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Abstract

In practical robot control systems, the input constraints problem, caused by a limited actuator output amplitude, is due to its own physical characteristics. In traditional control methods, this nonlinear problem is not paid enough attention, especially for robot systems with unknown disturbance. To handle this problem, adaptive compensate control for uncertain robot system with input constraints is proposed. It is combined with adaptive compensate control and hyperbolic tangent function to ensure that the uncertain robot system with input constraints is stable. The simulation results showed that the proposed method can make the uncertain robot system reach a stable state.

Keywords

Adaptive compensate control input constraints uncertain robot system hyperbolic tangent function

Introduction

With the continuous development of the robotics discipline, the research of robot control systems has become a research hotspot. In the field of robot control systems, there are also many practical problems needed to be solved urgently.^1–5

In the practical control system, the input constraints caused by the physical characteristics of the robot often leads to system divergence or even system collapse. Even if the system does not diverge, long-term high-intensity oscillations would cause structural damage to the robot system, resulting in failure.^6–13 However, the traditional control method does not pay much attention to this aspect. Thus, it is urgently needed to propose an effective control method to deal with this problem. Since the hyperbolic tangent function avoids the direct switching of the control input and eliminates the problem of system chattering, thereby it can be used to control the amplitude of input constraints.

On the other hand, in practice, owing to the environment in which the robot is located is extremely complex instead of ideal, there are various kinds of unknown disturbances. They have a great influence on the stability of the robot control system.^14–29 Unknown disturbances from the outside world must also be taken into account when designing the control system. Adaptive compensate control is used to compensate unknown disturbances to handle this problem.

Through the aforementioned discussion, an adaptive compensate control for uncertain robot system with input constraints is proposed. The hyperbolic tangent function is applied to control the amplitude of input constraints. Meanwhile, adaptive compensate control is used to compensate the unknown disturbances. The main compensation mechanisms and contributions of the proposed method are summarized as follows:

Considering the input constraints problem caused by the physical characteristics of the system, hyperbolic tangent function is used in the control system to solve this problem, which can make the system reach a stable state as well as the amplitude of input constraints can be controlled.

Taking into account the unknown disturbances of the robot system and using adaptive control to compensate. The stability of the system is greatly enhanced.

The rest of the arrangement is as follows. In section “The modeling of uncertain robot system and controller design,” the model and problem statement are given. In section “Simulation analysis,” the corresponding simulation proves that the proposed method is feasible. Conclusions will be summarized in section “Conclusion.”

The modeling of uncertain robot system and controller design

The modeling of uncertain robot system

The single-joint robot can be simplified into the following controlled objects

${\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = \frac{1}{J} u + d \end{matrix}$ (1)

where $x_{1}$ is the angle, $x_{2}$ is the angular velocity, u is the control input voltage, J is the moment of inertia, and d is the unknown disturbance.

Remark 1

Compared with equation (1), we could safely arrive that because usually the unknown disturbance is considered to be bounded for processing, there is a maximum. Setting $| d | \leq D$ , $x_{1}$ is the angle, $x_{2}$ is the angular velocity, u is the control input voltage, and J is the moment of inertia. Thus, the controlled objects of single joint-robot can be converted as equation (2)

Setting $| d | \leq D$ , equation (1) can be converted to

${\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} \leq \frac{1}{J} u + D \end{matrix}$ (2)

The instruction to take $x_{1}$ is $x_{1 d}$ , which is defined as

$e = x_{1} - x_{1 d}$ (3)

Derived from equations (2) and (3)

$\overset{\cdot}{e} = {\overset{\cdot}{x}}_{1} - {\overset{\cdot}{x}}_{1 d} = x_{2} - {\overset{\cdot}{x}}_{1 d}$ (4)

$\overset{\cdot\cdot}{e} = {\overset{\cdot}{x}}_{2} - {\overset{\cdot}{x}}_{1 d} \leq \frac{1}{J} u + D - {\overset{\cdot\cdot}{x}}_{1 d}$ (5)

Setting $e_{1} = e and e_{2} = \overset{\cdot}{e}$ , the model changes to

${\begin{matrix} {\overset{\cdot}{e}}_{1} = \overset{\cdot}{e} = e_{2} \\ {\overset{\cdot}{e}}_{2} = \overset{\cdot\cdot}{e} \leq \frac{1}{J} u + D - {\overset{\cdot\cdot}{x}}_{1 d} \end{matrix}$ (6)

Setting $v = (1 / J) u + \hat{D} - {\overset{\cdot\cdot}{x}}_{1 d}$ owing to $\tilde{D} = D - \hat{D}$ , the model turns into

${\begin{matrix} {\overset{\cdot}{e}}_{1} = e_{2} \\ {\overset{\cdot}{e}}_{2} \leq \frac{1}{J} u + D - {\overset{\cdot\cdot}{x}}_{1 d} = v + \tilde{D} \end{matrix}$ (7)

Controller design

The control law is designed as

$v = - ε \tanh (m e_{1} + n e_{2}) - η \tanh (n e_{2})$ (8)

where $ε, η, m, and n$ are constant, and $ε > 0, η > 0, m > 0, and n > 0$ .

Then it can achieve $e \to 0 and \overset{\cdot}{e} \to 0$ . The corresponding actual control law at this time is

$u = J (v + {\overset{\cdot\cdot}{x}}_{1 d})$ (9)

According to equation (8)

$| u | \leq J (ε + η + max {{\overset{\cdot\cdot}{x}}_{1 d}})$ (10)

Remark 2

Thereby controlling the constraints of the input, the magnitude of the control input limitation can be adjusted by $ε$ and $η$ .

The certification process is as follows:

Equation (7) can be converted to

${\begin{matrix} {\overset{\cdot}{e}}_{1} = e_{2} \\ {\overset{\cdot}{e}}_{2} \leq - ε \tanh (m e_{1} + n e_{2}) - η \tanh (n e_{2}) + \tilde{D} \end{matrix}$ (11)

Because of $c o s h (x) = (e^{- x} + e^{x}) / 2 \geq 1, \ln (c o s h (x)) \geq 0$ and while $x = 0$ , $\ln (\cosh (x)) = 0$ .

Defining a Lyapunov function

$V = ε \ln \cosh ((m e_{1} + n e_{2})) + η \ln (\cosh (n e_{2})) + \frac{1}{2} {me}_{2}^{2} + \frac{λ}{2} {\tilde{D}}^{2}$ (12)

The derivative of equation (12) can be simplified as

$\begin{matrix} \overset{\cdot}{V} = ε \frac{\sinh (m e_{1} + n e_{2})}{\cosh (m e_{1} + n e_{2})} (m {\overset{\cdot}{e}}_{1} + n {\overset{\cdot}{e}}_{2}) + η \frac{\sinh (n e_{2})}{\cosh (n e_{2})} n {\overset{\cdot}{e}}_{2} \\ + m e_{2} {\overset{\cdot}{e}}_{2} + λ \tilde{D} \overset{\cdot}{\tilde{D}} \\ = ε (m {\overset{\cdot}{e}}_{1} + n {\overset{\cdot}{e}}_{2}) \tanh (m e_{1} + n e_{2}) + η n \tan (n {\overset{\cdot}{e}}_{2}) \\ + m e_{2} {\overset{\cdot}{e}}_{2} + λ \tilde{D} \overset{\cdot}{\tilde{D}} \end{matrix}$ (13)

Analysis of stability

Setting $t_{1} = \tanh (m e_{1} + n e_{2}) and t_{2} = \tanh (n e_{2})$ , we can get

${\overset{\cdot}{e}}_{2} \leq - ε t_{1} - η t_{2} + \tilde{D}$ (14)

Derived from equations (13) and (14)

$\begin{matrix} \overset{\cdot}{V} \leq ε [m {\overset{\cdot}{e}}_{1} + n (- ε t_{1} - η t_{2} + \tilde{D})] t_{1} \\ + η n (- ε t_{1} - η t_{2} + \tilde{D}) t_{2} + m e_{2} (- ε t_{1} - η t_{2} + \tilde{D}) + λ \tilde{D} \overset{\cdot}{\tilde{D}} \\ \leq ε t_{1} m e_{2} - n ε^{2} {t_{1}}^{2} - η ε n t_{1} t_{2} + n ε t_{1} \tilde{D} - η n ε t_{1} t_{2} - η^{2} {nt}_{2}^{2} \\ + η n t_{2} \tilde{D} - m e_{2} ε t_{1} - m e_{2} η t_{2} + m e_{2} \tilde{D} + λ \tilde{D} \overset{\cdot}{\tilde{D}} \\ \leq - n {(ε t_{1} + η t_{2})}^{2} - m η e_{2} t_{2} \\ + \tilde{D} (n ε t_{1} + m e_{2} + η n t_{2} - λ \overset{\cdot}{\hat{D}}) \end{matrix}$ (15)

Setting $\overset{\cdot}{\hat{D}} = (n ε t_{1} + m e_{2} + η n t_{2}) / λ$ , then $\tilde{D} (n ε t_{1} + m e_{2} + η n t_{2} - λ \overset{\cdot}{\hat{D}}) = 0$

Because of $x \tanh (x) = (x ((e^{x} - e^{- x}) / (e^{- x} + e^{x}))) \geq 0$ , then $e_{2} t_{2} = e_{2} \tanh (n e_{2}) \geq 0$ , so it is obvious that $\overset{\cdot}{V} \leq 0$ . If and only if $e_{1} = e_{2} = 0$ , $\overset{\cdot}{V} = 0$ . According to the LaSalle invariance principle, it is understood that equation (11) is progressively stable, that is, when $t \to \infty$ , $e_{1} \to 0, and e_{2} \to 0$ . The convergence speed of the system depends on $ε, η, m, and n$ .

Owing to $\tanh (x) = ((e^{x} - e^{- x}) / (e^{- x} + e^{x})) \in [- 1, 1]$ , we can get

$| {\overset{\cdot}{e}}_{2} | = | - ε \tanh (m e_{1} + n e_{2}) - η \tanh (n e_{2}) | \leq ε + η$ (16)

Therefore, the proposed method can achieve a limited control input.

Simulation analysis

Since the controlled object is the model described by equation (11), the initial state is $[\begin{matrix} 0.5 & 0 \end{matrix}]$ , setting $J = 0$ , $x_{1 d} = \sin t$ , according to equations (8) and (9) to design the control law, setting $ε = 3, η = 3, m = 3,$ and $n = 3$ , according to equation (10), the amplitude of input constraints is $| u | \leq 10 (3 + 3 + 1) = 70$ . The simulation results are shown in Figures 1 –4.

Figure 1.

Angle response.

Figure 2.

Angle speed response.

Figure 3.

Control input.

Figure 4.

Angle tracking error response.

Remark 3

Through the analysis of the simulation results, we can know that the control method designed in this article is feasible. It can make the robot system with input constraints and unknown disturbance reach a steady state, and the amplitude of the control input is adjustable.

As is shown in Figures 1 and 2 that under the action of the control method proposed in this article, the actual angle and angular velocity of the robot system with input constraints problem and unknown disturbance are basically coincidental with the reference angle and the reference angular velocity after a certain time. As for Figure 4, it can be seen that the angle tracking error is within 1%. As can be seen from Figure 3, the control input of the system is $| u | \leq 70$ , and it can be guaranteed to fluctuate within a certain range.

Remark 4

These are all practical. The simulation results show that two contributions can be achieved.

Conclusion

In this article, adaptive compensate control for uncertain robot system with input constraints is proposed. In the actual robot control system, the input constraints problem is the most common type of nonlinear problem in this control system. Since the hyperbolic tangent function can effectively solve the system chattering problem and avoid the direct conversion of the control input, the proposed method combines the hyperbolic tangent function to solve this problem. At the same time, adaptive control is used to compensate for unknown disturbance of the robot system. After simulation, the proposed method can achieve the expected effect. When there are input constraints and unknown disturbance, the system can reach a stable state under the action of the proposed control method. However, how to handle the problem, how to accommodate the actuators with the output constraints problem, is needed to be investigated, which we will attempt to cope with in the future.

Footnotes

Handling Editor: Hamid Reza Karimi

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work has been supported by the National Natural Science Foundation of China Guangdong Joint Funds (grant nos.: U1701262 and U1801263) and partly supported by the Guangdong Provincial Key Laboratory of Cyber-Physical System (grant no.: 2016B030301008) and the Guangdong applied science and technology R&D special funds (grant no.: 2015B090922013) and the Guangdong Province Science and Technology Project (grant no.: 2016B090918017) and the Guangdong NC First Generation Project (grant no.: 2013B011302007) and the Guangzhou Science and Technology Project (grant no.: 201604016107).

ORCID iD

Guiyang Deng

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Adaptive compensate control for uncertain robot system with input constraints

Abstract

Keywords

Introduction

The modeling of uncertain robot system and controller design

The modeling of uncertain robot system

Remark 1

Controller design

Remark 2

Analysis of stability

Simulation analysis

Remark 3

Remark 4

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References