# Sliding Mode Control of Flexible Articulated Manipulator Based on Robust Observer

Authors: Yanghua Gao; Hailiang Lu
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

In this paper, a robust observer-based sliding mode control algorithm is proposed to address the modelling and measurement inaccuracies, load variations, and external disturbances of flexible articulated manipulators. Firstly, a sliding mode observer was designed with exponential convergence to observe system state accurately and to overcome the measuring difficulty of the state variables, unmeasurable quantities, and external disturbances. Next, a robust sliding mode controller was developed based on the observer, such that the output error of the system converges to zero in finite time. In this way, the whole system achieves asymptotic stability. Finally, the convergence conditions of the observer were theoretically analyzed to verify the convergence of the proposed algorithm, and simulation was carried out to confirm the effectiveness of the proposed method.


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## Body

## 1. Introduction

Flexible manipulators are increasingly applied in industrial and aerospace fields, such as welding robots, industrial production lines, mechanical arms of aircraft, and so on, owing to their energy efficiency, high speed, and low contact impact. More and more attention has been paid to the research of flexible manipulators, along with the development of aerospace technology, robotics, marine engineering, and industrial engineering. Flexible manipulators are now extensively used to comfort humans in different areas of work, which involves risky and tedious works such as painting, cutting, dispensing, material handling, machine tending, machining, and assembly. However, each flexible manipulator is an extremely complex, dynamic system with highly nonlinear, strongly coupled, and time-varying features. The system behaviors are complex and dynamic due to load variations, uncertain external perturbations, and inherent vibrations [1–3]. Hence, flexible manipulators can hardly be modelled or measured accurately, calling for a well-designed controller [4–8]. Against this backdrop, it is theoretically and practically significant to explore the response speed and control accuracy of trajectory tracking for the double-linked flexible-joint manipulator [9].To address the above problems, Lee et al. [10] designed an adaptive proportional-derivative (PD) controller to improve the trajectory tracking accuracy of the flexible-joint manipulator but did not consider the stability of the manipulator system. Lee and Lee [11] proposed a hybrid control strategy to optimize the design of the controller and generate hybrid trajectories. The strategy enhances the robustness of the flexible-joint manipulator system, yet it failed to take into account the trajectory tracking accuracy of the manipulator. Dong et al. [12] presented a fuzzy optimal control method for the design of a robust adaptive controller and demonstrated that the method ensures accurate and robust trajectory tracking of the flexible-joint manipulator. However, the manipulator’s response speed of trajectory tracking was not taken into consideration. Abd Latip et al. [13] automatically adjusted the control gain online with an adaptive proportional-integral-derivative (PID) controller, which supports the control of the single-link flexible manipulator even after the actuator failure.Ahanda et al. [14] addressed the robust adaptive control of a robotic manipulator under uncertain dynamics and joint space constraints and adopted command filters to overcome the time derivatives of virtual control, eliminating the need for differentiating the desired trajectory. In addition, a barrier Lyapunov function was introduced to handle joint space constraints, and a robust adaptive support vector regression architecture was employed to suppress filtering errors, approximation errors, and dynamic uncertainties. Based on unknown input observer (UIO), Wang et al. [15] put forward a novel funnel nonsingular terminal sliding mode control (FNTSMC) method for servomechanisms with unknown dynamics, e.g., nonlinear friction, uncertainties, and external disturbances. He et al. [16] created a full-state feedback neural network (NN) control to mitigate the uncertainties and enhance the robustness of the dynamic system of a flexible-joint manipulator. Through a Lyapunov stability analysis, it was demonstrated that the controller can ensure the stability of the flexible-joint manipulator system and guarantee the boundedness of system state variables, by choosing appropriate control gains. Rahmani and Belkheiri [17] came up with a novel approach for adaptive control of flexible multilink robots in the joint space, proved that the approach is valid for a class of highly uncertain systems with arbitrary but bounded dimensions, and realized trajectory tracking by developing a stable inversion for robot dynamics using only joint angle measurements. Guo et al. [18] investigated the repetitive motion planning (RMP) of robotic manipulators under the high precision of joint angle repeatability and end-effector motion and applied a special difference rule to discretize the existing RMP scheme with P-based formulation, yielding a novel pseudoinverse-based (P-based) RMP scheme for robotic manipulators.Through the above analysis, this paper proposes a sliding mode control strategy based on the robust observer. Firstly, a sliding mode robust observer was designed in light of the unmeasurable state, the modelling uncertainty, and the external disturbance moment of the flexible-joint manipulator. Next, a sliding mode controller was designed to track the positions of the first and second joints of the manipulator, aiming to realize the finite-time control of the system. Meanwhile, the convergence of the observer and controller was analyzed to present the convergence conditions. Finally, the effectiveness of the proposed method was verified through simulation.


## 2. Problem Description

The dynamics of the flexible-joint manipulator can be expressed as(1)Iθ¨+Kθ−θm+Mglsinq=0,Jθ¨m−Kθ−θm=u,where θ and θm are the angular positions of the link and rotor, respectively; I and J represent the rotational inertia of the link and rotor, respectively; K is the joint stiffness coefficient; M, g, and l are the link mass, gravitational acceleration at the link’s center of gravity, and joint length, respectively; and u is the motor torque input.Letx1=θ, x2=θ˙, x3=θm, and x4=θ˙m be state variables. Considering modelling uncertainty and external disturbance moments, the underdriven form of equation (1) can be obtained as(2)x˙1=x2,x˙2=a1x3+f1x1+Δ1t,x˙3=x4,x˙4=a2u+f2x1,x3+Δ2t,where a1=K/I; f1x1=−Mgl/I⋅sinx1−K/I⋅x1; a2=1/J; f2x1,x3=K/J⋅x1−x3; Δ1t and Δ2t are the uncertainty part and the external disturbance moment, respectively; and Δ1t≤ρ1 , Δ2t≤ρ2.The following lemma was introduced to facilitate the observer and controller stability analysis.Lemma 1 (see [1]).
ForV:0,∞∈R, the solution of V˙≤−αV+f with ∀t≥t0≥0 can be expressed as an inequality:(3)Vt≤e−αt−t0Vt0+∫t0te−αt−τfτdτ,where a is an arbitrary constant.


## 3. The Observer and Controller Design

### 3.1. The Observer Design

The observer ofx2 and x4 was designed as follows.To realizex2 and x4 observations, the following reconfiguration system was developed:(4)λ˙1=λ2+l1x1−λ1+D1x1−λ1,λ˙2=a1x3+f1x1+D¯2x1−λ1,λ˙3=λ1+l2x3−λ3+D3x3−λ3,λ˙4=a2u+f2x1,x3+D¯4x3−λ3,where l1, l2, D1, D¯2, D3, and D¯4 are the positive real numbers to be designed and λ1, λ2, λ3, and λ4 are meaningless intermediate state variables.Then, the observer was designed as(5)x^1=λ1,x^2=λ2+l1x1−λ1,x^3=λ3,x^4=λ4+l2x3−λ3,where x^i is the state estimation. The estimation error can be defined as(6)x˜i=xi−x^i.From equations (4)–(6), we have(7)x^˙1=λ2+l1x1−λ1+D1x1−λ1=x^2+D1x˜1,x^˙2=a1x3+f1x1+D¯2x1−λ1+l1x2−x^2−D1x˜1=a1x3+f1x1+l1x˜2+D¯2−l1D1x˜1,x^˙3=λ4+l2x3−λ3+D3x3−λ3=x^4+D3x˜3,x^˙4=a2u+f2x1,x3+D¯4x3−λ3+l2x4−x^4−D3x˜3=a2u+f2x1,x3+l2x˜4+D¯4−l2D3x˜3.Note thatD2=D¯2−l1D1 and D4=D¯4−l2D3. Then,(8)x^˙1=x^2+D1x˜1,x^˙2=a1x3+f1x1+l1x˜2+D2x˜1,x^˙3=x^4+D3x˜3,x^˙4=a2u+f2x1,x3+l2x˜4+D4x˜3.The following theorem was introduced to facilitate the proof of observer convergence.Theorem 1.
For system (2) and observer (5), if the initial conditions satisfy V0≤p, where p is any positive real number, there exists a condition that all the signals l1,l2,Dii=1,…,4 of the system are semiglobally consistent and bounded, and the observation error converges to an arbitrarily small residual set.Proof.
According to equations (5) and (6), the Lyapunov function is taken as(9)V=12∑i=14x˜i2.
The following can be derived from equation (9):(10)V˙=x˜1x2−x^2−D1x˜1+x˜2Δ1−l1x˜2−D2x˜1+x˜3x4−x^1−D3x˜3+x˜4Δ2−l2x˜4−D1x¯3=1−D2x˜1x˜2+1−D4x˜3x˜4−D1x˜12−l1x˜22−D3x˜32−l2x˜42+Δ1x˜2+Δ2x˜4.
TakingD2=D1=1 and the inequality ρi2/2+x˜j2/2≥ρi, x˜j≥Δix˜i, we have(11)V˙≤−D1x˜12−l1x˜22−D3x˜32−l2x˜42+ρ122+x˜222+ρ222+x˜422.
Inequality (11) can be rectified as(12)V˙≤−D1x˜12+D3x˜32+l1−12x˜22+l2−12x˜42+ρ122+ρ222.
Takingl1≥1/2+r, l2≥1/2+r, D1≥r, and D3≥r with r being the positive real number to be designed,(13)D1x˜12+D3x˜32+l1−12x˜22+l2−12x˜42≥rx˜12+x˜32+x˜22+x˜42.
Thus,(14)V˙≤−rx˜12+x32+x˜22+x˜42+ρ122+ρ222≤−2rV+Q,where Q=ρ12/2+ρ22/2.
According to Lemma1, the solution to inequality (14) is(15)Vt≤e−2rt−t0Vt0+Qe−2rt∫t0te2rτdτ=e−2rt−t0Vt0+Qe−2rt2re2rt−e2rt0=e−2rt−t0Vt0+Q2r1−e−2rt−t0.
That is,(16)Vt≤Q2r+Vt0−Q2re−2rt−t0.
Then, all the signals of the system are semiglobally bounded and satisfy(17)limt⟶∞Vt≤Q2r.Remark 1.
From equation (17), it can be inferred that the observation accuracy of the state depends on the upper bound Δ1t and the initial error Δ2t of the observer. When parameter r is infinitely large, the observation error will be arbitrarily small.Remark 2.
Without considering the modelling uncertaintyΔ1t=0 and the external disturbance moment Δ2t=0, ρ12/2+ρ22/2=0, that is, if V˙≤−2rV, then Vt≤e−2rt−t0Vt0. At this point, the observer converges exponentially.


### 3.2. Design and Analysis of Observer-Based Sliding Mode Controller

Observer-based sliding mode control is a new sliding mode control method in recent years. It solves the unknown disturbance problem directly from the sliding mode design side by purposefully designing the switching function and realizes the global nonsingular control of the system. At the same time, it inherits the finite-time convergence characteristics of sliding mode. Compared with the traditional sliding mode control, it can make the control system converge to the desired trajectory in finite time and has high steady-state accuracy. It is especially suitable for high-speed and high-precision control.Letx1 and x2 be the controlled targets of xd and x˙d, respectively. The design error can be expressed as(18)e1=x1−xd,e2=e˙1=x2−x˙d,e3=e¨1=x˙2−x¨d=a1x3+f1x1−x¨d,e4=e¨1=a1x˙3+f˙1x1−x⃛d=a1x4+f˙1x1−x⃛d.Then, the error of the observer can be expressed as(19)e^1=x^1−xd,e^2=x^2−x˙d,e^3=a1x^3+f^1x1−x¨d,e^4=a1x^4+f˙^1x1−x⃛d.From equations (18) and (19), we have(20)e˜1=e1−e^1=x˜1,e˜2=e2−e^2=x˜2,e˜3=e3−e^3=a1x˜3+f1x1−f^1x1,e˜4=e4−e^4=a1x˜4+f˙1x1−f˙^1x1.The design sliding mode function can be expressed as(21)s=c1e1+c2e2+c3e3+e4,where ci>0 is designed by i=1,2,3.Then,(22)s^=c1e˜1+c2e˜2+c3e˜3+e˜4=c1x˜1+c2x˜2+c3a1x˜3+f1x1−f^1x1+a1x˜4+f˙1x1−f˙^1x1.Then, the control law can be designed as(23)u=−1a1a2c1x^2−x˙d+c2a1x^3+f^1x1−x¨d+c3a1x^4+f˙^1x1−x⃛d+a1f^1x1,x3+f¨1^x1−x⃛d+ηs^,where η>0.Then, we have(24)s˙=c1e˙1+c2e˙2+c3e˙3+e˙1=c1x2−x˙d+c2a1x3+f1x1−x¨d+c3a1x4+f˙1x1−x⃛d+a1a2u+f1x1,x3+f¨1x1−x⃛d=c1x2−x˙d+c2a1x3+f1x1−x¨d+c3a1x4+f˙1x1−x⃛d−c1x^2−x˙d+c2a1x^3+f^1x1−x¨d+c3a1x^4+f˙^1x1−x⃛d+a1f^1x1,x3+f^1x1−x⃛d+ηs^+a1f1x1,x3+f¨1x1−x⃜d=c1x˜2+c2a1x˜3+c3a1x˜4+c2f1x1−f^1x1+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1−ηs−s˜.Take the Lyapunov function as(25)Vc=12s2.Then,(26)V˙c=ss˙=sc1x˜2+c2a1x˜3+c3a1x˜4+c2f1x1−f^1x1+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1−ηs−s˜=−ηs2+ηsc1x˜1+c2x˜2+c3a1x˜3+f1x1−f^1x1+a1x˜4+f˙1x1−f˙^1x1+sc1x˜2+c2a1x˜3+c3a1x˜4+c2f1x1−f^1x1+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1=−ηs2+sηc1x˜1+ηc2+c1x˜2+ηc3a1+c2a1x˜3+ηa1+c3a1x˜4+ηc3+c2f1x1−f^1x1+η+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1=−ηs2+sχx˜≤−ηs2+12s2+12χ2x˜=12−ηs2+12χ2x˜=1−2ηVc+12χ2x˜,where(27)χx˜=ηc1x˜1+ηc2+c1x˜2+ηc3a1+c2a1x˜3+ηa1+c3a1x˜1+ηc3+c2f1x1−f^1x1+η+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1.Since observer (5) converges exponentially, i.e., at time t⟶∞, x˜1 converges exponentially to x˜2, and x˜3 to x˜4. According to the Taylor series expansion of f1x1=−Mgl/I⋅sinx1−K/I⋅x1 and f2x1,x3=K/J⋅x1−x3, f1x1⟶f^1x1 converges exponentially to f˙1x1⟶f˙^1x1, f1x1,x3⟶f^1x1,x3. Thus, f¨1x1⟶f¨1^x1 also converges exponentially to 0.Considering the observer and the controller, the Lyapunov function of the closed loop is taken as(28)V=Vc+Vo.According to equation (28), we have(29)V˙=V˙0+V˙c≤−2rV0−2η−1Vc+12χ2x˜≤−η1V+χ⋅e−σ0t−t0,where η1=2r,2η−1max; χ⋅ is the class K function of x˜t0; and σ0>0.According to Lemma1, the solution to V˙≤−η1V+χ⋅e−σ0t−t0 can be expressed as an inequality:(30)Vt≤e−η1t−t0Vt0+χΔ∫t0te−η1t−τe−σ0τ−t0dτ=e−η1t−t0Vt0+χ⋅e−η1t+σ0t0∫t0te1η1τe−σ0τdτ=e−η1t−t0Vt0+χΔη1−σ0e−η1t+σ0t0eη1−σ0τ|t0t=e−η1t−t0Vt0+χΔη1−σ0e−η1t+σ0t0eη1−σ0t−eη1−σ0t0=e−η1t−t0Vt0+χ⋅η1−σ0e−σ0t−t0−e−η1t−t0.That is,limt⟶∞Vt≤0.SinceVt≥0, when t⟶∞, Vt=0, and Vt converges exponentially. The convergence accuracy depends on η1, i.e., r and η.Remark 3.
When the controller reaches the sliding mode surface, that is,s=0, we have e4=−c1e1−c2e2−c3e3. If E1=e1e2e3T and A=010001−c1−c2−c3, then E˙1=AE1. Through the design of c1, c2, and c3, A is Hurwitz zeta function. Thus, at time t⟶∞,E1=e1e2e3T⟶0. To make A as a Hurwitz zeta function, the real root part of the following equation must be negative:(31)A−λI=−λ100−λ1−c1−c2−c3−λ=λ2−c3−λ−c1−c2λ.
That is,−λ3−c3λ2−c2λ−c1=0. Taking the eigenvalue of −10, λ+103=0, from λ3+9λ2+27λ+27=0, we can obtain that λ3+c3λ2+c2λ+c1=0, c1=1000, c2=300, and c3=30. Hence, the convergence condition can be satisfied.


## 3.1. The Observer Design

The observer ofx2 and x4 was designed as follows.To realizex2 and x4 observations, the following reconfiguration system was developed:(4)λ˙1=λ2+l1x1−λ1+D1x1−λ1,λ˙2=a1x3+f1x1+D¯2x1−λ1,λ˙3=λ1+l2x3−λ3+D3x3−λ3,λ˙4=a2u+f2x1,x3+D¯4x3−λ3,where l1, l2, D1, D¯2, D3, and D¯4 are the positive real numbers to be designed and λ1, λ2, λ3, and λ4 are meaningless intermediate state variables.Then, the observer was designed as(5)x^1=λ1,x^2=λ2+l1x1−λ1,x^3=λ3,x^4=λ4+l2x3−λ3,where x^i is the state estimation. The estimation error can be defined as(6)x˜i=xi−x^i.From equations (4)–(6), we have(7)x^˙1=λ2+l1x1−λ1+D1x1−λ1=x^2+D1x˜1,x^˙2=a1x3+f1x1+D¯2x1−λ1+l1x2−x^2−D1x˜1=a1x3+f1x1+l1x˜2+D¯2−l1D1x˜1,x^˙3=λ4+l2x3−λ3+D3x3−λ3=x^4+D3x˜3,x^˙4=a2u+f2x1,x3+D¯4x3−λ3+l2x4−x^4−D3x˜3=a2u+f2x1,x3+l2x˜4+D¯4−l2D3x˜3.Note thatD2=D¯2−l1D1 and D4=D¯4−l2D3. Then,(8)x^˙1=x^2+D1x˜1,x^˙2=a1x3+f1x1+l1x˜2+D2x˜1,x^˙3=x^4+D3x˜3,x^˙4=a2u+f2x1,x3+l2x˜4+D4x˜3.The following theorem was introduced to facilitate the proof of observer convergence.Theorem 1.
For system (2) and observer (5), if the initial conditions satisfy V0≤p, where p is any positive real number, there exists a condition that all the signals l1,l2,Dii=1,…,4 of the system are semiglobally consistent and bounded, and the observation error converges to an arbitrarily small residual set.Proof.
According to equations (5) and (6), the Lyapunov function is taken as(9)V=12∑i=14x˜i2.
The following can be derived from equation (9):(10)V˙=x˜1x2−x^2−D1x˜1+x˜2Δ1−l1x˜2−D2x˜1+x˜3x4−x^1−D3x˜3+x˜4Δ2−l2x˜4−D1x¯3=1−D2x˜1x˜2+1−D4x˜3x˜4−D1x˜12−l1x˜22−D3x˜32−l2x˜42+Δ1x˜2+Δ2x˜4.
TakingD2=D1=1 and the inequality ρi2/2+x˜j2/2≥ρi, x˜j≥Δix˜i, we have(11)V˙≤−D1x˜12−l1x˜22−D3x˜32−l2x˜42+ρ122+x˜222+ρ222+x˜422.
Inequality (11) can be rectified as(12)V˙≤−D1x˜12+D3x˜32+l1−12x˜22+l2−12x˜42+ρ122+ρ222.
Takingl1≥1/2+r, l2≥1/2+r, D1≥r, and D3≥r with r being the positive real number to be designed,(13)D1x˜12+D3x˜32+l1−12x˜22+l2−12x˜42≥rx˜12+x˜32+x˜22+x˜42.
Thus,(14)V˙≤−rx˜12+x32+x˜22+x˜42+ρ122+ρ222≤−2rV+Q,where Q=ρ12/2+ρ22/2.
According to Lemma1, the solution to inequality (14) is(15)Vt≤e−2rt−t0Vt0+Qe−2rt∫t0te2rτdτ=e−2rt−t0Vt0+Qe−2rt2re2rt−e2rt0=e−2rt−t0Vt0+Q2r1−e−2rt−t0.
That is,(16)Vt≤Q2r+Vt0−Q2re−2rt−t0.
Then, all the signals of the system are semiglobally bounded and satisfy(17)limt⟶∞Vt≤Q2r.Remark 1.
From equation (17), it can be inferred that the observation accuracy of the state depends on the upper bound Δ1t and the initial error Δ2t of the observer. When parameter r is infinitely large, the observation error will be arbitrarily small.Remark 2.
Without considering the modelling uncertaintyΔ1t=0 and the external disturbance moment Δ2t=0, ρ12/2+ρ22/2=0, that is, if V˙≤−2rV, then Vt≤e−2rt−t0Vt0. At this point, the observer converges exponentially.


## 3.2. Design and Analysis of Observer-Based Sliding Mode Controller

Observer-based sliding mode control is a new sliding mode control method in recent years. It solves the unknown disturbance problem directly from the sliding mode design side by purposefully designing the switching function and realizes the global nonsingular control of the system. At the same time, it inherits the finite-time convergence characteristics of sliding mode. Compared with the traditional sliding mode control, it can make the control system converge to the desired trajectory in finite time and has high steady-state accuracy. It is especially suitable for high-speed and high-precision control.Letx1 and x2 be the controlled targets of xd and x˙d, respectively. The design error can be expressed as(18)e1=x1−xd,e2=e˙1=x2−x˙d,e3=e¨1=x˙2−x¨d=a1x3+f1x1−x¨d,e4=e¨1=a1x˙3+f˙1x1−x⃛d=a1x4+f˙1x1−x⃛d.Then, the error of the observer can be expressed as(19)e^1=x^1−xd,e^2=x^2−x˙d,e^3=a1x^3+f^1x1−x¨d,e^4=a1x^4+f˙^1x1−x⃛d.From equations (18) and (19), we have(20)e˜1=e1−e^1=x˜1,e˜2=e2−e^2=x˜2,e˜3=e3−e^3=a1x˜3+f1x1−f^1x1,e˜4=e4−e^4=a1x˜4+f˙1x1−f˙^1x1.The design sliding mode function can be expressed as(21)s=c1e1+c2e2+c3e3+e4,where ci>0 is designed by i=1,2,3.Then,(22)s^=c1e˜1+c2e˜2+c3e˜3+e˜4=c1x˜1+c2x˜2+c3a1x˜3+f1x1−f^1x1+a1x˜4+f˙1x1−f˙^1x1.Then, the control law can be designed as(23)u=−1a1a2c1x^2−x˙d+c2a1x^3+f^1x1−x¨d+c3a1x^4+f˙^1x1−x⃛d+a1f^1x1,x3+f¨1^x1−x⃛d+ηs^,where η>0.Then, we have(24)s˙=c1e˙1+c2e˙2+c3e˙3+e˙1=c1x2−x˙d+c2a1x3+f1x1−x¨d+c3a1x4+f˙1x1−x⃛d+a1a2u+f1x1,x3+f¨1x1−x⃛d=c1x2−x˙d+c2a1x3+f1x1−x¨d+c3a1x4+f˙1x1−x⃛d−c1x^2−x˙d+c2a1x^3+f^1x1−x¨d+c3a1x^4+f˙^1x1−x⃛d+a1f^1x1,x3+f^1x1−x⃛d+ηs^+a1f1x1,x3+f¨1x1−x⃜d=c1x˜2+c2a1x˜3+c3a1x˜4+c2f1x1−f^1x1+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1−ηs−s˜.Take the Lyapunov function as(25)Vc=12s2.Then,(26)V˙c=ss˙=sc1x˜2+c2a1x˜3+c3a1x˜4+c2f1x1−f^1x1+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1−ηs−s˜=−ηs2+ηsc1x˜1+c2x˜2+c3a1x˜3+f1x1−f^1x1+a1x˜4+f˙1x1−f˙^1x1+sc1x˜2+c2a1x˜3+c3a1x˜4+c2f1x1−f^1x1+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1=−ηs2+sηc1x˜1+ηc2+c1x˜2+ηc3a1+c2a1x˜3+ηa1+c3a1x˜4+ηc3+c2f1x1−f^1x1+η+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1=−ηs2+sχx˜≤−ηs2+12s2+12χ2x˜=12−ηs2+12χ2x˜=1−2ηVc+12χ2x˜,where(27)χx˜=ηc1x˜1+ηc2+c1x˜2+ηc3a1+c2a1x˜3+ηa1+c3a1x˜1+ηc3+c2f1x1−f^1x1+η+c3f˙1x1−f˙^1x1+a1f1x1,x3−f^1x1,x3+f¨1x1−f¨1^x1.Since observer (5) converges exponentially, i.e., at time t⟶∞, x˜1 converges exponentially to x˜2, and x˜3 to x˜4. According to the Taylor series expansion of f1x1=−Mgl/I⋅sinx1−K/I⋅x1 and f2x1,x3=K/J⋅x1−x3, f1x1⟶f^1x1 converges exponentially to f˙1x1⟶f˙^1x1, f1x1,x3⟶f^1x1,x3. Thus, f¨1x1⟶f¨1^x1 also converges exponentially to 0.Considering the observer and the controller, the Lyapunov function of the closed loop is taken as(28)V=Vc+Vo.According to equation (28), we have(29)V˙=V˙0+V˙c≤−2rV0−2η−1Vc+12χ2x˜≤−η1V+χ⋅e−σ0t−t0,where η1=2r,2η−1max; χ⋅ is the class K function of x˜t0; and σ0>0.According to Lemma1, the solution to V˙≤−η1V+χ⋅e−σ0t−t0 can be expressed as an inequality:(30)Vt≤e−η1t−t0Vt0+χΔ∫t0te−η1t−τe−σ0τ−t0dτ=e−η1t−t0Vt0+χ⋅e−η1t+σ0t0∫t0te1η1τe−σ0τdτ=e−η1t−t0Vt0+χΔη1−σ0e−η1t+σ0t0eη1−σ0τ|t0t=e−η1t−t0Vt0+χΔη1−σ0e−η1t+σ0t0eη1−σ0t−eη1−σ0t0=e−η1t−t0Vt0+χ⋅η1−σ0e−σ0t−t0−e−η1t−t0.That is,limt⟶∞Vt≤0.SinceVt≥0, when t⟶∞, Vt=0, and Vt converges exponentially. The convergence accuracy depends on η1, i.e., r and η.Remark 3.
When the controller reaches the sliding mode surface, that is,s=0, we have e4=−c1e1−c2e2−c3e3. If E1=e1e2e3T and A=010001−c1−c2−c3, then E˙1=AE1. Through the design of c1, c2, and c3, A is Hurwitz zeta function. Thus, at time t⟶∞,E1=e1e2e3T⟶0. To make A as a Hurwitz zeta function, the real root part of the following equation must be negative:(31)A−λI=−λ100−λ1−c1−c2−c3−λ=λ2−c3−λ−c1−c2λ.
That is,−λ3−c3λ2−c2λ−c1=0. Taking the eigenvalue of −10, λ+103=0, from λ3+9λ2+27λ+27=0, we can obtain that λ3+c3λ2+c2λ+c1=0, c1=1000, c2=300, and c3=30. Hence, the convergence condition can be satisfied.


## 4. Results and Discussion

### 4.1. The Simulation of the Observer

To verify the feasibility of the robust observer, a system was developed to run in an open loop and modelled considering the following external disturbance moments and modelling uncertainties:(32)θ˙1=ω1,ω˙1=a1θm+f1θ1+Δ1t,θ˙m=ωm,ω˙m=a2u+f2θ1,θm+Δ2t,where θ1, ω1, θm, and ωm are the position of rod 1, the angular velocity of rod 1, the position of rod m, and the angular velocity of rod m, respectively. The parameters were configured as follows: x=θ1ω1,…,θmωmT, Δ1t=sint, Δ2t=cost, J=1kg⋅m2, Mgl=5Nm, and K=40Nm/rad. Before simulation, the system state was initialized as x0=0.100.050T, and the observer state is initialized as λ0=0000T. The observer adopts the form of (4) and (5), with r=100. Based on l1≥1/2+r (5), l2≥1/2+r, D1≥r, and D2≥r, the following parameter values were selected: l1=l2=101, D2=D4=1.0, and D1=D3=101.The simulation structure of the observer is given in Figure1, and the simulation results are shown in Figures 2 and 3. Specifically, Figure 2 presents the flexible modes of the position states of the two joints and their derivatives (i.e., velocities), and Figure 3 displays the tracking errors of the states. It can be inferred that the proposed observer can completely observe each state of the system (as suggested by Figure 2) and fully track the states of the upper two joints after only 0.1 s (as indicated by the error curve in Figure 3, the errors are 0.001, 0.15, 0.0015, and 0.12 for Figures 3(a)–3(d)). Therefore, our method was proved to be fast and effective. Although there are disturbances in the system, i.e., Δ1t=sint and Δ2t=cost, the observation results show the anti-interference ability and good robustness of the proposed observer.Figure 1
Simulation structure of the observer.Figure 2
State estimations. (a)θ state estimation. (b) ω state estimation. (c) θm state estimation. (d) ωm state estimation.
(a)(b)(c)(d)Figure 3
State estimation errors. (a)θ estimation error. (b) ω estimation error. (c) θm estimation error. (d) ωm estimation error.
(a)(b)(c)(d)


### 4.2. The Simulation of the Control Algorithm

To verify the effectiveness of the proposed control algorithm, the system withΔ1t=0.15sint and Δ2t=0.23cost was taken as shown in equation (32), where x0=0.2000T and disturbance torque is λ0=0000T. The other parameters were kept the same as in simulation 1. The controller takes equation (23), with c1=1000, c2=300, c3=30, and η=1.5. The desired trajectory of joint 2 is θ1d=sin. The simulation results are shown in Figures 4 and 5. The former presents the angle and angular velocity of the second joint of the manipulator, and the latter exhibits the observed values of each state of the manipulator.Figure 4
Angle and angular velocity tracking with the proposed method. (a) Angle tracking. (b) Angular velocity tracking.
(a)(b)Figure 5
Angle and angular velocity tracking with PID control. (a) Angle tracking. (b) Angular velocity tracking.
(a)(b)As shown in Figure4, the system state was stabilized in a limited time, despite the presence of external disturbances and fault signals, indicating that the system converges well under this controller. Because the initial state of the system is x0=0.2000T and due to the existence of interference, there is a large error at the initial time. However, with the increase of control time, the system error decreases rapidly. Hence, our control method can effectively deal with the above problem. Figure 5 shows the results with PID controller; it can be seen that there is a large error in the position of PID control, and especially when the position reaches the maximum and minimum, the error is large.As shown in Figure6, our disturbance observer could observe the state information of the system with high accuracy and effectiveness. That is, the observed signals can be used in the controller design, which further illustrates the effectiveness of the method.Figure 6
Observation of each state of the manipulator.


## 4.1. The Simulation of the Observer

To verify the feasibility of the robust observer, a system was developed to run in an open loop and modelled considering the following external disturbance moments and modelling uncertainties:(32)θ˙1=ω1,ω˙1=a1θm+f1θ1+Δ1t,θ˙m=ωm,ω˙m=a2u+f2θ1,θm+Δ2t,where θ1, ω1, θm, and ωm are the position of rod 1, the angular velocity of rod 1, the position of rod m, and the angular velocity of rod m, respectively. The parameters were configured as follows: x=θ1ω1,…,θmωmT, Δ1t=sint, Δ2t=cost, J=1kg⋅m2, Mgl=5Nm, and K=40Nm/rad. Before simulation, the system state was initialized as x0=0.100.050T, and the observer state is initialized as λ0=0000T. The observer adopts the form of (4) and (5), with r=100. Based on l1≥1/2+r (5), l2≥1/2+r, D1≥r, and D2≥r, the following parameter values were selected: l1=l2=101, D2=D4=1.0, and D1=D3=101.The simulation structure of the observer is given in Figure1, and the simulation results are shown in Figures 2 and 3. Specifically, Figure 2 presents the flexible modes of the position states of the two joints and their derivatives (i.e., velocities), and Figure 3 displays the tracking errors of the states. It can be inferred that the proposed observer can completely observe each state of the system (as suggested by Figure 2) and fully track the states of the upper two joints after only 0.1 s (as indicated by the error curve in Figure 3, the errors are 0.001, 0.15, 0.0015, and 0.12 for Figures 3(a)–3(d)). Therefore, our method was proved to be fast and effective. Although there are disturbances in the system, i.e., Δ1t=sint and Δ2t=cost, the observation results show the anti-interference ability and good robustness of the proposed observer.Figure 1
Simulation structure of the observer.Figure 2
State estimations. (a)θ state estimation. (b) ω state estimation. (c) θm state estimation. (d) ωm state estimation.
(a)(b)(c)(d)Figure 3
State estimation errors. (a)θ estimation error. (b) ω estimation error. (c) θm estimation error. (d) ωm estimation error.
(a)(b)(c)(d)


## 4.2. The Simulation of the Control Algorithm

To verify the effectiveness of the proposed control algorithm, the system withΔ1t=0.15sint and Δ2t=0.23cost was taken as shown in equation (32), where x0=0.2000T and disturbance torque is λ0=0000T. The other parameters were kept the same as in simulation 1. The controller takes equation (23), with c1=1000, c2=300, c3=30, and η=1.5. The desired trajectory of joint 2 is θ1d=sin. The simulation results are shown in Figures 4 and 5. The former presents the angle and angular velocity of the second joint of the manipulator, and the latter exhibits the observed values of each state of the manipulator.Figure 4
Angle and angular velocity tracking with the proposed method. (a) Angle tracking. (b) Angular velocity tracking.
(a)(b)Figure 5
Angle and angular velocity tracking with PID control. (a) Angle tracking. (b) Angular velocity tracking.
(a)(b)As shown in Figure4, the system state was stabilized in a limited time, despite the presence of external disturbances and fault signals, indicating that the system converges well under this controller. Because the initial state of the system is x0=0.2000T and due to the existence of interference, there is a large error at the initial time. However, with the increase of control time, the system error decreases rapidly. Hence, our control method can effectively deal with the above problem. Figure 5 shows the results with PID controller; it can be seen that there is a large error in the position of PID control, and especially when the position reaches the maximum and minimum, the error is large.As shown in Figure6, our disturbance observer could observe the state information of the system with high accuracy and effectiveness. That is, the observed signals can be used in the controller design, which further illustrates the effectiveness of the method.Figure 6
Observation of each state of the manipulator.


## 5. Conclusions

The improvement of a robust observer-based sliding mode is improved, and the efficiency of the model is improved in this paper. Aiming at the problems of high nonlinearity, strong coupling, and external interference in the system, we firstly designed a state observer for the system through the auxiliary reconstruction system, solved the state observation problem of the system, clarified the convergence condition of the observer through theoretical analysis, and verified it through simulation. Then, the position and velocity tracking problem was tackled. Considering the external disturbance, a sliding mode control of the flexible-joint manipulator was derived based on the robust observer. The control method ensures that the system state can converge exponentially to zero in finite time under different inputs and outputs. The simulation results show that the observer can quickly observe the state variables of the system. Also, combined with the sliding mode controller, the system error can quickly converge to zero. The proposed control strategy is simple and easy to implement.



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*Source: 2440770-2022-01-04.xml*