# Trajectory Tracking Control of Robot Manipulators Based on U-Model

Authors: Xianghua Ma; Yang Zhao; Yiqun Di
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A new trajectory tracking control method based on the U-model is proposed to improve the trajectory tacking speed of robot manipulators. The U-model method is introduced to relieve the requirement of the dynamic mathematical model and make the design of trajectory tracking controller of robot manipulators simpler. To further improve the trajectory tacking speed, an improved iterative learning control algorithm is used to suppress the influence of the initial state error with less computation time. Experimental results show that the proposed control method is effective and practical for the trajectory tracking control of robot manipulators, especially with a high real-time requirement.


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## Body

## 1. Introduction

Nowadays, most robot manipulators need to possess the capacity of accurate and fast trajectory tracking. Trajectory tracking control is a key issue in the field of robot manipulator motion planning [1–3]. It aims to enable the joints or links of the robot manipulator to track the desired trajectory with ideal dynamic quality or to stabilize them in the specified position [4]. At present, there are two main ways to analyze and design the trajectory tracking control for robot manipulator [5]. The first one is with respect to a classical linear control method [6, 7]. This way is easy, and some classical control theory and methods can be used directly in some nonlinear properties which are approximated into a linear relationship or neglected. However, the trajectory tracking control of robot manipulators is a multi-input and multioutput (MIMO) control problem, and the motion equation of a robot manipulator with multiple degrees of freedom is highly coupled and complex [8]. If each joint is supposed to be independent and the inertia of each joint be constant, it may lead to inconsistent system damping and other unexpected situations in the entire workspace [9]. Therefore, there is the second way, which focuses on the nonlinear control method without the requirements of the above assumptions [10–12]. However, this nonlinear method also has some limits; for example, it needs to get the exact data of the structure and precise parameters when modelling the manipulator so as to correctly describe all these uncertainties, for example, the interaction among robot joints and the change in centroid.The initial state error has a great influence on the trajectory tracking speed. Iterative learning control method is used to rapidly suppress the influence of the initial state error. Hongfeng et al. [13] proposed an iterative learning trajectory tracking control algorithm for discrete nonlinear systems in which the initial states are not strictly identical to given expected values in track tracking. Jin [14] proposed a new iterative learning control method which can track the nonrepetitive motion trajectory without assuming that the initial state error of the system is unchanged. Hui et al. [15] proposed a data-driven iterative learning control method based on an extended state observer. The concept of ESO is introduced into the iteration direction. The random initial state and the disturbance are taken as the extended state of a whole. Theoretical analysis shows that the method is robust to initial displacements and perturbations of iterative changes.A reasonable learning gain matrix can improve the trajectory tracking speed. In order to obtain the optimal learning gain, many researches have been done including variable exponential gain method [16], fuzzy PID method [17], and the method combining neural network controller and compensation controller [18]. However, the complex control law and the learning law will bring a huge amount of computation time and affect the trajectory tracking speed.According to the above analysis, both the classical linear control method and the nonlinear control method have their advantages and also some limitations, respectively. If there is a way to combine the benefits of the classical linear and the nonlinear control methods, not only the analysis and design of trajectory tracking control can be easier but also the speed and accuracy of trajectory tracking control may be improved as well. U-Model belongs to a generic systematic approach, which converts the nonlinear polynomial model into a controller output-based time-varying polynomial model [19–23]. Inspired by this idea, the U-model theory is introduced into the trajectory tracking control, which can help understand the inner structure and parameters of the robot manipulator better and reduce the structure complexity of control system. Iterative learning control method is used with a new iterative learning control law, which can greatly decrease the computation time and further improve the trajectory tracking speed.The remainder of the paper is organized as follows. Section2 describes trajectory tracking U-control system in detail. Section 3 analyzes the asymptotic convergence of the control law and the stability of the system. In Section 4, simulation experiment is carried out to prove the effectiveness of the proposed method. Conclusions are given in Section 5.


## 2. Trajectory Tracking U-Control System

In this paper, a trajectory tracking U-control method is proposed, and the control system mainly includes robot manipulator and a U-controller, as shown in Figure1. The U-controller is composed with an iterative learning controller (ILC for abbreviation) and the U-model of a robot manipulator, where Gc is the mathematical model of the linear controller, Gp is the mathematical model of the robot manipulator, and Gp−1 is the dynamic inverse of robot manipulator model based on U-model theory. The desired output trajectory is denoted as ydt, and the system output is denoted as ykt at time t∈R+. k is the number of iterations.Figure 1
U-Control system framework.


### 2.1. Realization of U-Controller

Consider a robot manipulator withn DOF; its general dynamic model can be represented as(1)Mqq¨+Cq,q˙q˙+Gq+ua=u,where qn×1=q1,q2,…,qnT∈Rn is the angular displacement vector; q˙n×1 and q¨n×1 are angular velocity and angular acceleration vectors of each joint, respectively; u∈Rn×1 is the torque vector; Mq∈Rn×n represents the inertia term; Cq,q˙∈Rn×n is the Coriolis and centrifugal terms; Gq∈Rn is the gravitational term; and ua∈Rn×n is the unknown disturbance term.Based on the U-model theory, the mathematical modelGp of the robot manipulator (1) can be converted into the following U-model expression:(2)y¨t=−M−1ytCyt,y˙ty˙t+Gyt,y˙t+M−1ytut−M−1ytua,where yt=qt.In order to consistent with the U-model theoretical expression, equation (2) can be transformed as follows:(3)vkt=α0t+α1tukt+Et,where(4)vkt=y¨t,α0t=−M−1ytCyt,y˙ty˙t+Gyt,y˙t,α1t=M−1yt,Et=−M−1ytua.From the transformation of equation (2) to equation (3), it can be seen that the realization of the U-model is different from the general approximate linear transformation and the properties of the original robot manipulator dynamic model will not be changed.As seen from Figure1, the U-controller consists of ILC and the dynamic inverse controller Gp−1 based on the U-model. After the U-model expression Gp of the manipulator dynamic model (3) is obtained, the U-controller can be designed based on the U-model theory. The output of the ILC is shown as follows:(5)vkt=vk−1t+Gcet,where trajectory error et=ydt−ykt.According to the definition of U-model theory, in this trajectory tracking U-control system, the general mathematical express ofGp can be obtained by(6)vkMt=∑j=0MαjtukjNt+Et,M>N,where vkMt and ukNt are the Mth and Nth orders of derivatives of vkt and ukt, respectively. αjt is a time-varying coefficient of ukN−1t,…,ukt and vkM−1t,…,vkt.By solving the root of equation (6), the output ukt of the U-controller can be obtained and expressed as(7)ukt=FvkMt−∑j=0MαjtukjNt−Et,where F∗ is a root-solving algorithm. In order to improve the convergence speed of iterative algorithms, based on the Newton–Raphson iterative algorithm, the model of the U-controller is obtained as shown in equation (8), which can optimize the output of ILC so as to reduce the number of iterations and speed up the tracking convergence.The mathematical model of the U-controller can be expressed as(8)ukt=vkt−α0t−Etα1t.From the above equation, it can be seen that the output of the U-controller only needs to search a single value ofukt and obtain the output of ILC through fewer iterations, which greatly reduces the complexity of operation.


### 2.2. Iterative Learning Control Law

Iterative learning control algorithm is a common method for trajectory tracking control of manipulator. In order to improve the trajectory tracking speed of the manipulator, it is necessary to reduce the number of iterations required for the convergence while ensuring the convergence of the iterative learning control law. Therefore, both the initial state error and the convergence speed of the error are considered when the iterative learning control law is designed.Letxkt=qq˙T, the robot manipulator (1) can be rewritten into the following expression (9), and the displacement and velocity of each joint of the manipulator are used as the system output:(9)x˙kt=q˙q¨=fxkt,t+Btukt+Wkt,ykt=Ctxkt+Vkt,where Wkt and Vkt are state disturbance and output disturbance, respectively. k is the number of iterations.(10)fxkt,t=q˙−M−1Cq,q˙q˙+Gq,Bt=0M−1q,Wkt=0−M−1qua,Ct=I.It is assumed that the above robot manipulator system satisfies the following conditions.Assumption 1.
f
x
k
t
,
t satisfies the Lipschitz condition; that is, there exists a constant Lf Lf>0 that satisfies the following:(11)fxk+1t,t−fxkt,t≤Lfxk+1t−xkt.Assumption 2.
The adjacent state disturbance variation and the output disturbance variation are bounded, expressed as follows:(12)Wk+1t−Wkt≤bW,Vk+1t−Vkt≤bV.Assumption 3.
B
t and Ct are bounded.Assumption 4.
Expected trajectoryydt is continuous for all t∈0,T.
It is assumed that the system has a random initial error, which is denoted asekt=ydt−ykt, and the initial state of the kth iteration is xk0. The control law is as follows:(13)vkt=vk−1t+Γekt+Γe˙kt+ϕktXk0,t∈0,T,where Γ is the constant gain matrix and Gc=Γekt+e˙kt+ϕktXk0.
The learning law for the initial state is(14)ϕt=2akh1−akht,t∈0,hak,0,t∈hak,T,a>1,0<h<T,(15)Xk0=B0Γek0+xk0−xk+10,where a>1, 0<h<T.
From equation (14), the initial state error can only affect the trajectory tracking speed within the time interval 0,h/ak. After t=h/ak, the initial state error will be 0. With the increase in the number of iterations k, the time instant h/ak will tend to 0 quickly, which means the time the initial state error affects the trajectory tracking speed will become very short as well. It means that the trajectory tracking speed can be improved by the use of the proposed iterative learning control law.


## 2.1. Realization of U-Controller

Consider a robot manipulator withn DOF; its general dynamic model can be represented as(1)Mqq¨+Cq,q˙q˙+Gq+ua=u,where qn×1=q1,q2,…,qnT∈Rn is the angular displacement vector; q˙n×1 and q¨n×1 are angular velocity and angular acceleration vectors of each joint, respectively; u∈Rn×1 is the torque vector; Mq∈Rn×n represents the inertia term; Cq,q˙∈Rn×n is the Coriolis and centrifugal terms; Gq∈Rn is the gravitational term; and ua∈Rn×n is the unknown disturbance term.Based on the U-model theory, the mathematical modelGp of the robot manipulator (1) can be converted into the following U-model expression:(2)y¨t=−M−1ytCyt,y˙ty˙t+Gyt,y˙t+M−1ytut−M−1ytua,where yt=qt.In order to consistent with the U-model theoretical expression, equation (2) can be transformed as follows:(3)vkt=α0t+α1tukt+Et,where(4)vkt=y¨t,α0t=−M−1ytCyt,y˙ty˙t+Gyt,y˙t,α1t=M−1yt,Et=−M−1ytua.From the transformation of equation (2) to equation (3), it can be seen that the realization of the U-model is different from the general approximate linear transformation and the properties of the original robot manipulator dynamic model will not be changed.As seen from Figure1, the U-controller consists of ILC and the dynamic inverse controller Gp−1 based on the U-model. After the U-model expression Gp of the manipulator dynamic model (3) is obtained, the U-controller can be designed based on the U-model theory. The output of the ILC is shown as follows:(5)vkt=vk−1t+Gcet,where trajectory error et=ydt−ykt.According to the definition of U-model theory, in this trajectory tracking U-control system, the general mathematical express ofGp can be obtained by(6)vkMt=∑j=0MαjtukjNt+Et,M>N,where vkMt and ukNt are the Mth and Nth orders of derivatives of vkt and ukt, respectively. αjt is a time-varying coefficient of ukN−1t,…,ukt and vkM−1t,…,vkt.By solving the root of equation (6), the output ukt of the U-controller can be obtained and expressed as(7)ukt=FvkMt−∑j=0MαjtukjNt−Et,where F∗ is a root-solving algorithm. In order to improve the convergence speed of iterative algorithms, based on the Newton–Raphson iterative algorithm, the model of the U-controller is obtained as shown in equation (8), which can optimize the output of ILC so as to reduce the number of iterations and speed up the tracking convergence.The mathematical model of the U-controller can be expressed as(8)ukt=vkt−α0t−Etα1t.From the above equation, it can be seen that the output of the U-controller only needs to search a single value ofukt and obtain the output of ILC through fewer iterations, which greatly reduces the complexity of operation.


## 2.2. Iterative Learning Control Law

Iterative learning control algorithm is a common method for trajectory tracking control of manipulator. In order to improve the trajectory tracking speed of the manipulator, it is necessary to reduce the number of iterations required for the convergence while ensuring the convergence of the iterative learning control law. Therefore, both the initial state error and the convergence speed of the error are considered when the iterative learning control law is designed.Letxkt=qq˙T, the robot manipulator (1) can be rewritten into the following expression (9), and the displacement and velocity of each joint of the manipulator are used as the system output:(9)x˙kt=q˙q¨=fxkt,t+Btukt+Wkt,ykt=Ctxkt+Vkt,where Wkt and Vkt are state disturbance and output disturbance, respectively. k is the number of iterations.(10)fxkt,t=q˙−M−1Cq,q˙q˙+Gq,Bt=0M−1q,Wkt=0−M−1qua,Ct=I.It is assumed that the above robot manipulator system satisfies the following conditions.Assumption 1.
f
x
k
t
,
t satisfies the Lipschitz condition; that is, there exists a constant Lf Lf>0 that satisfies the following:(11)fxk+1t,t−fxkt,t≤Lfxk+1t−xkt.Assumption 2.
The adjacent state disturbance variation and the output disturbance variation are bounded, expressed as follows:(12)Wk+1t−Wkt≤bW,Vk+1t−Vkt≤bV.Assumption 3.
B
t and Ct are bounded.Assumption 4.
Expected trajectoryydt is continuous for all t∈0,T.
It is assumed that the system has a random initial error, which is denoted asekt=ydt−ykt, and the initial state of the kth iteration is xk0. The control law is as follows:(13)vkt=vk−1t+Γekt+Γe˙kt+ϕktXk0,t∈0,T,where Γ is the constant gain matrix and Gc=Γekt+e˙kt+ϕktXk0.
The learning law for the initial state is(14)ϕt=2akh1−akht,t∈0,hak,0,t∈hak,T,a>1,0<h<T,(15)Xk0=B0Γek0+xk0−xk+10,where a>1, 0<h<T.
From equation (14), the initial state error can only affect the trajectory tracking speed within the time interval 0,h/ak. After t=h/ak, the initial state error will be 0. With the increase in the number of iterations k, the time instant h/ak will tend to 0 quickly, which means the time the initial state error affects the trajectory tracking speed will become very short as well. It means that the trajectory tracking speed can be improved by the use of the proposed iterative learning control law.


## 3. Convergence Analysis

The stability of the trajectory tracking system requires that the actual output trajectory of the system converges to the desired trajectory under the action of the proposed U-controller; that is, the error between expected and actual trajectory will converge to 0 or within a boundary finally.Lemma 1.
For the robot manipulator described in equation (9), if it satisfies the following conditions,(16)ρ=supt∈0,T1−M−1qG<1,then the tracking error of the system will eventually converge to a boundary.Proof.
From equation (9), we can get(17)xk+1t−xkt=∫0tfxk+1s,s−fxks,sds+∫0tBsuk+1s−uksds+∫0tWk+1s−Wksds+xk+10−xk0.
According to equation (13), there is(18)∫0tBsvk+1s−vksds=∫0tBsΓeks+Γe˙ks+ϕksXk0ds=∫0tBsΓeksds+BtΓekt−∫0teksB˙sΓ+BsΓ˙ds+Xk0∫0tBsϕks−1ds+xk0−xk+10.
Then, equation (17) can turn into(19)xk+1t−xkt=∫0tfxk+1s,s−fxks,sds+∫0tWk+1s−Wksds+∫0tBsΓeksds+BtΓekt−∫0tektB˙sΓ+BsΓ˙ds+Xk0∫0tBsϕks−1ds.
According to the tracking error definitionekt=ydt−ykt,(20)ek+1t−ekt=ydt−yk+1t−ydt+ykt=−Ctxk+1t−xkt−Vk+1t−Vkt.
Substituting equation (19) into equation (20),(21)ek+1t=ekI−CBΓ−Ct∫0tfxk+1t,t−fxkt,tds+∫0tBsΓeksds−∫0teksB˙sΓ+BsΓ˙ds−Ct∫0tWk+1s−Wksds−CtXk0∫0tBsϕks−1ds−Vk+1t−Vkt.
According to equation (14), when t∈h/ak,T, we can get(22)∫0tBtϕkt−1ds=0.
According to Bellman–Gronwall lemma, equation (19), and assumptions (1) and (2), we have(23)xk+1t−xkt≤∫0tLfxk+1s−xksds+bWs0t+∫0tBsΓeksds+BtΓekt+∫0tB˙sΓ+BsΓ˙eksds.
Multiply both sides of equation (23) by e−λt, where λ>0, and take the maximum value of the right-hand side, and assume that t=tmax∈0,T:(24)xk+1t−xktλ≤Lfλxk+1t−xktλ+bWt+1+λλsuptBΓektλ+suptB˙Γ+BΓ˙λektλt=tmax.
LetLf<λ, and equation (24) can be simplified as(25)xk+1t−xktλ≤λbWtλ−Lf+1+λsuptBΓ+suptB˙Γ+BΓ˙λ−Lfektλ.
In the same way, equation (21) can turn into(26)ek+1tλ≤ektλρ+suptCtLfλxk+1t−xktλ+suptCtsuptBΓ+supB˙Γ+BΓλektλ+suptCtbWt+bV.
Substituting equation (25) into (26),(27)ek+1tλ≤Pektλ+Q,where(28)P=ρ+suptCtLf+1suptBΓ+suptB˙Γ+BΓ˙λ−Lf,Q=λsuptCtbWtλ−Lf+bv.
When the value ofλ is big enough, according to equation (16), we can get P≈ρ<1. When λ is chosen, Q becomes a constant. Further deform equation (27) into(29)ek+1tλ≤ρkektλ−Q1−M+Q1−M.
Therefore, we can have(30)ektλ≤ρk−1ek−1tλ−Q1−M+Q1−M.
With the number of iterationsk⟶∞, we have ρk−1⟶0ρ<1 and the following inequality:(31)limk⟶∞ektλ≤λsuptCtbWt1−ρλ−Lf+bV1−ρ.
From equation (31), it can be seen that, after time t=h/ak, the initial state error no longer affects the tracking error, and the tracking error is only determined by the degree of inaccuracy of the system model parameters and other external disturbances. For the value of the right part of inequality is constant, the tracking error of the system will eventually converge to a boundary.


## 4. Simulations

A 2-DOF planar robot manipulator is used to prove the effectiveness of the proposed control method. As shown in Figure2, the parameters are as follows: link 1: length l1 and mass m1, the distance between the center of mass of link 1 and base joint is lc1, and the inertia of link 1 is I1. Link 2: length l2 and mass m2; the distance between the center of mass of link 2 and joint 1 is lc2, and the inertia of link 2 is I2.Figure 2
2-DOF robot manipulator.The robot manipulator is modeled by the Lagrange method. The parameters of inertia matrixMq2×2, Coriolis and centrifugal matrix Cq,q˙2×2, and gravity term matrix Gq2×1 are set as follows:(32)M=mij2×2,m11=m1lc12+m2l12+lc22+2l1l2cosq2+I1+I2,m12=m21=m2lc22+l1lc2cosq2+l2,m22=m2lc22+I2,C=cij2×2,c11=hq˙2,c12=hq˙1+hq˙2,c21=−hq˙1,c22=0,h=−m2l1lc2sinq2,G=G1,G2T,G1=m1lc1+m2l1gcosq1+m2lc2gcosq1+q2,G2=m2lc2gcosq1+q2.The robot system parameters are(33)m1=m2=1,l1=l2=0.5,lc1=lc2=0.25,I1=I2=0.1,g=9.81.Set the expected trajectory of the system as(34)y1d=sin3t,y2d=cos3t.The constant gain matrix of iterative learning control law is set as(35)Γ=20000200.The remaining simulation parameters are initial inputu0t=00, h=0.1, and a=2. The initial state of the system is randomly generated by the random function.In order to illustrate the valid of the proposed U-controller, two experiments are carried out to show the difference of displacement errors rates of the two joints of the robot manipulator with and without U-model, respectively, and the results are shown in Figures3 and 4. x axle represents the number of iterations, and y axle represents the displacement errors of joint 1 and joint 2, respectively.Figure 3
Error convergence curve with U-model.Figure 4
Error convergence curve without U-model.From Figure3, we can see that when the U-model method is used, only one iteration is needed for each joint to make the displacement error converge to a boundary. However, at least two iterations for each joint are needed without the use of U-model; from the Figure 4, we can see that the angular displacement errors of joint 1 and joint 2 converge to 0 after three iterations. It means that the U-model control framework can reduce the number of iterations and effectively improve the tracking speed. However, there are displacement errors caused by the U-model, but the loss of tracking accuracy can be acceptable for a robot manipulator, especially when it is used in the fields with high real-time requirement.In order to show the valid of the proposed ILC law, two experiments are carried out using the U-controller with different ILC laws. The first ILC law is designed in this paper, and the other is the ILC only considering the initial state error. The number of iterationsk is 5. The trajectory tracking curves are shown in Figures 5 and 6. The blue solid line stands for the expected trajectory, and the red dotted line stands for the actual trajectory of the robot manipulator.Figure 5
Trajectory tracking result with the proposed ILC law with 5 iterations.
(a)
(b)Figure 6
Trajectory tracking result with the other ILC law with 5 iterations.
(a)
(b)From Figure5, we can see that the actual output trajectory has totally tracked the desired trajectory after 2.5 seconds, while in Figure 6, it takes about 4.5 seconds for the robot manipulator to track the desired trajectory. By the proposed ILC law, the speed of trajectory tracking is improved more obviously than other ILC law, which does not consider the convergence speed of the initial state error.With the increasingk, the benefit of the proposed ILC law will be more obvious. We set k=10 and repeat the above experiment with the tracking results, as shown in Figures 7 and 8. In Figure 7, by using the proposed method, the actual trajectory output is totally tracking the desired trajectory even from the beginning, while in Figure 8, there is still tracking error until 0.5 seconds later.Figure 7
Trajectory tracking result with the proposed ILC law with 10 iterations.
(a)
(b)Figure 8
Trajectory tracking result with the other ILC law with 10 iterations.
(a)
(b)


## 5. Conclusions

The proposed trajectory tracking U-control method has a simple structure and does not need to change the original nonlinear characteristics of the manipulator system. The iterative learning algorithm based on the initial state errors and the speed of error convergence is designed to effectively suppress the influence made by the initial errors and fasten the trajectory tracking. The U-model control framework can reduce the number of iterations of ILC law and further improve the speed of the trajectory tracking. The proposed trajectory tracking the U-control method is particularly suitable for real-time applications.



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