Bioinspired control design using cerebellar model articulation controller network for omnidirectional mobile robots

Abstract

As a learning mechanism that emulates the structure of the cerebellum, cerebellar model articulation controllers have been widely adopted in the control of robotic systems because of the fast learning ability and simple computational structure. In this article, a cerebellar model articulation controller–based neural network controller is developed for an omnidirectional mobile robot. With the powerful learning ability of cerebellar model articulation controller, a cerebellar model articulation controller neural network is constructed to learn the complex dynamics of the omnidirectional mobile robot such that the robot is controlled without a priori knowledge of the robot dynamics. In addition, to overcome the limitation of the neural network controller, a global control technique with a group of smooth switching functions is designed such that the global ultimately uniformly boundedness of cerebellar model articulation controller is achieved instead of conventional semi-global ultimately uniformly boundedness. Moreover, smooth decreasing boundary functions are synthesized into the controller to guarantee the transient control performance. Based on an omnidirectional mobile robot, numerical experiments have been conducted to demonstrate the effectiveness of the proposed cerebellar model articulation controller controller.

Introduction

Over the past decades, the mobile robots’ motion control has received much attention from both academia and industrial communities.^1–6 To overcome the nonholonomic constraints of the conventional two-wheeled mobile robots, omnidirectional mobile robots (OMRs) have been successfully developed, for example, steel wheel mobile robots, universal wheel mobile robots, Crawler mechanism, and orthogonal wheel mobile robots.^7,8 The omnidirectional mechanism enables the robots to control the rotation and translation of the mobile robot independently with significantly increased flexibility. Thus, the control of OMR has been widely studied by researchers.

An early study of the omnidirectional robot control was presented using the resolved acceleration control in by Watanabe. ⁷ Based on this work, the potential field-based motion planning controller was proposed to achieve the avoidance of obstacles. ⁸ Model predictive control (MPC) and sliding mode control have also been investigated to deal with the various unknown nonlinearities.^9–13 In the abovementioned works, the OMRs were well controlled with the full knowledge of the robot kinematic and dynamic models. However, in practice, the robot dynamics may not be available due to the modeling errors and environmental uncertainties. For example, the mass and inertia of payloads exerted on the mobile robot cannot always be known in advance, which may cause modeling uncertainties of robot control. This may result in the control performance degradation or even lead to unstable phenomenon when the robot is situated in the unstructured and dynamically changing environments. Therefore, effective modeling methods are required to enhance the control performance.

In the literature, research on bioinspired robot learning and control has received considerable attention since the biological creatures, mechanisms, and the underlying principles are likely to bring novel ideas to improve control performance of the robot in the complex environment.^14–17 In 1972, based on a cerebellum neurophysiological model, Albus ¹⁴ proposed a learning mechanism that emulates the structure and function of the cerebellum, called cerebellar model articulation controller (CMAC). In comparison to the conventional back-propagation neural networks (NNs), the CMAC is characterized by the properties of rapid learning, simple structure, computational cost efficiency, and readiness for implementation.^18–20

Inspired by the learning ability of CMAC, in this article, we develop a CMAC NN–based mobile robot controller without a priori of knowledge of robot dynamics. It has been reported that, by employing “continuous” activation cells like Gaussian functions, the CMAC NN has the ability to approximate any smooth and continuous nonlinear function with the desired precision. Thus, the NN controller has been widely applied in the nonlinear systems.^20–29 An NN was presented to control the continuous- and discrete-time robot systems by constructing a novel adaptation law. ²⁰ A recurrent fuzzy wavelet NN was employed to address the unknown dynamics of a mobile manipulator robot. ³⁰ An adaptive NN–based automatic motion control has been developed for the subsystem of a wheeled inverted pendulum in Yang et al., ³¹ where the NN was employed to model the mobile robot with external uncertainties.

Although the approximation ability of NN has been investigated in the literature, an assumption that the input of NN should remain in a compact set is commonly used. ³² For a CMAC, inputs of the NN are only valid in a compact set defining by the overlap of receptive fields and hence its approximation ability only holds in a compact domain. ³² If the inputs are outside the compact set, the learning ability of CMAC may be no longer valid and would lead to unsatisfactory and even unstable control performance. This compact set is hard to determine in advance, especially for the multiple inputs and multiple outputs system which has a large number of NN inputs. Sanner and Slotine ³³ first proposed an approach using a robust controller for driving the escaped transient back to the compact set to realize the global tracking stability, with example applications on simple linearizable systems. This design has been further extended to input–output linearizable systems in Tzirkel-Hancock and Fallside. ³⁴ To improve the robustness of the control performance when the inputs run out of the compact set, in this article, we develop a global CMAC control scheme using a set of sufficiently smooth switching functions, which includes an adaptive CMAC controller and a robust control scheme. In comparison to the conventional CMAC NN control, the proposed method does not require the initial neural inputs to remain in a compact set and can completely turn off the neural controller if the inputs are outside the compact set.

It should also be noted that, during the motion control of mobile robots, the motion constraints should be considered. In Kanjanawanishkul and Zell, ⁹ a control strategy was designed to handle the input constraints of a mobile robot system using optimization method. Quadratic programming (QP) combined with MPC was proposed to incorporate the constraints into the control designs. ⁴ In the abovementioned methods, however, the motion constraint control needs to use the robot dynamic model to solve the optimization problem, which may not be available in advance. In this article, we develop a control algorithm for motion constraints of the OMRs inspired by a barrier Lyapunov function (BLF) technique. The BLF can indirectly constrain the system tracking performance by setting the constraint functions to the error signals, which has been reported in the literature.^35–37 To ensure both position and velocity tracking performance of the mobile robots, a proper BLF is constructed for the motion constraint control of the OMR such that the enhanced transient performance of motion control can be achieved.

Compared with the previous results, the main contributions of this article are highlighted as follows:

1. Construction of an adaptive CMAC NN controller for the OMR without using information of the robot dynamics.

2. Integration of a transient performance function into the proposed control scheme using error transformations and the BLF technique. This guarantees that both the position and velocity states of the OMR can achieve the desired transient performance.

3. Realization of the global stability of the CMAC NN using a switch mechanism design. In comparison to the conventional NN control, the proposed method does not require the initial neural inputs to remain in a compact set such that the initial condition can be relaxed.

Problem formulation and system description

Problem formulation

In this article, we aim to design an omnidirectional robot controller such that the controller can simultaneously guarantee that (1) the position states and velocity states of the mobile robot do not violate prescribed bounds and (2) all system states remain globally uniformly ultimately bounded (GUUB) without using the knowledge of the robot dynamics.

System description

The structure of the omnidirectional wheeled mobile robot is shown in Figure 1. The omnidirectional wheel consists of a wheel and a number of rollers that allow the wheel to moving flexibly in two directions (along with the wheel and along with the roller) in a plane as shown in Figure 2. Consider an OMR moving in the world coordinate frame (WCF), and its kinematic can be given by

$$\stackrel{·}{x}=f(x,\stackrel{·}{q})$$

(1)

Figure 1.

The structure of an omnidirectional wheeled mobile robot.

Figure 2.

An overlook of the omnidirectional wheel.

where $x(t)=[x_w,y_w,\theta {]}^T$ denotes the potions and orientation state of mobile robot in WCF and $q$ is a vector of robot wheel’s velocities.

To enhance the control performance, the robot dynamics is useful since it can be used to deal with the nonlinear and time-varying term in the motor drive system. The mobile robot dynamics can be first defined in a robot moving coordinate frame (RCF) as ⁸

$$\begin{array}{c}{\stackrel{··}{x}}_m=b_1{\stackrel{·}{x}}_m+b_3\stackrel{·}{y}\stackrel{·}{\theta }-c_1({\tau }_1+{\tau }_2-2{\tau }_3)\hfill \\ {\stackrel{··}{y}}_m=b_1{\stackrel{·}{y}}_m-b_3\stackrel{·}{x}\stackrel{·}{\theta }+\sqrt{3}{c}_1({\tau }_1-{\tau }_2)\hfill \\ \stackrel{··}{\theta }=b_2\stackrel{·}{\theta }+c_2({\tau }_1+{\tau }_2+{\tau }_3)\hfill \end{array}$$

(2)

where

$$\begin{array}{c}{b}_1=\frac{-3\varrho }{(2m{r}^2+3I_w)},b_3=\frac{2m{r}^2}{(2m{r}^2+3I_w)}\hfill \\ \multicolumn{1}{c}{c_1=\frac{\mathrm{kr}}{(2m{r}^2+3I_w)},b_2=\frac{-3\rho l_c^2}{(I_v{r}^2+3I_w{l}_c^2)}}\\ \multicolumn{1}{c}{c_2=\frac{\mathrm{kr}{l}_c}{(I_v{r}^2+3I_w{l}_c^2)}}\end{array}$$

where $\stackrel{··}{(}·{)}_m$ and $\stackrel{·}{(}·{)}_m$ denote the acceleration and velocity of the mobile robot in the RCF, respectively, $m$ denotes the robot mass, $I_{\omega }$ is the inertia of wheel around its central axis, $I_v$ is the inertial moment of the robot, $\varrho$ is a parameter of viscous friction coefficient between the wheel and the ground, $r$ is the wheel’s radius, $k$ is the gain factor, and ${\tau }_i$ denotes the driving input torque.

The dynamics of mobile robot in the RCF can be transfered to the WCF through the following rotation transformation matrix

$$R_w=\left[\begin{array}{cc}\cos \theta \hfill & -\sin \theta \hfill \\ \multicolumn{1}{c}{\sin \theta }& \cos \theta \hfill \end{array}\right]$$

(3)

Using equation (3) and the relationship that $[{\stackrel{·}{x}}_w,{\stackrel{·}{y}}_w{]}^T=R_w[{\stackrel{·}{x}}_m,{\stackrel{·}{y}}_m{]}^T$ , we can convert the above dynamic formulations (2) to WCF as

$$\stackrel{··}{x}=A(\stackrel{·}{x})\stackrel{·}{x}+B(x)\tau$$

(4)

where

$$\begin{array}{c}A(x)=\left[\begin{array}{ccc}{b}_1\hfill & \frac{-3I_v\stackrel{·}{\theta }}{2M{r}^2+3I_w}\hfill & 0\hfill \\ \multicolumn{1}{c}{\frac{3I_v\stackrel{·}{\theta }}{2M{r}^2+3I_w}}& b_1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & b_2\hfill \end{array}\right]\hfill \\ B(x)=\left[\begin{array}{ccc}-c_1(\sqrt{3}s\theta -c\theta )\hfill & c_1(\sqrt{3}s\theta -c\theta )\hfill & 2c_{1}c\theta \hfill \\ \multicolumn{1}{c}{c_1(-\sqrt{3}c\theta -s\theta )}& c_1(\sqrt{3}c\theta -s\theta )\hfill & 2c_{1}s\theta \hfill \\ \multicolumn{1}{c}{c_2}& c_2\hfill & c_2\hfill \end{array}\right]\hfill \end{array}$$

and $\tau =[{\tau }_1,{\tau }_2,{\tau }_3]$ .

Noted that the matrix $B(x)$ can be linearly parameterized as $B(x)=M_{0}P(x)$ , where

$$\begin{array}{c}{M}_0=\mathrm{diag}(c_1,c_1,c_2)\hfill \\ P(x)=\left[\begin{array}{ccc}-\sqrt{3}s\theta +c\theta \hfill & \sqrt{3}s\theta -c\theta \hfill & c\theta \hfill \\ \multicolumn{1}{c}{-\sqrt{3}c\theta -s\theta }& \sqrt{3}c\theta -s\theta \hfill & s\theta \hfill \\ \multicolumn{1}{c}{1}& 1\hfill & 1\hfill \end{array}\right]\hfill \end{array}$$

where $c$ and $s$ are the abbreviations of $\cos (·)$ and $\sin (·)$ , respectively. Then equation (4) can be written in the following form

$$\begin{array}{c}M{\stackrel{·}{x}}_2+C(x_2,x_1)x_2+G+d(t)=P(x_1)\tau \hfill \\ {\stackrel{·}{x}}_1=x_2\hfill \end{array}$$

(5)

where $x_1=[x_w,y_w,\theta {]}^T$ denotes the position vector of the robot and $x_2=[{\stackrel{·}{x}}_w,{\stackrel{·}{y}}_w,\stackrel{·}{\theta }]$ denotes the velocity vector of the mobile robot. $M=M_0^{-1}\in {\mathbb{R}}^{3\times 3}$ is the inertial matrix, $C(x_2)=-M_0^{-1}A(x_2)\in {\mathbb{R}}^{3\times 3}$ denotes the vector of centripetal and Coriolis torques, $G(q)=[0,0,0{]}^T$ is the gravitational force vector, and $d(t)\in {\mathbb{R}}^3$ denotes the external disturbance which is bounded by $|d(t)|\le \overline{d}$ with $\overline{d}$ being a positive constant. The following assumption is considered for the OMR to facilitate the design and analysis of the controller.

Assumption 1

During the tracking procedure, the trajectory is well generated and the obstacle is perfectly avoided by the OMR.

Assumption 2

The matrix inverse of $P(x)$ always existed such that no singularity problem will occur during the control process.

Preliminaries

1. CMAC NN.^19,20

Figure 3 shows a basic structure of the CMAC NN proposed in Commuri et al., ²⁰ where the CMAC was designed to control a robot manipulator. In the following, let us briefly review the mathematical description of the CMAC NN. The CMAC can be used to approximate the unknown continuous function, $F(Z)=[f_1(Z),f_2(Z),\dots ,f_n(Z)]$ , where $Z\in {\mathbb{R}}^m$ denotes the m-dimensional input space. As shown in Figure 3, two components are involved in the CMAC NN to determine the value of the approximated nonlinear function $F(Z)$

$$R:Z\to C$$

(6)

$$P:C\to F$$

(7)

where

$$\begin{array}{c}\multicolumn{1}{c}{Z\mathrm{is}\mathrm{the}m-\mathrm{dimensional}\mathrm{input}\mathrm{space},F\mathrm{is}\mathrm{the}}\\ \multicolumn{1}{c}{n-\mathrm{dimensional}\mathrm{input}\mathrm{space},}\\ \multicolumn{1}{c}{C\mathrm{is}\mathrm{the}{N}_c-\mathrm{dimensional}\mathrm{association}\mathrm{space}}\end{array}$$

(8)

Figure 3.

Structure of a CMAC neural network.

and $R(·)$ denotes the mapping between the input vector and the association space, that is, $\alpha =R(Z)$ . The outputs are computed through $P(\alpha )$ using a projection of the association vector $\alpha$ onto a weight vector such that

$$f=P(\alpha )=W^T\alpha$$

(9)

It should be noted that $R(Z)$ can be represented by a multidimensional receptive field function such that each point in input $Z$ is assigned with an activation value. The receptive field basis functions of the association vector are chosen to be the Gaussian function as follows

$$h_{\mathrm{ik}}(\Vert z_i-{\mu }_{\mathrm{ik}}\Vert )=\exp \left[\frac{-{(z_i-{\mu }_{\mathrm{ik}})}^2}{{\vartheta }_{\mathrm{ik}}^2}\right],k=1,2\cdots ,l$$

(10)

where $l$ is the number of blocks of the association space, $h_{\mathrm{ik}}$ denotes the $k\mathrm{th}$ block associated with the input $z_i$ , ${\mu }_{\mathrm{ik}}$ denotes the receptive field’s center, and ${\vartheta }_{\mathrm{ik}}$ is the variance of the Gaussian function. Then, the multidimensional receptive field function can be described as

$$S(Z)=[s_1,s_2,\dots ,s_l{]}^T\in {\mathbb{R}}^l$$

(11)

where $s_k(Z,{\mu }_k,{\vartheta }_k)=\underset{i=1}{\overset{m}{\Pi }}{h}_{\mathrm{ik}}(z_i)$ , ${\mu }_k=[{\mu }_{1k},{\mu }_{2k},\dots ,{\mu }_{\mathrm{mk}}{]}^T$ , and ${\vartheta }_k=[{\vartheta }_{1k},{\vartheta }_{2k},\dots ,{\vartheta }_{\mathrm{mk}}{]}^T$ . The following lemma shows the approximation ability provided by the CMAC NN.

Lemma 1

For a continuous nonlinear function $F(z)$ , there exists an ideal weight value $W^{*}$ such that $F(z)$ can be uniformly approximated by a CMAC with the multiplication of the optimal weights $W^{*}$ and the association vector $S(Z)$ as ²⁰

$$F(Z)=W^{T}S(Z)+\epsilon$$

(12)

where $\epsilon$ is the NN construction error, satisfying $||\epsilon ||\le {\epsilon }_N$ , and ${\epsilon }_N$ is a small bounded positive value.

In this article, we will use the CMAC NN to approximate the uncertain dynamics of the OMR.

• 2. Useful function and key lemma. ³⁸

The following definition and lemma would be used to derive the conclusion.

Definition 1

A smooth switching function $\hslash (z)$ can be constructed as follows ³⁹

$$\hslash (z)=\{\begin{array}{cc}1,\hfill & \mathrm{if}|z|<d_1\hfill \\ \multicolumn{1}{c}{\frac{d_2^2-z^2}{d_2^2-d_1^2}{e}^{{\left(\frac{z^2-d_1^2}{\iota (d_2^2-d_1^2)}\right)}^2},}& \mathrm{if}{d}_1\le |z|\le d_2\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{if}|z|>d_2\hfill \end{array}$$

(13)

where $d_1$ , $d_2$ , and $\iota$ are the selected constants with $0<d_1<d_2$ and $\iota >0$ .

The switching function (13) is a smooth nonlinear function with the value field $[0,1]$ . The output of $\hslash (z)$ scaled to 1 within $|z|<d_1$ and is equal to 0 outside the specified region $(|z|>d_2)$ . In between the values $d_1$ and $d_2$ , $\hslash (z)$ is a decreased smooth function, the slope of which can be adjusted by the parameter $\iota$ . The switching function would be used to extend the compact set of CMAC inputs later.

Lemma 2

For any ${\sigma }_c>0$ and $e\in \mathbb{R}$ , the following inequality holds ³⁵

$$0\le |e|-e\tanh \left(\frac{e}{{\sigma }_c}\right)\le \kappa {\sigma }_c$$

(14)

with $\kappa$ being a constant such that $\kappa =e^{-(\kappa +1)}$ , that is, $\kappa =0.2785$ .

Lemma 3

For a given positive constant vector ${\chi }_v$ , the following inequality exists ³⁵

$$\log \frac{{\chi }_v^T{\chi }_v}{{\chi }_v^T{\chi }_v-e_v^T{e}_v}\le \frac{e_v^T{e}_v}{{\chi }_v^T{\chi }_v-e_v^T{e}_v}$$

(15)

where $e_v$ is any vector in the interval $||e_v||<||{\chi }_v||$ .

Controller design

Before proceeding the control design, let us define the tracking errors of OMR as

$$\begin{array}{c}{e}_p={[e_{p1},e_{p2},e_{p3}]}^T=x_1-x_d\hfill \\ \multicolumn{1}{c}{e_v={[e_{v1},e_{v2},e_{v3}]}^T=x_2-\eta }\end{array}$$

(16)

where $x_d\in {\mathbb{R}}^3$ is the reference trajectories specified in the WCF and $\eta$ is the virtual controller to be specified later.

Specified tracking performance

To guarantee that the tracking error $e_p$ satisfies the predefined transient performance, we need to design proper prescribed transient bounds at first. Consider a series of smooth decreasing performance functions $\xi (t)$ as follows

$${\xi }_i(t)=({\gamma }_{0i}-{\gamma }_{\infty i})e^{-{\sigma }_{i}t}+{\gamma }_{\infty i},i=1,2,3$$

(17)

where ${\gamma }_{0i}$ , ${\gamma }_{\infty i}$ , and ${\sigma }_i$ are selected positive constants. Obviously, ${\xi }_i(0)={\gamma }_{0i}$ , $\underset{t\to \infty }{\lim }{\xi }_i(t)={\gamma }_{\infty i}$ , and the function ${\xi }_i(t)$ is smooth, positive, and bounded. Thus, we can apply it to regulate the transient performance of the tracking errors such that $-{\beta }_1\xi (t)<e_p<{\beta }_2\xi (t)$ , where ${\beta }_1$ and ${\beta }_2$ are positive constants.

The control objective of OMR is to track the desired position and velocity trajectories specified in the world frame while simultaneously to guarantee that the position tracking errors would not violate the prescribed bounds, that is, ${\varphi }^{-}<e_p<{\varphi }^{+}$ , as shown in Figure 4, where ${\varphi }^{-}=-{\beta }_1\xi (t)$ and ${\varphi }^{+}={\beta }_2\xi (t)$ . In order to guarantee the desired control transient performance, the constraint functions are specified in eqaution (16) and the functions ${\varphi }_i^{+}$ and ${\varphi }_i^{-}$ are used to regulate the motion constraints of the mobile robot. Under the motion constraints, the maximum overshoot of the tracking error should be less than ${\beta }_2{\gamma }_{0i}$ , the maximum undershoot should be greater than ${\beta }_1{\gamma }_{0i}$ , and the maximum steady-state tracking error is constrained by ${\beta }_1{\gamma }_{\infty i}$ and ${\beta }_2{\gamma }_{\infty i}$ . The required control performance can be regulated by properly adjusting the parameters ${\gamma }_{0i}$ , ${\gamma }_{\infty i}$ , and ${\sigma }_i$ . In addition, the constraints of the velocity tracking errors are guaranteed by $-{\chi }_{\mathrm{vi}}<e_{\mathrm{vi}}<{\chi }_{\mathrm{vi}}$ , where ${\chi }_{\mathrm{vi}}$ is a positive constant. Define the argument errors by applying error transformation on $e_1$ as ⁴⁰

$$\begin{array}{c}{\zeta }_a={[{\zeta }_{a1},{\zeta }_{a2},{\zeta }_{a3}]}^T={\left[\begin{array}{ccc}\frac{e_{p1}}{{\varphi }_1^{-}},\hfill & \frac{e_{p2}}{{\varphi }_2^{-}},\hfill & \frac{e_{p3}}{{\varphi }_3^{-}}\hfill \end{array}\right]}^T\hfill \\ \multicolumn{1}{c}{{\zeta }_b={[{\zeta }_{b1},{\zeta }_{b2},{\zeta }_{b3}]}^T={\left[\begin{array}{ccc}\frac{e_{p1}}{{\varphi }_1^{+}},\hfill & \frac{e_{p2}}{{\varphi }_2^{+}},\hfill & \frac{e_{p3}}{{\varphi }_3^{+}}\hfill \end{array}\right]}^T}\end{array}$$

(18)

Figure 4.

An illustration of the transient performance and prescribed bounds.

Then we have

$${\zeta }_i={\delta }_i(e_p){\zeta }_{\mathrm{bi}}+(1-{\delta }_i(e_p)){\zeta }_{\mathrm{ai}}$$

(19)

where ${\zeta }_{\mathrm{ai}}$ and ${\zeta }_{\mathrm{bi}}$ are the $i\mathrm{th}$ elements of ${\zeta }_a$ and ${\zeta }_b$ , respectively, and ${\delta }_i(e_{\mathrm{pi}})=\{\begin{array}{cc}1,\hfill & e_{\mathrm{pi}}\ge 0\hfill \\ \multicolumn{1}{c}{0,}& e_{\mathrm{pi}}<0\hfill \end{array}$ .

Global CMAC NN controller design

In this section, we aim to develop a CMAC NN–based OMR controller such that the robot tracking can achieve the desired transient performance in the presence of the unknown system dynamics. The controller can be designed in the following two steps:

• Step 1: robot transient control.

Inspired by the work of Tee and Li, ³⁵ we construct an asymmetric time-varying barrier Lyapunov candidate as

$$V_1=\sum _{i=1}^n\frac{1}{2}\log \frac{1}{1-{\zeta }_i^2}$$

(20)

with $n=3$ . Then considering the definition in equation (19), the time differentiation of equation (20) yields

$${\stackrel{·}{V}}_1=\sum _{i=1}^n\left(\frac{1-{\delta }_i}{1-{\zeta }_{\mathrm{ai}}^2}{\zeta }_{\mathrm{ai}}{\stackrel{·}{\zeta }}_{\mathrm{ai}}\right)+\sum _{i=1}^n\left(\frac{{\delta }_i}{1-{\zeta }_{\mathrm{bi}}^2}{\zeta }_{\mathrm{bi}}{\stackrel{·}{\zeta }}_{\mathrm{bi}}\right)$$

(21)

Substituting equations (16) and (18) into equation (21) and in terms of the definitions of ${\zeta }_a$ and ${\zeta }_b$ , we have

$${\stackrel{.}{V}}_1={\displaystyle \sum _{i=1}^n\left(\frac{1-{\delta }_i}{(1-{\zeta }_{ai}^2){\varphi }_i^{-}}{\zeta }_{ai}\left({\stackrel{.}{e}}_{pi}-e_{pi}\frac{{\stackrel{.}{\varphi }}_i^{-}}{{\varphi }_i^{-}}\right)\right)}+{\displaystyle \sum _{i=1}^n\left(\frac{{\delta }_i}{(1-{\zeta }_{bi}^2){\varphi }_i^{+}}{\zeta }_{bi}\left({\stackrel{.}{e}}_{pi}-e_{pi}\frac{{\stackrel{.}{\varphi }}_i^{+}}{{\varphi }_i^{+}}\right)\right)}$$

(22)

Notice that ${\stackrel{·}{e}}_{\mathrm{pi}}={\stackrel{·}{x}}_{1i}-{\stackrel{·}{x}}_{\mathrm{di}}=e_{\mathrm{vi}}+{\eta }_i-{\stackrel{·}{x}}_{\mathrm{di}}$ ; substituting it into equation (22) yields

$${\stackrel{.}{V}}_1={\displaystyle \sum _{i=1}^n\left(\frac{(1-{\delta }_i){\zeta }_{ai}}{(1-{\zeta }_{ai}^2){\varphi }_i^{-}}\left(e_{vi}+{\eta }_i-{\stackrel{.}{x}}_{di}-e_{pi}\frac{{\stackrel{.}{\varphi }}_i^{-}}{{\varphi }_i^{-}}\right)\right)}+{\displaystyle \sum _{i=1}^n\left(\frac{{\delta }_i{\zeta }_{bi}}{(1-{\zeta }_{bi}^2){\varphi }_i^{+}}\left(e_{vi}+{\eta }_i-{\stackrel{.}{x}}_{di}-e_{pi}\frac{{\stackrel{.}{\varphi }}_i^{+}}{{\varphi }_i^{+}}\right)\right)}$$

(23)

Subsequently, the virtual controller can be designed as

$$\eta =[{\eta }_1,{\eta }_2,{\eta }_3{]}^T={\stackrel{·}{x}}_d-K_1{e}_p+\sigma (t)e_p$$

(24)

where $K_1=\mathrm{diag}[k_{11},k_{12},k_{13}]$ is the selected positive gain matrix, $\sigma =\mathrm{diag}[{\sigma }_1,{\sigma }_2,{\sigma }_3]$ and ${\sigma }_i(t)=\sqrt{{({\stackrel{·}{\varphi }}_i^{-}/{\varphi }_i^{-})}^2+{({\stackrel{·}{\varphi }}_i^{+}/{\varphi }_i^{+})}^2+{\varsigma }_i}$ with ${\varsigma }_i$ being a positive constant. Notice that ${\sigma }_i$ would satisfy the following inequality

$${\sigma }_i(t)+{\delta }_i\frac{{\stackrel{·}{\varphi }}_i^{-}}{{\varphi }_i^{-}}+(1-{\delta }_i)\frac{{\stackrel{·}{\varphi }}_i^{+}}{{\varphi }_i^{+}}\ge 0$$

(25)

Then the combination of equations (23)–(25) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_1=\sum _{i=1}^n(\frac{(1-{\delta }_i){\zeta }_{\mathrm{ai}}}{(1-{\zeta }_{\mathrm{ai}}^2){\varphi }_i^{-}}+\frac{{\delta }_i{\zeta }_{\mathrm{bi}}}{(1-{\zeta }_{\mathrm{bi}}^2){\varphi }_i^{+}})e_{\mathrm{vi}}}\\ \multicolumn{1}{c}{-\sum _{i=1}^n{k}_{1i}(\frac{{\delta }_i{\zeta }_{\mathrm{bi}}}{1-{\zeta }_{\mathrm{bi}}^2}\frac{e_{\mathrm{pi}}}{{\varphi }_i^{+}}+\frac{(1-{\delta }_i){\zeta }_{\mathrm{ai}}}{1-{\zeta }_{\mathrm{ai}}^2}\frac{e_{\mathrm{pi}}}{{\varphi }_i^{-}})}\\ \multicolumn{1}{c}{+\sum _{i=1}^n\left(\frac{(1-{\delta }_i){\zeta }_{\mathrm{ai}}}{(1-{\zeta }_{\mathrm{ai}}^2){\varphi }_i^{-}}(e_{\mathrm{pi}}{\sigma }_i-e_{\mathrm{pi}}\frac{{\stackrel{·}{\varphi }}_i^{-}}{{\varphi }_i^{-}})\right)}\\ \multicolumn{1}{c}{+\sum _{i=1}^n\left(\frac{{\delta }_i{\zeta }_{\mathrm{bi}}}{(1-{\zeta }_{\mathrm{bi}}^2){\varphi }_i^{+}}(e_{\mathrm{pi}}{\sigma }_i-e_{\mathrm{pi}}\frac{{\stackrel{·}{\varphi }}_i^{+}}{{\varphi }_i^{+}})\right)}\end{array}$$

(26)

Let us define

$${\varpi }_i=\frac{{\delta }_i}{{({\varphi }_i^{+})}^2-e_{\mathrm{pi}}^2}+\frac{1-{\delta }_i}{{({\varphi }_i^{-})}^2-e_{\mathrm{pi}}^2}$$

(27)

Substituting equation (27) into equation (26) and in terms of equation (18), we can obtain

$${\stackrel{.}{V}}_1\le {\displaystyle \sum _{i=1}^n\left(-\frac{k_{1i}{\zeta }_i^2}{(1-{\zeta }_i^2)}+\left(\frac{{\delta }_i}{{({\varphi }_i^{+})}^2-e_{pi}^2}+\frac{1-{\delta }_i}{{({\varphi }_i^{-})}^2-e_{pi}^2}\right)e_{pi}{e}_{vi}\right)}\le {\displaystyle \sum _{i=1}^n\left(-\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}+{\varpi }_i{e}_{pi}{e}_{vi}\right)}$$

(28)

Then the barrier Lyapunov candidate is defined as follows

$$V_2=V_1+\frac{1}{2}\sum _{i=1}^n\log \frac{{\chi }_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}+\frac{1}{2}{e}_v^{T}M{e}_v$$

(29)

The derivative of equation (29) with respect to time yields

$$\begin{array}{c}{\stackrel{·}{V}}_2={\stackrel{·}{V}}_1+\sum _{i=1}^n\frac{e_{\mathrm{vi}}{\stackrel{·}{e}}_{\mathrm{vi}}}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}+e_v^{T}M{\stackrel{·}{e}}_v\hfill \\ \multicolumn{1}{c}{\le \sum _{i=1}^n({\varpi }_i{e}_{\mathrm{pi}}{e}_{\mathrm{vi}}+\frac{e_{\mathrm{vi}}{\stackrel{·}{e}}_{\mathrm{vi}}}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2})+e_v^{T}M{\stackrel{·}{e}}_v}\\ \multicolumn{1}{c}{-\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)}\end{array}$$

(30)

Differentiating $e_v$ with respect to time and combining equations (5) and (16), we can obtain

$${\stackrel{·}{e}}_v=M^{-1}[P(x_1)\tau -C(x_2)x_2-G(x_1)-d(t)]-\stackrel{·}{\eta }$$

(31)

Applying the Moore–Penrose inverse of $e_v$ , we have

$$e_v^T(e_v^T{)}^{+}=\{\begin{array}{cc}0,\hfill & e_v=0\hfill \\ \multicolumn{1}{c}{1,}& \mathrm{elsewhere}\hfill \end{array}$$

(32)

Notice that in the case of $e_v=[0,0,0{]}^T$ the system is asymptotically stable according to the result of Yin et al. ⁴¹ In the case that $e_v$ is not equal to zero, assuming that the dynamic terms $M$ , $C(x_1,x_2)$ , and $G$ are known, the model-based controller can be designed as

$$\tau =P^{-1}(x_1)(C(x_2)x_2-G(x_1)+M\stackrel{·}{\eta }-\varpi e_p-{\left(e_v^T\right)}^{+}(\sum _{i=1}^n\frac{e_{\mathrm{vi}}(a-\stackrel{·}{\eta })}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}+\sum _{i=1}^n\frac{k_{\mathrm{vi}}{e}_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2})-K_2{e}_v)$$

(33)

where $K_2=\mathrm{diag}\{k_{21},k_{22},k_{23}\}$ , $k_{2i}$ and $k_{\mathrm{vi}}(i=1,2,3)$ are the positive constants, $\varpi =\mathrm{diag}({\varpi }_1,{\varpi }_2,{\varpi }_3)$ , and $a={\stackrel{·}{x}}_2$ . It should also be noted that the input transformation matrix $P$ in equation (5) is nonsingular, and thus its inverse can be employed. Substituting equations (31), (32), and (33) into equation (30), we can obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_2\le \sum _{i=1}^n({\varpi }_i{e}_{\mathrm{pi}}{e}_{\mathrm{vi}}+\frac{e_{\mathrm{vi}}{\stackrel{·}{e}}_{\mathrm{vi}}}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2})+e_v^{T}M{\stackrel{·}{e}}_v-\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)}\\ \multicolumn{1}{c}{=\sum _{i=1}^n({\varpi }_i{e}_{\mathrm{pi}}{e}_{\mathrm{vi}}+\frac{e_{\mathrm{vi}}{\stackrel{·}{e}}_{\mathrm{vi}}}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2})-\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)}\\ \multicolumn{1}{c}{+e_v^T(P(x_1)\tau -C(x_2)x_2-G(x_1)-d(t)-M\stackrel{·}{\eta })}\\ \multicolumn{1}{c}{=-\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)-\sum _{i=1}^n\frac{k_{\mathrm{vi}}{e}_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}-e_v^T{K}_2{e}_v-e_v^{T}d(t)}\end{array}$$

(34)

• Step 2: global CMAC NN control.

In practice, the robot dynamics $M$ , $C(x_2)$ , and $G$ may be unavailable, and thus the model-based control might not be directly applied in the controller design. Therefore, we aim to develop a CMAC NN–based OMR controller such that the robot can achieve the desired control performance in the presence of unknown dynamics. Using the universal approximation ability of the CMAC NN, the system dynamic can be formulated as

$$F(Z)=M\stackrel{·}{\eta }+C(x_2)x_2+G(x_1)=W^{*T}S(Z)+\epsilon$$

(35)

where $W^{*}$ is the optimal NN weight matrix, $S(Z)$ denotes the NN regressor basis vector, $Z=[x_1^T,x_2^T,{\stackrel{·}{\eta }}^T{]}^T$ is a vector of NN inputs, and $\epsilon$ is the construction error.

It should be noted that the NN approximation capability only holds on the compact set ${\Omega }_1$ . To enhance the robustness of the NN-based controller, the following NN control law is constructed

$$\begin{array}{c}\multicolumn{1}{c}{\tau =P^{-1}(x_1)(-\sum _{i=1}^n\left({\varpi }_i{e}_{\mathrm{pi}}\right)-{(e_v^T)}^{+}(\sum _{i=1}^n\frac{e_{\mathrm{vi}}(a-\stackrel{·}{\eta })}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}}\\ \multicolumn{1}{c}{+\sum _{i=1}^n\frac{k_{\mathrm{vi}}{e}_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2})+D(Z){\Phi }_a+(I-D(Z)){\Phi }_b-K_3{e}_v)}\end{array}$$

(36)

where $K_3=\mathrm{diag}\{k_{31},k_{32},k_{33}\}$ , $k_{3i}$ and $k_{\mathrm{vi}}(i=1,2,3)$ are the positive constants, and $\hat{W}=[{\hat{W}}_2^T,{\hat{W}}_2^T,{\hat{W}}_2^T{]}^T$ is the estimation of $W^{*}$ . $D(Z)=\mathrm{diag}(h_1(Z),h_2(Z),h_3(Z))$ , $h_i(Z)=\underset{j=1}{\overset{l_i}{\Pi }}\hslash (z_j)$ , $\hslash (·)$ has been defined in equation (13), and ${\Phi }_a$ and ${\Phi }_b$ are, respectively, the NN term and the robust term defined as

$$\{\begin{array}{c}{\Phi }_a=\hat{f}(Z)={\hat{W}}^{T}S(Z)\hfill \\ \multicolumn{1}{c}{{\Phi }_b=f^U(Z)\Xi (Z)}\end{array}$$

(37)

where $f^U=\mathrm{diag}(f_1^U,f_2^U,f_3^U)$ , $\Xi (Z)=[\tanh (f_1^U{e}_{v1}/{\sigma }_c),\tanh (f_2^U{e}_{v2}/{\sigma }_c),\tanh (f_3^U{e}_{v3}/{\sigma }_c){]}^T$ , and ${\sigma }_c$ is a positive constant. Then we can design the weight tuning law as

$$\stackrel{·}{\hat{W}}=-\Gamma (D(Z)S(Z)e_v-{\sigma }_w\hat{W})$$

(38)

where $\Gamma$ is a positive definitive matrix and ${\sigma }_w$ is a positive constant. The weight leaning law (equation (38)) was used to train the NN weight $W$ . The last term of equation (38) is the σ-modification (leakage) term which can be used to ensure that adaptive weights do not shift far away from the optimal weights. There are also some other modification methods like deadzone, e-modification and parameter projection, which can also be employed in the weight updating of NNs. Figure 5 shows the concept of global NN tracking.

Figure 5.

An overview of the global NN control concept. The blue line denotes the phase trajectory of the neural inputs and the blue and red dots denote its starting and end points, respectively. The red circle denotes the compact set of the CMAC input.

Remark 1

As shown in equation (36), the proposed controller consists of a robust controller ${\Phi }_i^b$ and a CMAC NN controller ${\Phi }_i^a$ . Using the switching mechanism as proposed in equation (13), if the neural input runs outside of the designed outer compact set ${\Omega }_2$ , the extra robust term ${\Phi }_i^b$ is activated and will guide the input state to the inner compact set ${\Omega }_1$ . When the neural input remains in the compact set ${\Omega }_1$ , the term ${\Phi }_i^a$ is turned on and the robust controller is shut off. If the neural inputs run in the domain between ${\Omega }_2$ and ${\Omega }_1$ , both ${\Phi }_i^a$ and ${\Phi }_i^b$ will work.

Stability analysis

Theorem 1

Consider the mobile robot system in equation (5) and the designed tracking errors (equation (16)), given the initial conditions such that ${\varphi }_i^{-}(0)<e_{\mathrm{pi}}(0)<{\varphi }_i^{+}(0)$ and $-{\chi }_v<e_{\mathrm{vi}}(0)<{\chi }_v$ , and employ the controller (equation (33)) with the adaptive law in equation (38). Then, the proposed control adaptive global NN control scheme can guarantee that (1) all the tracking errors asymptoticly converge to a small neighborhood around zero, (2) the signals of the system are globally ultimately uniformly bounded, and (3) the prescribed transient bounds and the state constraints are never violated.

Proof

Let us consider the following Lyapunov candidate

$$L=V_2+\frac{1}{2}\sum _{i=1}^n{{\tilde{W}}_i}^T{\Gamma }^{-1}{\tilde{W}}_i$$

(39)

Taking the time derivate of $L$ , we have

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{L}={\stackrel{·}{V}}_2+\sum _{i=1}^n{{\tilde{W}}_i}^T{\Gamma }^{-1}{\stackrel{·}{\hat{W}}}_i}\\ \multicolumn{1}{c}{=\sum _{i=1}^n({\varpi }_i{e}_{\mathrm{pi}}{e}_{\mathrm{vi}}+\frac{e_{\mathrm{vi}}{\stackrel{·}{e}}_{\mathrm{vi}}}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2})-\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)}\\ \multicolumn{1}{c}{+e_v^T(P(x_1)\tau -C(x_1)x_2-G(x_1)-d(t)-M\stackrel{·}{\eta })}\\ \multicolumn{1}{c}{+\sum _{i=1}^n{{\tilde{W}}_i}^T{\Gamma }^{-1}{\stackrel{·}{\hat{W}}}_i}\end{array}$$

(40)

Substituting the control law (equation (33)) as well as the NN adaptive law (equation (38)) into equation (40), we can obtain

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{L}=-\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)+\sum _{i=1}^n({\varpi }_i{e}_{\mathrm{pi}}{e}_{\mathrm{vi}}+\frac{e_{\mathrm{vi}}{\stackrel{·}{e}}_{\mathrm{vi}}}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2})}\\ \multicolumn{1}{c}{+e_v^T(P(x_1)\tau -W^{*T}S-d(t)-\epsilon )+\sum _{i=1}^n{{\tilde{W}}_i}^T{\Gamma }^{-1}{\stackrel{·}{\hat{W}}}_i}\\ \multicolumn{1}{c}{\le -\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)-\sum _{i=1}^n\frac{k_{\mathrm{vi}}{e}_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}-e_v^T{K}_3{e}_v-e_v^{T}d(t)}\\ \multicolumn{1}{c}{+e_v^{T}D(Z){\tilde{W}}^{T}S(Z)-e_v^T\epsilon +{\sigma }_w{\tilde{W}}^T\hat{W}}\\ \multicolumn{1}{c}{+\sum _{i=1}^n(e_{\mathrm{vi}}{f}_i^U-e_{\mathrm{vi}}{f}_i^U\tanh \left(\frac{f_i^U{e}_{\mathrm{vi}}}{{\sigma }_c}\right))}\\ \multicolumn{1}{c}{-\sum _{i=1}^n{\tilde{W}}_i^{T}D(Z)S(Z)e_v}\end{array}$$

(41)

Note that the following inequality holds for any ${\sigma }_{\mathrm{ci}}>0$ and $e_{\mathrm{vi}}\in \mathbb{R}$

$$0\le |e_{\mathrm{vi}}|-e_{\mathrm{vi}}\tanh \left(\frac{e_{\mathrm{vi}}}{{\omega }_o}\right)\le \kappa {\sigma }_{\mathrm{ci}}$$

(42)

where $\kappa$ is a constant specified in equation (2). Substituting equation (42) into equation (41), we can obtain

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{L}\le -\sum _{i=1}^n\left(\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}\right)-\sum _{i=1}^n\frac{k_{\mathrm{vi}}{e}_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}-e_v^T{K}_3{e}_v}\\ \multicolumn{1}{c}{-e_v^T\epsilon -e_v^{T}d(t)+{\sigma }_w{\tilde{W}}^T\hat{W}+\sum _{i=1}^n\kappa {\sigma }_{\mathrm{ci}}}\end{array}$$

(43)

According to Young’s inequality, we have

$$\begin{array}{c}{\tilde{W}}^T(W^{*}-\tilde{W})\le -\frac{1}{2}||\tilde{W}|{|}^2+\frac{1}{2}||W^{*}|{|}^2\hfill \\ -e_v^T\epsilon \le \frac{1}{2}{e}_v^T{e}_v+\frac{1}{2}{\epsilon }^T\epsilon \hfill \\ -e_v^{T}d(t)\le \frac{1}{2}{e}_v^T{e}_v+\frac{1}{2}{\overline{d}}^2\hfill \end{array}$$

(44)

Substituting the above inequalities into equation (43), we have

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{L}\le -\sum _{i=1}^n\frac{k_{1i}{\zeta }_i^2}{1-{\zeta }_i^2}-\sum _{i=1}^n\frac{k_{\mathrm{vi}}{e}_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}-e_v^T(K_3-I)e_v}\\ \multicolumn{1}{c}{+\frac{1}{2}{\epsilon }^T\epsilon +\frac{1}{2}{\overline{d}}^2-{\sigma }_w(\frac{1}{2}||\tilde{W}|{|}^2+\frac{1}{2}||W^{*}|{|}^2)+\sum _{i=1}^n\kappa {\sigma }_{\mathrm{ci}}}\end{array}$$

(45)

Consider the inequality in Lemma 3 and note that the fact ${\zeta }_i^2/(1-{\zeta }_i^2)>\log (1/(1-{\zeta }_i^2))$ holds where $\forall |{\zeta }_i|<1$ ; then we have

$$\begin{array}{c}\stackrel{·}{L}\le -e_v^T(K_3-I)e_v-\sum _{i=1}^n{k}_{1i}\log \frac{1}{1-{\zeta }_i^2}\hfill \\ -\sum _{i=1}^n{k}_{\mathrm{vi}}\log \frac{{\chi }_{\mathrm{vi}}^2}{{\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2}-\frac{1}{2}{\sigma }_w||\tilde{W}|{|}^2\hfill \\ +\frac{1}{2}{\epsilon }^T\epsilon +\frac{1}{2}{\overline{d}}^2+{\sigma }_w\frac{1}{2}||W^{*}|{|}^2+\sum _{i=1}^n\kappa {\sigma }_{\mathrm{ci}}\hfill \end{array}$$

(46)

From equation (46) and the definition in equations (20), (29), and (39), we can derive

$$\stackrel{·}{L}\le -{\rho }_{l}L+\mu$$

(47)

where ${\rho }_l=\min [\min 2(k_{1i}),\min (2k_{\mathrm{vi}}),(2{\sigma }_w/{\lambda }_{\max }({\Gamma }^{-1})),(2{\lambda }_{\min }(K_3-I)/{\lambda }_{\max }(M))]$ , $\mu =(1/2){\epsilon }^T\epsilon +(1/2){\overline{d}}^2+{\sigma }_w(1/2)||W^{*}|{|}^2+\sum _{i=1}^n\kappa {\sigma }_{\mathrm{ci}}$ , and ${\lambda }_{\max }(•)$ and ${\lambda }_{\min }(•)$ are the maximum and minimum eigenvalues of (•), respectively. By choosing the gains $k_{1i}$ , $k_{\mathrm{vi}}$ , $K_3$ , $\Gamma$ , and ${\sigma }_w$ to satisfy $k_{1i}>0$ , $k_{\mathrm{vi}}>0$ , $K_3>I$ , $\Gamma >0$ , and ${\sigma }_w>0$ , respectively, we obtain ${\rho }_l>0$ . Also, $\mu$ is bounded according to the boundedness of $\epsilon$ , $\overline{d}$ , $\Vert W^{*}\Vert$ , and ${\sigma }_{\mathrm{ci}}$ .

From the above analysis, we can obtain that $\stackrel{·}{L}$ is negative when the system states of $L$ are outside of a compact set $\{\Omega :L\le {\rho }_l/\mu \}$ . In other words, if the system signals run outside of the compact set $\Omega$ , then $\stackrel{·}{L}<0$ and the signals will enter $\Omega$ and remain inside the set for all the future time. Thus, the closed-loop system can achieve an uniformly ultimately bounded stability. In addition, according to the boundedness of $\log (1/(1-{\zeta }_i^2))$ and $\log ({\chi }_{\mathrm{vi}}^2/({\chi }_{\mathrm{vi}}^2-e_{\mathrm{vi}}^2))$ , we can further derive $|{\zeta }_i|<1$ and $|e_{\mathrm{vi}}|<|{\chi }_{\mathrm{vi}}|$ in terms of the result of Tee and Li. ³⁵ Hence, the prescribed transient bound control is guaranteed. This completes the proof.

Simulation study

To verify the effectiveness of our proposed control scheme, simulations are carried out based on an OMR. The kinematic model and dynamic model are given by equations (1) and (5). The robot modeling parameters are given in Table 1.

Table 1.

Setting the dynamic parameter values of the OMR.

Symbols	Values	Unit
Robot mass $m$	15.0	kg
Robot inertia $I_{\omega }$	2.090	kg m2
Wheel’s inertia $I_v$	0.0750	kg m2
Distant of wheel $l_c$	0.118	m
Wheel’s radius $r$	0.050	m
Gain factor $k$	0.80	–
Friction coefficient $\varrho$	0.30	–

OMR: omnidirectional mobile robot.

In the simulation, two groups of reference trajectories have been considered: (1) setpoint case—the robot is controlled to arrive at a referred location with the desired orientation and (2) path following case—the robot is controlled to follow a reference path.

Setpoint cases

In the setpoint tracking case, the OMR is controlled to achieve a desired position $x_d$ and to guarantee the system states to satisfy the prescribed motion constraint at the same time.

The initial positions and velocities of the mobile robot are set to be $x_1=[0,0,0]$ and ${\stackrel{·}{x}}_1=[0,0,0]$ , respectively. The desired position is set to be $x_d=[2.5,2,0.5]$ . The control gains $K_1$ and $K_3$ are selected as $K_1=\mathrm{diag}(3,3,3)$ and $K_3=\mathrm{diag}(3,3,3)$ . The parameters of the prescribed transient function of the error signals are chosen to be ${\gamma }_0=[3,3,1.57{]}^T$ , ${\gamma }_{\infty }=[0.3,0.3,0.5{]}^T$ , $\beta =[1,1,1{]}^T$ , $\sigma =[1.5,1.5,1.5{]}^T$ , and ${\chi }_v=[5,5,2]$ . The receptive field is selected as $h_{\mathrm{ik}}(z_i)=\exp [-(z_i-{\mu }_{\mathrm{ik}}{)}^2/{\vartheta }_{\mathrm{ik}}^2]$ to cover the input space $\{[-1,1][-1,1][-0.5,0.5][-3,3][-3,3][-3,3]\}$ with each input dimension being partitioned into three grids and the layer of each input dimension being selected as four. Here, the parameter is chosen as ${\vartheta }_{\mathrm{ik}}=0.1$ , the CMAC weights are initialized to zero, and adaptive gains are selected as $\Gamma =1$ and ${\sigma }_w=0.01$ .

The simulation results of the setpoint control are shown in Figures 6–10. We can see from Figure 6 that the setpoint has been successfully arrived with the desired position and orientation. The position tracking errors of the setpoint cases are shown in Figure 7, where the red dashed lines depict the upper and lower bounds of the prescribed motion constraints. We can see that the tracking errors have converged to zero and did not violate the prescribed bounds. The tracking performance of $e_v$ is depicted in Figure 8, where the red dashed line and the yellow dashed line denote the upper and lower bounds of the velocity constraints, respectively. We can see that the velocity tracking errors have not violated the prescribed bounds in all directions. Also, a comparative study has shown that the prescribed bounds have been passed through under a traditional proportional–integral–derivative (PID) controller. The approximation performance of CMAC is depicted in Figure 9. The control input of the proposed controller is shown in Figure 10, where the boundedness of the control torque is observed.

Figure 6.

The actual path of the mobile robot in phase plane (setpoint).

Figure 7.

Tracking errors $e_p$ of the mobile robot setpoint control.

Figure 8.

Tracking error $e_v$ of the mobile robot setpoint control.

Figure 9.

Approximation performance of CMAC neural network in the setpoint case.

Figure 10.

Control input of the OMR in the setpoint case.

Path following cases

In the path following case, the mobile robot is commanded to follow a straight line under the proposed controller. The desired trajectory is selected as $x_{d1}=1+0.2t$ , $x_{d2}=3-0.2t$ , $x_{d3}=0.5$ , ${\stackrel{·}{x}}_{d1}=0.2$ , ${\stackrel{·}{x}}_{d2}=-0.2$ , and ${\stackrel{·}{x}}_{d2}=0$ . The prescribed bounds are chosen to be ${\xi }_1(t)=(3.5-0.3)e^{-1.5t}+0.3$ , ${\xi }_2(t)=(3.5-0.3)e^{-1.5t}+0.3$ , and ${\xi }_3(t)=(1.57-0.5)e^{-1.5t}+0.5$ . The velocity tracking errors are constrained by ${\chi }_v=[3,5,2{]}^T$ . The control gains are chosen to be $K_1=\mathrm{diag}[5,5,5]$ and $K2=\mathrm{diag}[1,1,1]$ . The receptive field is selected as $h_{\mathrm{ik}}(z_i)=\exp [-(z_i-{\mu }_{\mathrm{ik}}{)}^2/{\vartheta }_{\mathrm{ik}}^2]$ to cover the input space $\{[-1,1][-1,1][-0.5,0.5][-3,3][-3,3][-3,3]\}$ with each input dimension being partitioned into three grids and the layer of each input dimension being selected as 4. Here, the parameters are chosen the same as those in the setpoint case.

The simulation results of the path tracking control are shown in Figures 11–15. From Figure 11, we can see that, after a short period of adjustment, the mobile robot has successfully tracked the desired trajectory and followed it very well. The position tracking errors of the path following case are shown in Figure 12, where the red dashed lines depict the upper and lower bounds of the prescribed motion constraint. Under the proposed controller, the prescribed constraints of the position tracking errors are not violated as shown in Figure 12. Also, the tracking performance of $e_v$ is depicted in Figure 13, where the red dashed line and the yellow dashed line denote the upper and lower bounds of the velocity constraints, respectively. We can see from the figure that the velocity tracking errors have converged to zero and have not violated the prescribed bounds. In comparison, the tracking errors have transcended the prescribed bounds under the PID control. Figure 14 shows the approximation performance of unknown dynamics under the proposed CMAC controller in the path following case. In the three subfigures of Figure 14, the output profiles of $F(Z)$ are illustrated by the blue dashed lines, and the outputs of CMAC are depicted by the red solid lines. We can see that that favorable approximation response is obtained after a short period of time, in spite of the errors observed at the beginning of the approximation. Also, after the tracking errors are converged, both the error dynamics and the outputs of CMAC converge to a small neighborhood of zero. The processing time of CMAC is very short since only limited weights are updated with a given input in each step. Figure 15 shows the control input of the proposed controller.

Figure 11.

A overview of the tracking performance of the path following case.

Figure 12.

Tracking errors $e_p$ of the mobile robot in the path following case.

Figure 13.

Tracking errors $e_v$ of the mobile robot in the path following case.

Figure 14.

Approximation performance of CMAC neural network in the path following case.

Figure 15.

Control input of the OMR in the path following case.

To further verify the robustness of the proposed control method, simulations are performed in the presence of surface friction forces and external disturbances during path following, that is, ${\tau }_d=[5\sin (t),4\cos (t)+e^{-t},2\cos (t)+1.5e^{-t}{]}^T$ . The simulation results are depicted as shown in Figures 16 and 17. As we can see, the mobile robot shows satisfactory tracking performance of position and velocity even in the presence of external disturbances. Also, although a small steady-state tracking error can be observed, the prescribed bounds are not violated. Thus, the effectiveness of our proposed control scheme has been demonstrated.

Figure 16.

Tracking performance of $e_p$ in the presence of external disturbances.

Figure 17.

Tracking performance of ${\tilde{x}}_2$ in the presence of external disturbances.

To further demonstrate the advantages of our proposed method, comparative studies have been carried out based on the proposed controller and a traditional CMAC controller ²⁰ in the path following case. Figure 18 shows the velocity tracking errors under which a CMAC with the prescribed bound control (red solid line) is implemented as well as a CMAC is employed alone (blue dashed line). We can see from the first two subfigures that, although both controllers present satisfactory control performance, the proposed controller has successfully prevented the states from violating the prescribed bounds. This has shown the effectiveness of the proposed control method.

Figure 18.

The tracking performance of $e_v$ under the proposed controller.

Conclusion

In this article, we have developed an adaptive global CMAC controller for an OMR with guaranteed prescribed transient performance. The prescribed bounds of the mobile robot were specified by the designed performance functions and the BLF technique was employed to integrate the prescribed bounds into the controller. The CMAC was employed in the control of the mobile robot for the compensation of the unknown system dynamics. Particularly, smooth switching functions were constructed to ensure the GUUB of CMAC neural inputs. In comparison to the conventional CMAC NN control, the proposed method does not require the initial neural inputs to remain in a compact set such that the condition can be relaxed. Simulations were carried out based on the model of the OMR, and two different types of trails were tested. The simulation results have shown that the proposed control scheme can achieve the desired transient performance even in the presence of the unknown disturbances. The future work includes real robot tests as well as the robot obstacle avoidance.