NEURAL NETWORK ADAPTIVE TRACKING CONTROL METHOD FOR JOINT ROBOTS

Abstract

The present disclosure discloses a neural network adaptive tracking control method for joint robots, which proposes two schemes: robust adaptive control and neural adaptive control, comprising the following steps: 1) establishing a joint robot system model; 2) establishing a state space expression and an error definition when taking into consideration both the drive failure and actuator saturation of the joint robot system; 3) designing a PID controller and updating algorithms of the joint robot system; and 4) using the designed PID controller and updating algorithms to realize the control of the trajectory motion of the joint robot. The present disclosure may solve the following technical problems at the same time: the drive saturation and coupling effect in the joint system, processing parameter uncertainty and non-parametric uncertainty, execution failure handling during the system operation, compensation for non-vanishing interference, and the like.

Description

TECHNICAL FIELD

The present disclosure relates to the technical field of joint robot system control which is highly nonlinear and influenced by external interference and uncertainty, in particular to the joint position tracking control of a rigid arm robot.

BACKGROUND

With the continuous progress of social science and technology, intelligent robots play an increasingly important role, which can complete various tasks instead of human beings in many complex situations.

However, we have never stopped studying the improvement of robot system performance. To be more dexterous and adaptive, the robot system must achieve control algorithms that are simpler in structure, more specialized and more powerful.

Early contributions to the development of joint robot system control schemes included those based entirely or partially on system models provided with feedforward compensation and nonlinear feedback techniques. Generally, however, the joint robot system is highly nonlinear in nature, a corresponding dynamic model for which is difficult to obtain accurately due to external interference and uncertainty.

SUMMARY

For this purpose, the present disclosure aims at proposing a neural network adaptive tracking control method for joint robots to solve the joint tracking control problem of a joint robot system troubled with declining and drive saturation, and to achieve the ideal tracking control.

In order to achieve the above purpose, the present disclosure provides the following scheme:

A neural network adaptive tracking control method for joint robots, which includes the following steps:

1) Establishing a joint robot system model:

D _q(q){umlaut over (q)}+C_q(q,{dot over (q)}){dot over (q)}+G_q(q)+τ({dot over (q)},t)=u_a

In the model mentioned above, q represents a position vector of the joint robot, {dot over (q)} represents a velocity vector of the joint robot, {umlaut over (q)} represents an acceleration vector of the joint robot action, u_a represents a control input of the joint robot system, the system parameter D_q(q) represents an inertia matrix of the joint robot system, the system parameter C_q(q,{dot over (q)}) represents a centrifugal matrix of the joint robot system, the system parameter G_q(q) represents a universal gravitation matrix of the joint robot system, and the system parameter τ({dot over (q)},t) represents uncertainty and interference factors of the joint robot system;

2) Establishing a state space expression and an error definition when taking into consideration both the drive failure and actuator saturation of the joint robot system:

u _a(t)=ρ(t)[Γ(0)+L(ξ)ν+ε(ν)]+ε(t)=ρ(t)L(ξ)ν+[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]

e=x ₁ −q*

ë={umlaut over (x)} ₁ −{umlaut over (q)}*=D _q ⁻¹(q)ρ(t)L(ξ)ν+D_q⁻¹(q)[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]+F(⋅)+Q(x₁,t)−{umlaut over (q)}*

In the above formulas, u_a(t) represents a system control input signal considering both drive failure and actuator saturation, Γ(0)+L(ξ)ν+ε(ν), represents a control signal in the case of actuator saturation, wherein ν represents an actual controller design quantity of the system, Γ(0)+L(ξ)ν represents a smooth function proposed according to the mean value theorem of ν, Γ(0) is a bounded matrix, L(ξ) is a non-negative positive definite matrix, ε(ν) is a bounded approximate error and represents an uncertain factor of the controller; ρ(t) represents a health coefficient of the driver, ε(t) represents an interference factor of the driver; e((or e(⋅)) represents a dynamic error of the system (e(⋅) is written as e for simplification in subsequent derivation), ë represents the second derivative of the dynamic error, wherein x₁=q represents a motion trajectory of the joint robot, {umlaut over (x)}₁ represents an acceleration of the joint robot motion, q* represents a given joint tracking trajectory; {umlaut over (q)}* represents an acceleration of the given joint tracking F(⋅)=D_q⁻¹(q)(C_q(q){dot over (q)}+G_q(q)), and Q(x₁,t)=D_q⁻¹(q)τ({dot over (q)},t).

3) Designing a PID controller and updating algorithms of the joint robot system:

The PID controller ν is expressed as

$v=-\left(k_{D0}+\Delta k_D\left(t\right)\right)\left(2\gamma e(·)+{\gamma }^2{\int }_0^{t}e(·)d\tau +\frac{\mathrm{de}(·)}{dt}\right)$

Wherein γ is a parameter that a designer can design at will, and k_D0 is a constant that is designed at the designer's option;

Wherein the updating algorithms consist of two algorithms as follows:

(1) Algorithm based on the robust adaptive control:

The robust adaptive algorithm is designed for automatically updating the controller parameters at an updating rate of:

$\begin{array}{cc}\Delta k_D\left(t\right)=\hat{c}{\phi }_0^2(·)\{\begin{array}{c}\stackrel{.}{\hat{c}}=-{\sigma }_0\hat{c}+{\sigma }_1{\phi }_0^2(·){E}^2\\ \hat{c}\left(0\right)\ge 0\end{array}& \end{array}$

Wherein, σ₀ and σ₁ are positive constants that the designer needs to design;

$\{\begin{array}{c}c=\max \left\{a_1,\frac{1}{2}{\gamma }_d\right\}\\ {\phi }_0(·)={\phi }_1(·)+\stackrel{.}{q}E\end{array},$

wherein ĉ is an estimated value of c; a₁=max {γ_da_f, γ_dγ², 2γ_dγ,γ_dx₂}, φ₁(⋅)=φ_f(⋅)+∥e∥+∥ė∥+1, wherein a_fφ_f(⋅) is a product of the constant a_f and the scalar function φ_f(⋅), representing the upper bound of the system uncertainty factor D_q⁻¹(q)[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]+F(⋅)+Q(x₁,t)−{dot over (q)}*, x₂ is the upper bound of an second derivative {umlaut over (q)}* of a given joint motion trajectory, γ_d is the upper bound of an system parameter D_q(q), and it is set that

$E=2\gamma e(·)+{\gamma }^2{\int }_0^{t}e(·)d\tau +\frac{\mathrm{de}(·)}{dt};$

(2) Algorithm based on the neural adaptive control:

The neural adaptive algorithm is designed for automatically updating the controller parameters at an updating rate of:

$\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\hat{b}}=-{\theta }_0\hat{b}+{\theta }_1{\psi }^2(·){E}^2\\ \hat{b}\left(0\right)\ge 0\end{array}\Delta k_D\left(t\right)=\hat{b}{\psi }^2(·)& \end{array}$

Wherein: θ₀ and θ₁ are positive constants that the designer needs to design; ψ(⋅)=∥S(⋅)∥+1, wherein S(⋅) is a primary function of a neural network, S(⋅) and a number of neurons are determined at the designer's will; b=max{∥W^T∥,m}, wherein {circumflex over (b)} is an estimated value of b, W^T is an ideal unknown weight, and m is the upper limit of an reconstruction error ∥η(⋅)∥ of the model;

$E=2\gamma e(·)+{\gamma }^2{\int }_0^{t}e(·)d\tau +\frac{\mathrm{de}(·)}{dt};$

4) Using the PID controller and the updating algorithms designed in step 3) for the joint robot system to control the trajectory motion of the joint robot.

The present disclosure is beneficial in:

The present disclosure provides a neural network adaptive tracking control method for joint robots, wherein algorithms based on robust adaptive control and neural adaptive control are respectively designed against a joint robot system troubled with declining and drive saturation, which are formed in a simple way of PID, have an advantage of low complexity, and may solve the following technical problems at the same time: the drive saturation and coupling effect in the joint system, processing parameter uncertainty and non-parametric uncertainty, execution failure handling during the system operation, compensation for non-vanishing interference, and the like. Also, the present disclosure is robust to external interference, adaptive to nonparametric uncertainty, and fault-tolerant to unpredictable drive failures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of the algorithm design control of the system. According to the present disclosure, the controller gain is adaptively adjusted by using the robust adaptive algorithm and the neural adaptive algorithm respectively, so that the joint motion trajectory of the controlled robot reaches an ideal trajectory.

FIG. 2 is a schematic diagram of actuator saturation, which includes an asymmetric and non-smooth saturation function and a smooth approximation function, and when ν reaches a certain value, the controller input u will reach a saturation status.

FIGS. 3 and 4 are respectively a simulation diagram of controller gain adjustment and a joint robot position tracking curve, which both adopt the robust adaptive control method in the embodiment to carry out the simulation control, wherein Δk_P, Δk_I, Δk_D respectively represent changes of three time-varying gains of the PID controller, and e₁, e₂, and e₃ respectively represent the trajectory errors of three joint motions of the robot.

FIGS. 5 and 6 are respectively a simulation diagram of controller gain adjustment and a joint robot position tracking curve, which both adopt the neural adaptive control method in the embodiment to carry out the simulation control, wherein Δk_P, Δk_I, Δk_D respectively represent changes of three time-varying gains of the PID controller, and e₁, e₂, and e₃ respectively represent the trajectory errors of three joint motions of the robot.

FIG. 7 is a diagram of the joint robot model.

DETAILED DESCRIPTION

The present disclosure will be further described with reference to figures and embodiments below to enable the implementation by those skilled in the art according to the text of the specification.

In this embodiment, the neural network adaptive tracking control method for joint robots, including the following steps:

1) Establishing a joint robot system model:

D _q(q){umlaut over (q)}+C_q(q,{dot over (q)}){dot over (q)}+G_q(q)+τ({dot over (q)},t)=u_a

In the model mentioned above, q represents a position vector of the joint robot, {dot over (q)} represents a velocity vector of the joint robot, {umlaut over (q)} represents an acceleration vector of the joint robot action, u_a represents a control input of the joint robot system, the system parameter D_q(q) represents an inertia matrix of the joint robot system, the system parameter C_q(q,{dot over (q)}) represents a centrifugal matrix of the joint robot system, the system parameter G_q(q) represents a universal gravitation matrix of the joint robot system, and the system parameter τ({dot over (q)},t) represents uncertainty and interference factors of the joint robot system;

2) Establishing a state space expression and an error definition when taking into consideration both the drive failure and actuator saturation of the joint robot system:

u _a(t)=ρ(t)[Γ(0)+L(ξ)ν+ε(ν)]+ε(t)=ρ(t)L(ξ)ν+[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]

e=x ₁ −q*

ë={umlaut over (x)} ₁ −{umlaut over (q)}*=D _q ⁻¹(q)ρ(t)L(ξ)ν+D_q⁻¹(q)[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]+F(⋅)+Q(x₁,t)−{umlaut over (q)}*

In the above formulas, u_a(t) represents a system control input signal considering both drive failure and actuator saturation, Γ(0)+L(ξ)ν+ε(ν), represents a control signal in the case of actuator saturation, wherein ν represents an actual controller design quantity of the system, Γ(0)+L(ξ)ν represents a smooth function proposed according to the mean value theorem of ν, Γ(0) is a bounded matrix, L(ξ) is a non-negative positive definite matrix, ε(ν) is a bounded approximate error and represents an uncertain factor of the controller; ρ(t) represents a health coefficient of the driver, ε(t) represents an interference factor of the driver; e((or e(⋅)) represents a dynamic error of the system (e(⋅) is written as e for simplification in subsequent derivation), ë represents the second derivative of the dynamic error, wherein x₁=q represents a motion trajectory of the joint robot, {umlaut over (x)}₁ represents an acceleration of the joint robot motion, q* represents a given joint tracking trajectory; {umlaut over (q)}* represents an acceleration of the given joint tracking F(⋅)=D_q⁻¹(q)(C_q(q){dot over (q)}+G_q(q)), and Q(x₁,t)=D_q⁻¹(q)τ({dot over (q)},t). The nonlinearity and uncertainty factors in the system set may be determined by the upper bound of the product of a constant and a scalar real-valued function, so as to obtain a robust adaptive control scheme; or the system is reconstructed through a neural network based on a radial primary function so as to produce the neural network adaptive control scheme.

3) Designing a PID controller and updating algorithms of the joint robot system:

The PID controller ν is expressed as

$v=-\left(k_{D0}+\Delta k_D\left(t\right)\right)\left(2\gamma e(·)+{\gamma }^2{\int }_0^{t}e(·)d\tau +\frac{\mathrm{de}(·)}{dt}\right)$

Wherein γ is a parameter that the designer can design at will, and k_D0 is a constant that is designed at the designer's option;

Wherein the updating algorithms consist of two algorithms as follows:

(1) Algorithm based on the robust adaptive control:

The robust adaptive algorithm is designed for automatically updating the controller parameters at an updating rate of:

$\begin{array}{cc}\Delta k_D\left(t\right)=\hat{c}{\phi }_0^2(·)\{\begin{array}{c}\stackrel{.}{\hat{c}}=-{\sigma }_0\hat{c}+{\sigma }_1{\phi }_0^2(·){E}^2\\ \hat{c}\left(0\right)\ge 0\end{array}& \end{array}$

Wherein, σ₀ and σ₁ are positive constants that the designer needs to design;

$\{\begin{array}{c}c=\max \left\{a_1,\frac{1}{2}{\gamma }_d\right\}\\ {\phi }_0(·)={\phi }_1(·)+\stackrel{.}{q}E\end{array},$

wherein ĉ is an estimated value of c; a₁=max {γ_da_f, γ_dγ², 2γ_dγ,γ_dx₂}, φ₁(⋅)=φ_f(⋅)+∥e∥+∥ė∥+1, wherein a_fφ_f(⋅) is a product of the constant a_f and the scalar function φ_f(⋅), representing the upper bound of the system uncertainty factor D_q⁻¹(q)[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]+F(⋅)+Q(x₁,t)−{dot over (q)}*, x₂ is the upper bound of an second derivative {umlaut over (q)}* of a given joint motion trajectory, γ_d is the upper bound of an system parameter D_q(q), and it is set that

$E=2\gamma e(·)+{\gamma }^2{\int }_0^{t}e(·)d\tau +\frac{\mathrm{de}(·)}{dt};$

(2) Algorithm based on the neural adaptive control:

The neural adaptive algorithm is designed for automatically updating the controller parameters at an updating rate of:

$\begin{array}{cc}\{\begin{array}{c}\stackrel{.}{\hat{b}}=-{\theta }_0\hat{b}+{\theta }_1{\psi }^2(·){E}^2\\ \hat{b}\left(0\right)\ge 0\end{array}\Delta k_D\left(t\right)=\hat{b}{\psi }^2(·)& \end{array}$

Wherein: θ₀ and θ₁ are positive constants that the designer needs to design; ψ(⋅)=∥S(⋅)∥+1, wherein S(⋅) is a primary function of a neural network, S(⋅) and a number of neurons are determined at the designer's will; b=max{∥W^T∥,m}, wherein {circumflex over (b)} is an estimated value of b, W^T is an ideal unknown weight, and m is the upper limit of an reconstruction error ∥η(⋅)∥ of the model;

$E=2\gamma e(·)+{\gamma }^2{\int }_0^{t}e(·)d\tau +\frac{\mathrm{de}(·)}{\mathrm{dt}};$

4) Using the PID controller and the updating algorithms designed in step 3) for the joint robot system to control the trajectory motion of the joint robot.

A detailed description will be provided below for the derivation processes of the PID controller and the updating algorithms designed in this embodiment.

A generalized error E is assumed to simplify the stability analysis of the PID controller, so we have

$E=2\gamma e(·)+{\gamma }^2{\int }_0^{t}e(·)d\tau +\frac{\mathrm{de}(·)}{\mathrm{dt}}$ $D_q\left(q\right)\stackrel{.}{E}=\rho \left(t\right)L\left(\xi \right)v+\left[\rho \left(t\right)\Gamma \left(0\right)+\rho \left(t\right)\varepsilon \left(v\right)+r\left(t\right)\right]+D_q\left(q\right)F(·)+D_q\left(q\right)Q\left(x_1,t\right)+D_q\left(q\right)\left({\gamma }^{2}e+2\gamma \stackrel{.}{e}-{\ddot{q}}^{*}\right)=J\left(x_1,t\right)v+I\left(x_1,t\right)$

Wherein: J(x₁,t)=ρ(t)L(ξ),

I(x₁,t)=[ρ(t)Γ(0)+ρ(t)ε(ν)+r(t)]+D_q(q)F(⋅)+D_q(q)Q(x₁,t)+D_q(q)(γ²e+2γė={umlaut over (q)}*)

To simplify the control design and stability analysis, the function is defined as follows:

Ψ(⋅)=I(x₁,t)+½D_qĖ

(1) Algorithm based on the robust adaptive control:

The nonlinearity and uncertainty factors in the system set may be determined by the upper bound of the product of a constant and a scalar real-valued function like:

I(x₁,t)≤γ_da_fφ_f(⋅)=γ_dγ²e+2γ_dγė−γ_d{umlaut over (q)}*≤a₁φ₁(⋅)

Wherein, γ_d is the upper bound of the system parameter D_q, a_fφ_f(⋅) is the upper bound of the system uncertainty factor D_q⁻¹[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]+F(⋅)+Q(x₁,t)−{dot over (q)}*, and

$\{\begin{array}{c}{a}_1=\max \left\{{\gamma }_d{a}_f,{\gamma }_d{\gamma }^2,2{\gamma }_d\gamma ,{\gamma }_d{\stackrel{\_}{x}}_2\right\}\\ {\phi }_1(·)={\phi }_f(·)+e+\stackrel{.}{e}+1\end{array}\hspace{1em}$

So that Ψ(⋅)≤a₁φ₁(⋅)+½γ_d∥{dot over (q)}∥∥E∥≤cφ₀(⋅)

Wherein

$\{\begin{array}{c}c=\max \left\{a_1,\frac{1}{2}{\gamma }_d\right\}\\ {\phi }_0(·)={\phi }_1(·)+\stackrel{.}{q}E\end{array}\hspace{1em}$

Therefore, the robust adaptive algorithm is designed for automatically updating the controller parameters at an updating rate of:

${\Delta k}_D\left(t\right)=\hat{c}{\phi }_0^2(·)$ $\{\begin{array}{c}\stackrel{.}{\hat{c}}=-{\sigma }_0\hat{c}+{\sigma }_1{\phi }_0^2(·){E}^2\\ \hat{c}\left(0\right)\ge 0\end{array}$

Wherein σ₀ and σ₁ are positive constants that the designer needs to design; and {tilde over (c)}=c−ĉ is selected as the error value of c.

Based on the design of the above controller and the selection of the update rate, by selecting the Lyapunov function

$V=\frac{1}{2}{E}^T{D}_{q}E+\frac{1}{2{\sigma }_1}{\tilde{c}}^2$

to correspondingly verify and analyze the designed controller, it can be proved that under the effect of the designed controller, all signals in the joint robot system will eventually converge to a global scope, thus ensuring that the tracking error of the system is globally consistent and bounded.

(3) Algorithm based on the neural adaptive control:

The system is reconstructed against the uncertainty factor of the function defined above by using the way of a neural network adaptive approximation, wherein it is set that

Ψ(⋅)=W^TS(⋅)=η(⋅)

Wherein the primary function S(⋅) and the number of neurons of the neural network are determined at the designer's will, so they satisfy

$\Psi (·)\le W^TS(·)+\eta (·)\le W^TS(·)+m\le b_{\psi }(·)$

Wherein

ψ(⋅)=∥S(⋅)∥+1

b=max{∥W^T∥,m}

∥η(⋅)∥≤m,∥W^T∥≤b, taking the time-varying nature of system parameters and the unknown weight of the system into consideration, we have chosen the estimated parameter b for design and system analysis, so that the design update rate is:

$\{\begin{array}{c}\stackrel{.}{\hat{b}}=-{\sigma }_0\hat{b}+{\sigma }_1{\phi }^2(·){E}^2\\ \hat{b}\left(0\right)\ge 0\end{array}{\Delta k}_D\left(t\right)=\hat{b}{\phi }^2(·)$

Wherein θ₀ and θ₁ are positive constants that the designer needs to design; and {tilde over (b)}=b−{circumflex over (b)} is selected as the error value of b

Based on the design of the above controller and the selection of the update rate, by selecting the Lyapunov function

$V=\frac{1}{2}{E}^T{D}_{q}E+\frac{1}{2{\theta }_1}\widetilde{b^2}$

to correspondingly verify and analyze the designed controller, it can be proved that under the effect of the designed controller, all signals in the system will eventually converge to a global scope, thus ensuring that the tracking error of the system is bounded, globally consistent and bounded.

The neural network adaptive tracking control method for joint robots provided in this embodiment may ensure that the system perfectly tracks the ideal trajectory in the case of drive failure and drive saturation, and at the same time realize the bounded tracking error. Compared with traditional PID controllers, this controller is relatively simple in structure, which may handle the drive saturation and coupling effect in the joint system to a better extent, the parameter uncertainty and non-parametric uncertainty, and the execution failure during the system running. In addition, this controller may compensate the non-vanishing interference, thereby greatly reducing the complexity of control algorithms in the prior art.

Finally, it is noted that the above embodiments are only for the purpose of illustrating the technical scheme of the present disclosure without limiting it. Although a detailed specification is given for the present disclosure by reference to preferred embodiments, those of ordinary skills in the art should understand that the technical schemes of the present disclosure can be modified or equivalently replaced without departing from the purpose and scope of the technical schemes thereof, which should be included in the scope of claims of the present disclosure.

Claims

1. A neural network adaptive tracking control method for joint robots, the neural network adaptive tracking control method comprising: 1) establishing a joint robot system model: Dq(q){umlaut over (q)}+Cq(q,{dot over (q)}){dot over (q)}+Gq(q)+τ({dot over (q)},t)=ua in the joint robot system model mentioned above, q represents a position vector of the joint robot, {dot over (q)} represents a velocity vector of the joint robot, {umlaut over (q)} represents an acceleration vector of the joint robot action, ua represents a control input of the joint robot system, the system parameter Dq(q) represents an inertia matrix of the joint robot system, the system parameter Cq(q,{dot over (q)}) represents a centrifugal matrix of the joint robot system, the system parameter Gq(q) represents a universal gravitation matrix of the joint robot system, and the system parameter τ({dot over (q)},t) represents uncertainty and interference factors of the joint robot system;2) establishing a state space expression and an error definition when taking into consideration both the drive failure and actuator saturation of the joint robot system: ua(t)=ρ(t)[Γ(0)+L(ξ)ν+ε(ν)]+ε(t)=ρ(t)L(ξ)ν+[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]e=x1−q* ë={umlaut over (x)}1−{umlaut over (q)}*=Dq−1(q)ρ(t)L(ξ)ν+Dq−1(q)[ρ(t)Γ(0)+ρ(t)ε(ν)+ε(t)]+F(⋅)+Q(x1,t)−{umlaut over (q)}* In the above formulas, ua(t) represents a system control input signal considering both drive failure and actuator saturation, Γ(0)+L(ξ)ν+ε(ν) represents a control signal in the case of actuator saturation, wherein ν represents an actual controller design quantity of the system, Γ(0)+L(ξ)ν represents a smooth function proposed according to the mean value theorem of ν, Γ(0) is a bounded matrix, L(ξ) is a non-negative positive definite matrix, ε(ν) is a bounded approximate error and represents an uncertain factor of the controller; ρ(t) represents a health coefficient of the driver, ε(t) represents an interference factor of the driver; e((or e(⋅)) represents a dynamic error of the system (e(⋅) is written as e for simplification in subsequent derivation), ë represents the second derivative of the dynamic error, wherein x1=q represents a motion trajectory of the joint robot, {umlaut over (x)}1 represents an acceleration of the joint robot motion, q* represents a given joint tracking trajectory; {umlaut over (q)}* represents an acceleration of the given joint tracking F(⋅)=Dq−1(q)(Cq(q){dot over (q)}+Gq(q)), and Q(x1,t)=Dq−1(q)τ({dot over (q)},t).3) designing a PID controller and updating algorithms of the joint robot system:the PID controller ν is expressed as