {\begin{array}{*{20}{l}} {{{\dot{e}}_1}}\\ {{{\dot{e}}_2}} \end{array}} \right] = A\left( \theta \right) \left[ {\begin{array}{*{20}{l}} v\\ \omega \end{array}} \right] - A\left( {{\theta _r}} \right) \left[ {\begin{array}{*{20}{l}} {{v_r}}\\ {{\omega _r}} \end{array}} \right] \end{aligned}$$
(15)

where

$$\begin{aligned} A\left( a \right) \buildrel \varDelta \over = \left[ {\begin{array}{*{20}{c}} {\cos a}&{}{ - l\sin a}\\ {\sin a}&{}{l\cos a} \end{array}} \right] \rightarrow {A^{ - 1}}\left( a \right) = \left[ {\begin{array}{*{20}{c}} {\cos a}&{}{\sin a}\\ { - \displaystyle \frac{{\sin a}}{l}}&{}{\displaystyle \frac{{\cos a}}{l}} \end{array}} \right] \end{aligned}$$
(16)

Define the stable sliding-mode surfaces

$$\begin{aligned} s = \left[ \begin{array}{l} {s_1}\\ {s_2} \end{array} \right] = \left[ \begin{array}{l} {{\dot{e}}_1} + {k_1}{e_1}\\ {{\dot{e}}_2} + {k_1}{e_2} \end{array} \right] = A\left( \theta \right) \left[ \begin{array}{l} v\\ \omega \end{array} \right] - A\left( {{\theta _r}} \right) \left[ \begin{array}{l} {v_r}\\ {\omega _r} \end{array} \right] + {k_1}\left[ \begin{array}{l} {e_1}\\ {e_2} \end{array} \right] \end{aligned}$$
(17)

in which \(k_1\) is a positive constant.

Let \(\left( {{{\bar{\tau }}_1},{{\bar{\tau }}_2}} \right) = \left( {{\tau _1} + {\tau _2},{\tau _1} - {\tau _2}} \right) \) and \(\left( {{p_1},{p_2}} \right) = \left( {\displaystyle \frac{1}{{mR}},\displaystyle \frac{L}{{IR}}} \right) \), the derivative of (17) becomes

$$\begin{aligned} \dot{s} = A\left( \theta \right) \left[ \begin{array}{l} {p_1}{{\bar{\tau }}_1}\\ {p_2}{{\bar{\tau }}_2} \end{array} \right] + A\left( \theta \right) B\left( \omega \right) \left[ \begin{array}{l} v\\ \omega \end{array} \right] - \left[ \begin{array}{l} {\varDelta _1}\\ {\varDelta _2} \end{array} \right] \end{aligned}$$
(18)

where

$$\begin{aligned} B\left( a \right) \mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {{k_1}}&{}{ - la}\\ {\displaystyle \frac{a}{l}}&{}{{k_1}} \end{array}} \right] ,\left[ \begin{array}{l} {\varDelta _1}\\ {\varDelta _2} \end{array} \right] = A\left( {{\theta _r}} \right) \left[ \begin{array}{l} {{\dot{v}}_r}\\ {{\dot{\omega }}_r} \end{array} \right] + A\left( {{\theta _r}} \right) B\left( {{\omega _r}} \right) \left[ \begin{array}{l} {v_r}\\ {\omega _r} \end{array} \right] \end{aligned}$$
(19)

To realize the input-output decoupling, define the new sliding-mode surfaces

$$\begin{aligned} \bar{s} = \left[ \begin{array}{l} {{\bar{s}}_1}\\ {{\bar{s}}_2} \end{array} \right] = {A^{ - 1}}\left( \theta \right) s \end{aligned}$$
(20)

Differentiating \(\bar{s}\) leads to

$$\begin{aligned} \dot{\bar{s}}= \left[ \begin{array}{l} p_1 {\bar{\tau }}_1 \\ p_2 {\bar{\tau }}_2\end{array}\right] +\left[ \begin{array}{l} \delta _{11} \\ \delta _{21}\end{array}\right] +\left[ \begin{array}{l} \delta _{12} \\ \delta _{22}\end{array}\right] \end{aligned}$$
(21)

where

$$\begin{aligned} \begin{array}{l} \left[ \begin{array}{l} {\delta _{11}}\\ {\delta _{21}} \end{array} \right] = B\left( \omega \right) \left[ \begin{array}{l} v\\ \omega \end{array} \right] + {{\dot{A}}^{ - 1}}\left( \theta \right) A\left( \theta \right) \left[ \begin{array}{l} v\\ \omega \end{array} \right] + {{\dot{A}}^{ - 1}}\left( \theta \right) {k_1}\left[ \begin{array}{l} {e_x} + l\cos \theta \\ {e_y} + l\sin \theta \end{array} \right] \\ \\ \left[ \begin{array}{l} {\delta _{12}}\\ {\delta _{22}} \end{array} \right] = - {A^{ - 1}}\left( \theta \right) \left[ \begin{array}{l} {\varDelta _1}\\ {\varDelta _2} \end{array} \right] - {{\dot{A}}^{ - 1}}\left( \theta \right) A\left( {{\theta _r}} \right) \left[ \begin{array}{l} {v_r}\\ {\omega _r} \end{array} \right] - {{\dot{A}}^{ - 1}}\left( \theta \right) {k_1}l\left[ \begin{array}{l} \cos {\theta _r}\\ \sin {\theta _r} \end{array} \right] \end{array} \end{aligned}$$
(22)

Theorem 1

Suppose that Assumption 1 establishes and the control parameters satisfy \({k_1}> 0,{\varepsilon _1}> 0,{\varepsilon _2} > 0\), the sliding-mode control law

$$\begin{aligned} \left\{ \begin{array}{l} \bar{\tau }_1=-\hat{\bar{p}}_1 \delta _{11}-\hat{\bar{p}}_1\delta _{12}-\mathrm{{sign}} \left( \bar{s}\right) \left( {\bar{p}'}_1 |\delta _{11}|+{\bar{p}'}_1 |\delta _{12}|+\varepsilon _1\right) \\ \bar{\tau }_2=-\hat{\bar{p}}_2 \delta _{21}-\hat{\bar{p}}_2\delta _{22}-\mathrm{{sign}} \left( \bar{s}\right) \left( {\bar{p}'}_2 |\delta _{21}|+{\bar{p}'}_2 |\delta _{22}|+\varepsilon _2\right) \end{array} \right. \end{aligned}$$
(23)

guarantees that \(\left( {{{\bar{s}}_1},{{\bar{s}}_2}} \right) \) converge to the origin in finite time, where \({\bar{p}_1} = p_1^{ - 1},{\bar{p}_2} = p_2^{ - 1}\) are unknown positive constants bounded by known constants \(\left( {{{\bar{p}}_{1M}},{{\bar{p}}_{1m}},{{\bar{p}}_{2M}},{{\bar{p}}_{2m}}} \right) \), i.e.

$$\begin{aligned} {\bar{p}_{1M}} \ge {\bar{p}_1} \ge {\bar{p}_{1m}}> 0,{\bar{p}_{2M}} \ge {\bar{p}_2} \ge {\bar{p}_{2m}} > 0 \end{aligned}$$
(24)

and

$$\begin{aligned} \left\{ \begin{array}{l} \hat{\bar{p}}_1=0.5\left( \bar{p}_{1m}+\bar{p}_{1M}\right) ,\hat{\bar{p}}_2=0.5\left( \bar{p}_{2m}+\bar{p}_{2M}\right) \\ \bar{p}'_1= \max \left( {|\bar{p}_1-\hat{\bar{p}}_1|}\right) =0.5\left( \bar{p}_{1M}-\bar{p}_{1m}\right) \\ \bar{p}'_2= \max \left( {|\bar{p}_2-\hat{\bar{p}}_2|}\right) =0.5\left( \bar{p}_{2M}-\bar{p}_{2m}\right) \end{array} \right. \end{aligned}$$
(25)

Proof

Choose \({V_1} = 0.5{\bar{p}_1}\bar{s}_1^2\) and \({V_\mathrm{{2}}}\mathrm{{ = 0}}\mathrm{{.5}}{\bar{p}_2}s_2^2\) as Lyapunov candidates functions and compute their derivatives along with the trajectory of closed-loop system (21)–(23) as

$$\begin{aligned} \begin{array}{lll} \dot{V}_1&{}=&{}\bar{s}_1\left[ \delta _{11}\left( \bar{p}_1- \hat{\bar{p}}_1\right) +\delta _{12}\left( \bar{p}_1- \hat{\bar{p}}_1\right) \right] -|\bar{s}_1|\left( p'_1|\delta _{11}|+p'_1|\delta _{12}|+\varepsilon _1\right) \\ &{} \le &{} |\bar{s}_1||\delta _{11}\left( \bar{p}_1- \hat{\bar{p}}_1\right) |+|\delta _{12}\left( \bar{p}_1- \hat{\bar{p}}_1\right) |-|\bar{s}_1|\left( p'_1|\delta _{11}|+p'_1|\delta _{12}|+\varepsilon _1\right) \\ &{} \le &{} -\varepsilon _1|\bar{s}_1|-|\bar{s}_1|\left( p'_1-|\bar{p}_1- \hat{\bar{p}}_1|\right) |\delta _{11}|-|\bar{s}_1|\left( p'_1-|\bar{p}_1- \hat{\bar{p}}_1|\right) |\delta _{12}|\\ &{} \le &{} -\varepsilon _1|\bar{s}_1| \\ \dot{V}_2&{}=&{}\bar{s}_2\left[ \delta _{21}\left( \bar{p}_2- \hat{\bar{p}}_2\right) +\delta _{22}\left( \bar{p}_2- \hat{\bar{p}}_2\right) \right] -|\bar{s}_2|\left( p'_2|\delta _{21}|+p'_2|\delta _{22}|+\varepsilon _2\right) \\ &{} \le &{} |\bar{s}_2||\delta _{21}\left( \bar{p}_2- \hat{\bar{p}}_2\right) |+|\delta _{22}\left( \bar{p}_2- \hat{\bar{p}}_2\right) |-|\bar{s}_2|\left( p'_2|\delta _{21}|+p'_2|\delta _{22}|+\varepsilon _2\right) \\ &{} \le &{} -\varepsilon _2|\bar{s}_2|-|\bar{s}_2|\left( p'_2-|\bar{p}_2- \hat{\bar{p}}_2|\right) |\delta _{21}|-|\bar{s}_2|\left( p'_2-|\bar{p}_2- \hat{\bar{p}}_2|\right) |\delta _{22}|\\ &{} \le &{} -\varepsilon _2|\bar{s}_2| \end{array} \end{aligned}$$
(26)

Let \({W_1} = \sqrt{2\bar{p}_1^{-1}{V_1}} = \left| {{{\bar{s}}_1}} \right| ,{W_2} = \sqrt{2\bar{p}_2^{ - 1}{V_2}} = \left| {{{\bar{s}}_2}} \right| \), we then obtain

$$\begin{aligned} {D^ + }{W_1} = \frac{{2\bar{p}_1^{ - 1}{{\dot{V}}_1}}}{{2\sqrt{2\bar{p}_1^{ - 1}{V_1}} }} \le - {\varepsilon _1}\bar{p}_1^{ - 1},{D^ + }{W_2} = \frac{{2\bar{p}_2^{ - 1}{{\dot{V}}_2}}}{{2\sqrt{2\bar{p}_2^{ - 1}{V_2}} }} \le - {\varepsilon _2}\bar{p}_2^{ - 1} \end{aligned}$$
(27)

Comparison principle can then be used to obtain the conservative estimation of converging time of \(\left( {{{\bar{s}}_1},{{\bar{s}}_\mathrm{{2}}}} \right) \) and we get

$$\begin{aligned} {\bar{s}_1}\left( t \right) = 0,{\bar{s}_2}\left( t \right) = 0,\forall t \ge {T_1} = \max \left\{ {{{\bar{p}}_{1M}}\frac{{\left| {{{\bar{s}}_1}\left( 0 \right) } \right| }}{{{\varepsilon _1}}},{{\bar{p}}_{2M}}\frac{{\left| {{{\bar{s}}_2}\left( 0 \right) } \right| }}{{{\varepsilon _2}}}} \right\} \end{aligned}$$
(28)

According to (21), we know that \(\left( {{s_1}\left( t \right) ,{s_2}\left( t \right) } \right) = \left( {0,0} \right) \) for \(t \ge {T_1}\). On the sliding surface \(\left( {{s_1}\left( t \right) ,{s_2}\left( t \right) } \right) = \left( {0,0} \right) \), the auxiliary position tracking errors \(\left( {{e_1},{e_2}} \right) \) will converge to zero exponentially.

Next, we show that the overall tracking error system is asymptotically stable because the zero-dynamics subsystem of (15), associated with \(e_{\theta }\), is asymptotically stable. Nulling \(\left( {{{\dot{e}}_1},{{\dot{e}}_2}} \right) \) in (15) gives rise to

$$\begin{aligned} \left[ \begin{array}{l} v\\ \omega \end{array} \right] = {A^{ - 1}}\left( \theta \right) A\left( {{\theta _r}} \right) \left[ \begin{array}{l} {v_r}\\ {\omega _r} \end{array} \right] \end{aligned}$$
(29)

Take out the angular velocity and write the dynamics of \({\dot{e}_\theta }\) as

$$\begin{aligned} {\dot{e}_\theta }\mathrm{{ = }}\dot{\theta }- {\dot{\theta }_r}\mathrm{{ = }}\omega - {\omega _r} = - \frac{{{v_r}}}{l}\sin {e_\theta } + {\omega _r}\left( {\cos {e_\theta } - 1} \right) \end{aligned}$$
(30)

Linearize (30) at \({e_\theta } = 0\), we obtain

$$\begin{aligned} \dot{e}_\theta = - \frac{v_r}{l}e_\theta \end{aligned}$$
(31)

which is exponentially stable under Assumption 1. So the overall closed-loop system is concluded locally asymptotically stable [11] and (7) establishes.

Replacing the unmeasurable variables \(\left( {{\theta _r},{v_r},{\omega _r},{{\dot{v}}_r},{{\dot{\omega }}_r}} \right) \) with their estimates \(\left( \hat{\theta }_r,\hat{v}_r,\hat{\omega }_r,\hat{\dot{v}}_r,\hat{\dot{\omega }}_r\right) \) in controller (23) leads to

$$\begin{aligned} \left\{ \begin{array}{l} \bar{\tau }_1=-\hat{\bar{p}}_1\delta _{11}-\hat{\bar{p}}_1 \hat{\delta }_{12}-\mathrm{{sign}} \left( \hat{\bar{s}}\right) \left( {\bar{p}'}_1 |\delta _{11}|+{\bar{p}'}_1 |\hat{\delta }_{12}|+\varepsilon _1\right) \\ \bar{\tau }_2=-\hat{\bar{p}}_2\delta _{21}-\hat{\bar{p}}_2\hat{\delta }_{22}-\mathrm{{sign}} \left( \hat{\bar{s}}\right) \left( {\bar{p}'}_2 |\delta _{21}|+{\bar{p}'}_2 |\hat{\delta }_{22}|+\varepsilon _2\right) \end{array} \right. \end{aligned}$$
(32)

where

$$\begin{aligned} \left\{ \begin{array}{l} \left[ \begin{array}{l} {{\hat{\delta }}_{12}}\\ {{\hat{\delta }}_{22}} \end{array} \right] = - {A^{ - 1}}\left( \theta \right) \left[ \begin{array}{l} {{\hat{\varDelta } }_1}\\ {{\hat{\varDelta } }_2} \end{array} \right] - {{\dot{A}}^{ - 1}}\left( \theta \right) A\left( {{{\hat{\theta }}_r}} \right) \left[ \begin{array}{l} {{\hat{v}}_r}\\ {{\hat{\omega }}_r} \end{array} \right] - {{\dot{A}}^{ - 1}}\left( \theta \right) {k_1}l\left[ \begin{array}{l} \cos {{\hat{\theta }}_r}\\ \sin {{\hat{\theta }}_r} \end{array} \right] \\ \left[ \begin{array}{l} {{\hat{\varDelta } }_1}\\ {{\hat{\varDelta } }_2} \end{array} \right] = A\left( {{{\hat{\theta }}_r}} \right) \left[ \begin{array}{l} {\hat{\dot{v}}_r}\\ {{\hat{\dot{\omega }} }_r} \end{array} \right] + A\left( {{{\hat{\theta }}_r}} \right) B\left( {{{\hat{\omega }}_r}} \right) \left[ \begin{array}{l} {{\hat{v}}_r}\\ {{\hat{\omega }}_r} \end{array} \right] \\ \left[ \begin{array}{l} {{\hat{\bar{s}}}_1}\\ {{\hat{\bar{s}}}_2} \end{array} \right] = {A^{ - 1}}\left( \theta \right) \left[ {A\left( \theta \right) \left[ \begin{array}{l} v\\ \omega \end{array} \right] - A\left( {{{\hat{\theta }}_r}} \right) \left[ \begin{array}{l} {{\hat{v}}_r}\\ {{\hat{\omega }}_r} \end{array} \right] + {k_1}\left[ \begin{array}{l} {{\hat{e}}_1}\\ {{\hat{e}}_2} \end{array} \right] } \right] \\ \left[ \begin{array}{l} {{\hat{e}}_1}\\ {{\hat{e}}_2} \end{array} \right] = \left[ \begin{array}{l} x - {x_r} + l\left( {\cos \theta - \cos {{\hat{\theta }}_r}} \right) \\ y - {y_r} + l\left( {\sin \theta - \sin {{\hat{\theta }}_r}} \right) \end{array} \right] \end{array} \right. \end{aligned}$$
(33)

Since the estimated variables \(\left( {{{\hat{\delta }}_{12}},{{\hat{\delta }}_{22}}} \right) \) converge to real ones in finite time, there exists \({T_2} > 0\) such that the performance of controller (32) equals to that of (23) for \(t \ge {T_2}\). Furthermore, we have

$$\begin{aligned} ({\bar{\tau }_1},{\bar{\tau }_\mathrm{{2}}}) \in {L_\infty }, \mathrm{{0}} \le t \le {T_\mathrm{{2}}} \end{aligned}$$
(34)

so that all states are bounded during transient process. Thus, the closed-loop system under control of (32) is also locally asymptotically stable.

4 Simulation Results

The model parameters of tracker robot are chosen from one real-wheeled mobile robot in the authors’ laboratory that satisfy

$$\begin{aligned} \left\{ \begin{array}{l} \left( {2 - \mathrm{{0}}\mathrm{{.2}}} \right) \mathrm{{kg}} \le m \le \left( {2 + \mathrm{{0}}\mathrm{{.2}}} \right) \mathrm{{kg}}\\ \left( {0.2 - \mathrm{{0}}\mathrm{{.02}}} \right) \mathrm{{m}} \le L \le \left( {0.2 + \mathrm{{0}}\mathrm{{.02}}} \right) \mathrm{{m}}\\ \left( {0.08 - \mathrm{{0}}\mathrm{{.008}}} \right) \mathrm{{kg}} \cdot {\mathrm{{m}}^2} \le I \le \left( {0.08 + \mathrm{{0}}\mathrm{{.008}}} \right) \mathrm{{kg}} \cdot {\mathrm{{m}}^2}\\ \left( {0.05 - \mathrm{{0}}\mathrm{{.005}}} \right) \mathrm{{m}} \le R \le \left( {0.05 + \mathrm{{0}}\mathrm{{.005}}} \right) \mathrm{{m}} \end{array} \right. \end{aligned}$$
(35)

which contribute to the inequalities

$$\begin{aligned} \left\{ \begin{array}{l} 0.121 = {{\bar{p}}_{1M}} \ge {{\bar{p}}_1} \ge {{\bar{p}}_{1m}} = 0.081\\ 0.027 = {{\bar{p}}_{2M}} \ge {{\bar{p}}_2} \ge {{\bar{p}}_{2m}} = 0.015 \end{array} \right. \end{aligned}$$
(36)

Let the position of reference robot be generated from an eight-shaped trajectory described by

$$\begin{aligned} {\dot{x}_r} = {g_r}\cos \left( {2{h_r}} \right) ,{\dot{y}_r} = {g_r}\sin \left( {{h_r}} \right) ,{\dot{h}_r} = {\varOmega _r} \end{aligned}$$
(37)

The initial states about robust differentiators are all set to zero, initial states and the rest parameters are

$$\begin{aligned} \left\{ \begin{array}{l} \left[ {x\left( 0 \right) ,y\left( 0 \right) ,\theta \left( 0 \right) } \right] = \left[ {0, - 2,0} \right] \\ \left[ {{x_r}\left( 0 \right) ,{y_r}\left( 0 \right) ,{\theta _r}\left( 0 \right) } \right] = [0, - 5,0]\\ l = 0.1,{k_1} = 0.5,{\varepsilon _1} = {\varepsilon _2} = 0.2\\ m = 1.8,L = 0.19,I = 0.081,R = 0.05\\ {\lambda _0} = 1.6,{\lambda _1} = 1.2,{\lambda _2} = 0.4,{\lambda _3} = 0.2\\ {g_r} = 0.5,{\varOmega _r} = 0.1 \end{array} \right. \end{aligned}$$
(38)

Define the tracking error function

$$\begin{aligned} {V_{tr}} = \sqrt{e_x^2 + e_y^2 + e_\theta ^2} \end{aligned}$$
(39)

The simulation results are all shown in Fig. 1.

Fig. 1
figure 1

The tracking trajectory and tracking error (39) under controller (32)

Full size image

Simulation results show that the robot has successfully catched up with the reference robot under proposed controller (33) and \(V_{tr}\) converges to zero asymptotically.

5 Conclusion

A robust sliding-mode controller with only position information of reference robot is obtained by combining sliding-mode control method with robust exact differentiators. Theoretical analysis shows that the overall closed-loop system is locally asymptotically stable. Numerical simulation results verify the efficiency of the propose controller. The proposed controller is robust to model parameters based on the sliding-model technique. The author would like to investigate the multiagent control problem of WMRs with uncertain model parameters and with only position information of neighbors in future work.