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Abstract

Aerial work platform is a special vehicle for carrying personnel to the appointed site in the air for operations. Therefore, the work platform requires high stability. This article proposes a sliding mode controller based on neural network for tracking control of folding-boom aerial work platform. Since the chattering caused by sliding mode controller with high-speed switching control may lead to system performance degradation, continuous control obtained from neural network system replaces discontinuous switching control to eliminate chattering. Furthermore, the whole system is proved to be stable by Lyapunov stability theorem. Finally, numerical results show that the designed controller can eliminate the chattering resulting from switching control in sliding mode controller and inhibit the vibration of work platform when there exists system uncertainty. Moreover, the controller is effective for the reduction of tracking error.

Keywords

Aerial work platform flexible multi-body dynamics trajectory tracking control neural network–based sliding mode control

Introduction

The folding-boom aerial work platform is a type of engineering vehicle which is used for enhancing the personnel to the designated place for installation and maintenance, as shown in Figure 1.¹ Therefore, it requires high stability for the security of people working on the platform. As the extensive use of light-long beam in the structure of arm system of aerial work platform, elastic deformation of beam cannot be ignored to guarantee work platform’s stable motion. For realizing trajectory tracking of aerial work platform, adaptive neural network controller is adopted in Jia et al.² and self-tuning fuzzy proportional–integral–derivative (PID) control scheme is proposed in Miao et al.³ However, the deformation of beam is not considered in the established model. Based on the theory of flexible multi-body dynamics and Lagrange’s equation, the model of folding-boom aerial work platform with flexible beam driven by hydraulic cylinder is established, and the vibration existed in flexible beam is shown in Hu et al.¹ Besides, the similar model is obtained, and fuzzy PID is used for the trajectory tracking of work platform in Meng,⁴ but this study only gives simulation results, and the stability of system is not proved. In addition, backstepping control method of work platform is proposed in Hu et al.⁵ based on the model in Hu et al.¹ However, this method can only be used for the accurate models. In fact, there exists parameter uncertainty in the model of folding-boom aerial work platform. Sliding mode control (SMC) has robustness to model uncertainty,⁶ which has been widely used for the control of nonlinear system. This article proposes an SMC method used for tracking control of aerial work platform. As a robust control scheme, SMC can make the state of the system move along the designed sliding mode surface using the switching control strategy. However, as the discontinuous switching property of SMC, it is difficult to make system state converge to equilibrium point along the sliding mode surface strictly. As a result, the chattering will be occurred. To reduce the chattering, several approaches have been proposed,^7–15 among which, neural network system is an alternative method.^13–15 Artificial neural network is derived from biological networks,¹⁶ which is a kind of dynamical complex network^17–18 being closely relevant to the graph theoretic methods. As neural network system has capability of approximating any nonlinear function with arbitrary precision on a compact set based on the universal approximation theorem, it can solve the chattering problem existed in SMC. In Sun et al.,¹⁹ a neural network–based sliding mode adaptive controller is proposed for robot manipulators; however, the link is considered as rigid. In addition, the similar control method is used for flexible links in Tang et al.,¹⁴ but the links are driven by DC motors instead of hydraulic cylinder used in aerial work platform. In this article, a neural network–based sliding mode control (NNSMC) for tracking control of aerial work platform with flexible beam driven by hydraulic cylinder is presented. The proposed controller combines universal approximation capability of neural network system with the robustness of sliding mode controller. Based on the Lyapunov stability theorem, NNSMC can guarantee the stability of the whole system. In addition, the simulation results show that the aerial work platform’s tracking error can be reduced, and the vibration can be inhibited effectively when there exist model parameter uncertainties.

Figure 1.

Scheme of folding-boom aerial work platform.

This article is organized as follows. In section “Flexible multi-body dynamic equations of work platform,” the dynamic equations of aerial work platform is introduced and analyzed. Then, a NNSMC is designed to attain the control objective in section “Design of NNSMC.” In section “Simulation results,” the simulation study for tracking control of aerial work platform is carried out, and finally concluding comment is provided in section “Conclusion.”

Flexible multi-body dynamic equations of work platform

The flexible multi-body dynamic equations of aerial work platform can be written as¹

{\begin{matrix} G \overset{\cdot\cdot}{θ} + U {\overset{\cdot}{θ}}^{2} + H \overset{\cdot\cdot}{q} + R = Q_{θ} \\ M \overset{\cdot\cdot}{q} + Nq + H^{T} \overset{\cdot\cdot}{θ} + V^{T} {\overset{\cdot}{θ}}^{2} = Q_{q} \end{matrix}

(1)

In equation (1), $Q_{q} = [\begin{matrix} Q_{3} & Q_{4} & Q_{5} & Q_{6} \end{matrix}]^{T}$ , $Q_{θ} = [\begin{matrix} Q_{1} & Q_{2} \end{matrix}]^{T}$ , $\overset{\cdot\cdot}{θ} = [\begin{matrix} {\overset{\cdot\cdot}{θ}}_{1} & {\overset{\cdot\cdot}{θ}}_{2} \end{matrix}]^{T}$ , ${\overset{\cdot}{θ}}^{2} = [\begin{matrix} {\overset{\cdot}{θ}}_{1}^{2} & {\overset{\cdot}{θ}}^{2}_{2} \end{matrix}]^{T}$ , $q = [\begin{matrix} q_{11} & q_{12} & q_{21} & q_{22} \end{matrix}]^{T}$ , and $\overset{\cdot\cdot}{q} = [\begin{matrix} {\overset{\cdot\cdot}{q}}_{11} & {\overset{\cdot\cdot}{q}}_{12} & {\overset{\cdot\cdot}{q}}_{21} & {\overset{\cdot\cdot}{q}}_{22} \end{matrix}]^{T}$ , in which $θ_{1}$ and $θ_{2}$ are the angles of beams 1 and 2 with respect to their horizontal planes; $q_{11}, q_{12}$ and $q_{21}, q_{22}$ are the deflection variables associated with the two former model functions of beams 1 and 2, respectively. G, M, and H are the mass matrices and can be expressed as follows

\begin{matrix} G = [\begin{matrix} (\frac{1}{3} m_{1} + m_{2} + m) {l_{1}}^{2} & (\frac{1}{2} m_{2} + m) l_{1} l_{2} \cos (θ_{1} - θ_{2}) \\ (\frac{1}{2} m_{2} + m) l_{1} l_{2} \cos (θ_{1} - θ_{2}) & (\frac{1}{3} m_{2} + m) {l_{2}}^{2} \end{matrix}] \\ M = [\begin{matrix} \frac{1}{2} m_{1} & 0 & 0 & 0 \\ 0 & \frac{1}{2} m_{1} & 0 & 0 \\ 0 & 0 & \frac{1}{2} m_{2} & 0 \\ 0 & 0 & 0 & \frac{1}{2} m_{2} \end{matrix}] \\ H = [\begin{matrix} \frac{m_{1} l_{1}}{π} & - \frac{1}{2} \frac{m_{1} l_{1}}{π} & \frac{2 m_{2} l_{1}}{π} \cos (θ_{1} - θ_{2}) & 0 \\ 0 & 0 & \frac{m_{2} l_{2}}{π} & - \frac{1}{2} \frac{m_{2} l_{2}}{π} \end{matrix}] \end{matrix}

N is the coefficient matrix corresponding to generalized coordinate q and is given by

N = [\begin{matrix} \frac{E I_{1} π^{4}}{2 {l_{1}}^{3}} & 0 & 0 & 0 \\ 0 & \frac{8 E I_{1} π^{4}}{{l_{1}}^{3}} & 0 & 0 \\ 0 & 0 & \frac{E I_{2} π^{4}}{2 {l_{2}}^{3}} & 0 \\ 0 & 0 & 0 & \frac{8 E I_{2} π^{4}}{{l_{2}}^{3}} \end{matrix}]

In addition, U and V are the coefficient matrices corresponding to ${\overset{\cdot}{θ}}^{2}$ and can be written as

\begin{matrix} U = [\begin{matrix} 0 & (\frac{1}{2} m_{2} + m) l_{1} l_{2} \sin (θ_{1} - θ_{2}) \\ - (\frac{1}{2} m_{2} + m) l_{1} l_{2} \sin (θ_{1} - θ_{2}) & 0 \end{matrix}] \\ V = [\begin{matrix} 0 & 0 & - \frac{2 m_{2} l_{1} \sin (θ_{1} - θ_{2})}{π} & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \end{matrix}

R is a column vector and is given by

R = {[\frac{2}{π} m_{2} l_{1} \sin (θ_{1} - θ_{2}) ({\overset{\cdot}{θ}}_{1} + {\overset{\cdot}{θ}}_{2}) {\overset{\cdot}{q}}_{21} + (\frac{1}{2} m_{1} + m_{2} + m) g l_{1} \cos θ_{1} (\frac{1}{2} m_{2} + m) g l_{2} \cos θ_{2}]}^{T}

Design of NNSMC

Design of SMC

Define generalized coordinates as $z = [\begin{matrix} θ & q \end{matrix}]^{T}$ , where $θ = [\begin{matrix} θ_{1} & θ_{2} \end{matrix}]^{T}$ and $q = [\begin{matrix} q_{11} & q_{12} & q_{21} & q_{22} \end{matrix}]^{T}$ , then $\overset{\cdot}{z} = [\begin{matrix} \overset{\cdot}{θ} & \overset{\cdot}{q} \end{matrix}]^{T}$ and $\overset{\cdot\cdot}{z} = [\begin{matrix} \overset{\cdot\cdot}{θ} & \overset{\cdot\cdot}{q} \end{matrix}]^{T}$ .Therefore, the model of aerial work platform can be described as

D (z) \overset{\cdot\cdot}{z} + C (z, \overset{\cdot}{z}) \overset{\cdot}{z} + B (z) = u

(2)

where

D (z) = [\begin{matrix} G & H \\ H^{T} & M \end{matrix}], C (z, \overset{\cdot}{z}) = [\begin{matrix} U \overset{\cdot}{Θ} & X \\ V^{T} \overset{\cdot}{Θ} & 0 \end{matrix}]

in which

\begin{array}{l} \dot{Θ} = [\begin{matrix} {\dot{θ}}_{1} & 0 \\ 0 & {\dot{θ}}_{2} \end{matrix}], \\ X = [\begin{matrix} 0 & 0 & \frac{2}{π} m_{2} l_{1} \sin (θ_{1} - θ_{2}) ({\dot{θ}}_{1} + {\dot{θ}}_{2}) & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \end{array}

In addition, $B (z)$ is given by

\begin{matrix} B (z) = [\begin{matrix} (\frac{1}{2} m_{1} + m_{2} + m) g l_{1} \cos θ_{1} & (\frac{1}{2} m_{2} + m) g l_{2} \cos θ_{2} \end{matrix} \\ {\begin{matrix} \frac{E I_{1} π^{4}}{2 {l_{1}}^{3}} q_{11} & \frac{8 E I_{1} π^{4}}{{l_{1}}^{3}} q_{12} & \frac{E I_{2} π^{4}}{2 {l_{2}}^{3}} q_{21} & \frac{8 E I_{2} π^{4}}{{l_{2}}^{3}} q_{22} \end{matrix}]}^{T} \end{matrix}

where $m_{1}, m_{2}, and m$ are the mass of beams 1, 2, and load, respectively; $l_{1} and l_{2}$ are the lengths of beams 1 and 2; g is the acceleration of gravity; E is the modulus of elasticity of the beam material, and $I_{1} and I_{2}$ are the moment of inertia of the cross section of beams 1 and 2. u is the control input and can be written as

\begin{matrix} u = {[\begin{matrix} u_{1} & u_{2} & u_{3} & u_{4} & u_{5} & u_{6} \end{matrix}]}^{T} \\ = {[\begin{matrix} Q_{θ}^{T} & Q_{q}^{T} \end{matrix}]}^{T} \\ = {[\begin{matrix} Q_{1} & Q_{2} & Q_{3} & Q_{4} & Q_{5} & Q_{6} \end{matrix}]}^{T} \end{matrix}

in which $Q_{θ} = [\begin{matrix} Q_{1} & Q_{2} \end{matrix}]^{T}$ and $Q_{q} = [\begin{matrix} Q_{3} & Q_{4} & Q_{5} & Q_{6} \end{matrix}]^{T}$ are the generalized forces corresponding to $θ = [\begin{matrix} θ_{1} & θ_{2} \end{matrix}]^{T}$ and $q = [\begin{matrix} q_{11} & q_{12} & q_{21} & q_{22} \end{matrix}]^{T}$ , respectively.

The model of folding-boom aerial work platform has following properties:

Property 1. The matrix $D (z)$ is symmetric positive definite and satisfies $0 < d_{m} \leq ‖ D (z) ‖ \leq d_{M}$ , $\forall z \in R^{n}$ , where $d_{m} and d_{M}$ denote the minimum and maximum eigenvalues of $D (z)$ , respectively.

Property 2. $\overset{\cdot}{D} (z) - 2 C (z, \overset{\cdot}{z})$ is a skew symmetric matrix, that is, $x^{T} [\overset{\cdot}{D} (z) - 2 C (z, \overset{\cdot}{z})] x = 0, \forall x \in R^{n}$ .

For convenience, $D (z)$ , $C (z, \overset{\cdot}{z})$ , and $B (z)$ are written as D, C, and B, respectively. In addition, define tracking error as

e = z - z_{d}

(3)

where $z_{d}$ is reference trajectory; then, the sliding surface can be chosen as

s = \overset{\cdot}{e} + λ e

(4)

where $λ = diag [\begin{matrix} λ_{1} & λ_{2} & λ_{3} & λ_{4} & λ_{5} & λ_{6} \end{matrix}]$ , in which $λ_{i}$ is positive constant.

Therefore, the tracking control is expressed as follows: design a control law u to make sliding mode occur on sliding surface (equation (4)), then tracking error (equation (3)) approaches zero asymptotically with a prescribed transient response.⁹ Define the reference state

{\overset{\cdot}{z}}_{r} = \overset{\cdot}{z} - s = {\overset{\cdot}{z}}_{d} - λ e, {\overset{\cdot\cdot}{z}}_{r} = \overset{\cdot\cdot}{z} - \overset{\cdot}{s} = {\overset{\cdot\cdot}{z}}_{d} - λ \overset{\cdot}{e}

Then, the control law u designed is shown below

u = \hat{u} - As - K sgn s

(5)

where

\hat{u} = \hat{D} {\overset{\cdot\cdot}{z}}_{r} + \hat{C} {\overset{\cdot}{z}}_{r} + \hat{B}

wherein $\hat{D}$ , $\hat{C}$ , and $\hat{B}$ are estimates of D, C, and B, respectively, which reflect the model uncertainties. $A = diag [\begin{matrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} & a_{6} \end{matrix}]$ is diagonal matrix, in which $a_{i}$ is positive constant. $K sgn s$ is switching control, in which $K = diag [\begin{matrix} K_{11} & K_{22} & K_{33} & K_{44} & K_{55} & K_{66} \end{matrix}]$ is gain matrix with constant $K_{ii} > 0$ . Besides, $sgn s_{i}$ is the sign function and is given by

sgn s_{i} = {\begin{matrix} 1 & s_{i} > 0 \\ 0 & s_{i} = 0 \\ - 1 & s_{i} < 0 \end{matrix}

Substituting equation (5) into equation (2) leads to

D \overset{\cdot}{s} = - (C + A) s + Δ f - K sgn s

(6)

where $Δ f$ is defined as the system uncertainties and is given by $Δ f = Δ D {\overset{\cdot\cdot}{z}}_{r} + Δ C {\overset{\cdot}{z}}_{r} + Δ B$ , in which $Δ D$ , $Δ C$ , and $Δ B$ can be expressed as $Δ D = \hat{D} - D$ , $Δ C = \hat{C} - C$ , and $Δ B = \hat{B} - B$ , respectively.

For the dynamic model (equation (2)) with sliding surface (equation (4)), the stability of overall system with sliding mode controller can be ensured by control law (equation (5)).¹¹ That is to say, the actual trajectory is able to follow the desired trajectory.

Although the SMC can guarantee stability of the overall system, there is chattering phenomenon on sliding surface since switching control $K sgn s$ is discontinuous function. Therefore, the control law (equation (5)) cannot guarantee that $lim_{t \to \infty} s = 0$ . In this article, a NNSMC is presented to eliminate the chattering.

Design of NNSMC

The radial basis function neural network (RBFNN), which has input layer, hidden layer, and output layer, is adopted in NNSMC. Figure 2 shows the structure of RBFNN.²⁰

Figure 2.

Radial basis function neural networks.

Let

\begin{array}{l} W = [\begin{matrix} w_{11} & w_{12} & \dots & w_{1 m} \\ w_{21} & w_{22} & \dots & w_{2 m} \\ ⋮ & ⋮ & ⋮ \\ w_{h 1} & w_{h 2} & \dots & w_{h m} \end{matrix}], \\ ϕ = [\begin{matrix} ϕ_{1} \\ ϕ_{2} \\ ⋮ \\ ϕ_{h} \end{matrix}], Y = [\begin{matrix} y_{1} \\ y_{2} \\ ⋮ \\ y_{m} \end{matrix}] \end{array}

then the output of RBFNN can be expressed as $Y = W^{T} ϕ$ , where W is the weight matrix of the RBFNN and $ϕ$ is radial basis function (RBF) vector.

As the universal approximation ability of RBFNN,²¹ the RBFNN with enough number of hidden layer neurons can be used to identify the $Δ f$ defined in equation (6) such that

Δ f = W^{* T} ϕ + ε_{1}

(7)

where $W^{*}$ is the optimization weights matrix, and $ε_{1}$ is the $6 \times 1$ approximation error vector.

To eliminate chattering in SMC, neural network output $\hat{k}$ is used for approximating the system uncertainties $Δ f$ and is given by

\hat{k} = {\hat{W}}^{T} ϕ

(8)

where the RBFNN weight matrix $\hat{W}$ can be adapted during learning process, and s is selected as neural network input. The control law can be chosen as follows

u = \hat{D} {\overset{\cdot\cdot}{z}}_{r} + \hat{C} {\overset{\cdot}{z}}_{r} + \hat{B} - As - \hat{k}

(9)

Substituting equation (9) into equation (2) leads to

D \overset{\cdot}{s} = - (C + A) s + Δ f - \hat{k}

(10)

Theorem 1

For the dynamic model (equation (2)) with the tracking error (equation (3)), the sliding surface is defined as equation (4), and the neural network–based sliding mode controller is designed as equations (9) and (8), then the actual trajectory can converge to the desired trajectory.

Proof

Define $\hat{Λ} = [\begin{matrix} 0 \\ \hat{W} \end{matrix}]$ , $Λ = [\begin{matrix} 0 \\ W^{*} \end{matrix}]$ with $‖ Λ ‖_{F} \leq Λ_{M}$ and $\tilde{Λ} = Λ - \hat{Λ}$ and then the selected Lyapunov candidate function is

V = \frac{1}{2} s^{T} Ds + \frac{1}{2} tr ({\tilde{Λ}}^{T} Σ^{- 1} \tilde{Λ})

(11)

where $Σ$ is a $12 \times 12$ matrix and can be written as $Σ = [\begin{matrix} 0 \\ Γ \end{matrix}]$ , in which $Γ$ is a $6 \times 6$ diagonal positive definite matrix. Since D is also a positive symmetric matrix, V is positive definite. Define $z = [\begin{matrix} 0 & s^{T} \end{matrix}]^{T}$ , the adaptive law of neural network weights can be designed as

\overset{\cdot}{\hat{Λ}} = Σ Φ z^{T} - n Σ ‖ z ‖ \hat{Λ}

(12)

in which $Φ = [\begin{matrix} 0 & ϕ^{T} \end{matrix}]^{T}$ is the $12 \times 1$ column vector, and n is the positive real number. The time derivative of equation (11) can be obtained as

\overset{\cdot}{V} = \frac{1}{2} [{\overset{\cdot}{s}}^{T} Ds + s^{T} \overset{\cdot}{D} s + s^{T} D \overset{\cdot}{s}] + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}})

(13)

As $\overset{\cdot}{D} - 2 C$ is a skew symmetric matrix, namely

s^{T} \overset{\cdot}{D} s = s^{T} (2 C) s

(14)

Substituting equation (14) into equation (13) leads to

\overset{\cdot}{V} = s^{T} [D \overset{\cdot}{s} + Cs] + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}})

(15)

Substituting equations (7), (8), and (10) into equation (15) gives

\begin{matrix} \overset{\cdot}{V} = s^{T} [- (C + A) s + Δ f - \hat{k} + Cs] + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}}) \\ = - s^{T} As + s^{T} (Δ f - \hat{k}) + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}}) \\ = - s^{T} As + s^{T} ((W^{* T} - {\hat{W}}^{T}) ϕ + ε_{1}) + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}}) \end{matrix}

Define the error of neural network weights as

{\tilde{W}}^{T} = W^{* T} - {\hat{W}}^{T}

then

\overset{\cdot}{V} = - s^{T} As + s^{T} ({\tilde{W}}^{T} ϕ + ε_{1}) + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}})

Define $K_{Z} = [\begin{matrix} 0 & 0 \\ 0 & A \end{matrix}]$ as $12 \times 12$ matrix and $ε = [\begin{matrix} 0 & ε_{1}^{T} \end{matrix}]^{T}$ as $12 \times 1$ column vector, and assume that $‖ ε ‖ < ε_{N}$ , where $ε_{N}$ is a constant, then

\begin{matrix} \overset{\cdot}{V} = - z^{T} K_{Z} z + z^{T} ε + z^{T} {\tilde{Λ}}^{T} Φ + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}}) \\ = - z^{T} K_{Z} z + z^{T} ε + tr ({\tilde{Λ}}^{T} Σ^{- 1} \overset{\cdot}{\tilde{Λ}} + {\tilde{Λ}}^{T} Φ z^{T}) \end{matrix}

Since $\overset{\cdot}{\tilde{Λ}} = - \overset{\cdot}{\hat{Λ}}$ , the time derivative of Lyapunov function can be expressed as

\overset{\cdot}{V} = - z^{T} K_{Z} z + z^{T} ε + n ‖ z ‖ tr ({\tilde{Λ}}^{T} (Λ - \tilde{Λ}))

In addition, according to Schwarz inequality

tr ({\tilde{Λ}}^{T} (Λ - \tilde{Λ})) \leq ‖ \tilde{Λ} ‖_{F} ‖ Λ ‖_{F} - ‖ \tilde{Λ} ‖_{F}^{2}

and the following inequality

z^{T} K_{Z} z \geq k_{min} ‖ z ‖^{2}

where $k_{min} > 0$ is the minimum eigenvalue of $K_{Z}$

\begin{matrix} \overset{\cdot}{V} \leq - k_{min} ‖ z ‖^{2} + ε_{N} ‖ z ‖ + n ‖ z ‖ ({‖ \tilde{Λ} ‖}_{F} {‖ Λ ‖}_{F} - ‖ \tilde{Λ} ‖_{F}^{2}) \\ = - ‖ z ‖ [k_{min} ‖ z ‖ - ε_{N} + n (‖ \tilde{Λ} ‖_{F}^{2} - {‖ \tilde{Λ} ‖}_{F} {‖ Λ ‖}_{F})] \\ \leq - ‖ z ‖ [k_{min} ‖ z ‖ - ε_{N} + n (‖ \tilde{Λ} ‖_{F}^{2} - {‖ \tilde{Λ} ‖}_{F} Λ_{M})] \end{matrix}

In order to ensure $\overset{\cdot}{V} < 0$ , the inequality below can be derived

k_{min} ‖ z ‖ - ε_{N} + n (‖ \tilde{Λ} ‖_{F}^{2} - {‖ \tilde{Λ} ‖}_{F} Λ_{M}) > 0

Therefore, the above inequality is tenable when $(ε_{N} + (n / 4) Λ_{M}^{2} / k_{min}) < ‖ z ‖$ . Thus, by adjusting the values of n and $k_{min}$ , $\overset{\cdot}{V} < 0$ can be ensured. Since $\overset{\cdot}{V} = 0$ only when $s = 0$ , the overall system is asymptotically stable based on Lyapunov stability theorem. This means that $lim_{t \to \infty} s = lim_{t \to \infty} (\overset{\cdot}{e} + λ e) = 0$ , which implies $lim_{t \to \infty} z = z_{d}$ and $lim_{t \to \infty} \overset{\cdot}{z} = {\overset{\cdot}{z}}_{d}$ . Thus, work platform’s actual trajectory can converge to the desired trajectory. That completes the proof.

Simulation results

The choices of simulation parameters and initial conditions are listed in Table 1.

Table 1.

Simulation parameters and initial conditions.

Parameter	Beam 1	Beam 2	Work platform
Length (m)	7.5	8.5
Mass (kg)	650	550	150
EI (N m²)	6 × 10⁸	5 × 10⁸
Initial angle (rad)	2.09	0.52
Initial angular velocity (rad/s)	0	0

Reference trajectory is given by

r (t) = {[\begin{matrix} 2 π / 3 & π t / 180 & 0 & 0 & 0 & 0 \end{matrix}]}^{T}

then

\begin{matrix} \overset{\cdot}{r} (t) = {[\begin{matrix} 0 & π / 180 & 0 & 0 & 0 & 0 \end{matrix}]}^{T} \\ \overset{\cdot\cdot}{r} (t) = {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{T} \end{matrix}

In equation (5), the control parameters are set as

\begin{matrix} A & = 50000 diag [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}] \\ λ & = 100 diag [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}] \\ K & = 100 diag [\begin{matrix} 5 & 5 & 1 & 1 & 1 & 1 \end{matrix}] \end{matrix}

And $\hat{D}$ , $\hat{C}$ , and $\hat{B}$ are chosen as 0.9D, 0.8C, and 0.7B, respectively, for demonstrating the robustness of SMC to parameter uncertainties.

In equation (9), the same control parameters are chosen for the convenience of comparing with SMC. Simultaneously, the parameters in equation (12) can be chosen as

n = 0.01, Γ = 100 diag [\begin{matrix} 1 & 1 & 1 & 1 & 1 & 1 \end{matrix}]

Gaussian basis functions are selected as

ϕ_{i} = \exp (- \frac{{‖ s ‖}^{2}}{2 \times {0.1}^{2}}), i = 1, 2, \dots, 10

where the center of Gaussian function $c_{i}$ is set at $0$ , the width $σ_{i}$ at $0.1$ . In addition, initial weights are set to zero. Figures 3 –10 are simulation results of NNSMC. Figures 11 –14 are simulation results of SMC.

Figure 3.

The tracking of $θ_{2}$ in the NNSMC.

Figure 4.

The tracking error of $θ_{2}$ in the NNSMC.

Figure 5.

The trajectory tracking of the work platform in the NNSMC.

Figure 6.

The trajectory tracking error of the work platform in the horizontal X direction in the NNSMC.

Figure 7.

Deflection variable $q_{11}$ varying with time in the NNSMC.

Figure 8.

The adjustment of the control input $u_{2}$ in the NNSMC.

Figure 9.

The output ${\hat{k}}_{2}$ of neural network in the NNSMC.

Figure 10.

Sliding surface of $θ_{2}$ in the NNSMC.

Figure 11.

The trajectory tracking of the work platform in the SMC.

Figure 12.

The trajectory tracking error of the work platform in the horizontal X direction in the SMC.

Figure 13.

The adjustment of the control input $u_{2}$ in the SMC.

Figure 14.

The switching control $K_{22} sgn (s_{2})$ in the SMC.

From Figures 3 –10, it can be seen that the proposed NNSMC for the uncertain model can realize the tracking control of work platform and suppress the vibration simultaneously. Figures 3 and 5 reflect that the actual trajectories of $θ_{2}$ and work platform are able to track the desired steadily, and simultaneously, the vibrations are eliminated effectively.

Figures 4 and 6 are the tracking errors of work platform on $θ_{2}$ and trajectory, respectively. From Figures 4 and 6, we can see that the tracking errors are very small, and the controller has good tracking performance.

Figure 7 demonstrates that vibration is effectively restrained. Deformation variable converges to small values, which means that the vibration existed in flexible beam is weakened effectively.

Figure 8 shows the adjustment of control input $u_{2}$ , which realizes tracking control of work platform. From Figure 8, we can see that the adjustment process of $u_{2}$ is smooth. The regulation process of the output ${\hat{k}}_{2}$ of neural network, which reflects the approximate process to the system uncertainties $Δ f_{2}$ , as shown in Figure 9.

As shown in Figure 10, the NNSMC in this article can make sliding surface s₂ converge to zero gradually. This reflects that tracking error of $θ_{2}$ can converge to zero asymptotically.

Figure 11 demonstrates that work platform can be driven to desired trajectory by SMC. Although tracking error in Figure 12 is small, there is high-frequency chattering in the control input $u_{2}$ during the period of 40–50 s, as shown in Figure 13. As the fast control is unrealistic for valve-controlling cylinder used for driving the beam, the implementation of SMC is very difficult.

Figure 14 is the adjusted process of switching control $K_{22} sgn (s_{2})$ , which reflects that the robustness to parameter disturbance is obtained by the high-speed switching control in SMC. As the unmodeled characteristics of system can be excited easily by the high-frequency input, the performance of control system will be degenerated. Therefore, it is difficult for the application of SMC in practical system.

With the comparison between Figures 3 –10 and Figures 11 –14, we conclude that the presented NNSMC can achieve good tracking performance and eliminate the chattering caused by switching control existed in SMC simultaneously.

Conclusion

In this article, SMC method is used for the model considering the flexible deformation of beam to reduce tracking error and suppress vibration of work platform when there exist system uncertainties. As the switching gain is needed for compensation of model uncertainties, the chattering phenomenon, which degenerates the system performance, is occurred. In order to eliminate the chattering, we propose NNSMC using continuous control acquired from neural network to replace discontinuous switching control. In the NNSMC, neural network output k can compensate the model uncertainties and the chattering is eliminated effectively. As the weight update law of neural network is obtained from Lyapunov method, the stability and convergence of the whole system can be assured. Simultaneously, the results of simulation demonstrate that the designed NNSMC is effective for eliminating chattering caused by switching control existed in SMC. Moreover, we can see that the good performance of NNSMC is achieved for work platform’s vibration restraining and tracking error reducing.

The main contribution of this research is the proposition of NNSMC for the model of aerial work platform with flexible beam. Both the theoretical proof and simulation results show that the controller is effective for trajectory tracking and vibration suppressing when there exist system uncertainties. However, the current results are limited in two aspects: (1) in order to apply to practice, the NNSMC should be converted into digital form further. (2) There exist difficulties in the construction of experimental platform to verify the effectiveness of NNSMC at present. These limitations will be improved in the future work.

Footnotes

Handling Editor: Francesco Massi

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 was supported by National Natural Science Foundation (NNSF) of China (Grants No. 61374054 and 61571064), by Fundamental Research Funds for the Central Universities (Grants No. 31920160003 and 31920170141), and by Natural Science Foundation of Inner Mongolia (Grant No. 2015MS0623).

References

Zhao

et al . Modeling and simulation of folding-boom aerial platform vehicle based on the flexible multi-body dynamics. In: Proceedings of 2010 IEEE international conference on intelligence control and information processing, Dalian, China, 13–15 August 2010, pp.798–802. New York: IEEE.

Jia

Liang

et al . Adaptive neural network control of an aerial work platform’s arm. In: Proceedings of the 10th world congress on intelligent control and automation, Beijing, China, 6–8 July 2012, pp.3567–3570. New York: IEEE.

Miao

Yuan

Song

et al . Folding-boom aerial working vehicle tracking and control. Chin J Constr Mach 2013; 4: 319–323.

Meng

SL.

Control of aerial platform vehicle operating arm based on the flexible multi-body dynamics. Chengdu, China: University of Electronic Science and Technology of China, 2013.

Zhao

et al . Backstepping controller design for the trajectory tracking control of work platform of folding-boom aerial platform vehicle. In: Proceedings of 2010 IEEE international conference on robotics and biomimetics, Tianjin, China, 14–18 December 2010, pp.1641–1644. New York: IEEE.

Hung

Gao

Hung

JC.

Variable structure control: a survey. IEEE T Ind Electron 1993; 40: 2–22.

Slotine

Applied nonlinear control. Englewood Cliffs, NJ: Prentice Hall, 1991.

Kaynak

Erbatur

Ertugrul

The fusion of computationally intelligent methodologies and sliding-mode control: a survey. IEEE T Ind Electron 2001; 48: 4–17.

Leung

TP.

A sliding mode controller with bound estimation for robot manipulators. IEEE T Robotic Autom 1993; 9: 208–214.

10.

Chen

Liu

Wang

. Control of robot manipulator using a fuzzy model-based sliding mode control scheme. In: Proceedings of the 33rd IEEE conference on decision and control, Lake Buena Vista, FL, 14–16 December 1994, pp.3506–3511. New York: IEEE.

11.

Guo

Woo

PY.

An adaptive fuzzy sliding mode controller for robotic manipulators. IEEE T Syst Man Cyb 2003; 33: 149–159.

12.

Sadaii

Talasaz

. A robust fuzzy sliding mode control for uncertain dynamic systems. In: Proceedings of IEEE international conference on systems, man and cybernetics, Hague, 10–13 October 2004, pp.2225–2230. New York: IEEE.

13.

Morioka

Wada

Sabanovic

et al . Neural network based chattering free sliding mode control. In: Proceedings of the 34th SICE annual conference, Hokkaido, Japan, 26–28 July 1995, pp.1303–1308. New York: IEEE.

14.

Tang

Sun

ZQ.

Neural network control of flexible-link manipulators using sliding mode. Neurocomputing 2006; 70: 288–295.

15.

Debbache

Golea

Neural network based adaptive sliding mode control of uncertain nonlinear systems. J Syst Eng Electron 2012; 23: 119–128.

16.

Stam

Reijneveld

JC.

Graph theoretical analysis of complex networks in the brain. Nonlinear Biomed Phys 2007; 1: 3.

17.

Liu

et al . Admissible output consensualization control for singular multi-agent systems with time delays. J Frankl Inst 2016; 353: 4074–4090.

18.

Cai

Diao

Khan

MJ.

A novel clustering method based on quasi-consensus motions of dynamical multi-agent systems. Complexity 2017; 4978613.

19.

Sun

Pei

Pan

et al . Neural network-based sliding mode adaptive control for robot manipulators. Neurocomputing 2011; 74: 2377–2384.

20.

Zhao

et al . Trajectory tracking control of omnidirectional wheeled mobile manipulators: robust neural network-based sliding mode approach. IEEE T Syst Man Cyb B 2009; 39: 788–799.

21.

Park

Sandberg

IW.

Universal approximation using radial-basis-function networks. Neural Comput 1991; 3: 246–257.

A neural network–based sliding mode controller of folding-boom aerial work platform

Abstract

Keywords

Introduction

Flexible multi-body dynamic equations of work platform

Design of NNSMC

Design of SMC

Design of NNSMC

Theorem 1

Proof

Simulation results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References