Multilayer Neurolearning of Measurement-Information-Poor Hydraulic Robotic Manipulators with Disturbance Compensation

Abstract

In order to further improve the tracking performance of multiple-degree-of-freedom serial electro-hydraulic robotic manipulators, a high-performance multilayer neurocontroller will be proposed. In detail, multilayer neural networks will be employed to approximate the smooth and non-smooth state-dependent modeling uncertainties. Meanwhile, extended state observers will be utilized to estimate matched and unmatched time-varying disturbances. Moreover, these estimated values will be incorporated into the synthesized controller to compensate for the modeling uncertainties. Significantly, the proposed controller without “explosion of complexity” is suitable for the scene where the joint angular velocities are not measurable. Additionally, the sensor measurement noises can be reduced and input saturation nonlinearity will be handled.

1. Introduction

Control of multiple-degree-of-freedom (n-DOF) robotic manipulators driven by the electro-hydraulic linear or rotary actuators is full of challenges. Since the nonlinear characteristics of the servo valve, friction nonlinearity, leakage nonlinearity, and so on, will make the closed-loop controller design more complicated [1,2,3,4]. Moreover, the coupled modeling uncertainties, input saturation and immeasurable states will reduce the expected control performance [5].

In [6], Bu and Yao have proposed an adaptive robust controller for hydraulic robotic arms with parametric uncertainties and uncertain nonlinearities. Moreover, Zhou et al. have extended this controller to underwater hydraulic manipulators [7]. A novel nonlinear mathematical model has been established in [8] to express a class of parallel-serial hydraulic robotic manipulators. By using the virtual decomposition method, Mattila et al. have proposed a contact force controller for heavy-duty hydraulic manipulators [9]. In addition, Rigatos et al. have proposed a H_∞ based optimal controller for hydraulic manipulators [10].

Notably, neural networks (NN) [11] and disturbance observers [12,13] have been employed to handle function uncertainties and disturbances. In [14], the researchers have developed an extended-state-observer-based adaptive robust controller for hydraulic manipulators [14]. Additionally, Ahn et al. have developed some tracking controllers [15,16,17] for hydraulic manipulators.

Through the above discussions, it can be seen that the modeling uncertainties and input saturation cannot be addressed simultaneously. Additionally, unmeasurable system states, signal measurement noises and “explosion of complexity” produced by the conventional backstepping design method also make it difficult to synthesize high-performance closed-loop controllers. Consequently, this paper will propose a high-performance multilayer neuroadaptive tracking controller for n-DOF serial electro-hydraulic robotic arms. In particular, the multilayer neural networks [18,19] will be utilized to approximate modeling uncertainties. Meanwhile, we will employ the extended state observers [20] to estimate time-varying disturbances. Significantly, the constructed algorithm without “explosion of complexity” is suitable for the scene where the joint angular velocities are not measurable. Specially, the sensor measurement noises can be weakened. And the input saturation nonlinearity will be handled. Notably, the major innovations are listed as follows:

• Smooth and non-smooth modeling uncertainties can be compensated.
• Matched and mismatched time-varying disturbances can be compensated.
• The proposed controller does not depend on the angular velocity measurements of the joints and is free of “explosion of complexity”.
• The proposed controller has the advantages of low noise sensitivity and can resist to input saturation.

Notations: i = 1, 2, 3; j = 1, 2; ι = 2, 3, …, n − 1 with n being the DOF of the system; $\widetilde{•}=•-\widehat{•}$ is the estimation error of $•$ with $\widehat{•}$ being its estimate; ||•|| is the standard Euclidean norm; tr(•) is the trace of a matrix $•$; σ_min(•) and σ_max(•) are the minimum and maximum eigenvalues of a matrix $•$, respectively.

2. Problem Formulation

This paper considers the motion control of n-DOF serial robotic manipulators with hydraulic cylinder/motor drives. In particular, the structure diagram of n-DOF serial robotic arms with hydraulic cylinder drives is given in Figure 1. Based on [6], the rigid-body dynamics of the hydraulic manipulators can be described as

$$J(q)\ddot{q}+V_c(q,\dot{q})\dot{q}+G(q)+F_f(\dot{q})+F_e(q,\dot{q})+f_d(t)=T_c(t)$$

(1)

where q(t)∈ℝⁿ is the joint angular displacement; J(q)∈ℝ^n×n is the nominal inertial matrix; $V_c(q, \dot{q})$∈ℝ^n×n is the centripetal–Coriolis matrix; G(q)∈ℝⁿ is the gravity vector; $F_{f }\left(\dot{q}\right)$∈ℝⁿ is the friction; $F_e(q, \dot{q})$∈ℝⁿ is the uncertainty; f_d(t)∈ℝⁿ is the disturbance; and T_c(t)∈ℝⁿ is the input torque.

The control torque is [6]

$$T_c={\displaystyle \frac{\partial y_L(q)}{\partial q}}(A_a{P}_a-A_b{P}_b)$$

(2)

where y_L(q) = [y_L1(q₁), y_L2(q₂), …, y_Ln(q_n)]^T∈ℝⁿ is the vector of the actuator displacements; A_a∈ℝ^n×n and A_b∈ℝ^n×n are the chamber effective areas or displacements of the hydraulic actuators; P_a∈ℝⁿ and P_b∈ℝⁿ are the chamber pressures.

The pressure dynamics of the hydraulic actuator can be established as

$$\begin{array}{l}{\dot{P}}_a={\displaystyle \frac{{\beta }_{ef}}{V_a(q)}}\left[Q_a-A_a{\displaystyle \frac{\partial y_L(q)}{\partial q}}\dot{q}-C_{tl}{P}_{ab}\right]+Q_{e1}(q,\dot{q},P_a,P_b)+Q_{d1}(t)\\ {\dot{P}}_b={\displaystyle \frac{{\beta }_{ef}}{V_b(q)}}\left[A_b{\displaystyle \frac{\partial y_L(q)}{\partial q}}\dot{q}-Q_b+C_{tl}{P}_{ab}\right]+Q_{e2}(q,\dot{q},P_a,P_b)+Q_{d2}(t)\end{array}$$

(3)

where β_ef∈ℝ is the effective bulk modulus of the hydraulic oil; V_a(q) = V_a0 + A_ay_L(q)∈ℝ^n×n and V_b(q) = V_b0 − A_by_L(q)∈ℝ^n×n are the total control volumes of the two hydraulic chambers, respectively, with V_a0∈ℝ^n×n and V_b0∈ℝ^n×n being initial control volumes of the corresponding chambers; Q_a∈ℝⁿ and Q_b∈ℝⁿ are the flow rates of the two chambers of hydraulic actuator, respectively; C_tl∈ℝ^n×n is the internal leakage coefficient; P_ab = P_a − P_b and P_L = A_aP_a − A_bP_b∈ℝⁿ;$Q_{e1}(q, \dot{q},{ P}_a,{ P}_b)=Q_{e1c}(q, \dot{q},{ P}_a,{ P}_b)+Q_{e1d}(P_a,{ P}_b)$∈ℝⁿ and $Q_{e2}(q, \dot{q},{ P}_a,{ P}_b)$ = ${ Q}_{e2c}(q, \dot{q},{ P}_a,{ P}_b)+Q_{e2d}(P_a,{ P}_b)$∈ℝⁿ are the uncertainties, specially, $Q_{e1c}(q, \dot{q},{ P}_a,{ P}_b)$ and $Q_{e2c}(q, \dot{q},{ P}_a,{ P}_b)$ are continuous uncertainties while $Q_{e1d}(P_a,{ P}_b)$ and $Q_{e2d}(P_a,{ P}_b)$ are discontinuous uncertainties; Q_d1(t)∈ℝⁿ and Q_d2(t)∈ℝⁿ are time-varying disturbances.

Q_a∈ℝⁿ and Q_b∈ℝⁿ are

$$\begin{array}{l}{Q}_a=C_d{\omega }_d{K}_g\sqrt{{\displaystyle \frac{2}{{\rho }_o}}}\left[w(u_{\mathrm{sat}})\sqrt{P_s-P_a}+w(-u_{\mathrm{sat}})\sqrt{P_a-P_r}\right]u_{\mathrm{sat}}(\tau )\\ Q_b=C_d{\omega }_d{K}_g\sqrt{{\displaystyle \frac{2}{{\rho }_o}}}\left[w(u_{\mathrm{sat}})\sqrt{P_b-P_r}+w(-u_{\mathrm{sat}})\sqrt{P_s-P_b}\right]u_{\mathrm{sat}}(\tau )\end{array}$$

(4)

where C_d∈ℝ^n×n, ω_d∈ℝ^n×n and K_g∈ℝ^n×n are diagonal discharge coefficient matrix, diagonal discharge coefficient matrix, and diagonal electrical gain matrix, respectively; ρ_o∈ℝ is the hydraulic oil density; u_sat(τ)∈ℝⁿ is the saturated input voltage with τ∈ℝⁿ being the actual input voltage applied to servo valve; w(u_sat)=[1+tanh(k_wu_sat)]/2∈ℝ^n×n with k_w∈ℝ^n×n being a positive diagonal gain matrix; P_s∈ℝⁿ and P_r∈ℝⁿ are the supply and return pressures, respectively.

u_sat(τ) is

$$u_{\mathrm{sat}}(\tau )=\mathrm{sat}(\tau )=\left\{\begin{array}{l}\overline{u}, \tau \ge \overline{u}\\ \tau , \underset{\_}{u}<\tau <\\ \underset{\_}{u}, \tau \le \underset{\_}{u}\end{array}\right.\overline{u}$$

(5)

where $\stackrel{\mathrm{-}}{u}$∈ℝⁿ and $\underset{\_}{u}$∈ℝⁿ are the upper and lower bounds of u, respectively.

Define the state vector ζ = [${\mathsf{\zeta }}_1^T$, ${\mathsf{\zeta }}_2^T$, ${\mathsf{\zeta }}_3^T$]^T $\triangleq$[q^T $, {\dot{q}}^T$, A_a$P_a^T$–A_b$P_b^T$]^T. The system dynamics can be established as

$$\begin{array}{l}{\dot{\zeta }}_1={\zeta }_2\\ {\dot{\zeta }}_2=J_{\mathrm{inv}}({\zeta }_1)\left[H({\zeta }_1){\zeta }_3+f_2({\zeta }_1,{\zeta }_2)+D_2(t)\right]\\ {\dot{\zeta }}_3=R_u{u}_{\mathrm{sat}}(\tau )+F_3({\zeta }_1,{\zeta }_2,P_a,P_b)+F_{dc}(P_a,P_b)+D_3(t)\end{array}$$

(6)

where

$$\begin{array}{c}{J}_{\mathrm{inv}}({\zeta }_1)=J^{-1}(q),H({\zeta }_1)={\displaystyle \frac{\partial y_L(q)}{\partial q}}\\ F_{dc}(P_a,P_b)=A_a{Q}_{e1d}(P_a,P_b)-A_b{Q}_{e2d}(P_a,P_b)\\ f_2({\zeta }_1,{\zeta }_2)=-V_c(q,\dot{q})\dot{q}-G(q)-F_f(\dot{q})-F_e(q,\dot{q})\\ D_2(t)=-f_d(t),D_3(t)=A_a{Q}_{d1}(t)-A_b{Q}_{d2}(t)\\ R_u={\beta }_{ef}{C}_d{\omega }_d{K}_g\sqrt{{\displaystyle \frac{2}{{\rho }_o}}}{\displaystyle \frac{A_a}{V_a(q)}}\left[w(u_{\mathrm{sat}})\sqrt{P_s-P_a}+w(-u_{\mathrm{sat}})\sqrt{P_a-P_r}\right]\\ +{\beta }_{ef}{C}_d{\omega }_d{K}_g\sqrt{{\displaystyle \frac{2}{{\rho }_o}}}{\displaystyle \frac{A_b}{V_b(q)}}\left[w(u_{\mathrm{sat}})\sqrt{P_b-P_r}+w(-u_{\mathrm{sat}})\sqrt{P_s-P_b}\right]\\ F_3({\zeta }_1,{\zeta }_2,P_a,P_b)=A_a{Q}_{e1c}(q,\dot{q},P_a,P_b)-\left[{\displaystyle \frac{A_a}{V_a(q)}}+{\displaystyle \frac{A_b}{V_b(q)}}\right]{\beta }_{ef}{C}_{tl}{P}_{ab}\\ -\left[{\displaystyle \frac{A_a^2}{V_a(q)}}+{\displaystyle \frac{A_b^2}{V_b(q)}}\right]{\beta }_{ef}{\displaystyle \frac{\partial y_L(q)}{\partial q}}\dot{q}-A_b{Q}_{e2c}(q,\dot{q},P_a,P_b)\end{array}$$

(7)

Control goal: Given any desired trajectory ζ_1d∈ℝⁿ, we aim to integrate a bounded τ(t) so that the output ζ₁ can track ζ_1d closely.

Property 1 

[21]. For any vector p∈ℝⁿ, there exist a constant $\underset{\_}{J}$∈ℝ⁺ and a function$\stackrel{\mathrm{-}}{J}$(q)∈ℝ⁺, so that

$$\underset{¯}{J}{‖p‖}^2\le p^{T}J(q)p\le \overline{J}(q){‖p‖}^2$$

(8)

Property 2 

[21]. The matrix ${\dot{J}\left(q\right)–2V}_c(q, \dot{q})$ is skew symmetric, i.e., for any vector p∈ℝⁿ

$$p^T\left[\dot{J}(q)-2V_c(q,\dot{q})\right]p=0$$

(9)

Assumption 1. 

The desired trajectory ζ_1d∈ℝⁿ is second-order differentiable.

Assumption 2. 

The joint angular velocity $\dot{q}$ is immeasurable.

Assumption 3. 

D₂(t) and D₃(t) as well as their first-order time derivatives are bounded.

3. Multilayer Neurocontroller with Disturbance Compensation

Firstly, we transform (6) as 

$$\begin{array}{l}{\dot{\zeta }}_1={\zeta }_2\\ {\dot{\zeta }}_2=J_{\mathrm{inv}}({\zeta }_{1d})H({\zeta }_{1d}){\zeta }_3+F_2({\zeta }_{1d},{\dot{\zeta }}_{1d})+J_{\mathrm{inv}}({\zeta }_{1d})D_2(t)+{\overline{N}}_2\\ {\dot{\zeta }}_3=R_u{u}_{\mathrm{sat}}(\tau )+F_3({\zeta }_{1d},{\dot{\zeta }}_{1d},P_a,P_b)+F_{dc}(P_a,P_b)+D_3(t)+{\overline{N}}_3\end{array}$$

(10)

where F₂(${\mathsf{\zeta }}_{1d}$, ${\dot{\mathsf{\zeta }}}_{1d}$) = J_inv(${\mathsf{\zeta }}_{1d}$)f₂(${\mathsf{\zeta }}_{1d}$, ${\dot{\mathsf{\zeta }}}_{1d}$) and

$$\begin{array}{l}{\overline{N}}_2=J_{\mathrm{inv}}({\zeta }_1)H({\zeta }_1){\zeta }_3-J_{\mathrm{inv}}({\zeta }_{1d})H({\zeta }_{1d}){\zeta }_3\\ +J_{\mathrm{inv}}({\zeta }_1)f_2({\zeta }_1,{\zeta }_2)-J_{\mathrm{inv}}({\zeta }_{1d})f_2({\zeta }_{1d},{\dot{\zeta }}_{1d})\\ +J_{\mathrm{inv}}({\zeta }_1)D_2(t)-J_{\mathrm{inv}}({\zeta }_{1d})D_2(t)\\ {\overline{N}}_3=F_3({\zeta }_1,{\zeta }_2,P_a,P_b)-F_3({\zeta }_{1d},{\dot{\zeta }}_{1d},P_a,P_b)\end{array}$$

(11)

Notably, ${\overline{N}}_2\le {\mathsf{\Theta }}_1‖{\zeta }_1-{\zeta }_{1d}‖+{\mathsf{\Theta }}_2‖{\zeta }_1-{\dot{\zeta }}_{1d}‖$ and ${\overline{N}}_3\le {\mathsf{\Theta }}_3‖{\zeta }_1-{\zeta }_{1d}‖+{\mathsf{\Theta }}_4‖{\zeta }_1-{\dot{\zeta }}_{1d}‖$ with ${\mathsf{\Theta }}_1$, ${\mathsf{\Theta }}_2$, ${\mathsf{\Theta }}_3$ and ${\mathsf{\Theta }}_4$ being some positive constants.

3.1. Multilayer Neuroadaptive Approximation

For any continuous nonlinear functions $F_2\left({\mathsf{\zeta }}_{1d}, {\dot{\mathsf{\zeta }}}_{1d}\right)$∈C^N(ℤ) and $F_3({\mathsf{\zeta }}_{1d}, {\dot{\mathsf{\zeta }}}_{1d},P_a,P_b)$∈C^N(ℤ) with the map ℤ→ℝ^N and any smooth function F_dc(P_a,P_b) except at a particular point where it has a finite jump and is continuous from the right, there exist weights and thresholds so that

$$\begin{array}{l}{F}_2({\zeta }_{1d},{\dot{\zeta }}_{1d})=W_2^T{\psi }_2\left(V_2^T{\overline{\zeta }}_{2d}\right)+{\tau }_2\left({\overline{\zeta }}_{2d}\right)\\ F_3({\zeta }_{1d},{\dot{\zeta }}_{1d},P_a,P_b)=W_3^T{\psi }_3\left(V_3^T{\overline{\zeta }}_{3d}\right)+{\tau }_3\left({\overline{\zeta }}_{3d}\right)\\ F_{dc}(P_a,P_b)=W_{dc}^T{\psi }_{dc}\left(V_{dc}^T{\overline{\zeta }}_{dc}\right)+{\tau }_{dc}\left({\overline{\zeta }}_{dc}\right)\end{array}$$

(12)

where V₂∈ℝ^{(N₁+1)×N₂}, V₃∈ℝ^{(N₃+1)×N₄} and V_dc∈ℝ^{(M₁+1)×M₂} are the bounded constant matrices of ideal weights for the first-to-second layers; W₂∈ℝ^(N₂+1)×N, W₃∈ℝ^(N₄+1)×N and W_dc∈ℝ^(M₂+1)×M are bounded constant matrices of ideal weights for the second-to-third layers; N₁, N₃ and M₁ are the number of neurons in the input layers; N₂, N₄ and M₂ are the number of neurons in the hidden layers; ${\psi }_2(•)$ and ${\psi }_3(•)$ are activation functions; ${\psi }_{\mathrm{dc}}(•)$ is a jump basis function; ${\overline{\mathsf{\zeta }}}_{2d}$, ${\overline{\mathsf{\zeta }}}_{3d}$ and ${\overline{\mathsf{\zeta }}}_{dc}$ are input vectors; additionally, ${\tau }_2\left({\overline{\mathsf{\zeta }}}_{2d}\right)$, ${\tau }_3\left({\overline{\mathsf{\zeta }}}_{3d}\right)$ and ${\tau }_{dc}\left({\overline{\mathsf{\zeta }}}_{dc}\right)$ are function reconstruction errors.

Based on (12), the smooth functions $F_2\left({\mathsf{\zeta }}_{1d}, {\dot{\mathsf{\zeta }}}_{1d}\right)$ and $F_3({\mathsf{\zeta }}_{1d}, {\dot{\mathsf{\zeta }}}_{1d},{ P}_a,{ P}_b)$ as well as the non-smooth function F_dc(P_a,P_b) can be approximated by the two-layer NN as [18,19]

$$\begin{array}{l}{\widehat{F}}_2({\zeta }_{1d},{\dot{\zeta }}_{1d})={\widehat{W}}_2^T{\psi }_2\left({\widehat{V}}_2^T{\overline{\zeta }}_{2d}\right)\\ {\widehat{F}}_3({\zeta }_{1d},{\dot{\zeta }}_{1d},P_a,P_b)={\widehat{W}}_3^T{\psi }_3\left({\widehat{V}}_3^T{\overline{\zeta }}_{3d}\right)\\ {\widehat{F}}_{dc}(P_a,P_b)={\widehat{W}}_{dc}^T{\psi }_{dc}\left(V_{dc}^T{\overline{\zeta }}_{dc}\right)\end{array}$$

(13)

3.2. Observer Design

Firstly, we extend the terms ${\tau }_2\left({\overline{\mathsf{\zeta }}}_{2d}\right)$ + J_inv(ζ_1d)D₂(t) and ${\tau }_3\left({\overline{\mathsf{\zeta }}}_{3d}\right)$ + D₃(t) as new states ζ_e2 and ζ_e3, respectively. Thus, (10) can be arranged as

$$\begin{array}{c}\left\{\begin{array}{l}{\dot{\zeta }}_1={\zeta }_2\\ {\dot{\zeta }}_2=J_{\mathrm{inv}}({\zeta }_{1d})H({\zeta }_{1d}){\zeta }_3+W_2^T{\psi }_2\left(V_2^T{\overline{\zeta }}_{2d}\right)+{\zeta }_{e2}+{\overline{N}}_2\\ {\dot{\zeta }}_{e2}={\Delta }_2(t)\end{array}\right.\\ \left\{\begin{array}{l}{\dot{\zeta }}_3=R_u{u}_{\mathrm{sat}}(\tau )+W_3^T{\psi }_3\left(V_3^T{\overline{\zeta }}_{3d}\right)\\ +W_{dc}^T{\psi }_{dc}\left(V_{dc}^T{\overline{\zeta }}_{dc}\right)+{\tau }_{dc}\left({\overline{\zeta }}_{dc}\right)+{\zeta }_{e3}+{\overline{N}}_3\\ {\dot{\zeta }}_{e3}={\Delta }_3(t)\end{array}\right.\end{array}$$

(14)

Based on (14) and [20], two sets of uncertainty observers can be constructed as

$$\begin{array}{c}\left\{\begin{array}{l}{\dot{\widehat{\zeta }}}_1={\widehat{\zeta }}_2+3{\omega }_{o1}\left({\zeta }_1-{\widehat{\zeta }}_1\right)\\ {\dot{\widehat{\zeta }}}_2={\overline{\zeta }}_3+{\widehat{W}}_2^T{\psi }_2\left({\widehat{V}}_2^T{\overline{\zeta }}_{2d}\right)+{\widehat{\zeta }}_{e2}+3{\omega }_{o1}^2{\tilde{\zeta }}_1\\ {\dot{\widehat{\zeta }}}_{e2}={\omega }_{o1}^3\left({\zeta }_1-{\widehat{\zeta }}_1\right)\end{array}\right.\\ \left\{\begin{array}{l}{\dot{\widehat{\zeta }}}_3=R_u{u}_{\mathrm{sat}}(\tau )+{\widehat{W}}_3^T{\psi }_3\left({\widehat{V}}_3^T{\overline{\zeta }}_{3d}\right)+{\widehat{\zeta }}_{e3}\\ +{\widehat{W}}_{dc}^T{\psi }_{dc}\left(V_{dc}^T{\overline{\zeta }}_{dc}\right)+2{\omega }_{o2}\left({\zeta }_3-{\widehat{\zeta }}_3\right)\\ {\dot{\widehat{\zeta }}}_{e3}={\omega }_{o2}^2\left({\zeta }_3-{\widehat{\zeta }}_3\right)\end{array}\right.\end{array}$$

(15)

where ${\overline{\mathsf{\zeta }}}_3$ = J_inv(ζ_1d)H(ζ_1d)ζ₃; ω_o1∈ℝ^n×n and ω_o2∈ℝ^n×n are positive gain matrices.

By introducing two state vectors as e_o = [${\mathrm{e}}_{\mathrm{o}1}^{\mathrm{T}}$, ${\mathrm{e}}_{\mathrm{o}2}^{\mathrm{T}}$, ${\mathrm{e}}_{\mathrm{o}3}^{\mathrm{T}}$]^T = [${\tilde{\mathsf{\zeta }}}_1^{\mathrm{T}}$, ${\tilde{\mathsf{\zeta }}}_2^{\mathrm{T}}/{\mathsf{\omega }}_{\mathrm{o}1}$, ${\tilde{\mathsf{\zeta }}}_{\mathrm{e}2}^{\mathrm{T}}/{\mathsf{\omega }}_{\mathrm{o}1}^2$]^T and ε_o = [${\mathsf{\epsilon }}_{\mathrm{o}1}^{\mathrm{T}}$, ${\mathsf{\epsilon }}_{\mathrm{o}2}^{\mathrm{T}}$]^T = [${\tilde{\mathsf{\zeta }}}_3^{\mathrm{T}}$, ${\tilde{\mathsf{\zeta }}}_{\mathrm{e}3}^{\mathrm{T}}/{\mathsf{\omega }}_{\mathrm{o}2}$]^T, we have

$${\dot{e}}_o={\omega }_{o1}{A}_o{e}_o+C_o{\displaystyle \frac{\left(R_{o1}+R_{o2}+{\overline{N}}_2\right)}{{\omega }_{o1}}}+H_o{\displaystyle \frac{{\Delta }_2(t)}{{\omega }_{o1}^T{\omega }_{o1}}}$$

(16)

$${\dot{\epsilon }}_o={\omega }_{o2}{B}_o{\epsilon }_o+D_o\left(N_{o1}+N_{o2}+N_{dc}+{\overline{N}}_4\right)+J_o{\displaystyle \frac{{\Delta }_3(t)}{{\omega }_{o2}}}$$

(17)

where $N_{dc}={\stackrel{\mathrm{~}}{W}}_{dc}^T{\psi }_{dc}\left(V_{dc}^T{\overline{\zeta }}_{dc}\right)$, ${\overline{N}}_4={\overline{N}}_3+{\mathsf{\tau }}_{dc}\left({\overline{\mathsf{\zeta }}}_{dc}\right)$ and

$$A_o=\left[\begin{array}{ccc}-3I& I& 0\\ -3I& 0& I\\ -I& 0& 0\end{array}\right], C_o=\left[\begin{array}{c}0\\ I\\ 0\end{array}\right], H_o=\left[\begin{array}{c}0\\ 0\\ I\end{array}\right]$$

(18)

$$B_o=\left[\begin{array}{cc}-2{\rm I}& {\rm I}\\ -{\rm I}& 0\end{array}\right], D_o=\left[\begin{array}{c}{\rm I}\\ 0\end{array}\right], J_o=\left[\begin{array}{c}0\\ {\rm I}\end{array}\right]$$

(19)

$$R_{o1}\triangleq {\tilde{W}}_2^T{\widehat{\psi }}_{2d}+{\widehat{W}}_2^T{{\widehat{\psi }}^{\prime }}_{2d}{\tilde{V}}_2^T{\overline{\zeta }}_{2d},R_{o2}\triangleq W_2^{T}O{\left({\tilde{V}}_2^T{\overline{\zeta }}_{2d}\right)}^2+{\tilde{W}}_2^T{{\widehat{\psi }}^{\prime }}_{2d}{\tilde{V}}_2^T{\overline{\zeta }}_{2d}$$

(20)

$$N_{o1}\triangleq {\tilde{W}}_3^T{\widehat{\psi }}_{3d}+{\widehat{W}}_3^T{{\widehat{\psi }}^{\prime }}_{3d}{\tilde{V}}_3^T{\overline{\zeta }}_{3d},N_{o2}\triangleq W_3^{T}O{\left({\tilde{V}}_3^T{\overline{\zeta }}_{3d}\right)}^2+{\tilde{W}}_3^T{{\widehat{\psi }}^{\prime }}_{3d}{\tilde{V}}_3^T{\overline{\zeta }}_{3d}$$

(21)

Since the matrices A_o and B_o are Hurwitz, there are positive definite matrices P_o and Q_o satisfying $A_o^T{P}_o+P_o{A}_o=-\mathbf{I}$ and $B_o^T{Q}_o+Q_o{B}_o=-\mathbf{I}$ [22,23,24]. Thus, P_o and Q_o can be calculated.

3.3. Controller Design

Define a set of error variables as [25,26]

$$\begin{array}{l}{e}_1 ={\zeta }_1-{\zeta }_{1d}, e_{j+1}={\zeta }_{j+1}-{\alpha }_{j,c}\\ z_1=e_1-{\phi }_1, z_{j+1}=e_{j+1}-{\phi }_{j+1}\end{array}$$

(22)

where φ_i∈ℝⁿ is the error compensation signal; and α_j,c∈ℝⁿ is the filtered value of the virtual control law α_j∈ℝⁿ via the following second-order filter [4]

$$\begin{array}{l}{\dot{\alpha }}_{j,c}={\alpha }_{j,dc}+2{\omega }_{cj}\left({\alpha }_j-{\alpha }_{j,c}\right)\\ {\dot{\alpha }}_{j,dc}={\omega }_{cj}^2\left({\alpha }_j-{\alpha }_{j,c}\right)\end{array}$$

(23)

in which α_j,dc∈ℝⁿ is the first-order time derivative of α_j,c; ω_cj∈ℝ^n×n is a positive gain matrix.

Denote ς_j = [α_j−α_j,c, $({\dot{\mathsf{\alpha }}}_{\mathrm{j}}–{\mathsf{\alpha }}_{\mathrm{j},\mathrm{dc}})/{\mathsf{\omega }}_{\mathrm{cj}}$]^T∈ℝⁿ, we have

$${\dot{\varsigma }}_j={\omega }_{cj}{A}_c{\varsigma }_i+H_c{\displaystyle \frac{{\alpha }_{j,dd}}{{\omega }_{cj}}}$$

(24)

where α_j,dd∈ℝⁿ is the second-order time derivative of α_j and

$$A_c=\left[\begin{array}{cc}-2I& I\\ -I& 0\end{array}\right], H_c=\left[\begin{array}{c}0\\ I\end{array}\right]$$

(25)

As A_c is Hurwitz, there is a positive definite matrix P_c = Q_o satisfying $A_c^T{P}_c+P_c{A}_c=-\mathbf{I}$ [23,24].

The error compensation signals are generated via

$$\begin{array}{l}{\dot{\phi }}_1=-k_1{\phi }_1+{\phi }_2+\left({\alpha }_{1,c}-{\alpha }_1\right)\\ {\dot{\phi }}_2=-k_2{\phi }_2+J_{inv}({\zeta }_{1d})H({\zeta }_{1d})\left[{\phi }_3+\left({\alpha }_{2,c}-{\alpha }_2\right)\right]\\ {\dot{\phi }}_3=-k_3{\phi }_3\end{array}$$

(26)

where k_i∈ℝ^n×n are positive diagonal feedback gain matrices.

Step 1: Differentiating z₁ along (10), (22) and (26) yields

$${\dot{z}}_1={\zeta }_2-{\dot{\zeta }}_{1d}+k_1{\phi }_1-{\phi }_2-\left({\alpha }_{1,c}-{\alpha }_1\right)={\alpha }_1+k_1{\phi }_1+z_2-{\dot{\zeta }}_{1d}$$

(27)

According to (27), α₁ can be designed as

$${\alpha }_1=-k_1{e}_1+{\dot{\zeta }}_{1d}$$

(28)

Substituting (28) into (27), it has

$${\dot{z}}_1=-k_1{z}_1+z_2$$

(29)

Step 2: Considering (10), (22) and (26), we can obtain ${\dot{z}}_2$ as

$${\dot{z}}_2=J_{\mathrm{inv}}({\zeta }_{1d})H({\zeta }_{1d})(z_3+{\alpha }_2)+W_2^T{\psi }_2\left(V_2^T{\overline{\zeta }}_{2d}\right)+{\zeta }_{e2}+{\overline{N}}_2-{\dot{\alpha }}_{1,c}+k_2{\phi }_2$$

(30)

Accordingly, α₂ can be designed as

$${\alpha }_2=J({\zeta }_{1d})H_{\mathrm{inv}}({\zeta }_{1d})\left[-k_2{\widehat{e}}_2-{\widehat{W}}_2^T{\psi }_2\left({\widehat{V}}_2^T{\overline{\zeta }}_{2d}\right)-{\widehat{\zeta }}_{e2}+{\dot{\alpha }}_{1,c}-z_1\right]$$

(31)

where ${\widehat{e}}_2={\widehat{\mathsf{\zeta }}}_2-{\mathsf{\alpha }}_{1,c}$.

Substituting (31) into (30) yields

$${\dot{z}}_2=-k_2{z}_2+k_2{\omega }_{o1}{e}_{o2}+R_{o1}+R_{o2}+{\overline{N}}_2+{\omega }_{o1}^2{e}_{o3}+J_{\mathrm{inv}}({\zeta }_{1d})H({\zeta }_{1d})z_3-z_1$$

(32)

Step 3: Differentiating z₃ along (10), (22) and (26) yields

$${\dot{z}}_3=R_u\tau +W_3^T{\psi }_3\left(V_3^T{\overline{\zeta }}_{3d}\right)+W_{dc}^T{\psi }_{dc}\left(V_{dc}^T{\overline{\zeta }}_{dc}\right)+{\tau }_{dc}\left({\overline{\zeta }}_{dc}\right)+{\zeta }_{e3}+{\overline{N}}_3-{\dot{\alpha }}_{2,c}+k_3{\phi }_3+\Delta u(\tau )$$

(33)

where Δu(τ) = R_uu_sat(τ)−R_uτ.

Based on (33), τ can be constructed as

$$\tau =R_{uinv}\left[-k_3\left(e_3-e_{\mathrm{sat}}\right)-{\widehat{W}}_3^T{\psi }_3\left({\widehat{V}}_3^T{\overline{\zeta }}_{3d}\right)-{\widehat{\zeta }}_{e3}-{\widehat{W}}_{dc}^T{\psi }_{dc}\left(V_{dc}^T{\overline{\zeta }}_{dc}\right)+{\dot{\alpha }}_{2,c}-J_{inv}({\zeta }_{1d})H({\zeta }_{1d})z_2\right]$$

(34)

where e_sat∈ℝⁿ is generated via [21]

$${\dot{e}}_{\mathrm{sat}}=\left\{\begin{array}{l}-k_e{e}_{\mathrm{sat}}-{\displaystyle \frac{\left|z_3^T\Delta u\right|+\frac{1}{2}\Delta u^T\Delta u}{{‖e_{\mathrm{sat}}‖}^2}}{e}_{\mathrm{sat}}+\Delta u, e_{\mathrm{sat}}\ge e_0 \\ 0, e_{\mathrm{sat}}< e_0\end{array}\right.$$

(35)

in which k_e∈ℝ^n×n is a positive diagonal feedback gain matrix and e₀∈ℝⁿ is a constant vector.

Substituting (34) into (33) yields

$${\dot{z}}_3=-k_3{z}_3+N_{o1}+N_{o2}+N_{dc}+{\omega }_{o2}{\epsilon }_{o2}+{\overline{N}}_4-J_{\mathrm{inv}}({\zeta }_{1d})H({\zeta }_{1d})z_2+k_3{e}_{\mathrm{sat}}+\Delta u(\tau )$$

(36)

3.4. Theoretical Results

Theorem 1. 

With the neural network weights tuned via

$$\begin{array}{l}{\dot{\widehat{W}}}_2=\mathrm{Proj}\left\{{\Upsilon }_{W2}\left[{\widehat{\psi }}_{2d}{\left({\widehat{z}}_2-{\displaystyle \frac{1}{2{\omega }_{o1}}}{e}_{o1}\right)}^T-{\gamma }_{W2}{\widehat{W}}_2\right]\right\}\\ {\dot{\widehat{V}}}_2=\mathrm{Proj}\left\{{\Upsilon }_{V2}\left[{\overline{\zeta }}_{2d}{\left({{\widehat{\psi }}^{\prime }}_{2d}^T{\widehat{W}}_2\left({\widehat{z}}_2-{\displaystyle \frac{1}{2{\omega }_{o1}}}{e}_{o1}\right)\right)}^T-{\gamma }_{V2}{\widehat{V}}_2\right]\right\}\\ {\dot{\widehat{W}}}_3=\mathrm{Proj}\left\{{\Upsilon }_{W3}\left[{\widehat{\psi }}_{3d}{\left(z_3+{\displaystyle \frac{1}{2}}{\epsilon }_{o1}\right)}^T-{\gamma }_{W3}{\widehat{W}}_3\right]\right\}\\ {\dot{\widehat{V}}}_3=\mathrm{Proj}\left\{{\Upsilon }_{V3}\left[{\overline{\zeta }}_{3d}{\left({{\widehat{\psi }}^{\prime }}_{3d}^T{\widehat{W}}_3\left(z_3+{\displaystyle \frac{1}{2}}{\epsilon }_{o1}\right)\right)}^T-{\gamma }_{V3}{\widehat{V}}_3\right]\right\}\\ {\dot{\widehat{W}}}_{dc}=\mathrm{Proj}\left\{{\Upsilon }_{dc}\left[{\psi }_{dc}{\left(z_3+{\displaystyle \frac{1}{2}}{\epsilon }_{o1}\right)}^T-{\gamma }_{dc}{\widehat{W}}_{dc}\right]\right\}\end{array}$$

(37)

where ϒ_W2, γ_W2, ϒ_V2, γ_V2, ϒ_W3, γ_W3, ϒ_V3, γ_V3, ϒ_dc and γ_dc  are positive adaptation rate matrices, and by reasonably choosing k_i, ω_oj and ω_cj, thus the proposed controller can guarantee that all system signals are bounded under the closed-loop operation.

Proof: 

See Appendix A. □

4. Comparative Verification

4.1. A 2-DOF Hydraulic Manipulator

To test the proposed controller, a 2-DOF serial hydraulic robotic manipulator will be employed, the 1th link of which is driven by a single-rod hydraulic cylinder and the other one is driven by a double-vane hydraulic motor. Particularly, for simplification, the hinges Z₀ and Z₁ in Figure 1 are assumed to be on the same horizontal line.

For this hydraulic manipulator, J(q), $V_c(q, \dot{q})$, and G(q) are expressed as

$$J(q)=\left[\begin{array}{cc}{J}_{11}& J_{12}\\ J_{21}& J_{22}\end{array}\right],V_c(q,\dot{q})=\left[\begin{array}{cc}{V}_{c11}& V_{c12}\\ V_{c21}& V_{c22}\end{array}\right],G(q)=\left[\begin{array}{c}{G}_1\\ G_2\end{array}\right]$$

(38)

where

$$\begin{array}{l}{J}_{11}=J_{I1}+J_{m1}{S}_{c1}^2+J_{m2}{S}_1^2+J_L{S}_1^2+J_{I2}\\ +J_{m2}{S}_{c2}^2+J_L{S}_2^2+2(J_2{S}_1{S}_{c2}+J_L{S}_1{S}_2)\cos (q_2)\\ J_{12}=J_{I2}+J_{m2}{S}_{c2}^2+J_L{S}_2^2+(J_2{S}_1{S}_{c2}+J_L{S}_1{S}_2)\cos (q_2)\\ J_{21}=J_{12},J_{22}=J_{I2}+J_{m2}{S}_{c2}^2+J_L{S}_2^2\\ V_{c11}=-(J_{m2}{S}_1{S}_{c2}+J_L{S}_1{S}_2){\dot{q}}_2\sin (q_2)\\ V_{c12}=-(J_{m2}{S}_1{S}_{c2}+J_L{S}_1{S}_2)({\dot{q}}_2-{\dot{q}}_1)\sin (q_2)\\ V_{c21}=(J_{m2}{S}_1{S}_{c2}{\dot{q}}_1-J_L{S}_1{S}_2{\dot{q}}_2)\sin (q_2),V_{c22}=0\\ G_1=(J_{m1}{S}_{c1}+J_{m2}{S}_1+J_L{S}_1)g\cos (q_1)\\ +(J_{m2}{S}_{c2}+J_L{S}_2)g\cos (q_1+q_2)\\ G_2=(J_{m2}{S}_{c2}+J_L{S}_2)g\cos (q_1+q_2)\end{array}$$

(39)

in which q_j is the angular position of the jth link; J_mj and J_L are the masses of the jth link and the load, respectively; S_j is the length of the jth link; S_cj is the length from the center of gravity position to the joint axis of the jth link; g is the gravitational acceleration; and J_Ij is the rotational inertia of the jth link around its own joint axis. Specifically, P_a = [P_a1, P_a2]^T, P_b = [P_b1, P_b2]^T, ζ₁ = [ζ₁₁, ζ₁₂]^T, ζ₂ = [ζ₂₁, ζ₂₂]^T, ζ₃ = [ζ₃₁, ζ₃₂]^T, ζ_e2 = [ζ_e21, ζ_e22]^T, ζ_e3 = [ζ_e31, ζ_e32]^T, C_tl = diag{C_tl11, C_tl22}, V_a = diag{V_a11, V_a22}, V_b = diag{V_b11, V_b22}, F₂ = [F₂₁, F₂₂]^T, F₃ = [F₃₁, F₃₂]^T and F_dc = [F_dc1, F_dc2]^T.

The System parameters are provided in

Table 1. System parameters.

Parameter	Value (Unit)	Parameter	Value (Unit)
Jm1	5.2 (kg)	L10	0.2 (m)
Jm2	6.2 (kg)	L11	0.2 (m)
JI1	2.25 (kg•m2)	L12	0.2 (m)
JI2	1.25 (kg•m2)	Ps	50 (bar)
JL	15.2 (kg)	Pr	0 (bar)
Sc1	0.2 (m)	βef	1.26 × 109 (Pa)
Sc2	0.15 (m)	Aa11	1 × 10−3 (m2)
S1	0.5 (m)	Ab11	5 × 10−4 (m2)
S2	0.2 (m)	Aa22	1.22 × 10−4 (m3/rad)
Ctl11, Ctl22	2.8 × 10−12 (m3/s/Pa)	Ab22	1.22 × 10−4 (m3/rad)
Cd11ωd11Kg11$\sqrt{{\displaystyle \frac{2}{{\mathsf{\rho }}_o}}}$	3.2 × 10−8 (m3/s/V/$\sqrt{{\mathrm{P}}_{\mathrm{a}}}$)	Cd22ωd22Kg22$\sqrt{{\displaystyle \frac{2}{{\mathsf{\rho }}_o}}}$	2.2 × 10−8 (m3/s/V/$\sqrt{{\mathrm{P}}_{\mathrm{a}}}$)

.

Three controllers will be employed to track the command ζ_1d = [q_1d, q_2d]^T = [0.42sin(1.25t)(1 − e^–0.5t) + 1.05rad, 0.22cos(2.2t)(1 − e^–0.5t) + 0.5rad]^T. Moreover, the initial position of the system is ζ₁(0) = [ζ₁₁(0), ζ₁₂(0)]^T = [π/3, 0.2]^T. Furthermore, the modeling uncertainties are artificially set as $F_e(q, \dot{q})$ = [0.5q₁${\dot{q}}_1$ + 2${\dot{q}}_1$cos(q₂) + 1.5${\dot{q}}_2$cos(q₂), 0.5q₂${\dot{q}}_2$+2q₂ + 1.5${\dot{q}}_2$]^T, f_d(t) = [200sin(t), 150sin(t)]^T, $Q_{e1}(q, \dot{q},{ P}_a,{ P}_b)$ = [1 × 10⁴q₁${\dot{q}}_1$ + 6 × 10⁻⁸(P_a1 − P_b1)² − 3.5 × 10⁻¹²β_ef$\mathrm{sign}(P_{a1}-P_{b1})\sqrt{\left|P_{a1}-P_{b1}\right|}$/V_a11, 1 × 10⁴q₂${\dot{q}}_2$ + 6 × 10⁻⁸(P_a2–P_b2)² − 3.5 × 10⁻¹²β_ef$\mathrm{sign}(P_{a2}-P_{b2})\sqrt{\left|P_{a2}-P_{b2}\right|}$/V_a22]^T, $Q_{e2}(q, \dot{q},{ P}_a,{ P}_b)$ = [8 × 10⁴q₁${\dot{q}}_1$+ 1 × 10⁻⁸(P_a1− P_b1)² + 3.5 × 10⁻¹²β_ef$\mathrm{sign}(P_{a1}-P_{b1})\sqrt{\left|P_{a1}-P_{b1}\right|}$/V_b11, 8 × 10⁴q₂${\dot{q}}_2$ + 1 × 10⁻⁸(P_a2–P_b2)² + 3.5 × 10 ^{− 12}β_efsign(P_a2 − P_b2)$\sqrt{\left|P_{a2}-P_{b2}\right|}$/V_b22]^T, Q_d1(t) = [1.2 × 10⁶sin(t), 1.2 × 10⁶sin(t)]^T and Q_d2(t) = [8 × 10⁶sin(t), 8 × 10⁶sin(t)]^T, respectively.

4.2. Comparative Results

(1)

C1: It is the proposed controller in Section 3. Its design parameters are picked in Table 2.

(2)

C2: It is same as C1 but without disturbance compensation. Notably, k₃ = diag{350, 350}.

(3)

C3: It is the backstepping based feedback controller.

Table 2. The design parameters of C1.

The gains of the controller	k1 = diag{80, 500}, k2 = diag{200, 500}, k3 = diag{300, 300}, ke = diag{5, 5}
The bandwidths of the observers	ωo1 = diag{350, 350}, ωo2 = diag{500, 500}
The activation functions of the MLNN	${\psi }_2(•)$ = tanh($•$), ${\psi }_3(•)$ = tanh($•$), ${\psi }_{dc}(•)$ = $\left\{\begin{array}{ll}{\left[1-e^{-\left(P_a-P_b\right)}\right]}^{{\mathsf{\theta }}_{dc}-1},& \mathrm{if} \left(P_a-P_b\right)>0\\ 0,& \mathrm{otherwise}\end{array}\right.$
The gains of the MLNN adaptive laws	ϒW2 = [1 × 102I11, 1 × 102I11]T, ϒV2 = [1 × 104I5, 1 × 104I5]T, ϒW3 = [8 × 101I14, 8 × 101I14]T, ϒV3 = [1 × 104I7, 1 × 104I7]T, ϒdc = [1 × 103I14, 5 × 102I14]T, γW2 = [1 × 10−1I11, 1 × 10−1I11]T, γV2 = [1 × 10−1I5, 1 × 10−1I5]T, γW3 = [1 × 10−1I14, 1 × 10−1I14]T, γV3 = [1 × 10−1I7, 1 × 10−1I7]T, γdc = [1 × 10−1I14, 1 × 10−1I14]T
The gains of the filters	ωc1 = diag{3 × 103, 3 × 103}, ωc2 = diag{3.2 × 103, 3.2 × 103}
Other parameters	e0 = [1 × 10−1, 1 × 10−1]T, $\overline{u}$ = [10, 10]T, $\underset{\_}{u}$ = [–10, –10]T

Table 2. The design parameters of C1.

The gains of the controller	k1 = diag{80, 500}, k2 = diag{200, 500}, k3 = diag{300, 300}, ke = diag{5, 5}
The bandwidths of the observers	ωo1 = diag{350, 350}, ωo2 = diag{500, 500}
The activation functions of the MLNN	${\psi }_2(•)$ = tanh($•$), ${\psi }_3(•)$ = tanh($•$), ${\psi }_{dc}(•)$ = $\left\{\begin{array}{ll}{\left[1-e^{-\left(P_a-P_b\right)}\right]}^{{\mathsf{\theta }}_{dc}-1},& \mathrm{if} \left(P_a-P_b\right)>0\\ 0,& \mathrm{otherwise}\end{array}\right.$
The gains of the MLNN adaptive laws	ϒW2 = [1 × 102I11, 1 × 102I11]T, ϒV2 = [1 × 104I5, 1 × 104I5]T, ϒW3 = [8 × 101I14, 8 × 101I14]T, ϒV3 = [1 × 104I7, 1 × 104I7]T, ϒdc = [1 × 103I14, 5 × 102I14]T, γW2 = [1 × 10−1I11, 1 × 10−1I11]T, γV2 = [1 × 10−1I5, 1 × 10−1I5]T, γW3 = [1 × 10−1I14, 1 × 10−1I14]T, γV3 = [1 × 10−1I7, 1 × 10−1I7]T, γdc = [1 × 10−1I14, 1 × 10−1I14]T
The gains of the filters	ωc1 = diag{3 × 103, 3 × 103}, ωc2 = diag{3.2 × 103, 3.2 × 103}
Other parameters	e0 = [1 × 10−1, 1 × 10−1]T, $\overline{u}$ = [10, 10]T, $\underset{\_}{u}$ = [–10, –10]T

Notably, all controller parameters of C2 and C3 are chosen to be same as those of C1, except for those already mentioned.

The tracking errors of the three controllers for the two links can be found in Figure 2 and Figure 3. From them, it can be easily seen that the proposed C1 controller has the best tracking performance. By comparing the results of C1 and C2, C1 and C3, or C2 and C3, the ability to reject matched and mismatched disturbances of C1 can be proved. Furthermore, the state estimation performance of Link 1 and Link 2 with C1 can be found in Figure 4 and Figure 5, respectively. As shown, the high-accuracy estimation performance of the proposed observers can be verified. Meanwhile, from Figure 6, Figure 7, Figure 8 and Figure 9, bounded and regular disturbance and function estimation can further demonstrate the effectiveness of the designed observers. Additionally, from Figure 10, the control torques of Link 1 and Link 2 with C1 are both regular and meanwhile bounded.

5. Conclusions

This paper proposes a novel high-performance multilayer neuroadaptive algorithm for trajectory tracking control of n-DOF electro-hydraulic robotic manipulators. Notably, the developed controller can simultaneously compensate for smooth and non-smooth modeling uncertainties via multilayer neural networks. Meanwhile, matched and mismatched time-varying disturbances can be compensated by using the design technology of extended state observer. Moreover, the proposed algorithm can be suitable for the scene where the joint angular velocities are not measurable and can be free of the “explosion of complexity” generated by the conventional backstepping design method. Furthermore, the proposed controller has the advantages of anti-noise and anti-input-saturation, which puts it in front of the existing control algorithms.