# Linear Active Disturbance Rejection Control of a Two-Degrees-of-Freedom Manipulator

Authors: Dawei Liu; Qinhe Gao; Zhixiang Chen; Zhihao Liu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper presents linear active disturbance rejection control (LADRC) for a two-degrees-of-freedom (2-DOF) manipulator system to achieve trajectory tracking. The system is widely used in engineering applications and exhibits the characteristics of high nonlinearity, strong coupling, and large uncertainty with two inputs and two outputs. First, the problem of dynamic coupling in the model of the 2-DOF manipulator is addressed by considering the dynamic coupling, model uncertainties, and external disturbances as total disturbances. Second, a linear extended state observer is designed to estimate the total disturbances, while a linear state error feedback control law is designed to compensate these disturbances. The main contribution is that the stability of the closed-loop system with two inputs and two outputs is analyzed, and the relationship between the performance of the closed-loop system and the controller parameters is established. The joint simulation of SolidWorks and Matlab/Simulink is conducted. The simulation and experimental results clearly indicate the superiority of LADRC over the PID for trajectory tracking and dynamic performance.


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## Body

## 1. Introduction

With the rapid development of automation, manipulators are widely used in industries for applications such as assembling, picking, painting, and welding. Therefore, the trajectory tracking control of a manipulator has been widely studied. One challenge in achieving an acceptable performance (such as small errors, good trajectory, and disturbance rejection) might arise from uncertainty, high nonlinearity, and dynamic coupling in robotic systems [1, 2].Focusing on the trajectory tracking control problem of a manipulator system, various control strategies have been studied in related fields, such as PD control with feedforward compensation [3, 4], adaptive control [5–7], fuzzy control [8, 9], neural network control [10–13], and sliding mode control [14]. Chen et al. [3] proposed a modified PD feedforward compensation control. By using feedforward compensation in the PD control structure, the control accuracy of the manipulator is effectively improved. Kuc and Han [5] combined adaptive control with PID control and proposed an adaptive PID learning controller that was based on feedforward input learning. Ouyang et al. [6] proposed an adaptive switching learning PD control (ASL-PD), which was a combination of the feedback PD control law with a gain switching technique and the feedforward learning control law with the input torque profile based on interative learning. According to the ASL-PD method, the convergence rate of manipulator trajectory tracking is effectively increased. The control of an uncertain robotic manipulator with input saturation was studied by Tran and Sam [7]. They introduced a model reference adaptive control-like (MRAC-like) to deal with the problems of input saturation, unknown input scaling, and disturbances. Lin et al. [9] proposed a variable structure control method to compensate for the uncertain part of the robotic manipulator and achieved the elimination of variable structure chattering by introducing the fuzzy control method. In order to improve the tracking precision of the flexible-link manipulator, Xu [12] proposed the DOB-based composite neural control to deal with unknown dynamics and time-varying disturbances. Considering the influence of gravity change, Liu et al. [14] designed a sliding mode controller (SMC) for a space manipulator. To solve the influence of the uncertain part of the manipulator, several studies combined the neural network with the PD control, fuzzy control, adaptive control, and sliding mode control compensating for the uncertain part [4, 8, 11, 15]. However, the structure of the controllers became complicated as a result of the increase in the number of neural networks.The abovementioned control strategies have had a certain effect on solving the uncertainty and nonlinearity of the manipulator; however, the problem of dynamic coupling has not been solved effectively. There are certain shortcomings in each control strategy influencing the performance of the systems [16]:(1)
A contradiction between system rapidity and overshoot in PID control, which has a limited effect on nonlinear characteristics and time-varying disturbances(2)
The adaptive control method requires the control object model to be sufficiently accurate and the tuning of the controller parameters is difficult(3)
The structure of fuzzy control and neural network control is complicated and the controller parameters are hard to tuneThe combination of the control algorithms can make up for the disadvantages of each other to a certain extent; however, the problem of the complex structure of controllers cannot be solved.Active disturbance rejection control (ADRC) was proposed by Han in 1995. It is almost independent of an accurate and detailed dynamic model of the plant and has strong robustness to nonlinear, uncertainties, and external disturbances [17]. The most prominent feature of ADRC is that all the uncertainties of the plant and external disturbances are considered as total disturbances, which are estimated by extended state observer (ESO) and compensated by linear state error feedback control law (NSEFCL). The traditional ADRC has many parameters, which makes the controller tuning complex. To simplify the tuning, Gao proposed Linear ADRC (LADRC) and used linear ESO (LESO) [18, 19]. ADRC is widely applied because of its excellent performance of disturbance rejection and the convenience of engineering realization [20–24]. In recent years, remarkable progress has been made in the theoretical basis of ADRC. Zhao and Guo [25, 26] analyzed the stability of nonlinear active disturbance rejection control (NLADRC). Xue et al. [27–30] analyzed the performance of LADRC and provided theoretical references for an LADRC of a two-degrees-of-freedom (2-DOF) manipulator.The rest of the contents are organized as follows. First, the mathematical model of the 2-DOF manipulator is established, which is based on the model decoupling. Section3 plans the trajectory of a regular octagonal for the manipulator. The LADRC controller based on the decoupling system is presented. Boundedness of the estimation error and the closed-loop tracking error are then established. Furthermore, the estimation error and the closed-loop tracking error monotonously decrease with the two observers’ and the two controllers’ bandwidths, respectively. In addition, the joint simulation of SolidWorks and Matlab/Simulink is conducted. Furthermore, the simulation and the experiment demonstrate the better trajectory tracking and dynamic performance of LADRC than that of PID.


## 2. Modeling and Trajectory Planning of a 2-DOF Manipulator System

### 2.1. Modeling

The simplified model of the 2-DOF manipulator is depicted in Figure1.Figure 1
Simplified model of the 2-DOF manipulator.The dynamic model can be derived using the Euler–Lagrange dynamic equation, which comprises two coupled nonlinear second-order differential equations. Defineθ=θ1,θ2T, and the model can be represented as(1)M11θ¨1+M12θ¨2+N1θ,θ˙+G1θ=u1+d1,M21θ¨1+M22θ¨2+N2θ,θ˙+G2θ=u2+d2,where(2)M11=a1+a2cosθ2,M12=M21=a3+12a2cosθ2,M22=a3,N1θ,θ˙=−a2sinθ2θ˙1θ˙2+12θ˙22,N2θ,θ=12a2sinθ2θ˙12,G1θ=a4sinθ1+a5sinθ1+θ2,G2θ=a5sinθ1+θ2,a1=13m1l12+13m2l22+m2l12,a2=m2l2l1,a3=13m2l22,a4=12m1l1g+m2l1g,a5=12m2l2g,where l1 and l2 represents the length of the 2 rods, θ1andθ2 are the joint angles shown in Figure 1, m1andm2 represent the mass of the 2 rods, u1andu2 are the control laws of rod 1 and rod 2, respectively, and d1andd2 represent the external disturbance of 2 rods, respectively.


### 2.2. Trajectory Planning

The trajectory planning and desired signal generation of the manipulator must be achieved before designing the LADRC, and they can be achieved as follows.Assuming that the reachable area of the end effector isx,y∈ℝ2x2+y2≤l1+l22, trajectory planning allows the end effector to run a desired trajectory in the area where the controlled objects are the motors that drive the two link rods to rotate.Assuming that the end effector moves from the initial pointA=xA,yA=0,−l1−l2 to another point B=xr,yr, the displacement in the x and y directions can be written as(3)lx=xr−xA=xr,ly=yr−yA=yr+l1+l2.From point A to point B, the motion inx and y directions is planned using the method of quintic polynomial [31]. Thus, the displacement, velocity, and acceleration in the x and y directions can be represented in the quintic form as(4)lx=a11+a12⋅t+a13⋅t2+a14⋅t3+a15⋅t4+a16⋅t5,vx=a12+2⋅a13⋅t+3⋅a14⋅t2+4⋅a15⋅t3+5⋅a16⋅t4,ax=2⋅a13+6⋅a14⋅t+12⋅a15⋅t2+20⋅a16⋅t3,ly=a21+a22⋅t+a23⋅t2+a24⋅t3+a25⋅t4+a26⋅t5,vy=a22+2⋅a23⋅t+3⋅a24⋅t2+4⋅a25⋅t3+5⋅a26⋅t4,ay=2⋅a23+6⋅a24⋅t+12⋅a25⋅t2+20⋅a26⋅t3,where a1=A−1⋅b1,a2=A−1⋅b2,(5)A=1toto2to3to4to5012to3to24to35to40026to12to220to31tftf2tf3tf4tf5012tf3tf24tf35tf40026tf12tf220tf3,b1=xA00xr00,b2=yA00yr00,where t0 is the time at the initial point A, tf is time to reach the point B, xA and yA represent the positions of the end effector in x and y directions at the initial point, respectively, and xr and yr are the positions of the x and the y directions at the point of arrival.From (3) and (4), the planned trajectory can be written as(6)xrt=xA+lxt=a11+a12⋅t+a13⋅t2+a14⋅t3+a15⋅t4+a16⋅t5,yrt=yA+lyt=a21+a22⋅t+a23⋅t2+a24⋅t3+a25⋅t4+a26⋅t5−l1−l2,and from (4) and (6), we can obtain the desired signal.According to the structure and coordinate system definition of the 2-DOF manipulator, we have(7)x=l1sinθ1+l2sinθ1+θ2,y=−l1cosθ1−l2cosθ1+θ2.From (7), the corresponding Jacobian matrix J can be calculated as(8)J=l1cosθ1+l2cosθ1+θ2l2cosθ1+θ2l1sinθ1+l2sinθ1+θ2l2sinθ1+θ2,and let v=x˙y˙T,a=x¨y¨T,ω=θ˙1θ˙2T,andγ=θ¨1θ¨2T; we have(9)v=Jω,a¯=Jγ,where(10)a¯=x¨+l1sinθ1θ˙1+l2sinθ1+θ2θ˙1+θ˙2θ˙1+l2sinθ1+θ2θ˙1+θ˙2θ˙2y¨−l1cosθ1θ˙1+l2cosθ1+θ2θ˙1+θ˙2θ˙1−l2cosθ1+θ2θ˙1+θ˙2θ˙2.From (9), the desired velocity signal of the 2-DOF manipulator can be written as(11)ωr=J−1vr,γr=J−1a¯r,and the desired angle signal can be written as(12)θrt=∫0tωrτdτ.Considering the case whereJ is irreversible, assuming(13)J=l1l2sinθ2=0,we can obtain θ2=2kπ,k∈ℕ, when J is irreversible. This situation must be avoided. Hence, when θ2−2kπ≤δ∈ℝ+, letting θ˙2=c2∈ℝ, the system will avoid the irreversible region of matrix J. In trajectory planning, we should avoid reaching the irreversible region, which is the outer circle and the inner circle of the collinear point of the two link rods. The irreversible region can be written as(14)x,y∈ℝ2x2+y2=l1+l22∪x,y∈ℝ2x2+y2=l1−l22.To avoid any calculation error, (11) is modified as(15)ωr=J−1vr+αxr−x,where α is error gain.The desired signal generation principle of the 2-DOF manipulator is depicted in Figure2. The desired trajectory planned in this study is a regular octagonal.Figure 2
Desired signal generation principle of the 2-DOF manipulator.


## 2.1. Modeling

The simplified model of the 2-DOF manipulator is depicted in Figure1.Figure 1
Simplified model of the 2-DOF manipulator.The dynamic model can be derived using the Euler–Lagrange dynamic equation, which comprises two coupled nonlinear second-order differential equations. Defineθ=θ1,θ2T, and the model can be represented as(1)M11θ¨1+M12θ¨2+N1θ,θ˙+G1θ=u1+d1,M21θ¨1+M22θ¨2+N2θ,θ˙+G2θ=u2+d2,where(2)M11=a1+a2cosθ2,M12=M21=a3+12a2cosθ2,M22=a3,N1θ,θ˙=−a2sinθ2θ˙1θ˙2+12θ˙22,N2θ,θ=12a2sinθ2θ˙12,G1θ=a4sinθ1+a5sinθ1+θ2,G2θ=a5sinθ1+θ2,a1=13m1l12+13m2l22+m2l12,a2=m2l2l1,a3=13m2l22,a4=12m1l1g+m2l1g,a5=12m2l2g,where l1 and l2 represents the length of the 2 rods, θ1andθ2 are the joint angles shown in Figure 1, m1andm2 represent the mass of the 2 rods, u1andu2 are the control laws of rod 1 and rod 2, respectively, and d1andd2 represent the external disturbance of 2 rods, respectively.


## 2.2. Trajectory Planning

The trajectory planning and desired signal generation of the manipulator must be achieved before designing the LADRC, and they can be achieved as follows.Assuming that the reachable area of the end effector isx,y∈ℝ2x2+y2≤l1+l22, trajectory planning allows the end effector to run a desired trajectory in the area where the controlled objects are the motors that drive the two link rods to rotate.Assuming that the end effector moves from the initial pointA=xA,yA=0,−l1−l2 to another point B=xr,yr, the displacement in the x and y directions can be written as(3)lx=xr−xA=xr,ly=yr−yA=yr+l1+l2.From point A to point B, the motion inx and y directions is planned using the method of quintic polynomial [31]. Thus, the displacement, velocity, and acceleration in the x and y directions can be represented in the quintic form as(4)lx=a11+a12⋅t+a13⋅t2+a14⋅t3+a15⋅t4+a16⋅t5,vx=a12+2⋅a13⋅t+3⋅a14⋅t2+4⋅a15⋅t3+5⋅a16⋅t4,ax=2⋅a13+6⋅a14⋅t+12⋅a15⋅t2+20⋅a16⋅t3,ly=a21+a22⋅t+a23⋅t2+a24⋅t3+a25⋅t4+a26⋅t5,vy=a22+2⋅a23⋅t+3⋅a24⋅t2+4⋅a25⋅t3+5⋅a26⋅t4,ay=2⋅a23+6⋅a24⋅t+12⋅a25⋅t2+20⋅a26⋅t3,where a1=A−1⋅b1,a2=A−1⋅b2,(5)A=1toto2to3to4to5012to3to24to35to40026to12to220to31tftf2tf3tf4tf5012tf3tf24tf35tf40026tf12tf220tf3,b1=xA00xr00,b2=yA00yr00,where t0 is the time at the initial point A, tf is time to reach the point B, xA and yA represent the positions of the end effector in x and y directions at the initial point, respectively, and xr and yr are the positions of the x and the y directions at the point of arrival.From (3) and (4), the planned trajectory can be written as(6)xrt=xA+lxt=a11+a12⋅t+a13⋅t2+a14⋅t3+a15⋅t4+a16⋅t5,yrt=yA+lyt=a21+a22⋅t+a23⋅t2+a24⋅t3+a25⋅t4+a26⋅t5−l1−l2,and from (4) and (6), we can obtain the desired signal.According to the structure and coordinate system definition of the 2-DOF manipulator, we have(7)x=l1sinθ1+l2sinθ1+θ2,y=−l1cosθ1−l2cosθ1+θ2.From (7), the corresponding Jacobian matrix J can be calculated as(8)J=l1cosθ1+l2cosθ1+θ2l2cosθ1+θ2l1sinθ1+l2sinθ1+θ2l2sinθ1+θ2,and let v=x˙y˙T,a=x¨y¨T,ω=θ˙1θ˙2T,andγ=θ¨1θ¨2T; we have(9)v=Jω,a¯=Jγ,where(10)a¯=x¨+l1sinθ1θ˙1+l2sinθ1+θ2θ˙1+θ˙2θ˙1+l2sinθ1+θ2θ˙1+θ˙2θ˙2y¨−l1cosθ1θ˙1+l2cosθ1+θ2θ˙1+θ˙2θ˙1−l2cosθ1+θ2θ˙1+θ˙2θ˙2.From (9), the desired velocity signal of the 2-DOF manipulator can be written as(11)ωr=J−1vr,γr=J−1a¯r,and the desired angle signal can be written as(12)θrt=∫0tωrτdτ.Considering the case whereJ is irreversible, assuming(13)J=l1l2sinθ2=0,we can obtain θ2=2kπ,k∈ℕ, when J is irreversible. This situation must be avoided. Hence, when θ2−2kπ≤δ∈ℝ+, letting θ˙2=c2∈ℝ, the system will avoid the irreversible region of matrix J. In trajectory planning, we should avoid reaching the irreversible region, which is the outer circle and the inner circle of the collinear point of the two link rods. The irreversible region can be written as(14)x,y∈ℝ2x2+y2=l1+l22∪x,y∈ℝ2x2+y2=l1−l22.To avoid any calculation error, (11) is modified as(15)ωr=J−1vr+αxr−x,where α is error gain.The desired signal generation principle of the 2-DOF manipulator is depicted in Figure2. The desired trajectory planned in this study is a regular octagonal.Figure 2
Desired signal generation principle of the 2-DOF manipulator.


## 3. Design of LADRC

### 3.1. Problem Description

Before designing an LADRC, we describe its mathematical model in (1). Letting NG1=N1+G1andNG2=N2+G2, (1) becomes(16)M11M22−M12M21θ¨1+M22NG1−M12NG2=M22u1−M12u2+M22d1−M12d2,M11M22−M12M21θ¨2+M11NG2−M21NG1=−M21u1+M11u2−M21d1+M11d2.From (16), the descriptions (i)-(ii) are made:(i)
The conditionΔ=M11M22−M12M21>0 can be verified according to the definition of the parameters in (1). The control input u1 and u2 both have an effect on θ1 and θ2. Hence, the two link rods are coupled.(ii)
When the parameters of the system contain uncertain factors and no parameters in (16) are known accurately, the model-based control design will no longer apply. Considering the coupling part of the two link rods as disturbance, the disturbances in each subsystem are estimated and compensated as per the control principle of the designed LADRC. Thus, θ1 and θ2 can be controlled by u1 and u2, respectively.


### 3.2. Controller Design

From (16), we have(17)θ¨1=−M22NG1−M12NG2−M12u2+M22d1−M12d2M11M22−M12M21+M22M11M22−M12M21−b1u1+b1u1,θ¨2=−M11NG2−M21NG1−M21u1−M21d1+M11d2M11M22−M12M21+M11M11M22−M12M21−b2u2+b2u2.Assuming the total disturbances of rod 1 and rod 2 as(18)f1θ,u1,u2,d1,d2=−M22NG1+M12NG2M11M22−M12M21−M12u2−M22d1+M12d2M11M22−M12M21+M22M11M22−M12M21−b1u1,f2θ,u1,u2,d1,d2=−M11NG2−M21NG1M11M22−M12M21−M21u1+M21d1−M11d2M11M22−M12M21+M11M11M22−M12M21−b2u2,(17) can be rewritten as(19)θ¨1=f1θ,u2,d1,d2+b1u1,θ¨2=f2θ,u1,d1,d2+b2u2.(19) is then rewritten as(20)ξ¨=fξ,w+b1u1,ς¨=gς,m+b2u2,where w is the sum of d1andd2 and the coupling control u2, m is the sum of d1andd2 and the coupling control u1, and b1 and b2 are the nominal values of the control gain of rod 1 and rod 2, respectively. Assuming fandg are differentiable and that h=f˙andn=g˙, (20) can be written in an augmented state-space form:(21)ξ˙1=ξ2,ξ˙2=ξ3+b1u1,ξ˙3=hξ,w,y=ξ1,ς˙1=ς2,ς˙2=ς3+b2u2,ς˙3=nς,m,z=ς1,where ξ=ξ1ξ2ξ3T∈ℝ3, ς=ς1ς2ς3T∈ℝ3 are the states of the system, and u1,u2∈R, y,z∈R are the input and output of the system, respectively.LADRC is designed for (21) and the LESO is given as(22)ξ^˙1=ξ^2+ω0α1ξ1−ξ^1,ξ^˙2=ξ^3+ω02α2ξ1−ξ^1+b1u1,ξ^˙3=ω03α3ξ1−ξ^1,ς^˙1=ς^2+ω1α1ς1−ς^1,ς^˙2=ς^3+ω12α2ς1−ς^1+b2u2,ς^˙3=ω13α3ς1−ς^1,where ω0,ω1>0 and α1,α2,andα3 are selected such that the characteristic polynomial s3+α1s2+α2s+α3 is Hurwitz.Then, the ADRC control law is given as(23)u1=ϕ2rt+k11r1−ξ^1+k12r2−ξ^2−ξ^3/b1,u2=ϕ2st+k21s1−ς^1+k22s2−ς^2−ς^3/b2,where r is the desired signal of ξ, s is the desired signal of ς, and the values are assigned and bounded. Let(24)r˙1=r2,r˙2=ϕ2rt,s˙1=s2,s˙2=ϕ2st.r
˙
1
,
r
˙
2
,
s
˙
1
,
and
s
˙
2 are also bounded and k11,k12,k21,andk22 are the controller gains selected to make s2+k12s+k11 and s2+k22s+k21 Hurwitz.


## 3.1. Problem Description

Before designing an LADRC, we describe its mathematical model in (1). Letting NG1=N1+G1andNG2=N2+G2, (1) becomes(16)M11M22−M12M21θ¨1+M22NG1−M12NG2=M22u1−M12u2+M22d1−M12d2,M11M22−M12M21θ¨2+M11NG2−M21NG1=−M21u1+M11u2−M21d1+M11d2.From (16), the descriptions (i)-(ii) are made:(i)
The conditionΔ=M11M22−M12M21>0 can be verified according to the definition of the parameters in (1). The control input u1 and u2 both have an effect on θ1 and θ2. Hence, the two link rods are coupled.(ii)
When the parameters of the system contain uncertain factors and no parameters in (16) are known accurately, the model-based control design will no longer apply. Considering the coupling part of the two link rods as disturbance, the disturbances in each subsystem are estimated and compensated as per the control principle of the designed LADRC. Thus, θ1 and θ2 can be controlled by u1 and u2, respectively.


## 3.2. Controller Design

From (16), we have(17)θ¨1=−M22NG1−M12NG2−M12u2+M22d1−M12d2M11M22−M12M21+M22M11M22−M12M21−b1u1+b1u1,θ¨2=−M11NG2−M21NG1−M21u1−M21d1+M11d2M11M22−M12M21+M11M11M22−M12M21−b2u2+b2u2.Assuming the total disturbances of rod 1 and rod 2 as(18)f1θ,u1,u2,d1,d2=−M22NG1+M12NG2M11M22−M12M21−M12u2−M22d1+M12d2M11M22−M12M21+M22M11M22−M12M21−b1u1,f2θ,u1,u2,d1,d2=−M11NG2−M21NG1M11M22−M12M21−M21u1+M21d1−M11d2M11M22−M12M21+M11M11M22−M12M21−b2u2,(17) can be rewritten as(19)θ¨1=f1θ,u2,d1,d2+b1u1,θ¨2=f2θ,u1,d1,d2+b2u2.(19) is then rewritten as(20)ξ¨=fξ,w+b1u1,ς¨=gς,m+b2u2,where w is the sum of d1andd2 and the coupling control u2, m is the sum of d1andd2 and the coupling control u1, and b1 and b2 are the nominal values of the control gain of rod 1 and rod 2, respectively. Assuming fandg are differentiable and that h=f˙andn=g˙, (20) can be written in an augmented state-space form:(21)ξ˙1=ξ2,ξ˙2=ξ3+b1u1,ξ˙3=hξ,w,y=ξ1,ς˙1=ς2,ς˙2=ς3+b2u2,ς˙3=nς,m,z=ς1,where ξ=ξ1ξ2ξ3T∈ℝ3, ς=ς1ς2ς3T∈ℝ3 are the states of the system, and u1,u2∈R, y,z∈R are the input and output of the system, respectively.LADRC is designed for (21) and the LESO is given as(22)ξ^˙1=ξ^2+ω0α1ξ1−ξ^1,ξ^˙2=ξ^3+ω02α2ξ1−ξ^1+b1u1,ξ^˙3=ω03α3ξ1−ξ^1,ς^˙1=ς^2+ω1α1ς1−ς^1,ς^˙2=ς^3+ω12α2ς1−ς^1+b2u2,ς^˙3=ω13α3ς1−ς^1,where ω0,ω1>0 and α1,α2,andα3 are selected such that the characteristic polynomial s3+α1s2+α2s+α3 is Hurwitz.Then, the ADRC control law is given as(23)u1=ϕ2rt+k11r1−ξ^1+k12r2−ξ^2−ξ^3/b1,u2=ϕ2st+k21s1−ς^1+k22s2−ς^2−ς^3/b2,where r is the desired signal of ξ, s is the desired signal of ς, and the values are assigned and bounded. Let(24)r˙1=r2,r˙2=ϕ2rt,s˙1=s2,s˙2=ϕ2st.r
˙
1
,
r
˙
2
,
s
˙
1
,
and
s
˙
2 are also bounded and k11,k12,k21,andk22 are the controller gains selected to make s2+k12s+k11 and s2+k22s+k21 Hurwitz.


## 4. Stability Analysis

### 4.1. Stability Analysis of LESO

Letξ˜i=ξi−ξ^iandς˜=ςi−ς^i,i=1,2,3. From (21) and (22), the observer estimation error can be written as(25)ξ˜˙1=ξ˜˙2−ω0α1ξ˜1,ξ˜˙2=ξ˜˙3−ω02α2ξ˜1,ξ˜˙3=hξ,w−ω03α3ξ˜1,ς˜˙1=ς˜˙2−ω1α1ς˜1,ς˜˙2=ς˜˙3−ω12α2ς˜1,ς˜˙3=nς,m−ω13α3ς˜1.Now, letεi=ξ˜i/ω0i−1andχi=ς˜i/ω1i−1,i=1,2,3, and (25) can be rewritten as(26)ε˙=ω0Aε+Bhξ,wω02,χ˙=ω1Aχ+Bnς,mω12,where A=−α110−α201−α300andB=001T.Theorem 1.
Assuminghξ,w and nς,m is bounded, there exists constant σi>0,ρi>0 and a finite T1>0, such that ξ˜it≤σi,ς˜it≤ρi,i=1,2,3, ∀t≥T1>0, ω0>0, and ω1>0. Furthermore, σi=O1/ω0k and ρi=O1/ω1k, for some positive integer k.Proof.
Solving (26), it follows that(27)εt=eω0Atε0+∫0teω0At−τBhξτ,wω02dτ,χt=eω1Atχ0+∫0teω1At−τBnχτ,mω12dτ,and let(28)pt=∫0teω0At−τBhξτ,wω02dτ,qt=∫0teω1At−τBnςτ,mω12dτ,as hξτ,w and nςτ,m be bounded, i.e., hξτ,w≤δ, nςτ,m≤δ, where δ is a positive constant. For i=1,2,3, we have(29)pit≤δω03A−1Bi+A−1eω0AtBi,qit≤δω13A−1Bi+A−1eω1AtBi.
ForA and B defined in (26)(30)A−1=00−1α310−α1α301−α2α3,A−1Bi≤ν,where ν=maxi=2,31/α3,αi−1/α3. As A is Hurwitz, there exists a finite time T1>0 such that(31)eω0Atij≤1ω03,eω1Atij≤1ω13,for all t≥T1, i,j = 1,2,3. Hence,(32)eω0AtBi≤1ω03,eω1AtBi≤1ω13,for all t≥T1, i,j = 1,2,3.
LetA−1=g11g12g13g21g22g23g31g32g33, eω0At=d11d12d13d21d22d23d31d32d33, and eω1At=v11v12v13v21v22v23v31v32v33, we have (33)A−1eω0AtBi≤μω03,A−1eω1AtBi≤μω13,for all t≥T1, i,j = 1,2,3, where μ=maxi=2,31/α3,1+αi−1/α3.
From (29), (30), and (33), we obtain(34)pit≤δνω03+δμω06.
From (29), (30), and (33), we obtain(35)qit≤δνω13+δμω16,for all t≥T1, i,j = 1,2,3.
Letεsum0=ε10+ε20+ε30 and χsum0=χ10+χ20+χ30, and it follows that (36)eω0Atε0i≤εsum0ω03,eω1Atχ0i≤χsum0ω13,for all t≥T1, i,j = 1,2,3.
From (27), one has(37)εit≤eω0Atε0i+pit,χit≤eω1Atχ0i+qit.
Letξ˜sun0=ξ˜10+ξ˜20+ξ˜30 and ς˜sum0=ς˜10+ς˜20+ς˜30. According to εi=ξ˜i/ω0i−1, χi=ς˜i/ω1i−1, and (34)–(37), we have(38)ξ˜it≤ξ˜sum0ω03+δνω04−i+δμω07−i=σi,ς˜it≤ς˜sum0ω13+δνω14−i+δμω17−i=ρi,for all t≥T1, i,j = 1,2,3. According to (38), the estimation error of LESO (22) is bounded and the upper bound monotonously decreases with the two observers’ bandwidth.


### 4.2. Stability Analysis of LADRC

From (23), we have(39)u1=ϕ2rt+k11r1−ξ^1+k12r2−ξ^2−ξ^3/b1,u2=ϕ2st+k21s1−ς^1+k22s2−ς^2−ς^3/b2,where k11,k12,k21,andk22 are the controller gains selected to make s2+k12s+k11 and s2+k22s+k21 Hurwitz. The closed-loop system becomes(40)y¨=f−ξ^3+k11r1−ξ^1+k12r2−ξ^2+ϕ2rt,z¨=g−ς^3+k21s1−ς^1+k22s2−ς^2+ϕ2st.Consider(41)η˙t=Nηt+gt,where ηt=η1t,η2tT∈ℝ2, gt=g1t,g2tT∈ℝ2, and N is a 2×2 matrix.Lemma 1.
IfN is Hurwitz and limt⟶∞gt=0, then limt⟶∞ηt=0; Zheng et al. [32] provided detailed proof for the lemma.Theorem 2.
Assuminghξ,w and nς,m is bounded, there exist λi>0,βi>0 and a finite time T5>0 such that eit≤λi, e¯it≤βi,i=1,2, ∀t≥T5>0, ω0>0,ω1>0, ωc1>0,ωc2>0. Furthermore, ρi=O1/ωc1j and βi=O1/ωc2j, j∈+N, and ei=ri−ξi and e¯i=si−ςi, i=1,2.Proof.
From (23), we have(42)u1=k11e1+ξ˜1+k12e2+ξ˜2−ξ3−ξ˜3+ϕ2rt/b1,u2=k21e¯1+ς˜1+k22e¯2+ς˜2−ς3−ς˜3+ϕ2st/b2,and it follows that(43)e˙1=r˙1−ξ˙1=r2−ξ2=e2,e˙2=r˙2−ξ˙2=r3−ξ3+b1u1,=−k11e1+ξ˜1−k12e2+ξ˜2−ξ˜3.e¯˙1=s˙1−ς˙1=s˙2−ς2=e¯2,e¯˙2=s˙2−ς˙2=s˙3−ς3+b2u2,=−k21e¯1+ς˜1−k22e¯2+ς˜2−ς˜3.
Lete=e1e2T, ξ˜=ξ˜1ξ˜2ξ˜3T, e¯=e¯1e¯2T, and ς˜=ς˜1ς˜2ς˜3T; then, (44)e˙t=Aeet+Aξ˜ξ˜t,e¯˙t=Ae¯e¯t+Aς˜ς˜t,where(45)Ae=01−k11−k12,Aξ˜=000−k11−k12−1,Ae¯=01−k21−k22,Aς˜=000−k21−k22−1,since k11,k12,k21,andk22 are selected such that the characteristic polynomials s2+k12s+k11 and s2+k22s+k21 are Hurwitz and Ae and Ae¯ are Hurwitz. For tunning simplicity, we let s2+k12s+k11=s+ωc12 and s2+k22s+k21=s+ωc22, where ωc1>0,ωc2>0, k11=ωc12, k12=2ωc2,k21=ωc22,andk22=2ωc2 [32]. This implies that ωc1 and ωc2, which are controller bandwidths of rod 1 and rod 2, respectively, the only tunning parameters to be adjusted for the controller.
Solving the equatione˙t=Aeet+Aξ˜ξ˜t in (44), we obtain(46)et=eAete0+∫0teAet−τAξ˜ξ˜τdτ.
According to the equatione˙t=Aeet+Aξ˜ξ˜t and Theorem 1, we have(47)Aξ˜ξ˜τi=1=0,Aξ˜ξ˜τ2≤kd1σi=γ,∀t>T1,where kd1=1+k11+k12.
DefiningΨ=00⋯0γT and letting φt=∫0teAet−τAξ˜ξ˜τdτ, it follows that (48)φit≤Ae−1Ψi+Ae−1eAetΨi,Ae−1Ψ1=γk11=γωc12,Ae−1Ψ2=0.
SinceAe is Hurwitz, there exists a finite time T4>0 such that(49)eAetij≤1ωc13,for all t≥T4,i,j=1,2.
LetT5=maxT1,T4. It follows that(50)eAetΨi≤γωc13,for all t≥T5, i=1,2, and(51)Ae−1eAetΨi≤1+∑i=22k1iωc12γωc13|i=1,γωc13|i=2,for all t≥T5. From (48) and (51), we obtain(52)φit≤γωc13+1+∑i=22k1iωc12γωc13|i=1,γωc13|i=2,for all t≥T5. Let eAet=o11o12o21o22,es0=e10+e20. It follows that(53)eAete0i≤es0ωc13,for all t≥T5, i=1,2. From (46), we have(54)eit≤eAete0i+φit.
According to (47) and (52)–(54), we have(55)eit≤es0ωc13+kd1σiωc12+1+∑i=22k1ikd1σiωc15|i=1,es0+kd1σiωc13|i=2,≤υ,for all t≥T5, i=1,2, where(56)υi=maxes0ωc13+kd1σiωc12+1+∑i=22k1ikd1σiωc15,es0+kd1σiωc13.
Similarly, we can obtain(57)e¯it≤e¯s0ωc23+kd2ζiωc22+1+∑i=22k2ikd2ζiωc25|i=1,e¯s0+kd2ζiωc23|i=2,≤β,where(58)βi=maxe¯s0ωc23+kd2ζiωc22+1+∑i=22k2ikd2ζiωc25,e¯s0+kd2ζiωc23.
According to (55) and (57), we can observe that the error of the desired signal r1 and the system output ξ1 and the error of desired signal s1 and the system output ς1 are bounded, and the upper bound monotonously decreases with the two controllers’ bandwidth.Corollary 1.
Assuminghξ,w is globally Lipschitz with respect to ξ and nς,m is globally Lipschitz with respect to ς, there exist constants ω0>0,ω1>0,ωc1>0,andωc2>0, such that the closed-loop system (40) is asymptotically stable.Proof.
From (44), we know(59)e˙t=Aeet+Aξ˜ξ˜t,e¯˙t=Ae¯e¯t+Aς˜ς˜t.
According to Theorem1 postulated by [27], limt⟶∞Aξ˜ξ˜t=0 and limt⟶∞Aς˜ς˜t=0, if hξ,w is globally Lipschitz with respect to ξ and nς,m is globally Lipschitz with respect to ς. Furthermore, according to Lemma 1, since Ae and Ae¯ is Hurwitz and limt⟶∞Aξ˜ξ˜t=0,limt⟶∞Aς˜ς˜t=0, such that limt⟶∞eit=0, limt⟶∞e¯it=0, i=1,2, the closed-loop system (40) is asymptotically stable.


#### 4.2.1. Simulation

In this section, a simulation study on a 2-DOF manipulator is conducted to demonstrate the strong robust and high precision control effect of LADRC. The 3D model of the system is established by SolidWorks and is illustrated in Figure3. Furthermore, by generating the XML file of the model in SolidWorks and accessing the file in Matlab, we can obtain the Simulink model of the 2-DOF manipulator, as shown in Figure 4. The simulation of control is obtained from Matlab/Simulink.Figure 3
3D model of the 2-DOF manipulator.Figure 4
Simulink model of the 2-DOF manipulator.In the simulation, the LADRC and PID are applied for controlling plant (20) with the same regular octagon trajectory. The controller parameters are listed in Table 1.Table 1
LADRC and PID controller parameters in simulation.
LADRC
PID
Rod 1
ω
c
1
ω
0
b
1
α
1
α
2
α
3
k
p
1
k
i
1
k
d
1
35
500
20
3
3
1
300
50
0.5
Rod 2
ω
c
2
ω
1
b
2
α
1
α
2
α
3
k
p
2
k
i
2
k
d
2
30
500
55
3
3
1
150
40
0.5


#### 4.2.2. Simulation of Increasing Payload

Upon adding a 1.5 kg payload to the end effector, the controller parameters of LADRC and PID remain constant. The tracking curve and tracking error between LADRC and PID are compared in Figure5. From Figures 5(a), 5(c), and 5(e), we can observe that although the payload is increased, LADRC and PID can make the end effector track the octagonal trajectory. However, LADRC has a more accurate tracking performance both in x and y directions. From Figures 5(b) and 5(d), we can observe that the control of LADRC has smaller tracking errors and less tracking error fluctuations.Figure 5
Simulation comparison of tracking performance between LADRC and PID. (a) Comparison of tracking trajectory in thex-direction. (b) Comparison of tracking errors in the x-direction. (c) Comparison of tracking trajectory in the y-direction. (d) Comparison of tracking errors in the y-direction. (e) Tracking comparison of regular octagonal trajectory.
(a)
(b)
(c)
(d)
(e)


### 4.3. Simulation of Sudden Disturbances

When the end effector has no payload, we add step signals of the same amplitude to several angles and edges of the octagonal trajectory, respectively. From Figure6, we can conclude that compared with PID control, the maximum mutation of the tracking response of the LADRC method is smaller, the response fluctuation is smaller, and the time of the response recovery to the desired trajectory is shorter despite the position of the disturbances in the octagonal trajectory. The tracking errors depicted in Figure 7 also show that when the disturbances occur suddenly, the LADRC method has smaller maximum tracking errors, less tracking deviation fluctuation, and shorter deviation convergence times. Thus, the simulation results show that LADRC has better robustness and better control performance than PID control.Figure 6
Simulation comparison of tracking performance between LADRC and PID during a sudden disturbance. (a) Comparison of trajectory tracking with disturbance at the angles of octagonal in thex-direction. (b) Comparison of trajectory tracking with disturbance at the edges of octagonal in the x-direction. (c) Comparison of trajectory tracking with disturbance at the angles of octagonal in the y-direction. (d) Comparison of trajectory tracking with disturbance at the edges of octagonal in the y-direction. (e) Comparison of trajectory tracking with disturbance at the angles of octagonal. (f) Comparison of trajectory tracking with disturbance at the edges of octagonal.
(a)
(b)
(c)
(d)
(e)
(f)Figure 7
Simulation comparison of tracking errors between LADRC and PID during sudden disturbance. (a) Comparison of tracking errors with a disturbance at the angles of octagonal in thex-direction. (b) Comparison of tracking errors with a disturbance at the edges of octagonal in the x-direction. (c) Comparison of tracking errors with disturbance at the angles of octagonal in the y-direction. (d) Comparison of tracking errors with disturbance at the edges of octagonal in the y-direction.
(a)
(b)
(c)
(d)


## 4.1. Stability Analysis of LESO

Letξ˜i=ξi−ξ^iandς˜=ςi−ς^i,i=1,2,3. From (21) and (22), the observer estimation error can be written as(25)ξ˜˙1=ξ˜˙2−ω0α1ξ˜1,ξ˜˙2=ξ˜˙3−ω02α2ξ˜1,ξ˜˙3=hξ,w−ω03α3ξ˜1,ς˜˙1=ς˜˙2−ω1α1ς˜1,ς˜˙2=ς˜˙3−ω12α2ς˜1,ς˜˙3=nς,m−ω13α3ς˜1.Now, letεi=ξ˜i/ω0i−1andχi=ς˜i/ω1i−1,i=1,2,3, and (25) can be rewritten as(26)ε˙=ω0Aε+Bhξ,wω02,χ˙=ω1Aχ+Bnς,mω12,where A=−α110−α201−α300andB=001T.Theorem 1.
Assuminghξ,w and nς,m is bounded, there exists constant σi>0,ρi>0 and a finite T1>0, such that ξ˜it≤σi,ς˜it≤ρi,i=1,2,3, ∀t≥T1>0, ω0>0, and ω1>0. Furthermore, σi=O1/ω0k and ρi=O1/ω1k, for some positive integer k.Proof.
Solving (26), it follows that(27)εt=eω0Atε0+∫0teω0At−τBhξτ,wω02dτ,χt=eω1Atχ0+∫0teω1At−τBnχτ,mω12dτ,and let(28)pt=∫0teω0At−τBhξτ,wω02dτ,qt=∫0teω1At−τBnςτ,mω12dτ,as hξτ,w and nςτ,m be bounded, i.e., hξτ,w≤δ, nςτ,m≤δ, where δ is a positive constant. For i=1,2,3, we have(29)pit≤δω03A−1Bi+A−1eω0AtBi,qit≤δω13A−1Bi+A−1eω1AtBi.
ForA and B defined in (26)(30)A−1=00−1α310−α1α301−α2α3,A−1Bi≤ν,where ν=maxi=2,31/α3,αi−1/α3. As A is Hurwitz, there exists a finite time T1>0 such that(31)eω0Atij≤1ω03,eω1Atij≤1ω13,for all t≥T1, i,j = 1,2,3. Hence,(32)eω0AtBi≤1ω03,eω1AtBi≤1ω13,for all t≥T1, i,j = 1,2,3.
LetA−1=g11g12g13g21g22g23g31g32g33, eω0At=d11d12d13d21d22d23d31d32d33, and eω1At=v11v12v13v21v22v23v31v32v33, we have (33)A−1eω0AtBi≤μω03,A−1eω1AtBi≤μω13,for all t≥T1, i,j = 1,2,3, where μ=maxi=2,31/α3,1+αi−1/α3.
From (29), (30), and (33), we obtain(34)pit≤δνω03+δμω06.
From (29), (30), and (33), we obtain(35)qit≤δνω13+δμω16,for all t≥T1, i,j = 1,2,3.
Letεsum0=ε10+ε20+ε30 and χsum0=χ10+χ20+χ30, and it follows that (36)eω0Atε0i≤εsum0ω03,eω1Atχ0i≤χsum0ω13,for all t≥T1, i,j = 1,2,3.
From (27), one has(37)εit≤eω0Atε0i+pit,χit≤eω1Atχ0i+qit.
Letξ˜sun0=ξ˜10+ξ˜20+ξ˜30 and ς˜sum0=ς˜10+ς˜20+ς˜30. According to εi=ξ˜i/ω0i−1, χi=ς˜i/ω1i−1, and (34)–(37), we have(38)ξ˜it≤ξ˜sum0ω03+δνω04−i+δμω07−i=σi,ς˜it≤ς˜sum0ω13+δνω14−i+δμω17−i=ρi,for all t≥T1, i,j = 1,2,3. According to (38), the estimation error of LESO (22) is bounded and the upper bound monotonously decreases with the two observers’ bandwidth.


## 4.2. Stability Analysis of LADRC

From (23), we have(39)u1=ϕ2rt+k11r1−ξ^1+k12r2−ξ^2−ξ^3/b1,u2=ϕ2st+k21s1−ς^1+k22s2−ς^2−ς^3/b2,where k11,k12,k21,andk22 are the controller gains selected to make s2+k12s+k11 and s2+k22s+k21 Hurwitz. The closed-loop system becomes(40)y¨=f−ξ^3+k11r1−ξ^1+k12r2−ξ^2+ϕ2rt,z¨=g−ς^3+k21s1−ς^1+k22s2−ς^2+ϕ2st.Consider(41)η˙t=Nηt+gt,where ηt=η1t,η2tT∈ℝ2, gt=g1t,g2tT∈ℝ2, and N is a 2×2 matrix.Lemma 1.
IfN is Hurwitz and limt⟶∞gt=0, then limt⟶∞ηt=0; Zheng et al. [32] provided detailed proof for the lemma.Theorem 2.
Assuminghξ,w and nς,m is bounded, there exist λi>0,βi>0 and a finite time T5>0 such that eit≤λi, e¯it≤βi,i=1,2, ∀t≥T5>0, ω0>0,ω1>0, ωc1>0,ωc2>0. Furthermore, ρi=O1/ωc1j and βi=O1/ωc2j, j∈+N, and ei=ri−ξi and e¯i=si−ςi, i=1,2.Proof.
From (23), we have(42)u1=k11e1+ξ˜1+k12e2+ξ˜2−ξ3−ξ˜3+ϕ2rt/b1,u2=k21e¯1+ς˜1+k22e¯2+ς˜2−ς3−ς˜3+ϕ2st/b2,and it follows that(43)e˙1=r˙1−ξ˙1=r2−ξ2=e2,e˙2=r˙2−ξ˙2=r3−ξ3+b1u1,=−k11e1+ξ˜1−k12e2+ξ˜2−ξ˜3.e¯˙1=s˙1−ς˙1=s˙2−ς2=e¯2,e¯˙2=s˙2−ς˙2=s˙3−ς3+b2u2,=−k21e¯1+ς˜1−k22e¯2+ς˜2−ς˜3.
Lete=e1e2T, ξ˜=ξ˜1ξ˜2ξ˜3T, e¯=e¯1e¯2T, and ς˜=ς˜1ς˜2ς˜3T; then, (44)e˙t=Aeet+Aξ˜ξ˜t,e¯˙t=Ae¯e¯t+Aς˜ς˜t,where(45)Ae=01−k11−k12,Aξ˜=000−k11−k12−1,Ae¯=01−k21−k22,Aς˜=000−k21−k22−1,since k11,k12,k21,andk22 are selected such that the characteristic polynomials s2+k12s+k11 and s2+k22s+k21 are Hurwitz and Ae and Ae¯ are Hurwitz. For tunning simplicity, we let s2+k12s+k11=s+ωc12 and s2+k22s+k21=s+ωc22, where ωc1>0,ωc2>0, k11=ωc12, k12=2ωc2,k21=ωc22,andk22=2ωc2 [32]. This implies that ωc1 and ωc2, which are controller bandwidths of rod 1 and rod 2, respectively, the only tunning parameters to be adjusted for the controller.
Solving the equatione˙t=Aeet+Aξ˜ξ˜t in (44), we obtain(46)et=eAete0+∫0teAet−τAξ˜ξ˜τdτ.
According to the equatione˙t=Aeet+Aξ˜ξ˜t and Theorem 1, we have(47)Aξ˜ξ˜τi=1=0,Aξ˜ξ˜τ2≤kd1σi=γ,∀t>T1,where kd1=1+k11+k12.
DefiningΨ=00⋯0γT and letting φt=∫0teAet−τAξ˜ξ˜τdτ, it follows that (48)φit≤Ae−1Ψi+Ae−1eAetΨi,Ae−1Ψ1=γk11=γωc12,Ae−1Ψ2=0.
SinceAe is Hurwitz, there exists a finite time T4>0 such that(49)eAetij≤1ωc13,for all t≥T4,i,j=1,2.
LetT5=maxT1,T4. It follows that(50)eAetΨi≤γωc13,for all t≥T5, i=1,2, and(51)Ae−1eAetΨi≤1+∑i=22k1iωc12γωc13|i=1,γωc13|i=2,for all t≥T5. From (48) and (51), we obtain(52)φit≤γωc13+1+∑i=22k1iωc12γωc13|i=1,γωc13|i=2,for all t≥T5. Let eAet=o11o12o21o22,es0=e10+e20. It follows that(53)eAete0i≤es0ωc13,for all t≥T5, i=1,2. From (46), we have(54)eit≤eAete0i+φit.
According to (47) and (52)–(54), we have(55)eit≤es0ωc13+kd1σiωc12+1+∑i=22k1ikd1σiωc15|i=1,es0+kd1σiωc13|i=2,≤υ,for all t≥T5, i=1,2, where(56)υi=maxes0ωc13+kd1σiωc12+1+∑i=22k1ikd1σiωc15,es0+kd1σiωc13.
Similarly, we can obtain(57)e¯it≤e¯s0ωc23+kd2ζiωc22+1+∑i=22k2ikd2ζiωc25|i=1,e¯s0+kd2ζiωc23|i=2,≤β,where(58)βi=maxe¯s0ωc23+kd2ζiωc22+1+∑i=22k2ikd2ζiωc25,e¯s0+kd2ζiωc23.
According to (55) and (57), we can observe that the error of the desired signal r1 and the system output ξ1 and the error of desired signal s1 and the system output ς1 are bounded, and the upper bound monotonously decreases with the two controllers’ bandwidth.Corollary 1.
Assuminghξ,w is globally Lipschitz with respect to ξ and nς,m is globally Lipschitz with respect to ς, there exist constants ω0>0,ω1>0,ωc1>0,andωc2>0, such that the closed-loop system (40) is asymptotically stable.Proof.
From (44), we know(59)e˙t=Aeet+Aξ˜ξ˜t,e¯˙t=Ae¯e¯t+Aς˜ς˜t.
According to Theorem1 postulated by [27], limt⟶∞Aξ˜ξ˜t=0 and limt⟶∞Aς˜ς˜t=0, if hξ,w is globally Lipschitz with respect to ξ and nς,m is globally Lipschitz with respect to ς. Furthermore, according to Lemma 1, since Ae and Ae¯ is Hurwitz and limt⟶∞Aξ˜ξ˜t=0,limt⟶∞Aς˜ς˜t=0, such that limt⟶∞eit=0, limt⟶∞e¯it=0, i=1,2, the closed-loop system (40) is asymptotically stable.


### 4.2.1. Simulation

In this section, a simulation study on a 2-DOF manipulator is conducted to demonstrate the strong robust and high precision control effect of LADRC. The 3D model of the system is established by SolidWorks and is illustrated in Figure3. Furthermore, by generating the XML file of the model in SolidWorks and accessing the file in Matlab, we can obtain the Simulink model of the 2-DOF manipulator, as shown in Figure 4. The simulation of control is obtained from Matlab/Simulink.Figure 3
3D model of the 2-DOF manipulator.Figure 4
Simulink model of the 2-DOF manipulator.In the simulation, the LADRC and PID are applied for controlling plant (20) with the same regular octagon trajectory. The controller parameters are listed in Table 1.Table 1
LADRC and PID controller parameters in simulation.
LADRC
PID
Rod 1
ω
c
1
ω
0
b
1
α
1
α
2
α
3
k
p
1
k
i
1
k
d
1
35
500
20
3
3
1
300
50
0.5
Rod 2
ω
c
2
ω
1
b
2
α
1
α
2
α
3
k
p
2
k
i
2
k
d
2
30
500
55
3
3
1
150
40
0.5


### 4.2.2. Simulation of Increasing Payload

Upon adding a 1.5 kg payload to the end effector, the controller parameters of LADRC and PID remain constant. The tracking curve and tracking error between LADRC and PID are compared in Figure5. From Figures 5(a), 5(c), and 5(e), we can observe that although the payload is increased, LADRC and PID can make the end effector track the octagonal trajectory. However, LADRC has a more accurate tracking performance both in x and y directions. From Figures 5(b) and 5(d), we can observe that the control of LADRC has smaller tracking errors and less tracking error fluctuations.Figure 5
Simulation comparison of tracking performance between LADRC and PID. (a) Comparison of tracking trajectory in thex-direction. (b) Comparison of tracking errors in the x-direction. (c) Comparison of tracking trajectory in the y-direction. (d) Comparison of tracking errors in the y-direction. (e) Tracking comparison of regular octagonal trajectory.
(a)
(b)
(c)
(d)
(e)


## 4.2.1. Simulation

In this section, a simulation study on a 2-DOF manipulator is conducted to demonstrate the strong robust and high precision control effect of LADRC. The 3D model of the system is established by SolidWorks and is illustrated in Figure3. Furthermore, by generating the XML file of the model in SolidWorks and accessing the file in Matlab, we can obtain the Simulink model of the 2-DOF manipulator, as shown in Figure 4. The simulation of control is obtained from Matlab/Simulink.Figure 3
3D model of the 2-DOF manipulator.Figure 4
Simulink model of the 2-DOF manipulator.In the simulation, the LADRC and PID are applied for controlling plant (20) with the same regular octagon trajectory. The controller parameters are listed in Table 1.Table 1
LADRC and PID controller parameters in simulation.
LADRC
PID
Rod 1
ω
c
1
ω
0
b
1
α
1
α
2
α
3
k
p
1
k
i
1
k
d
1
35
500
20
3
3
1
300
50
0.5
Rod 2
ω
c
2
ω
1
b
2
α
1
α
2
α
3
k
p
2
k
i
2
k
d
2
30
500
55
3
3
1
150
40
0.5


## 4.2.2. Simulation of Increasing Payload

Upon adding a 1.5 kg payload to the end effector, the controller parameters of LADRC and PID remain constant. The tracking curve and tracking error between LADRC and PID are compared in Figure5. From Figures 5(a), 5(c), and 5(e), we can observe that although the payload is increased, LADRC and PID can make the end effector track the octagonal trajectory. However, LADRC has a more accurate tracking performance both in x and y directions. From Figures 5(b) and 5(d), we can observe that the control of LADRC has smaller tracking errors and less tracking error fluctuations.Figure 5
Simulation comparison of tracking performance between LADRC and PID. (a) Comparison of tracking trajectory in thex-direction. (b) Comparison of tracking errors in the x-direction. (c) Comparison of tracking trajectory in the y-direction. (d) Comparison of tracking errors in the y-direction. (e) Tracking comparison of regular octagonal trajectory.
(a)
(b)
(c)
(d)
(e)


## 4.3. Simulation of Sudden Disturbances

When the end effector has no payload, we add step signals of the same amplitude to several angles and edges of the octagonal trajectory, respectively. From Figure6, we can conclude that compared with PID control, the maximum mutation of the tracking response of the LADRC method is smaller, the response fluctuation is smaller, and the time of the response recovery to the desired trajectory is shorter despite the position of the disturbances in the octagonal trajectory. The tracking errors depicted in Figure 7 also show that when the disturbances occur suddenly, the LADRC method has smaller maximum tracking errors, less tracking deviation fluctuation, and shorter deviation convergence times. Thus, the simulation results show that LADRC has better robustness and better control performance than PID control.Figure 6
Simulation comparison of tracking performance between LADRC and PID during a sudden disturbance. (a) Comparison of trajectory tracking with disturbance at the angles of octagonal in thex-direction. (b) Comparison of trajectory tracking with disturbance at the edges of octagonal in the x-direction. (c) Comparison of trajectory tracking with disturbance at the angles of octagonal in the y-direction. (d) Comparison of trajectory tracking with disturbance at the edges of octagonal in the y-direction. (e) Comparison of trajectory tracking with disturbance at the angles of octagonal. (f) Comparison of trajectory tracking with disturbance at the edges of octagonal.
(a)
(b)
(c)
(d)
(e)
(f)Figure 7
Simulation comparison of tracking errors between LADRC and PID during sudden disturbance. (a) Comparison of tracking errors with a disturbance at the angles of octagonal in thex-direction. (b) Comparison of tracking errors with a disturbance at the edges of octagonal in the x-direction. (c) Comparison of tracking errors with disturbance at the angles of octagonal in the y-direction. (d) Comparison of tracking errors with disturbance at the edges of octagonal in the y-direction.
(a)
(b)
(c)
(d)


## 5. Experiment

Here, the proposed LADRC is further examined through an experiment, which is conducted on a 2-DOF manipulator, as shown in Figure8. The system comprises two motors, two reducers, two rods, GTHD servodrivers, and a GT-800-SV motion control card. The nominal values of the system parameters are given in Table 2.Figure 8
Experimental platform of a 2-DOF manipulator.Table 2
Nominal values of the 2-DOF manipulator system parameters.
Parameters
m
1 (kg)
m
2 (kg)
l
1 (m)
l
2 (m)
Value
0.85
0.15
0.3
0.2The sampling period in the experiment is set as 0.001 s and the solver in Matlab/Simulink environment is set as odel (Euler).First, the experiment of an increasing payload is conducted and the experiment of sudden disturbances is then conducted without payload, i.e., the step signals of the same amplitude are suddenly added at certain angles and edges of the octagonal trajectory, respectively. The controller parameters in the experiment are listed in Table3.Table 3
LADRC and PID controller parameters in experiment.
LADRC
PID
Rod 1
ω
c
1
ω
0
b
1
α
1
α
2
α
3
k
p
1
k
i
1
k
d
1
40
200
900
3
3
1
1600
50
80
Rod 2
ω
c
2
ω
1
b
2
α
1
α
2
α
3
k
p
2
k
i
2
k
d
2
50
200
1000
3
3
1
2500
50
100


### 5.1. Experiment of Increasing Payload

Upon adding a 1.5 kg payload to the end effector, the controller parameters of LADRC and PID remain constant. Experimental results are shown in Figure9. From Figures 9(a), 9(c), and 9(e), we can observe that when the payload is increased, LADRC and PID can also make the end effector track the octagonal trajectory. However, LADRC has a more accurate tracking performance despite its position in x or y directions. Figures 9(b) and 9(d) show that the control of LADRC has smaller tracking errors and less tracking error fluctuations than PID.Figure 9
Experimental comparison of tracking performance between LADRC and PID. (a) Comparison of tracking trajectory in thex-direction. (b) Comparison of tracking errors in the x-direction. (c) Comparison of tracking trajectory in the y-direction. (d) Comparison of tracking errors in the y-direction. (e) Tracking comparison of regular octagonal trajectory.
(a)
(b)
(c)
(d)
(e)


### 5.2. Experiment of Sudden Disturbances

When the end effector has no payload, we add a step signal of 20 V to the several angles and edges of the octagonal trajectory. The experimental results are plotted in Figures10 and 11. From Figure 10, we can observe that despite the position of the disturbances in the octagonal trajectory, the LADRC method has a smaller maximum trajectory deviation, faster response, smaller tracking fluctuations, and shorter tracking error convergence times when compared to PID. The tracking errors depicted in Figure 11 also show that when the disturbances occur suddenly, the LADRC method has smaller maximum tracking errors, less tracking deviation fluctuation, and shorter deviation convergence times. The experimental results verify that the control performance and robustness of LADRC are better than PID.Figure 10
Experimental comparison of the tracking performance between LADRC and PID. during sudden disturbance. (a) Comparison of trajectory tracking with disturbance at the angles of octagonal in thex-direction. (b) Comparison of trajectory tracking with disturbance at the edges of octagonal in the x-direction. (c) Comparison of trajectory tracking with disturbance at the angles of octagonal in the y-direction. (d) Comparison of trajectory tracking with disturbance at the edges of octagonal in the y-direction (e) Comparison of trajectory tracking with disturbance at the angles of octagonal. (f) Comparison of trajectory tracking with disturbance at the edges of octagonal.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Experimental comparison of tracking errors between LADRC and PID during sudden disturbance. (a) Comparison of tracking errors with disturbance at the angles of octagonal in thex-direction. (b) Comparison of tracking errors with disturbance at the edges of octagonal in the x-direction. (c) Comparison of tracking errors with disturbance at the angles of octagonal in the y-direction. (d) Comparison of tracking errors with disturbance at the edges of octagonal in the y-direction.
(a)
(b)
(c)
(d)


## 5.1. Experiment of Increasing Payload

Upon adding a 1.5 kg payload to the end effector, the controller parameters of LADRC and PID remain constant. Experimental results are shown in Figure9. From Figures 9(a), 9(c), and 9(e), we can observe that when the payload is increased, LADRC and PID can also make the end effector track the octagonal trajectory. However, LADRC has a more accurate tracking performance despite its position in x or y directions. Figures 9(b) and 9(d) show that the control of LADRC has smaller tracking errors and less tracking error fluctuations than PID.Figure 9
Experimental comparison of tracking performance between LADRC and PID. (a) Comparison of tracking trajectory in thex-direction. (b) Comparison of tracking errors in the x-direction. (c) Comparison of tracking trajectory in the y-direction. (d) Comparison of tracking errors in the y-direction. (e) Tracking comparison of regular octagonal trajectory.
(a)
(b)
(c)
(d)
(e)


## 5.2. Experiment of Sudden Disturbances

When the end effector has no payload, we add a step signal of 20 V to the several angles and edges of the octagonal trajectory. The experimental results are plotted in Figures10 and 11. From Figure 10, we can observe that despite the position of the disturbances in the octagonal trajectory, the LADRC method has a smaller maximum trajectory deviation, faster response, smaller tracking fluctuations, and shorter tracking error convergence times when compared to PID. The tracking errors depicted in Figure 11 also show that when the disturbances occur suddenly, the LADRC method has smaller maximum tracking errors, less tracking deviation fluctuation, and shorter deviation convergence times. The experimental results verify that the control performance and robustness of LADRC are better than PID.Figure 10
Experimental comparison of the tracking performance between LADRC and PID. during sudden disturbance. (a) Comparison of trajectory tracking with disturbance at the angles of octagonal in thex-direction. (b) Comparison of trajectory tracking with disturbance at the edges of octagonal in the x-direction. (c) Comparison of trajectory tracking with disturbance at the angles of octagonal in the y-direction. (d) Comparison of trajectory tracking with disturbance at the edges of octagonal in the y-direction (e) Comparison of trajectory tracking with disturbance at the angles of octagonal. (f) Comparison of trajectory tracking with disturbance at the edges of octagonal.
(a)
(b)
(c)
(d)
(e)
(f)Figure 11
Experimental comparison of tracking errors between LADRC and PID during sudden disturbance. (a) Comparison of tracking errors with disturbance at the angles of octagonal in thex-direction. (b) Comparison of tracking errors with disturbance at the edges of octagonal in the x-direction. (c) Comparison of tracking errors with disturbance at the angles of octagonal in the y-direction. (d) Comparison of tracking errors with disturbance at the edges of octagonal in the y-direction.
(a)
(b)
(c)
(d)


## 6. Conclusions

In this study, the decoupled mathematical model of the 2-DOF manipulator is established and the desired trajectory is planned using the quintic polynomial method. LESO is adopted to estimate the total disturbances comprising dynamic decoupling, internal uncertainties, and external disturbances. Furthermore, a linear state error feedback control law is designed to compensate for the total disturbances and to establish the stability of the closed-loop system. In addition, the stability of the proposed LESO and the closed-loop system is analyzed. The relationship between the performance of the closed-loop system and the controller parameters is also demonstrated. Furthermore, the comparative simulations and experiments verify the better performance and stronger robustness of the proposed LADRC than that of PID.



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*Source: 6969207-2020-05-06.xml*