Adaptive finite-time fuzzy command filtered controller design for uncertain robotic manipulators

Abstract

In this article, an adaptive fuzzy control command filtered control approach, which is capable of achieving finite-time trajectory tracking control of uncertain robotic manipulators, is proposed by using the backstepping control technique. To obtain a finite-time estimation of a virtual control input and its first-order derivative, a second-order finite-time command filter is designed. Based on the backstepping control technique, an adaptive fuzzy controller that guarantees not only that the tracking errors tend to an arbitrary small region in finite time but also that all signals in the closed-loop system keep bounded is established. Finally, the effectiveness of the proposed method is demonstrated by simulation on a two-link robotic manipulator.

Introduction

Since the early 1980s, control of robotic manipulators, which is a meaningful work and has wide practical applications, has received more and more attention of engineers and scholars from various fields. The model of the robotic manipulators is highly nonlinear, highly complicated, and coupled. ^{1 –7} In the early research of robots control, the prior knowledge of the system model are needed. A lot of control methods, for example, active control, inverse-based control, backstepping control, have been proposed for robotic manipulators with accurate system model. Nevertheless, in real-world applications, robots usually encounter system uncertainties, such as, sensor errors, modeling errors, external disturbances, parameters drift, which will damage the control performance if they are not well handled ?. ^{8 –16} It is well-known that fuzzy logic system (FLS) has been well used in controlling uncertain nonlinear systems because it does not need an accurate system model, and it can take advantage of human expert knowledge. The effectiveness of adaptive fuzzy control (AFC) has been shown in much literature. ^{17 –33} To reduce the fuzzy approximation errors as well as external disturbances, AFC method was usually cooperated with some robust compensator, such as sliding mode control (SMC) and $H_{\infty }$ control. Recently, a lot of AFC methods have been proposed for robotic manipulators, which can be divided into two different types. In the first AFC approach, the robotic’s mathematical model was fully unknown. According to the literature, ^{34 –36} FLSs were utilized to model system dynamics. It has been indicated that this method can eliminate chattering phenomenon. However, under this method, the structure of the AFC is complicated, and the computation burden was very large. Some other AFC or adaptive neural network control methods for manipulators can be refereed to the literature. ^{34,37 –40} In another type of AFC approach, the prior knowledge of the robotic systems should be known. ^41,42 It should be pointed out that under this method, the stability of the controlled system is hard to be theoretically discussed.

On the other hand, the finite-time control approach, which has fast responses, high tracking accuracy, and very good robustness, has been investigated extensively in recent years. ^{2,43 –45} Initial works about finite-time control mainly focused on discontinuous dynamic systems which can deteriorate the system transient performance. ^43,46 To develop finite-time stability for more general actual nonlinear systems, Bhat and Bernstein ⁴⁷ studied finite-time control of multidimensional continuous autonomous systems. Finite-time synchronization control of a class of neural networks with time delay using feedback technique was investigated byHu et al. ⁴⁶ Up to now, finite-time control approach has been used in control switched systems, ⁴⁸ impulsive dynamical systems, ⁴⁹ robotic manipulators, ^43,46 and so on. It should be mentioned that there are only few results of finite-time control of uncertain robotic manipulators due to the fact that it is very hard to cancel the effect of the system uncertainties in the finite-time controller design, and the stability analysis of the controlled robotic system is very complicated. Thus, how to design controller for uncertain robotic manipulators guaranteeing the finite-time convergence of the tracking error is an interesting but challenging work.

Adaptive backstepping control has been a powerful technique for designing controller for strict-feedback nonlinear systems. ^33,50,51 However, the problems “explosion of terms” and “functions should be linear” which are hard to be solved effectively usually occur in this method. To overcome the first problem, the dynamic surface control was introduced by Swaroop et al.. ⁵² It should be mentioned that the approximation errors that produced by a first-order dynamic surface was omitted, and the control performance was decreased. For this reason, command filtered control (CFC) was proposed by Farrell and colleagues. ^53,54 In the CFC, the approximation errors of the command filters were handled by some compensated signals. To deal with the second problem, the CFC can be used together with the AFC. However, how to design finite-time adaptive fuzzy CFC method for dynamical systems is still an open problem.

Motivated by the above discussion, in this work, an adaptive finite-time fuzzy command filtered control (AFTFCFC) is proposed for robotic manipulators to obtain finite-time tracking. FLSs are employed to approximate the unknown nonlinear functions as well as external disturbances. A second-order command filter that can guarantee the finite-time approximation of the virtual control inputs and their derivatives in the backstepping design is proposed. Thus, the problem “explosion of terms” can be solved. To handle the approximation of the finite-time command filter, compensated signals are designed. The stability analysis is discussed using Lyapunov stability criterion. The main contribution of our work can be summarized as: (1) an AFTFCFC method that can guarantee the finite-time convergence of the tracking error to an sufficiently small region is established, and the proposed method has self-tuning ability; (2) compared with the conventional backstepping control method, the “explosion of terms” problem is solved effectively by proposing a second-order finite-time command filter; and (3) based on the theoretical derivation and the simulation results, the proposed method has good robustness.

The structure of this work is as follows. The second section gives the description of FLS and some lemmas that are valid for the subsequent stability analysis. The description of the robotic manipulators and control problem is presented in the third section. The AFTFCFC design and stability analysis is given in the fourth section. Effectiveness of the proposed method is demonstrated by simulation results in the fifth section. Conclusions are described in the final section.

Preliminaries

Description of the FLS

An FLS usually has four parts, that is, the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier, which can be expressed as ⁵⁵

$\widehat{f}(\mathit{x})=\frac{\sum _{j\in J}{\theta }_j{\mu }_j(\mathit{x})}{\sum _{j\in J}{\mu }_j(\mathit{x})}$

1

with $\widehat{f}$ being the output, $x(t)=\hspace{0.17em}{\left[x_1(t),\hspace{0.17em}{x}_2(t),\hspace{0.17em}\dots ,\hspace{0.17em}{x}_n(t)\right]}^{\text{T}}\in C^1\left[\mathcal{I},\text{Ω}\right]\cap {\mathcal{R}}^n$ being the input. Denote $\mathit{\theta }(t)={\left[{\theta }_1(t),\cdots ,{\theta }_N(t)\right]}^{\text{T}}$ and $\psi (x(t))={\left[q_1(x(t)),q_2(x(t)),\cdots ,q_N(x(t))\right]}^{\text{T}}$ , where $N$ is the number of fuzzy rules and

$q_j(x(t))=\frac{{\theta }_j(t)}{\sum _{s\in J}{\mu }_s(\mathit{x})}$

2

Then, equation (1) can be rearranged as follows:

$\widehat{f}\left(\mathit{x}\right)={\mathit{\theta }}^{\text{T}}\psi \left(x\right)$

3

The FLS has the following property.

Lemma 1

Suppose that $f:\text{Ω}\mapsto \mathcal{R}$ is a Lipschitz continuous function. ⁵⁶ For any $\mathit{x}\in C^1\left[\mathcal{I},\text{Ω}\right]\cap {\mathcal{R}}^n$ and $\epsilon >0$ , there always exists a FLS in the form of equation (3) such that

$\underset{t\in \mathcal{I}}{\text{sup}}|f(\mathit{x})-{\mathit{\theta }}^{\text{T}}\psi (\mathit{x})|\le \epsilon$

4

Some useful results

To facilitate the stability analysis, we need the following results.

Lemma 2

Let $V(\mathit{y})\in {\mathcal{C}}^1\cap {\mathcal{R}}^{+}$ satisfying $\dot{V}(\mathit{y})+\alpha V^{\beta }(\mathit{y})$ with $\text{y}\in \text{Ω}\subset {\mathcal{R}}^n$ where $\text{Ω}$ is an open set and $\alpha \in {\mathcal{R}}^{+}$ , $\beta \in (0,1)$ . Then, one can find an open set ${\text{Ω}}_1\subset {\mathcal{R}}^n$ , such that any $V(\mathit{y})$ started from ${\text{Ω}}_1$ will reach $V(\mathit{y})\equiv 0$ in finite time. Furthermore, let $T_0$ be the time needed to arrive at $V(\mathit{y})\equiv 0$ , we have

$T_0\le \frac{V^{1-\beta }(\mathit{y}(0))}{\alpha \left(1-\beta \right)}$

5

Lemma 3

For any scalars ${\kappa }_1,{\kappa }_2\in {\mathcal{R}}^{+}$ and $\beta \in (0,1)$ , the finite-time stability can be guaranteed by

$\dot{V}\left(\mathit{y}\right)+{\kappa }_{1}V\left(\mathit{y}\right)+{\kappa }_2{V}^{\beta }\left(\mathit{y}\right)\le 0$

6

where $V(\mathit{y})$ is a Lyapunov function with $\mathit{y}\in \text{Ω}\subset {\mathcal{R}}^n$ . In addition, the settling time satisfies

$T_0\le t_0+\frac{\frac{1}{{\kappa }_1(1-\beta )}}{\text{ln}\left[{\kappa }_1{V}^{1-\beta }(t_0)+{\kappa }_2\right]-\text{ln}{\kappa }_2}$

7

Lemma 4

Consider the following Levant differentiator ^50,56 :

$\{\begin{array}{l}{\dot{\mathit{\eta }}}_1(t)=\mathit{\upsilon }(t),\hfill \\ \mathit{\upsilon }(t)=-b_1\mathbf{\text{Λsign}}({\mathit{\eta }}_1-\mathit{y})+{\mathit{\eta }}_2\hfill \\ {\dot{\mathit{\eta }}}_2(t)=-b_2\text{sign}({\mathit{\eta }}_2-\mathit{\upsilon }),\hfill \end{array}$

8

with $\mathit{y}\in \text{Ω}\subset {\mathcal{R}}^n$ being the input signal and $b_1,b_2\in {\mathcal{R}}^{+}$ being design parameters. If the design parameters are given properly, after some finite time, the following results hold:

${\mathit{\eta }}_1=\mathit{y},\mathit{\upsilon }=\dot{\mathit{y}}$

9

Problem description

The dynamic equation of an $n$ -link robotic manipulator is give as ^{38 –40,41}

$\mathit{M}(\mathit{\xi })\ddot{\mathit{\xi }}+\mathit{C}(\mathit{\xi },\dot{\mathit{\xi }})\dot{\mathit{\xi }}+\mathit{G}(\mathit{\xi })+\mathit{F}(\dot{\mathit{\xi }})=\mathit{u}(t)+\mathit{d}(t)$

10

with $\mathit{\xi }={\left[{\xi }_1,{\xi }_2,\cdots ,{\xi }_n\right]}^{\text{T}}\in {\mathcal{R}}^n$ denoting the joint position, $\dot{\mathit{\xi }}$ and $\ddot{\mathit{\xi }}$ representing the joint velocity and acceleration vectors, $\mathit{M}(\mathit{\xi })\in {\mathcal{R}}^{n\times n}$ standing for the positive definite inertia matrix satisfying ${\mu }_1\mathit{I}\le \mathit{M}(\mathit{\xi })\le {\mu }_2\mathit{I}$ ( ${\mu }_1,{\mu }_2$ are positive constants), $\mathit{C}(\mathit{\xi },\dot{\mathit{\xi }})$ denoting the centrifugal and Coriolis matrix, $\mathit{G}(\mathit{\xi })\in {\mathcal{R}}^{n\times n}$ being gravity force, $\mathit{F}(\dot{\mathit{\xi }})$ representing the friction vector, $\mathit{u}(t)\in {\mathcal{R}}^n$ being the control input, and $\mathit{d}(t)\in \mathcal{R}$ being the external disturbance. $\dot{\mathit{M}}(\mathit{\xi })-2\mathit{C}(\mathit{\xi },\dot{\mathit{\xi }})$ is a skew symmetric matrix, that is, ${\mathit{y}}^T\left[\dot{\mathit{M}}(\mathit{\xi })-2\mathit{C}(\mathit{\xi },\dot{\mathit{\xi }})\right]\mathit{y}=0$ , where $\mathit{y}\in {\mathcal{R}}^n$ .

Denote ${\mathit{x}}_1(t)=\mathit{\xi },{\mathit{x}}_2(t)=\dot{\mathit{\xi }}$ , and take system uncertainty into consideration, the robotic dynamic equation (10) is rearranged as

$\{\begin{array}{l}{\dot{\mathit{x}}}_1(t)={\mathit{x}}_2(t),\hfill \\ {\dot{\mathit{x}}}_2(t)={\mathit{M}}^{-1}(\mathit{x})\mathit{u}(t)+{\mathit{M}}^{-1}(\mathit{x})\mathit{d}(t)+f(\mathit{x})\hfill \end{array}$

11

where $\mathit{x}={\left[{\mathit{x}}_1^{\text{T}},{\mathit{x}}_2^{\text{T}}\right]}^{\text{T}}\in {\mathcal{R}}^{2n}$ , $f(\mathit{x})={\mathit{M}}^{-1}(\mathit{x})\left[-\mathit{C}({\mathit{x}}_1,{\mathit{x}}_2){\mathit{x}}_2-\mathit{G}({\mathit{x}}_1)-\mathit{F}({\mathit{x}}_2)\right]\in {\mathcal{R}}^n$ .

Let the smooth desired signal be ${\mathit{x}}_{\mathit{d}}\in {\mathcal{R}}^n$ . We aim to achieve the finite-time trajectory tracking, that is to say, for a given ${\mathit{x}}_{\mathit{d}}$ , to design proper backstepping control inputs ${\alpha }_1(t)$ (the virtual control input) and $\mathit{u}(t)$ with adaptation laws of the fuzzy parameters such that the system variable ${\mathit{x}}_1$ track ${\mathit{x}}_{\mathit{d}}$ in some finite-time and guarantee the boundedness of all the signals in the controlled system.

AFTFCFC design and stability analysis

To proceed, define the tracking error as ${\mathit{e}}_1(t)={\mathit{x}}_1(t)-{\mathit{x}}_{\mathit{d}}(t)$ . Then we have

$\begin{array}{c}{\dot{\mathit{e}}}_1={\mathit{x}}_2-{\dot{\mathit{x}}}_{\mathit{d}}\\ =\mathit{\alpha }+\widehat{\alpha }-\mathit{\alpha }+{\mathit{x}}_2-\widehat{\alpha }-{\dot{\mathit{x}}}_{\mathit{d}}\\ =\mathit{\alpha }+\widehat{\alpha }-\mathit{\alpha }+{\mathit{e}}_2-{\dot{\mathit{x}}}_{\mathit{d}}\end{array}$

12

where $\mathit{\alpha }\in {\mathcal{R}}^n$ is the virtual control to be designed, ${\mathit{e}}_2={\mathit{x}}_2-{\widehat{\alpha }}_1$ is the tracking error, and $\widehat{\alpha }\in {\mathcal{R}}^n$ is the estimation of $\mathit{\alpha }$ . In the conventional CFC for example, in, ^15,54,57 in every backstepping step, an error compensation system is given to reduce the effect of the virtual control input estimation error $\widehat{\alpha }-\mathit{\alpha }$ . However, the finite-time tracking performance can not be guaranteed by using aforementioned control methods. To overcome this limitation, here we will introduce a finite-time error compensation signal as

${\dot{z}}_1=-k_1{z}_1+\widehat{\alpha }-\mathit{\alpha }+z_2-c_1\text{sign}(z_1)$

13

where $z_2$ is also a compensation signal that will be defined in the next step, $k_1,c_1\in {\mathcal{R}}^{+}$ are design parameters, and the initial condition of $z$ is chosen as $z(\mathit{0})=0$ . To approximate the virtual control input $\mathit{\alpha }$ and its derivative, let us use the following finite-time command filter ⁵⁰ :

$\{\begin{array}{l}{\dot{\mathit{\eta }}}_1(t)=\mathit{\upsilon }(t),\hfill \\ \mathit{\upsilon }(t)=-b_1\mathbf{\text{Λsign}}({\mathit{\eta }}_1-\mathit{\alpha })+{\mathit{\eta }}_2\hfill \\ {\dot{\mathit{\eta }}}_2(t)=-b_2\text{sign}({\mathit{\eta }}_2-\mathit{\upsilon })\hfill \end{array}$

14

where $\mathbf{\text{Λ}}=\text{diag}\left(\sqrt{|{\mathit{\eta }}_{11}-{\mathit{\alpha }}_1|},\sqrt{|{\mathit{\eta }}_{12}-{\mathit{\alpha }}_2|},\cdots ,\sqrt{|{\mathit{\eta }}_{1n}-{\mathit{\alpha }}_n|}\right)$ . Thus, the outputs of the command filter in equation (14) can be used to approximate the the virtual control input $\mathit{\alpha }$ and its derivative, that is, $\widehat{\alpha }={\mathit{\eta }}_1$ and $\dot{\widehat{\alpha }}=\mathit{\upsilon }$ . Thus, the compensated tracking error is given as

$\{\begin{array}{l}{\tilde{\mathit{e}}}_1={\mathit{e}}_1-{\mathit{z}}_1\hfill \\ {\tilde{\mathit{e}}}_2={\mathit{e}}_2-{\mathit{z}}_2\hfill \end{array}$

15

where ${\mathit{e}}_2$ is defined as ${\mathit{e}}_2={\mathit{x}}_2-\widehat{\alpha }$ . The virtual control input is designed as

$\mathit{\alpha }=-k_1{\mathit{e}}_1+{\dot{\mathit{x}}}_{\mathit{d}}-a_1{\tilde{\mathit{e}}}_1^{\beta }$

16

where $a_1\in {\mathcal{R}}^{+}$ is a design parameter, $\beta \in (0,1)$ is a constant, and ${\tilde{\mathit{e}}}_1^{\beta }$ is defined as ${\tilde{\mathit{e}}}_1^{\beta }={\left[{\tilde{e}}_{11}^{\beta },{\tilde{e}}_{12}^{\beta },\cdots ,{\tilde{e}}_{1n}^{\beta }\right]}^{\text{T}}\in {\mathcal{R}}^n$ . Let the Lyapunov function be

$V_1=\frac{1}{2}{\tilde{\mathit{e}}}_1^T{\tilde{\mathit{e}}}_1$

17

and its derivative with respect to time is

$\begin{array}{c}{\dot{V}}_1={\tilde{\mathit{e}}}_1^{\text{T}}{\dot{\tilde{\mathit{e}}}}_1(t)\\ ={\tilde{\mathit{e}}}_1^{\text{T}}\left[\mathit{\alpha }+\widehat{\alpha }-\mathit{\alpha }+{\mathit{e}}_2-{\dot{\mathit{x}}}_{\mathit{d}}+k_1{\mathit{z}}_1-\widehat{\alpha }+\mathit{\alpha }-{\mathit{z}}_2+c_1\text{sign}({\mathit{z}}_1)\right]\\ ={\tilde{\mathit{e}}}_1^{\text{T}}\left[-k_1{\mathit{e}}_1-a_1{\tilde{\mathit{e}}}_1^{\beta }+{\mathit{e}}_2+k_1{\mathit{z}}_1-{\mathit{z}}_2+c_1\text{sign}({\mathit{z}}_1)\right]\\ =-k_1{\tilde{\mathit{e}}}_1^{\text{T}}{\tilde{\mathit{e}}}_1-a_1{\tilde{\mathit{e}}}_1^{\text{T}}{\tilde{\mathit{e}}}_1^{\beta }+{\tilde{\mathit{e}}}_1^{\text{T}}{\mathit{e}}_2-{\tilde{\mathit{e}}}_1^T{\mathit{z}}_2+c_1{\tilde{\mathit{e}}}_1^{\text{T}}\text{sign}({\mathit{z}}_1)\\ =-k_1{\tilde{\mathit{e}}}_1^{\text{T}}{\tilde{\mathit{e}}}_1-a_1{\tilde{\mathit{e}}}_1^{\text{T}}{\tilde{\mathit{e}}}_1^{\beta }+{\tilde{\mathit{e}}}_1^{\text{T}}{\tilde{\mathit{e}}}_2+c_1{\tilde{\mathit{e}}}_1^{\text{T}}\text{sign}({\mathit{z}}_1)\end{array}$

18

It follows from equations (11) and (15) that

${\dot{\tilde{\mathit{e}}}}_2={\mathit{M}}^{-1}(\mathit{x})\mathit{u}(t)+\text{Θ}(\mathit{x})-{\dot{\widehat{\alpha }}}_1-{\dot{\mathit{z}}}_2$

19

where $\mathbf{\Theta }(\mathit{x})={\mathit{M}}^{-1}(\mathit{x})\mathit{d}(t)+f(\mathit{x})={\left[{\Theta }_1(\mathit{x}),{\Theta }_2(\mathit{x}),\cdots ,{\Theta }_n(\mathit{x})\right]}^{\text{T}}\in {\mathcal{R}}^n$ , $\dot{\mathit{\alpha }}$ is generated by the finite-time command filter (equation (14)), and ${\dot{\mathit{z}}}_2$ will be defined later. Noting that $\mathbf{\Theta }(\mathit{x})$ is an unknown continuous function, it can be approximated by the FLS (equation (3)) as

${\widehat{\Theta }}_i={\mathit{\theta }}_i^{\text{T}}{\psi }_i(\mathit{x})$

20

Let ${\mathit{\theta }}_i^{*}$ be the optimal parameter defined by

${\mathit{\theta }}_i^{*}=\text{arg}\underset{{\mathit{\theta }}_i}{\text{min}}\left[\underset{\mathit{x}}{\text{sup}}|{\widehat{\Theta }}_i-{\Theta }_i|\right]$

21

Let the optimal parameter estimation and the optimal fuzzy approximation error be

${\tilde{\theta }}_i={\mathit{\theta }}_i-{\mathit{\theta }}_i^{*}$

22

and

${\epsilon }_i={\widehat{\Theta }}_i^{*}-{\Theta }_i$

23

respectively. Assume: $|{\epsilon }_i|\le {\overline{\epsilon }}_i,$ with ${\overline{\epsilon }}_i\in {\mathcal{R}}^{+}$ . Thus, we have

$\begin{array}{c}{\widehat{\Theta }}_i-{\Theta }_i={\mathit{\theta }}_i^{\text{T}}{\psi }_i(\mathit{x})-{\Theta }_i\\ ={\mathit{\theta }}_i^{\text{T}}{\psi }_i(\mathit{x})-{\mathit{\theta }}_i^{*\text{T}}{\psi }_i(\mathit{x})+{\mathit{\theta }}_i^{*\text{T}}{\psi }_i(\mathit{x})-{\Theta }_i\\ ={\tilde{\theta }}_i^{\text{T}}{\psi }_i(\mathit{x})-{\epsilon }_i\end{array}$

24

Let $\mathit{\theta }(t)={\left[{\mathit{\theta }}_1^{\text{T}}(t),{\mathit{\theta }}_2^{\text{T}}(t),\cdots ,{\mathit{\theta }}_n^{\text{T}}(t)\right]}^{\text{T}}\in {\mathcal{R}}^{N\times n}$ , ${\mathit{\theta }}^{\text{T}}\psi (\mathit{x})={\left[{\mathit{\theta }}_1^{\text{T}}{\psi }_1(\mathit{x}),{\mathit{\theta }}_2^{\text{T}}{\psi }_2(\mathit{x}),\cdots ,{\mathit{\theta }}_n^{\text{T}}{\psi }_n(\mathit{x})\right]}^{\text{T}}$ , ${\tilde{\theta }}^{\text{T}}\psi (\mathit{x})={\left[{\tilde{\theta }}_1^{\text{T}}{\psi }_1(\mathit{x}),{\tilde{\theta }}_2^{\text{T}}{\psi }_2(\mathit{x}),\cdots ,{\tilde{\theta }}_n^{\text{T}}{\psi }_n(\mathit{x})\right]}^{\text{T}}$ and $\epsilon ={\left[{\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n\right]}^{\text{T}}$ , then, equation (24) implies that

$\begin{array}{l}\widehat{\mathbf{\Theta }}-\mathbf{\Theta }={\tilde{\theta }}^{\text{T}}\psi \left(\mathit{x}\right)-\mathit{\epsilon }\hfill \end{array}$

25

The compensated signal ${\mathit{z}}_2$ is given as

${\dot{\mathit{z}}}_2=-k_2{\mathit{z}}_2-{\mathit{z}}_1-c_2\text{sign}({\mathit{z}}_2)$

26

where $k_2,c_2\in {\mathcal{R}}^{+}$ are two design parameters. Thus, the controller can be given as

$\mathit{u}=\mathit{M}(\mathit{x})\left[-k_2{\mathit{e}}_2+\dot{\widehat{\alpha }}-{\widehat{\mathit{\theta }}}^{\text{T}}\psi (\mathit{x})-\sigma \text{sign}({\tilde{\mathit{e}}}_2)-a_2{\tilde{\mathit{e}}}_2^{\beta }\right]$

27

with $\sigma ,a_2\in {\mathcal{R}}^{+}$ being two design parameters satisfying $\sigma \ge \text{max}\left\{{\overline{\epsilon }}_1,{\overline{\epsilon }}_2,\cdots ,{\overline{\epsilon }}_n\right\}$ . Substituting equations (25) to (27) into equation (19) gives

$\begin{array}{c}{\dot{\tilde{\mathit{e}}}}_2=-k_2{\mathit{e}}_2-{\widehat{\mathit{\theta }}}^{\text{T}}\psi (\mathit{x})-\sigma \text{sign}({\tilde{\mathit{e}}}_2)-a_2{\tilde{\mathit{e}}}_2^{\beta }+\mathbf{\Theta }(\mathit{x})-{\dot{\mathit{z}}}_2\\ =-k_2{\mathit{e}}_2-{\tilde{\theta }}^{\text{T}}\psi (\mathit{x})+\epsilon -\sigma \text{sign}({\tilde{\mathit{e}}}_2)-a_2{\tilde{\mathit{e}}}_2^{\beta }+k_2{\mathit{z}}_2-{\tilde{\mathit{e}}}_1+c_2\text{sign}({\mathit{z}}_2)\\ =-k_2{\tilde{\mathit{e}}}_2-{\tilde{\theta }}^{\text{T}}\psi (\mathit{x})+\epsilon -\sigma \text{sign}({\tilde{\mathit{e}}}_2)-a_2{\tilde{\mathit{e}}}_2^{\beta }-{\tilde{\mathit{e}}}_1+c_2\text{sign}({\mathit{z}}_2)\end{array}$

28

Multiplying ${\tilde{\mathit{e}}}_2^{\text{T}}$ to both sides of equation (28) gives

$\begin{array}{c}{\tilde{\mathit{e}}}_2^{\text{T}}{\dot{\tilde{\mathit{e}}}}_2=-k_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2-{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{\theta }}}^{\text{T}}\psi (\mathit{x})+{\tilde{\mathit{e}}}_2^{\text{T}}\epsilon -\sigma {\tilde{\mathit{e}}}_2^{\text{T}}\text{sign}({\tilde{\mathit{e}}}_2)-a_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2^{\beta }-{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_1+c_2{\tilde{\mathit{e}}}_2^{\text{T}}\text{sign}({\mathit{z}}_2)\\ =-k_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2-{\displaystyle \sum _{i=1}^n}{\mathit{e}}_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\psi }_i(\mathit{x})+{\displaystyle \sum _{i=1}^n}{\tilde{\mathit{e}}}_{2i}{\epsilon }_i-\sigma {\displaystyle \sum _{i=1}^n}\left|{\tilde{\mathit{e}}}_{2i}\right|-a_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2^{\beta }-{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_1+c_2{\tilde{\mathit{e}}}_2^{\text{T}}\text{sign}({\mathit{z}}_2)\\ \le -k_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2-{\displaystyle \sum _{i=1}^n}{\mathit{e}}_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\psi }_i(\mathit{x})+{\displaystyle \sum _{i=1}^n}\left|{\tilde{\mathit{e}}}_{2i}\right|{\overline{\epsilon }}_i-\sigma {\displaystyle \sum _{i=1}^n}\left|{\tilde{\mathit{e}}}_{2i}\right|-a_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2^{\beta }-{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_1+c_2{\tilde{\mathit{e}}}_2^{\text{T}}\text{sign}({\mathit{z}}_2)\\ \le -k_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2-{\displaystyle \sum _{i=1}^n}{\mathit{e}}_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\psi }_i(\mathit{x})-a_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2^{\beta }-{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_1+c_2{\tilde{\mathit{e}}}_2^{\text{T}}\text{sign}({\mathit{z}}_2)\end{array}$

29

Define the Lyapunov function as

$V_2=V_1+\frac{1}{2}{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2$

30

It follows from equations (18), (29), and (30) that

$\begin{array}{c}{\dot{V}}_2=-k_1{\tilde{\mathit{e}}}_1^{\text{T}}{\tilde{\mathit{e}}}_1-a_1{\tilde{\mathit{e}}}_1^{\text{T}}{\tilde{\mathit{e}}}_1^{\beta }+c_1{\tilde{\mathit{e}}}_1^{\text{T}}\text{sign}({\mathit{z}}_1)-k_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2-\sum _{i=1}^n{\mathit{e}}_{2i}{\tilde{\theta }}_i^{\text{T}}{\psi }_i(\mathit{x})-a_2{\tilde{\mathit{e}}}_2^{\text{T}}{\tilde{\mathit{e}}}_2^{\beta }+c_2{\tilde{\mathit{e}}}_2^{\text{T}}\text{sign}({\mathit{z}}_2)\\ =-\sum _{j=1}^2{k}_j{\tilde{\mathit{e}}}_j^{\text{T}}{\tilde{\mathit{e}}}_j-\sum _{j=1}^2{a}_j{\tilde{\mathit{e}}}_j^{\text{T}}{\tilde{\mathit{e}}}_j^{\beta }+\sum _{j=1}^2{c}_j{\tilde{\mathit{e}}}_j^{\text{T}}\text{sign}({\mathit{z}}_j)-\sum _{i=1}^n{\mathit{e}}_{2i}{\tilde{\theta }}_i^{\text{T}}{\psi }_i(\mathit{x})\\ =\sum _{j=1}^2\sum _{i=1}^n\left[-k_j{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}+c_j{\tilde{\mathit{e}}}_{ji}\text{sign}({\mathit{z}}_j)\right]-\sum _{i=1}^n{\mathit{e}}_{2i}{\tilde{\theta }}_i^{\text{T}}{\psi }_i(\mathit{x})\end{array}$

31

The adaptation law for ${\mathit{\theta }}_i$ is given as

${\dot{\mathit{\theta }}}_i={\gamma }_{1i}{\mathit{e}}_{2i}{\psi }_i(\mathit{x})-{\gamma }_{1i}{\gamma }_{2i}{\mathit{\theta }}_i$

32

where ${\gamma }_{1i},{\gamma }_{2i}\in {\mathcal{R}}^{+}$ are design constants.

Now we can conclude the above theoretical derivation as the following theorem.

Theorem 1

Consider the robotic manipulator system (equation (10)) or its equivalent form (equation (11)). Let the virtual control input $\mathit{\alpha }$ be as in equation (7). $\widehat{\alpha }$ (the estimation of $\mathit{\alpha }$ ) and its derivative is obtained by solving the finite-time command filter (equation (14)). The compensated tracking errors are defined in equations (13) and (26). Then, the proposed controller (equation (27)) accompanied with the adaptation law (equation (32)) guarantees not only that the tracking error ${\mathit{e}}_1$ tends to a small region of zero in some finite time but also the finite-time boundedness of all signals in the closed-loop system.

Proof

Let the Lyapunov function be

$V=V_2+\sum _{i=1}^n\frac{1}{2{\gamma }_{1i}}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i$

33

Then, based on equations (31) to (33), we obtain

$\begin{array}{c}\dot{V}\le {\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\left[-k_j{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}+c_j{\tilde{\mathit{e}}}_{ji}\text{sign}({\mathit{z}}_j)\right]-{\displaystyle \sum _{i=1}^n}{\mathit{e}}_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\psi }_i(\mathit{x})+{\displaystyle \sum _{i=1}^n}\frac{1}{{\gamma }_{1i}}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\dot{\mathit{\theta }}}_i\\ ={\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\left[-k_j{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}+c_j{\tilde{\mathit{e}}}_{ji}\text{sign}({\mathit{z}}_j)\right]-{\displaystyle \sum _{i=1}^n}{\gamma }_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\mathit{\theta }}_i\\ \le {\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\left[-\frac{2k_j-c_j}{2}{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}\right]-{\displaystyle \sum _{i=1}^n}{\gamma }_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\mathit{\theta }}_i+{\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\frac{c_j}{2}\\ ={\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\left[-\frac{2k_j-c_j}{2}{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}\right]-{\displaystyle \sum _{i=1}^n}{\gamma }_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}({\tilde{\mathit{\theta }}}_i+{\mathit{\theta }}_i^{*})+{\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\frac{c_j}{2}\\ ={\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\left[-\frac{2k_j-c_j}{2}{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}\right]-{\displaystyle \sum _{i=1}^n}{\gamma }_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\tilde{\mathit{\theta }}}_i+{\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\frac{c_j}{2}+{\displaystyle \sum _{i=1}^n}{\gamma }_{2i}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\mathit{\theta }}_i^{*}\\ \le {\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\left[-\frac{2k_j-c_j}{2}{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}\right]-{\displaystyle \sum _{i=1}^n}\frac{3{\gamma }_{2i}}{4}{\tilde{\mathit{\theta }}}_i^{\text{T}}{\tilde{\mathit{\theta }}}_i+{\displaystyle \sum _{j=1}^2}{\displaystyle \sum _{i=1}^n}\frac{c_j}{2}+{\displaystyle \sum _{i=1}^n}{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\end{array}$

34

Further, using equation (34), we obtain

$\begin{array}{l}\dot{V}\le \sum _{j=1}^2\sum _{i=1}^n\left[-\frac{2k_j-c_j}{2}{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}\right]-\sum _{i=1}^n{\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}-\sum _{i=1}^n\frac{3{\gamma }_{2i}}{4}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i\\ +\sum _{i=1}^n{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}+\sum _{j=1}^2\sum _{i=1}^n\frac{c_j}{2}+\sum _{i=1}^n{\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^T{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}\mathrm{.}\end{array}$

35

If ${\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}\ge 1$ , it is easy to know that

$\begin{array}{l}{\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}-\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\le \frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i-\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}={\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\end{array}$

36

On the other hand, if ${\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^T{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}<1$ ,

$\begin{array}{l}{\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}-\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}<1-\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\le 1+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\end{array}$

37

It follows from equations (36) and (37) that

${\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}-\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\le 1+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}$

38

Substituting equation (38) into equation (35) gives

$\begin{array}{l}\dot{V}\le \sum _{j=1}^2\sum _{i=1}^n\left[-\frac{2k_j-c_j}{2}{\tilde{\mathit{e}}}_{ji}^2-a_j{\tilde{\mathit{e}}}_{ji}^{\beta +1}\right]-\sum _{i=1}^n{\left(\frac{{\gamma }_{2i}}{2}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i\right)}^{\frac{1}{2}(\beta +1)}-\sum _{i=1}^n\frac{{\gamma }_{2i}}{4}{\tilde{\theta }}_i^{\text{T}}{\tilde{\theta }}_i+\sum _{i=1}^n\left[1+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\right]+\sum _{j=1}^2\sum _{i=1}^n\frac{c_j}{2}\\ \le -q_{1}V-q_2{V}^{\frac{\beta +1}{2}}+q_3\end{array}$

39

where

$\begin{array}{l}{q}_1=\text{min}\left\{\frac{2k_1-c_1}{2},\frac{2k_2-c_2}{2},\cdots ,\frac{2k_n-c_n}{2},\frac{{\gamma }_{\text{min}}}{2}\right\},\\ q_2=\text{min}\left\{2^{\frac{\beta +1}{2}}{a}_1{\mathrm{,2}}^{\frac{\beta +1}{2}}{a}_2,\cdots {\mathrm{,2}}^{\frac{\beta +1}{2}}{a}_n,{\gamma }_{\text{min}}^{\frac{\beta +1}{2}}\right\},\end{array}$

and

$q_3=\sum _{i=1}^n\left[1+{\gamma }_{2i}{\mathit{\theta }}_i^{\text{*T}}{\mathit{\theta }}_i^{*}\right]+\sum _{j=1}^2\sum _{i=1}^n\frac{c_j}{2}$

are three positive constants with ${\gamma }_{\text{min}}=\text{min}\left\{{\gamma }_1,{\gamma }_2,\cdots ,{\gamma }_n\right\}$ . Then, equation (39) can be rearranged as

$\dot{V}\le -\left(q_1-\frac{q_3}{2V}\right)V-\left(q_2-\frac{q_3}{2V^{\frac{\beta +1}{2}}}\right)V^{\frac{\beta +1}{2}}$

40

According to equation (40) and Lemma 3, if $k_i$ and $c_i$ are chosen such that $k_i>\frac{1}{2}{c}_i$ , then the signal $\tilde{\mathit{e}}(t)$ will tend to the region

$\parallel \tilde{\mathit{e}}(t)\parallel \le \text{max}\left\{\sqrt{\frac{q_3}{q_1}},\sqrt{2{\left(\frac{q_3}{2q_2}\right)}^{\frac{\beta +1}{2}}}\right\}$

41

in some finite time. Noting that $\mathit{e}=\tilde{\mathit{e}}+\mathit{z}$ , where $\mathit{z}=[{\mathit{z}}_1,{\mathit{z}}_2{]}^{\text{T}}\in {\mathcal{R}}^{2n}$ , the rest task consists in to prove the finite-time boundedness of the compensated signal $\mathit{z}(t)$ . Let define the following Lyapunov function:

$V_3(t)=\frac{1}{2}{\mathit{z}}^{\text{T}}\mathit{z}$

42

Using equations (13) and (26), the derivative of $V_3$ is given as

$\begin{array}{c}{\dot{V}}_3={\mathit{z}}_1^{\text{T}}{\dot{\mathit{z}}}_1+{\mathit{z}}_2^{\text{T}}{\dot{\mathit{z}}}_2\\ =-k_1{\mathit{z}}_1^{\text{T}}{\mathit{z}}_1+{\mathit{z}}_1^{\text{T}}(\widehat{\alpha }-\mathit{\alpha })+{\mathit{z}}_1^{\text{T}}{\mathit{z}}_2-c_1{\mathit{z}}_1^{\text{T}}\text{sign}({\mathit{z}}_1)-k_2{\mathit{z}}_2^{\text{T}}{\mathit{z}}_2-{\mathit{z}}_2^{\text{T}}{\mathit{z}}_1-c_2{\mathit{z}}_2^{\text{T}}\text{sign}({\mathit{z}}_2)\\ \le -\sum _{j=1}^2\sum _{i=1}^n{k}_j{\mathit{z}}_{ji}^2-\sum _{j=1}^2\sum _{i=1}^n{c}_j{\mathit{z}}_{ji}\text{sign}({\mathit{z}}_{ji})+{\mathit{z}}_1^{\text{T}}(\widehat{\alpha }-\mathit{\alpha })\\ \le -\sum _{j=1}^2\sum _{i=1}^n{k}_j{\mathit{z}}_{ji}^2-\sum _{j=1}^2\sum _{i=1}^n{c}_j|{\mathit{z}}_{ji}|+\sum _{j=1}^2\sum _{i=1}^n{\rho }_j{\mathit{z}}_{ji}{\tilde{\mathit{\alpha }}}_{ji}\end{array}$

43

where ${\rho }_1=1,{\rho }_2=0$ , ${\tilde{\mathit{\alpha }}}_{1i}={\widehat{\alpha }}_{1i}-{\mathit{\alpha }}_{1i}$ , ${\tilde{\mathit{\alpha }}}_{2i}=0$ for $i=1,2,\cdots ,n$ .

From equation (14) and Lemma 4, we know that ${\tilde{\mathit{\alpha }}}_{1i}$ is finite-time bounded, that is, there exists a positive constant ${\delta }_i$ such that $|{\tilde{\mathit{\alpha }}}_{1i}|\le {\delta }_i$ . Thus, we have

$\begin{array}{c}{\dot{V}}_3\le -\sum _{j=1}^2\sum _{i=1}^n{k}_j{\mathit{z}}_{ji}^2-\sum _{j=1}^2\sum _{i=1}^n{c}_j|{\mathit{z}}_{ji}|+\sum _{j=1}^2\sum _{i=1}^n{\rho }_j{\delta }_i|{\mathit{z}}_{ji}|\\ \le \underset{\_}{k}{V}_3-\underset{\_}{c}\sqrt{V_3}+\overline{\rho }\overline{\delta }\sqrt{V_3}\\ =\underset{\_}{k}{V}_3-(\underset{\_}{c}-\overline{\rho }\overline{\delta })\sqrt{V_3}\end{array}$

44

with $\underset{\_}{k}=2\text{min}\left\{k_i\right\}$ , $\underset{\_}{c}=\text{min}\left\{c_i\right\}$ , $\overline{\rho }=\text{max}\{{\rho }_1,{\rho }_2\}$ , and $\overline{\delta }=\text{max}\left\{{\delta }_i\right\}$ . Consequently, the finite-time stability of $\tilde{\mathit{e}}$ can be guaranteed. This completes the proof.▪

Remark 1

In the proposed method, to achieve the finite-time stability, the sign function is used in the compensated signals (equations (13) and (26)), the finite-time command filter (equation (14)), the virtual control input (equation (7)), and the control input (equation (27)). Thus, the chattering phenomenon which is harmful to control performance will occur. To solve this problem, we can replace this discontinuous function by some continuous function, such as $\text{arctan}(\cdot )$ . According to Theorem 1, we know that the tracking error and the compensated tracking error will tend to a small region of the origin in some finite time. If we use the aforementioned continuous function to replace the sign function, the convergence radius will be larger.

Remark 2

In the proof of Theorem 1, we give the conclusion that $|{\tilde{\mathit{\alpha }}}_{1i}|\le {\delta }_i$ . In fact, according to Lemma (4), $|{\tilde{\mathit{\alpha }}}_{1i}|$ will be zero in finite time. However, when the system is affected by bounded external noise, as stated in the literature, ^50,56 the signal $|{\tilde{\mathit{\alpha }}}_{1i}|$ will be finite-time bounded.

Remark 3

In this article, using the finite-time command filter (equation (14)), we need not to calculate the derivative of the virtual control input $\mathit{\alpha }$ . The finite-time command filter (equation (14)) can obtain the estimation of $\mathit{\alpha }$ and its first-order derivative in some finite time. The conventional CFC approach, such as in the literature, ^53,54,58 has no such ability. In fact, in the cited works, ^53,54,58 a first-order command filter was used, which can be written as

$\dot{\mathit{\eta }}(t)=-b_1({\mathit{\eta }}_1-\mathit{\alpha })$

45

where $b_1\in {ℝ}^{+}$ is a sufficiently large parameter, $\mathit{\alpha }$ is the virtual control input. Thus, $\mathit{\eta }$ and $\dot{\mathit{\eta }}$ are used to approximate $\mathit{\alpha }$ and $\dot{\mathit{\alpha }}$ , respectively. From the above analysis, it is clear that by in comparison with the first-order filter (equation (45)), the proposed finite-time command filter (equation (14)) has faster convergence speed. In addition, the comparison results are also presented in the next section between the two methods.

Remark 4

It should be mentioned that many control methods can be used to control robotic manipulators, such as, proportional–integral–derivative (PID) control, SMC, $H_{\infty }$ control. In fact, the finite-time convergence of the tracking error can not be guaranteed by the PID control approach. In addition, it is very hard for PID controller designing when the robotic manipulators subject to dynamical uncertainties. The fast convergence of the tracking error and well robustness can be guaranteed by the SMC, where only the matched system uncertainties can be considered, that is, the SMC can not be properly used to control robotic manipulators with unmatched system uncertainties. With respect to $H_{\infty }$ control, the complicated Riccati equation should be well resolved. However, the proposed method ensure not only the finite-time convergence of the tracking error, but also the good ability to handle the unmatched system uncertainties in robotic manipulators.

Simulation results

To indicate the effectiveness of the proposed AFTFCFC method, simulation results will be given by controlling a 2-link robotic manipulator that is shown in Figure 1.

Figure 1.

A two-link robot.

The details of system model (10) are given as follows

$\begin{array}{c}\mathit{F}(\dot{\mathit{\xi }})=[0.4{\dot{\xi }}_1\mathrm{,0.4}{\dot{\xi }}_2{]}^{\text{T}},\mathit{M}(\mathit{\xi })=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{12}& M_{22}\end{array}\right],\\ \mathit{C}(\mathit{\xi },\dot{\mathit{\xi }})=\left[\begin{array}{c}-\frac{1}{2}{a}_2{l}_1{l}_2(2{\dot{\xi }}_1{\dot{\xi }}_2+{\dot{\xi }}_2^2)\text{sin}{\xi }_2\\ \frac{1}{2}{a}_2{l}_1{l}_2{\dot{\xi }}_1^2\text{sin}{\xi }_2\end{array}\right],\\ \mathit{G}(\mathit{\xi })=\left[\begin{array}{c}\left(\frac{1}{2}{a}_1+a_2\right)g{l}_1\text{cos}{\xi }_2+\frac{1}{2}{a}_2{l}_{2}g\text{cos}({\xi }_1+{\xi }_2)\\ \frac{1}{2}{a}_{2}g{l}_2\text{cos}({\xi }_1+{\xi }_2)\end{array}\right],\\ \mathit{d}(t)=\left[\begin{array}{c}\frac{1}{2}\text{sin}t+\frac{1}{2}\text{cos}t\\ \frac{1}{2}\text{sin}t-\frac{1}{2}\text{cos}t\end{array}\right]\end{array}$

where $M_{11}=\left(\frac{1}{4}{a}_1+a_2\right)l_1^2+\frac{1}{4}{a}_2{l}_2^2+a_2{l}_1{l}_2\text{cos}{\xi }_2$ , $M_{12}=\frac{1}{4}{a}_2{l}_2^2+\frac{1}{2}{a}_2{l}_1{l}_2\text{cos}{\xi }_2$ , $M_{22}=\frac{1}{4}{a}_2{l}_2^2$ , $a_1,a_2$ are the mass, $l_1,l_2$ denote the length. In the simulation, $a_1=4,a_2=2,l_1=2l_2=2$ .

Define $\mathit{x}={\left[{\mathit{x}}_1^{\text{T}},{\mathit{x}}_2^{\text{T}}\right]}^{\text{T}}\in {\mathcal{R}}^4$ , where ${\mathit{x}}_1(t)={\left[{\xi }_1,{\xi }_2\right]}^{\text{T}},{\mathit{x}}_2(t)={\left[{\dot{\xi }}_1,{\dot{\xi }}_2\right]}^{\text{T}}$ . The desired signal is ${\mathit{x}}_{\mathit{d}}={\left[\frac{1}{2}\text{sin}\left(\frac{t}{2}\right),\frac{1}{2}\text{cos}\left(\frac{t}{2}\right)\right]}^{\text{T}}$ . The design parameters are chosen as: $b_1=b_2=1,k_1=k_2=1.5,{\gamma }_{11}={\gamma }_{12}=5,{\gamma }_{21}={\gamma }_{22}=0.005,c_1=c_2=1$ . The initial condition is $\mathit{x}(0)={\left[1.5,0.2,0,1.2\right]}^{\text{T}}$ .

There are two FLSs are used in the simulation. Each FLS uses ${\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3$ and ${\mathit{x}}_4$ as its inputs. For each input, let us define three Gaussian membership functions distributed on interval $[-2.5,2.5]$ . Thus, we will use $3^4=81$ fuzzy rules in the simulation. The initial conditions of the FLSs are chosen randomly.

Simulation results are indicated in Figures 2 to 10. The tracking performance of joints 1 and 2 is shown in Figures 2 and 3, respectively. The tracking errors and their derivatives are given in Figures 4 and 5, respectively. It can be seen that the systems variables track the desired signal in a short time and the tracking performance is well. The control inputs are presented in Figure 6, where chattering phenomenon can be seen because the noncontinuous sign function has been used in the controller. The virtual control inputs ${\mathit{\alpha }}_1,{\mathit{\alpha }}_2$ and their estimation ${\widehat{\alpha }}_1,{\widehat{\alpha }}_2$ are presented in Figures 7 and 8. It can be seen that the proposed finite-time second-order command filter 14 has very good approximation ability and arrives and the target value in some finite time. Finally, the fuzzy parameters ${\mathit{\theta }}_1$ and ${\mathit{\theta }}_2$ are given in Figures 9 and 10, respectively.

Figure 2.

Tracking of joint 1.

Figure. 3.

Tracking of joint 2.

Figure 4.

${\tilde{\mathit{e}}}_{11}$ and ${\tilde{\mathit{e}}}_{12}$ .

Figure 5.

${\dot{\stackrel{\sim }{\mathit{e}}}}_{11}$ and ${\dot{\stackrel{\sim }{\mathit{e}}}}_{12}$ .

Figure 6.

Control inputs ${\mathit{u}}_1$ and ${\mathit{u}}_2$ .

Figure 7.

Virtual control input ${\mathit{\alpha }}_1$ and its estimation ${\widehat{\alpha }}_1$ .

Figure 8.

Virtual control input ${\mathit{\alpha }}_2$ and its estimation ${\widehat{\alpha }}_2$ .

Figure 9.

Fuzzy parameter ${\mathit{\theta }}_1$ .

Figure 10.

Fuzzy parameter ${\mathit{\theta }}_2$ .

Noting that the discontinuous function is used in the proposed AFTFCFC, just as indicated by Figures. 5 and 6, the chattering phenomenon does exist. To solve this problem, we can replace the discontinuous function $\text{sign}(\cdot )$ in equations (13), (26), (14), (7), and (27) by $\text{arctan}(20\cdot )$ , and the simulation results are presented in Figure 11.

Figure 11.

Simulation results without chattering, in (a) compensated tracking errors; (b) control inputs; (c) virtual input ${\mathit{\alpha }}_1$ and its estimation; (d) virtual input ${\mathit{\alpha }}_2$ and its estimation.

On the other hand, to show the effectiveness of the proposed method, a comparison between the conventional first-order command filtered control (FOCFC) method that introduced in previous studies ^53,54,58 and our’s will be given. For fair comparison, to construct the conventional FOCFC, we only replace the proposed finite-time filter (equation (8)) by the first-order (equation (45)) and keep all the other parameters as same as above. The simulation results are shown in Figure 12. It can be seen that compared with the conventional FOCFC, the proposed AFTFCFC has faster convergence speed and better tracking performance.

Figure 12.

Comparison between the conventional FOCFC and the proposed method, in (a) compensated tracking error ${\tilde{\mathit{e}}}_{11}$ ; (b) compensated tracking error ${\stackrel{\sim }{\mathit{e}}}_{12}$ ; (c) control input ${\mathit{u}}_1$ ; (d) control input ${\mathit{u}}_2$ ; (e) ${\widehat{\alpha }}_1$ , and (f) ${\widehat{\alpha }}_2$ . FOCFC: first-order command filtered control.

Conclusions

In this study, to obtain finite-time trajectory tracking control of robotic manipulators, an adaptive fuzzy command filtered control method is introduced. Unlike the backstepping control design, in our approach, the virtual control input and its derivative is estimated by a second-order finite-time command filter. It has been proven that the proposed adaptive fuzzy controller can guarantee that the states of the robotic manipulator track the desired signal in finite time. The simulation test on a two-link robotic manipulator shows the advantages of the proposed scheme: the proposed controller does not need the prior knowledge of the system and also has good robustness. Noting that the proposed approach guarantees that the tracking error tends to a small region in finite time. How to get finite-time asymptotical stability of the controlled robotic manipulators is one of our future research directions.