# Backstepping Sliding Mode Robust Control for a Wire-Driven Parallel Robot Based on a Nonlinear Disturbance Observer

Authors: Yuqi Wang; Qi Lin; Lei Zhou; Xinxin Shi; Lei Wang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Based on a nonlinear disturbance observer, a backstepping sliding mode robust control is proposed for a wire-driven parallel robot (WDPR) system used in the wind tunnel test to dominate the motion of the end effector. The control method combines both the merits of backstepping control and sliding mode robust control. The WDPR is subject to different types of disturbances, and these disturbances will affect the motion precision of the end effector. To overcome these problems, a nonlinear disturbance observer (NDO) is designed to reject such disturbances. In this study, the design method of the nonlinear disturbance observer does not require the reliable dynamic model of the WDPR. Moreover, the design method can be used not only in the WDPR but also in other parallel robots. Then, a backstepping design method is adopted and a sliding mode term is introduced to construct a desired controller, and the disturbances are compensated in the controller to reduce the switching gain and guarantee the robustness. For the sake of verifying the stabilization of the closed-loop system, the Lyapunov function is constructed to analyze the stabilization of the system. Finally, the feasibility and validity of the proposed control scheme are proved through both simulation and experimental results.


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## Body

## 1. Introduction

The wind tunnel test is a flight condition test, which simulates an aircraft in a real atmospheric environment. And it is the most important method to predict the aerodynamic performance of the aircraft. In the wind tunnel test, model support plays an extremely important effect in the study of aerodynamics and the development of new aircraft. The traditional rigid support mode, due to the support frame, causes distortion of the flow around the model, which makes the model test results different from the aerodynamic characteristics of the real aircraft. To improve the data accuracy of the wind tunnel test, the change of support mode of the wind tunnel model is one of the urgent problems to be solved.In the last few years, wire (cable)-driven parallel robots have developed rapidly. Researchers have done a lot of research and analysis for wire (cable)-driven parallel robots. Chen et al. [1] designed a 7-DOF cable-driven humanoid arm and proposed the kinematic calibration issues in order to improve its motion control accuracy. Zi et al. [2] analyzed a cable parallel manipulator with and without hybrid-driven planar five-bar mechanism (HDPM). Yuan et al. [3] presented an original approach to analyze the modal interaction and illustrated this approach by numerical investigations and experimental analyses. Gao et al. [4] presented a novel bio-inspired cable-driven parallel robot with a flexible spine and investigated inverse kinematics and optimized the cable placements to minimize the actuating cable force. Aflakian et al. [5] presented an experimental study on the modeling and control for underconstrained cable-driven parallel robots (CDPRs). Heo et al. [6] improved the dynamic stability of CDPRs by means of frequency-based variable constraints over the whole workspace.However, wire (cable)-driven parallel robots have multiple properties such as strong coupling, nonlinearity, and time variant [7]; these may cause difficulties in achieving accuracy control. For the sake of solving these problems, various control methods have been developed rapidly. For the fully constrained cable-driven parallel robot, Khosravi and Taghirad [8] proposed a robust PID control. However, this method does not consider the external disturbance and the verification experiment is only for a planar CDPM. In 2015, Babaghasabha et al. [9] also presented an adaptive robust sliding mode controller for the fully constrained cable-driven parallel robot. Although the method had been verified in practice, it had increased the complexity of the system and the difficulty of physical realization. Kraus et al. [10] presented a position-force hybrid control. However, the actual trajectory performance is not good because the external disturbance is not considered. For a cable-driven parallel system, Wang et al. [11] studied the problem of type-2 fuzzy adaptive inverse control. However, due to the limitation of adaptive inverse control for the complex nonlinear system, interval type-2 fuzzy nonlinear models are considered, thus increasing the complexity of system control. For a fully constrained planar cable robot, Babaghasabha et al. [12] proposed an adaptive robust controller on the basis of the singular perturbation theory. However, the method was only verified by simulation on a planar cable-driven parallel robot. And there is no specific explanation and verification whether it can be used in space parallel robots. Jabbari Asl and Yoon et al. [13] also designed a robust tracking controller for fully constrained CDPRs. However, the error between the desired path and robot’s trajectories in the XaYa plane is large according to the simulation results.In the perspective of wind tunnel tests, some researchers have sensitively captured the advantages and application prospects of wire (cable)-driven parallel robots. Therefore, in recent years, they have proposed that wire (cable)-driven parallel robots can be introduced into the wind tunnel test, which provides a new method for the supporting technology of the aircraft model [14, 15]. However, as concluded from the previous studies, the above control methods cannot solve the control problem of WDPR in the wind tunnel test very well. Because during the operation of wire (cable)-driven parallel robots, external disturbances are usually unavoidable. To control the pose of the end-effector during the operation, this study proposes a backstepping sliding mode robust control based on a nonlinear disturbance observer, thus improving the robustness and accuracy of the control method. The main contribution of this study is that the backstepping sliding mode robust control method is applied to a 6-DOF 8-wire-driven parallel robot which has been developed by us and used in the wind tunnel test. And the advantages and significance of the proposed control method is that it can effectively deal with the nonlinearities, joint frictions, external disturbances, and dynamic uncertainties of the WDPR.The rest of this study is organized as follows. In Section2, the problem formulation is described. The designed nonlinear disturbance observer is presented in Section 3. Moreover, the design of backstepping sliding mode robust control and Lyapunov stability analysis are addressed in Section 4. In Sections 5 and 6, the simulation results and the prototype experimental results are given, respectively. Finally, the conclusion is drawn in Section 7.


## 2. Problem Formulation

In this section, we will give the composition, working principle, dynamic equations, properties on the WDPR, and an initial disturbance observer, which play a central effect in designing the nonlinear disturbance observer and the controller.


### 2.1. System Description

The WDPR and its wire structure are shown in Figure1. In this study, the WDPR is designed for the low-speed wind tunnel test. It is a nonlinear parallel system with multiinput and multioutput (MIMO). And it suffers from different types of disturbances, for instance, frictions between the traction wires and pulleys, parameter uncertainties, unknown payloads, and unmodeled dynamics. The WDPR is composed of the end effector (standard dynamics model (SDM)), a drive system, a control system, a vision measurement system, frame, and so on. Through the pulley hinge point (Bi, i = 1, 2, 3, …, 8), the connection point (Pi, i = 1, 2, 3, …, 8) on the end-effector of the moving platform is connected with one end of the traction wire. Through controlling the length variation of eight wires, the end effector is made to do 6-DOF motion of 3 positions and 3 attitude angles in space. The structure schematic diagram is shown in Figure 2.Figure 1
WDPR and its wire structure.
(a)
(b)Figure 2
Structure schematic diagram of the WDPR.The dynamic modeling of the WDPR is mainly composed of two parts: the dynamic modeling of the end effector and the dynamic modeling of the drive system. And the dynamic modeling of the end effector is the design foundation of motion control. The two dynamic equations are shown as follows:(1)M0θ¨m+C0θ˙m+μT=τ+τ1,MXX¨+NX,X˙−wg−we=−JTT,where M0 is the inertia matrix equivalent to the driver, C0 is the viscous friction coefficient matrix equivalent to the driver, μ is the transmission coefficient of the ball screw, θm∈R8×1 is the motor angle vector, T∈R8×1 is the wire tension vector, τ∈R8×1 is the output torque vector of the driver, τ1∈R8×1 is the vector of external disturbances, MX∈R6×6 is the inertial matrix of the end-effector, X=Xp,Yp,Zp,ϕ,θ,ψT is the pose of the end-effector, XP,YP,ZPT is the coordinate of point P relative to the static platform, ϕ,θ,ψT is the attitude angle of the end-effector, and ϕ, θ, and ψ are the roll angle, pitch angle, and yaw angle, respectively; X˙∈R6×1 is the pose velocity of the end-effector, J∈R8×6 is a Jacobi matrix, NX,X˙∈R6×1 is a nonlinear coriolis centrifugal matrix, wg∈R6×1 is a gravitational vector of the end-effector, wg=0,0,mg,0,0,0T, and we∈R6×1 is subject to external dynamic loads at the end-effector, and we=fe;τe. fe and τe are the aerodynamic and dynamic moments acting on the end-effector, respectively. If the dynamic analysis is carried out under the condition of 0 wind speed, then we=0:(2)MX=mI3×303×303×3AGH,NX,X˙=03×1AGH˙Q˙+HQ˙×AGHQ˙6×1,H=cosθcosψ−sinψ0cosθsinψcosψ0−sinθ01,Q˙=ϕ˙θ˙ψ˙.Here, AG is the inertia matrix of the end-effector, and Q˙ is the attitude angular velocity vector.The relationship between the pose velocityX˙, the wire length L, and the motor angle θm is denoted by the following equations:(3)L˙=μ⋅θ˙m=JGX˙,(4)L¨=μ⋅θ¨m=J˙GX˙+JG˙X˙+JGX¨,where G∈R6×6 is a transition matrix based on the attitude angle, G=I3×303×303×3H.According to equations (1), (3), and (4), we obtain the whole dynamic equation of the WDPR with 6 DOF:(5)MX−1μ2⋅JTM0JGX¨−1μ2JTM0J˙G+JTM0JG˙+JTC0JGX˙=−1μ⋅JTτ+τ1+wg+we−NX,X˙.Equation (5) is simplified as the following equation:(6)−μMX−1μ2⋅JTM0JGX¨+1μJTM0J˙G+JTM0JG˙+JTC0JGX˙+μwg=JTτ+τd,where τd=JTτ1+μNX,X˙−μwe.Let(7)AX=−μMX−1μ2⋅JTM0JG,BX=1μJTM0J˙G+JTM0JG˙+JTC0JG.Equation (6) is simplified as the following equation:(8)AXX¨+BXX˙+μwg=JTτ+τd.The WDPR has several properties. In this study, these properties will be used when designing the nonlinear disturbance observer and the controller. These properties are listed as follows.Property 1.
The inertial matrixMX of the end-effector is symmetric and positive definite. And its norm is bounded:(9)M−1X=M−TX>0,MX=MTX>0,M0 and G are symmetric and positive definite; therefore,(10)−μMX−1μ2⋅JTM0JG=−μMX−1μ2⋅JTM0JGT,that is,(11)A−1X=A−TX>0,AX=AXT>0.
In the same way,BX and B−1X are symmetric and positive definite by verification.Property 2.
The Jacobi matrixJ meets the following properties:(12)JTJT+=I6×6,J+J=I6×6.Property 3.
The transition matrixG meets the following properties:(13)GG−1=I6×6,G−1G=I6×6.where I6×6 is the identity matrix.
In practical engineering application, it is difficult to acquire complete information of the WDPR because of parameter uncertainties and external disturbances. Therefore, assume that the whole dynamic equation of the WDPR is as follows:(14)AX+ΔAXX¨+BX+ΔBXX˙+μwg=JTτ+τd,where ΔAX and ΔBX are parameter uncertainties present in nonlinear systems.
Then, the total disturbance vector is defined as(15)DX,X˙,X¨=τd−ΔAXX¨−ΔBXX˙.
According to the above definition, the effects of all dynamic uncertainties, external disturbances, and joint frictions are concentrated in a single disturbance vectorDX,X˙,X¨. It is seen from equation (8) that(16)AXX¨+BXX˙+μwg=JTτ+DX,X˙,X¨.


### 2.2. Initial Disturbance Observer

Since the total disturbanceDX,X˙,X¨ will adversely affect the precision of the end-effector of the WDPR, a disturbance observer is an ordinary means and can be used to reject these disturbances. A disturbance observer equates the difference between the actual object and the nominal model caused by the external disturbance and the change of model parameters to control input, that is, the equivalent disturbance is observed, and the equivalent compensation is introduced into control to realize the complete control of the disturbance. A basic idea of designing the observer is to modify the estimation by the difference between estimated output and actual output. Hence, equation (16) can be written as(17)DX,X˙,X¨=AXX¨+BXX˙+μwg−JTτ.Therefore, according to the dynamic model of equation (17), the initial disturbance observer can be designed as follows:(18)D^˙X,X˙,X¨=KXDX,X˙,X¨−D^X,X˙,X¨=−KXD^X,X˙,X¨+KXDX,X˙,X¨=−KXD^X,X˙,X¨+KXAXX¨+BXX˙+μwg−JTτ,where KX is the gain matrix of the observer and D^X,X˙,X¨ is the estimated value of DX,X˙,X¨.In general, there is no prior information about the derivative of the disturbanceDX,X˙,X¨; therefore, it is assumed that the variation of the disturbance is slow relative to the dynamic characteristics of the observer, that is, DX,X˙,X¨=0. Then, let the estimation error of the nonlinear disturbance observer be(19)D˜X,X˙,X¨=DX,X˙,X¨−D^X,X˙,X¨.Combining equation (19) with the initial disturbance observer (18) yields(20)D˜˙X,X˙,X¨=D˙X,X˙,X¨−D^˙X,X˙,X¨=D^˙X,X˙,X¨=−D^˙X,X˙,X¨=−KXDX,X˙,X¨−D^X,X˙,X¨=−KXD˜X,X˙,X¨.That is, the observation error satisfies the constraint conditions:(21)D˜˙X,X˙,X¨+KXD˜X,X˙,X¨=0.


## 2.1. System Description

The WDPR and its wire structure are shown in Figure1. In this study, the WDPR is designed for the low-speed wind tunnel test. It is a nonlinear parallel system with multiinput and multioutput (MIMO). And it suffers from different types of disturbances, for instance, frictions between the traction wires and pulleys, parameter uncertainties, unknown payloads, and unmodeled dynamics. The WDPR is composed of the end effector (standard dynamics model (SDM)), a drive system, a control system, a vision measurement system, frame, and so on. Through the pulley hinge point (Bi, i = 1, 2, 3, …, 8), the connection point (Pi, i = 1, 2, 3, …, 8) on the end-effector of the moving platform is connected with one end of the traction wire. Through controlling the length variation of eight wires, the end effector is made to do 6-DOF motion of 3 positions and 3 attitude angles in space. The structure schematic diagram is shown in Figure 2.Figure 1
WDPR and its wire structure.
(a)
(b)Figure 2
Structure schematic diagram of the WDPR.The dynamic modeling of the WDPR is mainly composed of two parts: the dynamic modeling of the end effector and the dynamic modeling of the drive system. And the dynamic modeling of the end effector is the design foundation of motion control. The two dynamic equations are shown as follows:(1)M0θ¨m+C0θ˙m+μT=τ+τ1,MXX¨+NX,X˙−wg−we=−JTT,where M0 is the inertia matrix equivalent to the driver, C0 is the viscous friction coefficient matrix equivalent to the driver, μ is the transmission coefficient of the ball screw, θm∈R8×1 is the motor angle vector, T∈R8×1 is the wire tension vector, τ∈R8×1 is the output torque vector of the driver, τ1∈R8×1 is the vector of external disturbances, MX∈R6×6 is the inertial matrix of the end-effector, X=Xp,Yp,Zp,ϕ,θ,ψT is the pose of the end-effector, XP,YP,ZPT is the coordinate of point P relative to the static platform, ϕ,θ,ψT is the attitude angle of the end-effector, and ϕ, θ, and ψ are the roll angle, pitch angle, and yaw angle, respectively; X˙∈R6×1 is the pose velocity of the end-effector, J∈R8×6 is a Jacobi matrix, NX,X˙∈R6×1 is a nonlinear coriolis centrifugal matrix, wg∈R6×1 is a gravitational vector of the end-effector, wg=0,0,mg,0,0,0T, and we∈R6×1 is subject to external dynamic loads at the end-effector, and we=fe;τe. fe and τe are the aerodynamic and dynamic moments acting on the end-effector, respectively. If the dynamic analysis is carried out under the condition of 0 wind speed, then we=0:(2)MX=mI3×303×303×3AGH,NX,X˙=03×1AGH˙Q˙+HQ˙×AGHQ˙6×1,H=cosθcosψ−sinψ0cosθsinψcosψ0−sinθ01,Q˙=ϕ˙θ˙ψ˙.Here, AG is the inertia matrix of the end-effector, and Q˙ is the attitude angular velocity vector.The relationship between the pose velocityX˙, the wire length L, and the motor angle θm is denoted by the following equations:(3)L˙=μ⋅θ˙m=JGX˙,(4)L¨=μ⋅θ¨m=J˙GX˙+JG˙X˙+JGX¨,where G∈R6×6 is a transition matrix based on the attitude angle, G=I3×303×303×3H.According to equations (1), (3), and (4), we obtain the whole dynamic equation of the WDPR with 6 DOF:(5)MX−1μ2⋅JTM0JGX¨−1μ2JTM0J˙G+JTM0JG˙+JTC0JGX˙=−1μ⋅JTτ+τ1+wg+we−NX,X˙.Equation (5) is simplified as the following equation:(6)−μMX−1μ2⋅JTM0JGX¨+1μJTM0J˙G+JTM0JG˙+JTC0JGX˙+μwg=JTτ+τd,where τd=JTτ1+μNX,X˙−μwe.Let(7)AX=−μMX−1μ2⋅JTM0JG,BX=1μJTM0J˙G+JTM0JG˙+JTC0JG.Equation (6) is simplified as the following equation:(8)AXX¨+BXX˙+μwg=JTτ+τd.The WDPR has several properties. In this study, these properties will be used when designing the nonlinear disturbance observer and the controller. These properties are listed as follows.Property 1.
The inertial matrixMX of the end-effector is symmetric and positive definite. And its norm is bounded:(9)M−1X=M−TX>0,MX=MTX>0,M0 and G are symmetric and positive definite; therefore,(10)−μMX−1μ2⋅JTM0JG=−μMX−1μ2⋅JTM0JGT,that is,(11)A−1X=A−TX>0,AX=AXT>0.
In the same way,BX and B−1X are symmetric and positive definite by verification.Property 2.
The Jacobi matrixJ meets the following properties:(12)JTJT+=I6×6,J+J=I6×6.Property 3.
The transition matrixG meets the following properties:(13)GG−1=I6×6,G−1G=I6×6.where I6×6 is the identity matrix.
In practical engineering application, it is difficult to acquire complete information of the WDPR because of parameter uncertainties and external disturbances. Therefore, assume that the whole dynamic equation of the WDPR is as follows:(14)AX+ΔAXX¨+BX+ΔBXX˙+μwg=JTτ+τd,where ΔAX and ΔBX are parameter uncertainties present in nonlinear systems.
Then, the total disturbance vector is defined as(15)DX,X˙,X¨=τd−ΔAXX¨−ΔBXX˙.
According to the above definition, the effects of all dynamic uncertainties, external disturbances, and joint frictions are concentrated in a single disturbance vectorDX,X˙,X¨. It is seen from equation (8) that(16)AXX¨+BXX˙+μwg=JTτ+DX,X˙,X¨.


## 2.2. Initial Disturbance Observer

Since the total disturbanceDX,X˙,X¨ will adversely affect the precision of the end-effector of the WDPR, a disturbance observer is an ordinary means and can be used to reject these disturbances. A disturbance observer equates the difference between the actual object and the nominal model caused by the external disturbance and the change of model parameters to control input, that is, the equivalent disturbance is observed, and the equivalent compensation is introduced into control to realize the complete control of the disturbance. A basic idea of designing the observer is to modify the estimation by the difference between estimated output and actual output. Hence, equation (16) can be written as(17)DX,X˙,X¨=AXX¨+BXX˙+μwg−JTτ.Therefore, according to the dynamic model of equation (17), the initial disturbance observer can be designed as follows:(18)D^˙X,X˙,X¨=KXDX,X˙,X¨−D^X,X˙,X¨=−KXD^X,X˙,X¨+KXDX,X˙,X¨=−KXD^X,X˙,X¨+KXAXX¨+BXX˙+μwg−JTτ,where KX is the gain matrix of the observer and D^X,X˙,X¨ is the estimated value of DX,X˙,X¨.In general, there is no prior information about the derivative of the disturbanceDX,X˙,X¨; therefore, it is assumed that the variation of the disturbance is slow relative to the dynamic characteristics of the observer, that is, DX,X˙,X¨=0. Then, let the estimation error of the nonlinear disturbance observer be(19)D˜X,X˙,X¨=DX,X˙,X¨−D^X,X˙,X¨.Combining equation (19) with the initial disturbance observer (18) yields(20)D˜˙X,X˙,X¨=D˙X,X˙,X¨−D^˙X,X˙,X¨=D^˙X,X˙,X¨=−D^˙X,X˙,X¨=−KXDX,X˙,X¨−D^X,X˙,X¨=−KXD˜X,X˙,X¨.That is, the observation error satisfies the constraint conditions:(21)D˜˙X,X˙,X¨+KXD˜X,X˙,X¨=0.


## 3. Nonlinear Disturbance Observer Design

Due to the uncertainties of the WDPR and external disturbance (the total disturbance), for the sake of reducing the impact of the total disturbance on the WDPR and improving the control precision of the system, a nonlinear disturbance observer (NDO) is designed to approach the system disturbance. Based on the advantages of references [16, 17], first, the auxiliary variable is designed as follows:(22)z=D^X,X˙,X¨−QX,X˙,where z∈R6×1 and QX,X˙ is a function vector.Let(23)Q˙X,X˙=KXAXX¨.Then,(24)z˙=D^˙X,X˙,X¨−Q˙X,X˙=D^˙X,X˙,X¨−KXAXX¨,D^˙X,X˙,X¨=−KXD^X,X˙,X¨+KXAXX¨+BXX˙+μwg−JTτ,z˙=D^˙X,X˙,X¨−Q˙X,X˙=KXAXX¨+BXX˙+μwg−JTτ−KXD^X,X˙,X¨−KXAXX¨,=KXAXX¨+KXBXX˙+KXμwg−KXJTτ−KXD^X,X˙,X¨−KXAXX¨,=KXBXX˙+KXμwg−KXJTτ−KXD^X,X˙,X¨.The design nonlinear disturbance observer is as follows:(25)z˙=KXBXX˙+KXμwg−KXJTτ−KXD^X,X˙,X¨,D^X,X˙,X¨=z+QX,X˙.Then,(26)z˙=KXBXX˙+KXμwg−KXJTτ−KXz+QX,X˙=KXBXX˙+KXμwg−KXJTτ−KXz−KXQX,X˙,=KXBXX˙+μwg−JTτ−QX,X˙−KXz.According to the above analysis, sinceD˜˙X,X˙,X¨=D˙X,X˙,X¨−D^˙X,X˙,X¨=−D^˙X,X˙,X¨, D˙X,X˙,X¨=0, there are(27)D˜˙X,X˙,X¨=−z˙−Q˙X,X˙.Equations (23) and (25) are substituted into equation (27) and we obtain(28)D˜˙X,X˙,X¨=−z˙−Q˙X,X˙=−KXBXX˙+μwg−JTτ−QX,X˙+KXz−KXAXX¨,=KXz+QX,X˙−KXAXX¨+BXX˙+μwg−JTτ.Let function vectorQX,X˙=PX˙ and design the gain matrix KX=PA−1X, where P is an invertible matrix; then, the observation error equation is as follows:(29)D˜˙X,X˙,X¨+PA−1XD˜X,X˙,X¨=0,where P is obtained through Linear Matrix Inequality (LMI), so that the gain matrix KX is obtained and the estimated value D^X,X˙,X¨ approximates the disturbance by exponent.Proof.
For the sake of verifying whether the design nonlinear disturbance observer is convergent, a Lyapunov function is designed as follows:(30)V0=D˜TX,X˙,X¨P−TAXP−1D˜X,X˙,X¨.SinceAX is a symmetric and positive definite matrix, and the matrix P is invertible, the matrix P−TAXP−1 is also positive definite. Thus, the Lyapunov function V0 is positive definite.The Lyapunov function derivation is obtained with respect to timet, which is shown below:(31)V˙0=D˜˙TX,X˙,X¨P−TAXP−1D˜X,X˙,X¨+D˜TX,X˙,X¨P−TA˙XP−1D˜X,X˙,X¨+D˜TX,X˙,X¨P−TAXP−1D˜˙X,X˙,X¨.From the error equation of the observer, we obtain(32)D˜˙TX,X˙,X¨=−KXD˜X,X˙,X¨T=−PA−1XD˜X,X˙,X¨T=−D˜TX,X˙,X¨A−TXPT.Hence, the following can be obtained:(33)V˙0=−D˜TX,X˙,X¨A−TXPTP−TAXP−1D˜X,X˙,X¨+D˜TX,X˙,X¨P−TA˙XP−1D˜X,X˙,X¨+D˜TX,X˙,X¨P−TAXP−1D˜˙X,X˙,X¨=−D˜TX,X˙,X¨P−1D˜X,X˙,X¨+D˜TX,X˙,X¨P−TA˙XP−1D˜X,X˙,X¨−D˜TX,X˙,X¨P−TAXP−1PA−1XD˜X,X˙,X¨=−D˜TX,X˙,X¨P−1D˜X,X˙,X¨+D˜TX,X˙,X¨P−TA˙XP−1D˜X,X˙,X¨−D˜TX,X˙,X¨P−TD˜X,X˙,X¨=−D˜TX,X˙,X¨P−1−P−TA˙XP−1+P−TD˜X,X˙,X¨.To makeV˙0<0, P−1−P−TA˙XP−1+P−T needs to satisfy a positive definite condition. Let(34)P−1−P−TA˙XP−1+P−T≥Π,where Π>0 and Π is a symmetric positive definite matrix.Then,(35)V˙0≤−D˜TX,X˙,X¨ΠD˜X,X˙,X¨.There isΠ′>0 that makes V˙0≤−D˜TX,X˙,X¨ΠD˜X,X˙,X¨=−Π′V0.From equation (34), the nonlinear disturbance observer exponentially converges, and the convergence accuracy depends on the value of parameter Π. The larger the value Π is, the faster the convergence rate is and the higher the precision is.The design of the nonlinear disturbance observer can ultimately be attributed to solving the value of the invertible matrixP. It is known that the value of P can be obtained by using Linear Matrix Inequality (LMI).Therefore, letZ=P, ZT=PT, and Z=P are substituted into equation (34) and we obtain(36)PTP−1P−PTP−TA˙XP−1P+PTP−TP≥PTΠP.SincePTP−T=I6×6 and P−1P=I6×6, equation (35) can be simplified as(37)PT−A˙X+P≥PTΠP.That is,ZT−A˙X+Z≥ZTΠZ, ZT+Z−ZTΠZ≥A˙X.SetA˙X≤η, then A˙X≤ηI6×6. If the above equation is tenable, sufficient conditions for establishment of equation (37) are(38)ZT+Z−ZTΠZ≥ηI6×6.That is,ZT+Z−ZTΠZ−ηI6×6≥0.Theorem 1.
The Schur complement property of matrices can transform nonlinear matrix inequalities into LMI problems. Schur complement theorem: assuming thatC1 is a positive definite matrix, then A1−B1C1−1B1T≥0 is equivalent to A1B1TB1C1≥0.
Hence, equation (37) is equivalent to ZT+Z−ηI6×6ZTZΠ−1≥0.
By using the YALMIP toolbox in MATLAB to getP, the validity of the solution to the inequality depends on η and Π. The smaller the values of η and Π are, the easier it is to get the effective solution.


## 4. Backstepping Sliding Mode Robust Control Design

The backstepping design method is a systematical controller synthesis method for uncertain systems. It is a regression design method combining the selection of Lyapunov functions with the design of the controller. The backstepping method has been applied extensively because of the unique superiority in dealing with the nonlinear control problem [18–23]. It begins with the lowest order differential equations of the system, introduces the concept of virtual control, and designs the virtual control to meet the requirements step by step. Finally, the real control law is designed. The advantage of sliding mode control is that it can get over the uncertainties of the system, which has strong robustness to the disturbance and unmodeled dynamics, and has good effect especially on the control of the nonlinear system.Therefore, based on the above mentioned designed nonlinear disturbance observer to observe the total disturbanceDX,X˙,X¨, the backstepping control method is adapted and the sliding mode term is introduced to overcome disturbance and enhance the robustness of the controller.Firstly, letX1=X and X2=X˙, and the dynamic equation (16) can be simplified as follows:(39)X˙1=X2,X˙2=A−1X−BXX2+JTτ+DX,X˙,X¨−μwg.Set the ideal pose to beXd and take the tracking error as e=X−Xd; then, e˙=X−X˙d; that is, e˙=X2−X˙d. The following sliding mode function is defined:(40)s=e˙+Γe,where Γ=diagλ1λ2⋯λ6,λi>0,i=1,2,…,6.Letz1=e and define z2=s=e˙+Γe=z˙1+Γz1, where z2 is the virtual control item, z˙1=z2−Γz1.Proof.
A Lyapunov function is defined as(41)V1=12eTe=12z1Tz1.Then,(42)V˙1=z1Tz˙1=z1Tz2−Γz1=z1Tz2−z1TΓz1.Define the switching function as(43)s1=κz1+z2=κe+e˙+Γe=e˙+κ+Γe=z˙1+κ+Γz1,where κ=diagξ1ξ2⋯ξ6,ξi>0,i=1,2,…,6.Letϑ=κ+Γ, since ϑ=κ+Γ=diagλ1+ξ1λ2+ξ2⋯λ6+ξ6,λi+ξi>0,i=1,2,…,6; if s1=0, then z1=0, z2=0, and V˙1≤0.Proof.
For the sake of making the control law free of acceleration termX¨, X˙r=X˙−s1 is defined; then, the derivative of Xr is(44)X˙r=X˙−s1=X˙−e˙+κ+Γe=X˙−e˙−κ+Γe=X˙d−κ+Γe,X¨r=X¨d−κ+Γe˙.Therefore, it is guaranteed thatX¨r does not contain the acceleration item X¨.As a result,s˙1=X¨−X¨d+ϑe˙, then(45)s˙1=X¨−X¨r=X¨−X¨d+κ+Γe˙=X¨−X¨d+ϑe˙=A−1X−BXX2+JTτ+DX,X˙,X¨−μwg−X¨r,AXs˙1=AXA−1X−BXX2+JTτ+DX,X˙,X¨−μwg−X¨r=−BXX2+JTτ+DX,X˙,X¨−μwg−AXX¨r.Define the following Lyapunov function as follows:(46)V2=V1+12s1TAXs1=12eTe+12s1TAXs1=12z1Tz1+12s1TAXs1.The derivative ofV2 is(47)V˙2=V˙1+s1TAXs˙1+12s1TA˙Xs1=z1Tz2−z1TΓz1+s1TAXκz˙1+z˙2+12s1TA˙Xs1=z1Tz2−z1TΓz1+s1T−BXX2+JTτ+DX,X˙,X¨−μwg−AXX¨r+12s1TA˙Xs1.According to equations (47) and (39), a backstepping sliding mode robust control law τ is designed as follows:(48)τ=JT+AXX¨d−ϑz2−Γz1+BXX˙d−ϑz1+μwg−χsgns1−D^X,X˙,X¨=JT+AXX¨d−ϑe˙+BXX˙d−ϑe+μwg−χsgns1−D^X,X˙,X¨=JT+AXX¨r+BXX˙r+μwg−χsgns1−D^X,X˙,X¨,where χ=diagε1ε2⋯ε6,εi>0,i=1,2,…,6.The design control law is substituted into equation (47), and we obtain(49)V˙2=V˙1+s1TAXs˙1+12s1TA˙Xs1=z1Tz2−z1TΓz1+s1TAXA−1X−BXX2+JTJT+AXX¨r+BXX˙r+μwg−χsgns1−D^X,X˙,X¨+DX,X˙,X¨−μwg−X¨r+12s1TA˙Xs1=z1Tz2−z1TΓz1+s1TAXA−1X−BXX2+AXX¨r+BXX˙r+μwg−χsgns1−D^X,X˙,X¨+DX,X˙,X¨−μwg−X¨r+12s1TA˙Xs1=z1Tz2−z1TΓz1+s1TAXA−1X−BXX˙+BXX˙r−χsgns1+D˜X,X˙,X¨+12s1TA˙Xs1=z1Tz2−z1TΓz1+s1TAXA−1X−BXs1−χsgns1+D˜X,X˙,X¨+12s1TA˙Xs1=z1Tz2−z1TΓz1−s1TAXA−1XBXs1−s1TAXA−1Xχsgns1+s1TAXA−1XD˜X,X˙,X¨+12s1TA˙Xs1=z1Tz2−z1TΓz1−s1TBXs1−s1Tχsgns1+s1TD˜X,X˙,X¨+12s1TA˙Xs1,V˙2≤z1Tz2−z1TΓz1−s1TXs1−s1Tχsgns1+12s1TA˙Xs1+s1TKPs1=z1Tz2−z1TΓz1−s1TBX−A˙X2+KPs1−s1Tχsgns1=−z1TΓz1−z1Tz2+s1TBX−A˙X2+KPs1−s1Tχsgns1,where AXA−1X=I6×6, KP=diagK1K2⋯K6,andKi>0,i=1,2,…,6.TakeY as shown in equation (50) as follows:(50)Y=Γ+κTBX−A˙X2+KPκBX−A˙X2+KPκ−12BX−A˙X2+KPκ−12BX−A˙X2+KP.LetxT=z1Tz2T, then(51)xTYx=xTΓ+κTBX−A˙X2+KPκBX−A˙X2+KPκ−12BX−A˙X2+KPκ−12BX−A˙X2+KPx=z1Tz2TΓ+κTBX−A˙X2+KPκBX−A˙X2+KPκ−12BX−A˙X2+KPκ−12BX−A˙X2+KPz1z2=z1TΓ+κTBX−A˙X2+KPκ+z2TBX−A˙X2+KPκ−12z1TBX−A˙X2+KPκ−12+z2TBX−A˙X2+KPz1z2,=z1TΓ+κTBX−A˙X2+KPκz1+z2TBX−A˙X2+KPκ−12z1+z1TBX−A˙X2+KPκ−12z2+z2TBX−A˙X2+KPz2=z1TΓ+κTBX−A˙X2+KPκz1+2z1TBX−A˙X2+KPκ−12z2+z2TBX−A˙X2+KPz2=z1TΓz1−z1Tz2+z1TκTBX−A˙X2+KPκz1+2z1TBX−A˙X2+KPκz2+z2TBX−A˙X2+KPz2=z1TΓz1−z1Tz2+κz1+z2TBX−A˙X2+KPκz1+z2=z1TΓz1−z1Tz2+s1TBX−A˙X2+KPs1.IfY is a positive definite matrix, xTYx>0; then, V˙2≤−xTYx−s1Tχsgns1≤0.According to the positive definite discriminant method of block matrix, suitable values (Γ, κ, KP) are selected to ensure that Y is a positive definite matrix; therefore, V˙2≤0.Proof.
Design the Lyapunov function of the closed-loop system as(52)V=V2+V0=V1+12s1TAXs1+V0.
The derivative ofV can be derived as(53)V˙=V˙2+V˙0=V˙1+s1TAXs˙1+12s1TA˙Xs1+V˙0.Because ofV˙2≤0 and V˙0≤0, V˙≤0.Therefore, the backstepping sliding mode robust control system is asymptotically stable.


## 5. Simulation Experiments

For the sake of verifying the effectiveness of the proposed control approach, simulation experiments are implemented on the WDPR, as shown in Figure1. The WDPR is used to do 6-DOF motion for the simulation experiments. The desired pose of the end-effector is set as follows:(54)X=0.2×sint;0.2×sint;0.2×sint−0.582;pi/6×sint;pi/6×sint;pi/6×sint.Taking MATLAB/SIMULINK as the simulation software, according to the task nature and motion characteristics of WDPR, a simulation main program diagram of the backstepping sliding mode robust control is established.The coordinate points of WDPR: the positions of Pi (the connecting point of the end-effector) and Bi (the hinge point of the pulley) (i = 1∼8) are as shown in Table 1.Table 1
Positions of Pi and Bi.
Symbol
Coordinate (mm)
P1
(−208, 78, −1)T
P2
(−208, −78, −1)T
P3
(0, −27.7, −10.1)T
P4
(0, 27.7, −10.1)T
P5
(0, 27.7, 10.1)T
P6
(−208, 78, 1)T
P7
(−208, −78, 1)T
P8
(0, −27.7, 10.1)T
B1
(200, 415, −1280)T
B2
(200, −415, −1280)T
B3
(−300, −308, −1250)T
B4
(−300, 308, −1250)T
B5
(−315, 315, −70)T
B6
(160, 415, −70)T
B7
(160, −415, −70)T
B8
(−315, −315, −70)T
The parameter values in Table2 are used for simulation analysis.The mass of the end-effector (aircraft model) is 1.028 kg;Y is a positive definite matrix when Γ, κ, and KP and take the values in Table 2. When η=2, χ=5I6×6, P=7.3361I6×6, Π=0.01∗diag57981015, and KP=103∗diag311120.150.120.25, the system is stable. In the simulation experiments, the ranges of pose error are set as follows: ±0.005 m and ±0.05°. The ranges of pose speed error are set as ±0.5 m/s and ±5°/s. The range of disturbance observation error is set as ±0.05. The simulation results are shown in Figures 3–11.Table 2
Simulation parameter table.
Parameter
Value
M
0
7∗ 10−5I8×8
C
0
0.005I8×8
m
1.028 kg
μ
0.0025 pi
Γ
15I6×6
g
9.82 m/s2
κ
5I6×6
η
2
χ
5I6×6
P
7.3361I6×6
Π
0.01
∗
diag
5
7
9
8
10
15
K
P
10
3
∗
diag
3
11
12
0
.
15
0
.
12
0
.
25Figure 3
Pose curve.Figure 4
Pose error curve.Figure 5
Pose speed curve.Figure 6
Pose speed error curve.Figure 7
Disturbance observation results.Figure 8
Disturbance observation error.Figure 9
Control input of motors.Figure 10
Wire tension.Figure 11
Wire length.According to the simulation results of the pose, it can be concluded from Figure3 that the curves of the theoretical pose and the actual pose coincide well with continuous 10 s; it can be seen from Figure 4 that the tracking error of the pose is basically 0, and the tracking effect is good.It can be concluded from Figure5 that the curve coincidence degree between the theoretical pose speed and the actual pose speed also tends to be consistent within continuous 10 s. As can be seen from Figure 6, the tracking error of pose speed tends to be 0 and the control precision is very high.From the simulation results in Figure7, it can be seen that the estimated disturbance observed by the nonlinear disturbance observer is extremely high in coincidence degree with the external disturbance; it can be concluded from Figure 8 that the observation error of the nonlinear disturbance observer is basically 0, which realizes the real-time estimation and real-time compensation of external disturbance.It can be seen from Figure9 that control inputs of the eight motors all vary within a certain range, which conforms to the physical significance of the design.As can be seen from Figures10 and 11, due to the six-degrees-of-freedom motion, there is no regularity in the variation of wire length and wire force, but the variation of eight-wire length and the variation of wire tension are within a certain range. The variation of wire length is smooth, and the variation of wire tension is gentle, which reduces the risk of broken wires.The simulation results show that the equivalent disturbance can be observed through the proposed nonlinear disturbance observer, and the equivalent compensation is introduced into designed sliding mode control, which realizes the effective control of the disturbance. The tracking effect of the WDPR is good, and the control precision is high, which proves that the designed backstepping sliding mode robust control is practicable and dependable and satisfies motion tracking trajectory for the WDPR.


## 6. Prototype Experiments and Results

In order to further verify the correctness and effectiveness of the designed control law, the experimental verification is carried out on the WDPR prototype. The experimental mechanism is shown in Figure1. The experimental mechanism mainly consists of control software and hardware system. The control software is programmed in C++ language on Visual Studio. The hardware system mainly includes an industrial personal computer (IPC), a motion controller, servo system, and mechanical structure. The motion controller is IMAC-HX, and it is powered by DC regulated power supply of DRP-240-24, as shown in Figure 12.Figure 12
IMAC controller.In order to validate the designed control method, first, the servo driver is set to the torque mode, and the connection of the hardware device and the configuration of the software environment are completed. In view of the established dynamic model and the robust control, the dynamic control software is designed to complete the control experiment of WDPR prototype. The control software mainly includes control main program, trajectory planning, inverse kinematics model, dynamic model, and other modules. Since the dynamic control software involves a large number of matrix operations for modeling and control, Eigen mathematical function library is used to deal with the operation of matrix vectors. The dynamic control software invokes the class library MFC to design the visual interface of the program. The software interface of WDPR prototype experiment mechanism is designed by class library MFC.In the control process, the Jacobian matrixJ and its pseudoinverse matrix J+ involve a lot of matrix operations, and the calculation speed is too slow to realize real-time online computation. Therefore, through MATLAB simulation, the Jacobian matrix J and its pseudoinverse matrix J+ are calculated offline and invoked in real-time during the control to improve the control efficiency of the WDPR prototype mechanism. The performance of the existing hardware devices cannot meet the requirements of complex motion experiments. Therefore, the feasibility and control property of the proposed control law is verified through a single DOF motion experiment. The theoretical pose of the end-effector is set as follows: Xd=00−0.5820pi/6×sint0T. The experimental results of the WDPR prototype are shown in Figures 13–17.Figure 13
Pose tracking curve.Figure 14
Pose tracking error curve.Figure 15
Actual angular velocity of #1 motor.Figure 16
Absolute error of #1 motor (angular velocity).Figure 17
Output torque of #1 motor.In Figures13 and 14, the experimental results show that the experimental pose curves of the end-effector are smooth and continuous. And the error between the theoretical pose and the experimental pose is about ±0.05°. Taking #1 motor as an example, it can be seen from Figures 15 and 16 that the trend of the actual angular velocity of #1 motor is correct and the absolute error of #1 motor is ±0.1 rad/s. And it can be seen from Figure 17 that the actual output torque of #1 motor is within the range of rated torque (1.27 Nm). The above experimental results are satisfactory. Therefore, it can be concluded that the proposed backstepping sliding mode robust control is correct and effective, and it can be used on the prototype for the wind tunnel test.


## 7. Conclusions

In this study, the dynamic model for WDPR used in the wind tunnel test is constructed. According to its dynamic model, a backstepping sliding mode robust control is designed based on a nonlinear disturbance observer. Through using the designed nonlinear disturbance observer, the equivalent disturbance is observed. And the equivalent compensation is introduced into the designed backstepping sliding mode robust control, which achieves complete suppression of the disturbance. The Lyapunov second method is used to prove the correctness of the designed control strategy and analyze the stability of the closed-loop system. The simulation results show that the robust control strategy has high control precision, and the designed control scheme is feasible. Finally, the effectiveness of the proposed control scheme is further proved through prototype experiment. The designed backstepping sliding mode robust control has certain innovation, guidance, and practicality for the application of WDPR prototype in the wind tunnel test. The future research direction is to improve the hardware performance of WDPR and carry out composite motion and six degree of freedom motion experiments on the prototype to further verify the feasibility of the designed control law.



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*Source: 3146762-2020-03-24.xml*