# Research on a LADRC Strategy for Trajectory Tracking Control of Delta High-Speed Parallel Robots

Authors: Cheng Liu; Yanming Cheng; Dejun Liu; Guohua Cao; Ilkyoo Lee
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In order to better track the planned trajectory of Delta high-speed parallel robot, this paper proposes a dynamics control strategy for Delta high-speed parallel robots based on the linear active disturbance rejection control (LADRC) strategy which realizes decoupling control through observing and compensating the coupling and internal and external disturbances between the three joints. Firstly, the structure and dynamics model of the Delta high-speed parallel robot are analyzed, respectively. Secondly, the control scheme of the Delta high-speed parallel robot dynamic LADRC strategy is constructed, and then, the system stability is analyzed. Taking a representative 8-shaped space helical variance trajectory as a given input of the system and a triangular wave as an external disturbance as given disturbance input of the system, simulations are carried out to demonstrate the effectiveness of the proposed LADRC strategy; results indicate that the system with the LADRC strategy has a good quick and precise real-time trajectory tracking and strong robustness.


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## Body

## 1. Introduction

The parallel robot has the advantages of high speed, lightweight, and strong flexibility, which is complementary to the serial robot. Thereby, an increasing number of researches were carried out by numerous research institutions at home and abroad [1, 2]. At present, the parallel robot has not been widely used inasmuch as it has a difficult design of the parallel mechanism, difficult kinematics solution, complex trajectory planning, difficult trajectory tracking control, and other problems [2–6]. In this paper, Delta parallel robot is chosen as a study object; the control of Delta parallel robot can be divided into two types: one is kinematics control, which ignores the centripetal force, Coriolis force, and all kinds of disturbances of the robot. It directly controls the rotation angle of the servo motor through the given rotation angle of trajectory planning, which is mainly used in the low-speed parallel robot; the other is dynamics control. Under the condition of high-speed motion, if the centripetal force, Coriolis force, and all kinds of disturbances are ignored, the operation accuracy of the robot will decrease, and the joints of the robot will vibrate. Therefore, the dynamics control method is designed to improve the dynamic response of the robot system and effectively control all kinds of forces and disturbances in high-speed motion, which realize that the control of the robot in high-speed motion accuracy is of great importance. The controls of three joints of Delta parallel robot are coupled with each other, and the control object has nonlinear characteristics. Therefore, its control has always been a difficult issue in the field of parallel robots; the quality of control strategy directly affects the quality of trajectory tracking, the speed, and operation accuracy of the parallel robot.The existing control strategies that have been applied to the parallel robot include PID control [7–11], calculated torque control [12–17], and sliding mode variable structure control [18–20]. These methods have higher requirements on the model of the control object, and the operating conditions and operating environment are deterministic. In order to improve the control performance, some scholars combined the intelligent control strategy with the traditional control strategy on control of the Delta parallel robot [21–28].It is generally accepted that classical PID control has good robustness and reliability, and it is easy to be realized. It plays an important role in single input and single output applications. The outputs of three joints of Delta parallel robot are coupled with each other, and the control is nonlinear. The classical PID control is difficult to ensure high-precision and high-speed trajectory tracking and has poor robustness. Therefore, scholars at home and abroad are committed to researching the improved PID control method.In literature [7–11], starting from the robot joint or joint drive servo system, PD control, PD + speed feed-forward, PD + position feed-forward, a nonlinear combination of PID parameters, online optimization of PID parameters, fuzzy self-tuning PID, and other controllers as well as various dynamic compensation controllers are used to enable the robot joints to track the given trajectory curve. The method of calculating torque control is to simplify the dynamic model of each joint and carry out decoupling control, which can achieve linear control of each joint. In literature [12–17], the dynamics control of the robot is realized based on the computational torque control method which achieves a good trajectory tracking control. In order to improve the tracking performance of the system, various improvements and compensations are made to predict or calculate the torque in the literature, and the robustness of the system is analyzed. In the nonlinear control system, the sliding mode variable structure control strategy is very popular with researchers, but the controller itself has the problems of chattering and easy to be disturbed. In literature [18–20], a new adaptive variable structure controller, a new smooth sliding mode control algorithm, and a combination of REF neural network and sliding mode control algorithm are used to solve the chattering problem of the traditional sliding mode control, with strong robustness. Since the advent of artificial intelligence control technology in the 1960s, an increasing number of control systems have been added to the elements of artificial intelligence. There are many active figures of artificial intelligence control technology in control of robot trajectory tracking of the robot. Experts and scholars have applied various neural networks, predictive control, fuzzy algorithm, expert system, and other technologies to control of robot trajectory tracking of the robot, which achieve good results. Moreover, the performance of the controller is improved by combining the artificial intelligence method with PID control, calculation torque method, and sliding mode variable structure control, such as the achievements in literature [11, 20, 27–29]. However, the aforementioned methods excessively depend on the model and are difficult to be realized due to the complexity of calculation. Consequently, these methods cannot meet the requirements of high-precision and high-speed control of high-speed parallel robot.In literature [30], the method of combining the Active Disturbance Rejection Controller (ADRC) with Generalized Proportional Integral (GPI) was adopted. ADRC has the advantage of being model independent, and Generalized Proportional Integral Observer (GPIO) was used instead of Extended State Observer (ESO), which has a better performance comparing with the calculated torque. However, the entire system has many parameters to be tuned, which can achieve good results at a specific load time. Nevertheless, the performance of the system decreases when the load changes, and the parameters cannot be optimized.In the last few years, Professor Gao Ziqiang proposed the linear active disturbance rejection control method [31] (LADRC) which mainly deals with the linearization of the Extended State Observer and the error feedback combined controller, which is convenient for stability analysis and parameter tuning by using the frequency domain method. The LADRC has been widely used in a wider range owing to its advantages of simple parameter tuning and strong anti-interference ability, and more achievements have been achieved in some literature [32–38]. Motivated by the aforementioned analysis, this paper proposes an LADRC strategy to apply to the dynamic stability control of Delta high-speed parallel robot for the first time. Simulation results demonstrate that the system with LADRC strategy has the performance of a good trajectory tracking and strong robustness.


## 2. Structure and Dynamics Modeling of Delta High-Speed Parallel Robots

### 2.1. Structure of Delta High-Speed Parallel Robots

In this paper, the structure sketch of Delta high-speed parallel robot is depicted in Figure1. A1A2A3 is a fixed platform, B1B2B3 is a mobile platform, Ai is a rotating joint, Bi and Ci are spherical joints, AiCi is an active arm, and CiBi is a driven arm, i = 1, 2, 3. O is the center of the fixed platform, and P is the center of the mobile platform. R is the outer circle radius of the fixed platform, and r is the outer circle radius of the mobile platform. Both A1A2A3 and B1B2B3 are regular triangles.Figure 1
Mechanism sketch of three-degree-of-freedom Delta high-speed parallel robots.


### 2.2. Dynamics Modeling of Delta High-Speed Parallel Robots

According to the literature [30], the dynamics equation of delta high-speed parallel robot can be expressed by(1)Mθθ¨+Cθ,θ˙θ˙+Gθ=τ.Add friction force and disturbance to formula (1), and formula (2) can be obtained as(2)Mθθ¨+Cθ,θ˙θ¨+Gθ+Fθ˙+τd=τ,where θ∈Rn is the rotation angle of the three joints of Delta high-speed parallel robot; Mθ∈Rnxn is inertia matrix of the three joints of Delta high-speed parallel robot; Cθ,θ˙ is centripetal force and Coriolis force; Gθ∈Rn is the gravity term; Fθ˙∈Rn is friction; τd is disturbance term; and τ is servo input.Formula (2) is transformed into formula (3), as(3)θ¨=−M−1Cθ,θ˙θ˙−M−1Gθ−M−1Fθ˙−M−1θτd+M−1τ.


## 2.1. Structure of Delta High-Speed Parallel Robots

In this paper, the structure sketch of Delta high-speed parallel robot is depicted in Figure1. A1A2A3 is a fixed platform, B1B2B3 is a mobile platform, Ai is a rotating joint, Bi and Ci are spherical joints, AiCi is an active arm, and CiBi is a driven arm, i = 1, 2, 3. O is the center of the fixed platform, and P is the center of the mobile platform. R is the outer circle radius of the fixed platform, and r is the outer circle radius of the mobile platform. Both A1A2A3 and B1B2B3 are regular triangles.Figure 1
Mechanism sketch of three-degree-of-freedom Delta high-speed parallel robots.


## 2.2. Dynamics Modeling of Delta High-Speed Parallel Robots

According to the literature [30], the dynamics equation of delta high-speed parallel robot can be expressed by(1)Mθθ¨+Cθ,θ˙θ˙+Gθ=τ.Add friction force and disturbance to formula (1), and formula (2) can be obtained as(2)Mθθ¨+Cθ,θ˙θ¨+Gθ+Fθ˙+τd=τ,where θ∈Rn is the rotation angle of the three joints of Delta high-speed parallel robot; Mθ∈Rnxn is inertia matrix of the three joints of Delta high-speed parallel robot; Cθ,θ˙ is centripetal force and Coriolis force; Gθ∈Rn is the gravity term; Fθ˙∈Rn is friction; τd is disturbance term; and τ is servo input.Formula (2) is transformed into formula (3), as(3)θ¨=−M−1Cθ,θ˙θ˙−M−1Gθ−M−1Fθ˙−M−1θτd+M−1τ.


## 3. Study on LADRC Strategy

### 3.1. Scheme of LADRC

The mechanical structure of the Delta high-speed parallel robot is complex, and the three joints are coupled with each other. The control structure is shown in Figure2. X1s,X2s, and X3s are inputs of Delta high-speed parallel robot. Y1s, Y2s, and Y3s are outputs of the Delta high-speed parallel robot. When one Xis is changed, at the same time, the other two Xis will follow it, and three Yis will change. Thereby, in order to realize the trajectory tracking control of Delta high-speed parallel robot, three joints need to be input at the same time and decoupled each other.Figure 2
Control system structure of Delta high-speed parallel robot.When the parallel robot is running, the decomposed load on each joint changes nonlinearly with the change of pose and angular acceleration, which cannot be expressed linearly. Therefore, PID control cannot meet the requirements of the high-speed and high-precision control system of Delta high-speed parallel robot. According to the motion requirements and nonlinear characteristics of Delta high-speed parallel robot, this paper adopts the LADRC strategy to control Delta high-speed parallel robot. An 8-shaped space helical variance trajectory as the input signal is used to verify the trajectory tracking of the control strategy.Figure3 shows the LADRC strategy. The LADRC strategy has strong robustness, in that many uncertain factors are omitted in dynamics modeling. In practical control, these are uncertain factors of control, while the LADRC strategy does not depend on the establishment of the dynamics model, which circumvents the error problem of the dynamics model of the parallel robot. Control idea is to control each joint separately; the coupling parameters between the joints, the omitted factors in modeling, and the disturbance in actual operation are all treated as the disturbance to compensate. Finally, the high-precision trajectory tracking control of Delta high-speed parallel robot is realized.Figure 3
Delta high-speed parallel robot based on LADRC.


### 3.2. Design of System Controller

The dynamics of Delta high-speed parallel robot are expressed by formula (3) which can be converted into(4)θ¨=−a1θ˙−a2θ+f0θ,θ˙,ω,t+bμ,where μ is the input of the system; θ is the output of the system; −a1θ˙−a2θ is known modeling dynamics of object; f0θ,θ˙,ω,t is unknown modeling dynamics of objects and sum of external disturbances; b is the uncertain control gain; and b0 is the approximate value of uncertain control gain b; letfθ,θ˙,ω,t=f⋅−a1θ˙−a2θ+f0θ,θ˙,ω,t+b−b0μ which is called “total disturbance.”Letx1=θ, x2=θ˙, and x3=f⋅. Formula (4) expresses state equation of the second-order objects which can be written as(5)x˙1=x2,x˙2=x3+b0μ,x˙3=h+f˙.The state equation of the LESO (Linear Extended State Observer) designed for formula (3) is expressed as follows:(6)e=z1−θ,z˙1=z2−β1f1e,z˙2=z3−β2f2e+b0μ,z˙3=−β3f3e,(7)μ=−z3+μ0b0,where L is the gain of LESO; L=β1β2β3T, fie=z1−θi=1,2,3, and β1, β2, and β3 are selected appropriate gains of LESO, respectively; LESO can track all variables including z1=θ, z2=θ˙, and z3=f⋅ in the system expressed through formula (6) in real time.Linear control law can be expressed by(8)u0=kpv−z1+kdv˙−z2.According to formula (6),(9)z1=β1s2+β2s+β3s3+β1s2+β2s+β3θ+b0ss3+β1s2+β2s+β3μ,z2=β2s+β3ss3+β1s2+β2s+β3θ+b0s+β1ss3+β1s2+β2s+β3μ,z3=β3s2s3+β1s2+β2s+β3θ−b0β3s3+β1s2+β2s+β3μ.According to formula (8),(10)μ0=1b0kpv−z1+kdv˙−z2−z3.Substitutez1,z2, and z3into formula (11) as follows:(11)μ=1b0s3+β1s2+β2s+β3s3+β1+kds2+β2+kp+kdβ1skp+kdsvkpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3s3+β1s2+β2s+β3.


#### 3.2.1. Stability Analysis

From equation (13), the single closed-loop structure diagram is shown in Figure 4.Figure 4
The single closed-loop structure diagram.One has(12)G1s=kp+kds,(13)G2s=s3+β1s2+β2s+β3s3+β1+kds2+β2+kp+kdβ1s,(14)Hs=kpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3s3+β1s2+β2s+β3.From the structure diagram in Figure4, the closed-loop transfer function can be obtained as(15)Gbs=G1sG2sGs/b01+G2sGs/b0Hs.(1) The Scenario of Precisely Known Controlled Object. It can be seen from the above analysis that dynamics model formula (3) of the parallel robot is established by omitting the moment of inertia and the friction between joints; thereby, the model is considered to be known.Theorem 1.
When the model of the controlled object is known accurately, the differential tracker does not affect the stability of the system but only affects the zero point of the system. Reasonable selection ofb0, kp, kd, β1, β2, and β3 can make the system stable.Proof.
If the object model is known precisely, its transfer function is(16)Gs=k0s2+a1s+a2.
Substituting formulae (14)–(18) into formula (17), one obtains(17)Gbs=kp+kdss3+β1s2+β2s+β3k0b0s3+β1+kds2+β2+kp+kdβ1ss2+a1s+a2+k0kpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3.
Closed-loop characteristic equation is presented as(18)Ds=b0s5+b0β1+kd+a1s4+b0β2+kp+kdβ1+a1β1+kd+a2s3+b0a1β2+kp+kdβ1+b0a2β1+kd+k0kpβ1+kdβ2+β3s2+b0a2β2+kp+kdβ1+k0kpβ2+kdβ3s+k0kpβ3.
The following constraint can be specified:(19)D0=b0,D1=b0β1+kd+a1,D2=b0β2+kp+kdβ1+a1β1+a1kd+a2,D3=b0a1β2+kp+kdβ1+b0a2β1+kd+k0kpβ1+kdβ2+β3,D4=b0a2β2+kp+kdβ1+k0kpβ2+kdβ3,D5=k0kpβ3.
From Routh criterion, the stability of the system can be obtained as(20)s5D0D2D4,s4D1D3D5,s3B31B32,s2B41B42,s1B51,s0B61,B31=D1D2−D0D3D1,B32=D1D4−D0D5D1,B41=B31D3−D1B32B31,B42=D5,B51=B41B32−B31D5B41,B61=D5.
(2) The Scenario of Unknown Parameters of the Controlled Object. When Delta high-speed parallel robot runs at a high speed, the coupling relationships between joints are significantly relevant to the effects of the moment of inertia and friction between joints. They will result in the system chatter in the process of operation and ultimately destroy the stability of the system and decrease trajectory tracking performance if these impact factors were not taken into account. Therefore, in the design of the LADRC strategy, these factors are taken into consideration as unknown model parameters of the system when carrying out modeling.
The stability of the system model is proved as follows. Let the nominal model of the object beGn, and then the actual object is Gs=Gn1+δGs; δG is the perturbation of nominal model and meets δGjω≤δGω¯;δGω¯ is bounded the uncertainty of multiplicative norm.
Owing to the closed-loop characteristic equation1+G2s1/b0GsHs=0, the following expression is obtained as(21)1+G2s1b0Gns1+δGHs=0.
According to robust stability criterion, for arbitraryω, the inequality satisfying formula (21) is as follows:(22)b0+G2sGnsHs+G2sGnsHsδGs=0,δGω¯<ΔGω=b0+G2sGnsHsG2sGnsHs.
The conditions given by formula (22) can stabilize the system.
It can be seen that selectingb0, kp, kd, β1, β2, and β3 properly can make the system stable, which can guarantee that the system has the stability and certain robustness.
It can be concluded from the above stability proof that the LADRC strategy not only realizes the dynamics control of the trajectory tracking of Delta high-speed parallel robot but also guarantees that the uncertain factors of the model will not affect the stability of the system at high-speed operation.


#### 3.2.2. Design of Delta High-Speed Parallel Robots

The controller structure of the LADRC strategy is shown in Figure5.Figure 5
Structure diagram of LADRC for the second-order system.As shown in Figure5, the LADRC strategy only needs to be set by setting up stability conditions of LESO which are β1>0, β2>0, and β3>0 and meet β1β2>β3.As depicted in Figure2, Delta high-speed parallel robot is composed of three inputs X1s, X2s, and X3s and three outputs Y1s, Y2s, and Y3s, which are coupled with each other and can be constructed as shown in the following formula:(23)Y1s=G11sX1s+G12sX2s+G13sX3s,Y2s=G21sX1s+G22sX2s+G23sX3s,Y3s=G31sX1s+G32sX2s+G33sX3s.According to formulas (6) and (23), the system equation of Delta high-speed parallel robot can be obtained as expressed in(24)θ¨1=f1θ1,θ˙1,μ1t,t+k1μ1,θ¨2=f2θ2,θ˙2,μ2t,t+k2μ2,θ¨3=f1θ3,θ˙3,μ3t,t+k3μ3,where f1θ1,θ˙1,μ1t,t, f2θ2,θ˙2,μ2t,t, and f3θ3,θ˙3,μ3t,t are equivalent comprehensive disturbance which are the comprehensive functions of the coupling term between the three axes, the uncertainty term in high-speed motion, and various disturbance terms in the field. The control structure block diagram is shown in Figure 6.Figure 6
Simulation structure block diagram.


## 3.1. Scheme of LADRC

The mechanical structure of the Delta high-speed parallel robot is complex, and the three joints are coupled with each other. The control structure is shown in Figure2. X1s,X2s, and X3s are inputs of Delta high-speed parallel robot. Y1s, Y2s, and Y3s are outputs of the Delta high-speed parallel robot. When one Xis is changed, at the same time, the other two Xis will follow it, and three Yis will change. Thereby, in order to realize the trajectory tracking control of Delta high-speed parallel robot, three joints need to be input at the same time and decoupled each other.Figure 2
Control system structure of Delta high-speed parallel robot.When the parallel robot is running, the decomposed load on each joint changes nonlinearly with the change of pose and angular acceleration, which cannot be expressed linearly. Therefore, PID control cannot meet the requirements of the high-speed and high-precision control system of Delta high-speed parallel robot. According to the motion requirements and nonlinear characteristics of Delta high-speed parallel robot, this paper adopts the LADRC strategy to control Delta high-speed parallel robot. An 8-shaped space helical variance trajectory as the input signal is used to verify the trajectory tracking of the control strategy.Figure3 shows the LADRC strategy. The LADRC strategy has strong robustness, in that many uncertain factors are omitted in dynamics modeling. In practical control, these are uncertain factors of control, while the LADRC strategy does not depend on the establishment of the dynamics model, which circumvents the error problem of the dynamics model of the parallel robot. Control idea is to control each joint separately; the coupling parameters between the joints, the omitted factors in modeling, and the disturbance in actual operation are all treated as the disturbance to compensate. Finally, the high-precision trajectory tracking control of Delta high-speed parallel robot is realized.Figure 3
Delta high-speed parallel robot based on LADRC.


## 3.2. Design of System Controller

The dynamics of Delta high-speed parallel robot are expressed by formula (3) which can be converted into(4)θ¨=−a1θ˙−a2θ+f0θ,θ˙,ω,t+bμ,where μ is the input of the system; θ is the output of the system; −a1θ˙−a2θ is known modeling dynamics of object; f0θ,θ˙,ω,t is unknown modeling dynamics of objects and sum of external disturbances; b is the uncertain control gain; and b0 is the approximate value of uncertain control gain b; letfθ,θ˙,ω,t=f⋅−a1θ˙−a2θ+f0θ,θ˙,ω,t+b−b0μ which is called “total disturbance.”Letx1=θ, x2=θ˙, and x3=f⋅. Formula (4) expresses state equation of the second-order objects which can be written as(5)x˙1=x2,x˙2=x3+b0μ,x˙3=h+f˙.The state equation of the LESO (Linear Extended State Observer) designed for formula (3) is expressed as follows:(6)e=z1−θ,z˙1=z2−β1f1e,z˙2=z3−β2f2e+b0μ,z˙3=−β3f3e,(7)μ=−z3+μ0b0,where L is the gain of LESO; L=β1β2β3T, fie=z1−θi=1,2,3, and β1, β2, and β3 are selected appropriate gains of LESO, respectively; LESO can track all variables including z1=θ, z2=θ˙, and z3=f⋅ in the system expressed through formula (6) in real time.Linear control law can be expressed by(8)u0=kpv−z1+kdv˙−z2.According to formula (6),(9)z1=β1s2+β2s+β3s3+β1s2+β2s+β3θ+b0ss3+β1s2+β2s+β3μ,z2=β2s+β3ss3+β1s2+β2s+β3θ+b0s+β1ss3+β1s2+β2s+β3μ,z3=β3s2s3+β1s2+β2s+β3θ−b0β3s3+β1s2+β2s+β3μ.According to formula (8),(10)μ0=1b0kpv−z1+kdv˙−z2−z3.Substitutez1,z2, and z3into formula (11) as follows:(11)μ=1b0s3+β1s2+β2s+β3s3+β1+kds2+β2+kp+kdβ1skp+kdsvkpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3s3+β1s2+β2s+β3.


### 3.2.1. Stability Analysis

From equation (13), the single closed-loop structure diagram is shown in Figure 4.Figure 4
The single closed-loop structure diagram.One has(12)G1s=kp+kds,(13)G2s=s3+β1s2+β2s+β3s3+β1+kds2+β2+kp+kdβ1s,(14)Hs=kpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3s3+β1s2+β2s+β3.From the structure diagram in Figure4, the closed-loop transfer function can be obtained as(15)Gbs=G1sG2sGs/b01+G2sGs/b0Hs.(1) The Scenario of Precisely Known Controlled Object. It can be seen from the above analysis that dynamics model formula (3) of the parallel robot is established by omitting the moment of inertia and the friction between joints; thereby, the model is considered to be known.Theorem 1.
When the model of the controlled object is known accurately, the differential tracker does not affect the stability of the system but only affects the zero point of the system. Reasonable selection ofb0, kp, kd, β1, β2, and β3 can make the system stable.Proof.
If the object model is known precisely, its transfer function is(16)Gs=k0s2+a1s+a2.
Substituting formulae (14)–(18) into formula (17), one obtains(17)Gbs=kp+kdss3+β1s2+β2s+β3k0b0s3+β1+kds2+β2+kp+kdβ1ss2+a1s+a2+k0kpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3.
Closed-loop characteristic equation is presented as(18)Ds=b0s5+b0β1+kd+a1s4+b0β2+kp+kdβ1+a1β1+kd+a2s3+b0a1β2+kp+kdβ1+b0a2β1+kd+k0kpβ1+kdβ2+β3s2+b0a2β2+kp+kdβ1+k0kpβ2+kdβ3s+k0kpβ3.
The following constraint can be specified:(19)D0=b0,D1=b0β1+kd+a1,D2=b0β2+kp+kdβ1+a1β1+a1kd+a2,D3=b0a1β2+kp+kdβ1+b0a2β1+kd+k0kpβ1+kdβ2+β3,D4=b0a2β2+kp+kdβ1+k0kpβ2+kdβ3,D5=k0kpβ3.
From Routh criterion, the stability of the system can be obtained as(20)s5D0D2D4,s4D1D3D5,s3B31B32,s2B41B42,s1B51,s0B61,B31=D1D2−D0D3D1,B32=D1D4−D0D5D1,B41=B31D3−D1B32B31,B42=D5,B51=B41B32−B31D5B41,B61=D5.
(2) The Scenario of Unknown Parameters of the Controlled Object. When Delta high-speed parallel robot runs at a high speed, the coupling relationships between joints are significantly relevant to the effects of the moment of inertia and friction between joints. They will result in the system chatter in the process of operation and ultimately destroy the stability of the system and decrease trajectory tracking performance if these impact factors were not taken into account. Therefore, in the design of the LADRC strategy, these factors are taken into consideration as unknown model parameters of the system when carrying out modeling.
The stability of the system model is proved as follows. Let the nominal model of the object beGn, and then the actual object is Gs=Gn1+δGs; δG is the perturbation of nominal model and meets δGjω≤δGω¯;δGω¯ is bounded the uncertainty of multiplicative norm.
Owing to the closed-loop characteristic equation1+G2s1/b0GsHs=0, the following expression is obtained as(21)1+G2s1b0Gns1+δGHs=0.
According to robust stability criterion, for arbitraryω, the inequality satisfying formula (21) is as follows:(22)b0+G2sGnsHs+G2sGnsHsδGs=0,δGω¯<ΔGω=b0+G2sGnsHsG2sGnsHs.
The conditions given by formula (22) can stabilize the system.
It can be seen that selectingb0, kp, kd, β1, β2, and β3 properly can make the system stable, which can guarantee that the system has the stability and certain robustness.
It can be concluded from the above stability proof that the LADRC strategy not only realizes the dynamics control of the trajectory tracking of Delta high-speed parallel robot but also guarantees that the uncertain factors of the model will not affect the stability of the system at high-speed operation.


### 3.2.2. Design of Delta High-Speed Parallel Robots

The controller structure of the LADRC strategy is shown in Figure5.Figure 5
Structure diagram of LADRC for the second-order system.As shown in Figure5, the LADRC strategy only needs to be set by setting up stability conditions of LESO which are β1>0, β2>0, and β3>0 and meet β1β2>β3.As depicted in Figure2, Delta high-speed parallel robot is composed of three inputs X1s, X2s, and X3s and three outputs Y1s, Y2s, and Y3s, which are coupled with each other and can be constructed as shown in the following formula:(23)Y1s=G11sX1s+G12sX2s+G13sX3s,Y2s=G21sX1s+G22sX2s+G23sX3s,Y3s=G31sX1s+G32sX2s+G33sX3s.According to formulas (6) and (23), the system equation of Delta high-speed parallel robot can be obtained as expressed in(24)θ¨1=f1θ1,θ˙1,μ1t,t+k1μ1,θ¨2=f2θ2,θ˙2,μ2t,t+k2μ2,θ¨3=f1θ3,θ˙3,μ3t,t+k3μ3,where f1θ1,θ˙1,μ1t,t, f2θ2,θ˙2,μ2t,t, and f3θ3,θ˙3,μ3t,t are equivalent comprehensive disturbance which are the comprehensive functions of the coupling term between the three axes, the uncertainty term in high-speed motion, and various disturbance terms in the field. The control structure block diagram is shown in Figure 6.Figure 6
Simulation structure block diagram.


## 3.2.1. Stability Analysis

From equation (13), the single closed-loop structure diagram is shown in Figure 4.Figure 4
The single closed-loop structure diagram.One has(12)G1s=kp+kds,(13)G2s=s3+β1s2+β2s+β3s3+β1+kds2+β2+kp+kdβ1s,(14)Hs=kpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3s3+β1s2+β2s+β3.From the structure diagram in Figure4, the closed-loop transfer function can be obtained as(15)Gbs=G1sG2sGs/b01+G2sGs/b0Hs.(1) The Scenario of Precisely Known Controlled Object. It can be seen from the above analysis that dynamics model formula (3) of the parallel robot is established by omitting the moment of inertia and the friction between joints; thereby, the model is considered to be known.Theorem 1.
When the model of the controlled object is known accurately, the differential tracker does not affect the stability of the system but only affects the zero point of the system. Reasonable selection ofb0, kp, kd, β1, β2, and β3 can make the system stable.Proof.
If the object model is known precisely, its transfer function is(16)Gs=k0s2+a1s+a2.
Substituting formulae (14)–(18) into formula (17), one obtains(17)Gbs=kp+kdss3+β1s2+β2s+β3k0b0s3+β1+kds2+β2+kp+kdβ1ss2+a1s+a2+k0kpβ1+kdβ2+β3s2+kpβ2+kdβ3s+kpβ3.
Closed-loop characteristic equation is presented as(18)Ds=b0s5+b0β1+kd+a1s4+b0β2+kp+kdβ1+a1β1+kd+a2s3+b0a1β2+kp+kdβ1+b0a2β1+kd+k0kpβ1+kdβ2+β3s2+b0a2β2+kp+kdβ1+k0kpβ2+kdβ3s+k0kpβ3.
The following constraint can be specified:(19)D0=b0,D1=b0β1+kd+a1,D2=b0β2+kp+kdβ1+a1β1+a1kd+a2,D3=b0a1β2+kp+kdβ1+b0a2β1+kd+k0kpβ1+kdβ2+β3,D4=b0a2β2+kp+kdβ1+k0kpβ2+kdβ3,D5=k0kpβ3.
From Routh criterion, the stability of the system can be obtained as(20)s5D0D2D4,s4D1D3D5,s3B31B32,s2B41B42,s1B51,s0B61,B31=D1D2−D0D3D1,B32=D1D4−D0D5D1,B41=B31D3−D1B32B31,B42=D5,B51=B41B32−B31D5B41,B61=D5.
(2) The Scenario of Unknown Parameters of the Controlled Object. When Delta high-speed parallel robot runs at a high speed, the coupling relationships between joints are significantly relevant to the effects of the moment of inertia and friction between joints. They will result in the system chatter in the process of operation and ultimately destroy the stability of the system and decrease trajectory tracking performance if these impact factors were not taken into account. Therefore, in the design of the LADRC strategy, these factors are taken into consideration as unknown model parameters of the system when carrying out modeling.
The stability of the system model is proved as follows. Let the nominal model of the object beGn, and then the actual object is Gs=Gn1+δGs; δG is the perturbation of nominal model and meets δGjω≤δGω¯;δGω¯ is bounded the uncertainty of multiplicative norm.
Owing to the closed-loop characteristic equation1+G2s1/b0GsHs=0, the following expression is obtained as(21)1+G2s1b0Gns1+δGHs=0.
According to robust stability criterion, for arbitraryω, the inequality satisfying formula (21) is as follows:(22)b0+G2sGnsHs+G2sGnsHsδGs=0,δGω¯<ΔGω=b0+G2sGnsHsG2sGnsHs.
The conditions given by formula (22) can stabilize the system.
It can be seen that selectingb0, kp, kd, β1, β2, and β3 properly can make the system stable, which can guarantee that the system has the stability and certain robustness.
It can be concluded from the above stability proof that the LADRC strategy not only realizes the dynamics control of the trajectory tracking of Delta high-speed parallel robot but also guarantees that the uncertain factors of the model will not affect the stability of the system at high-speed operation.


## 3.2.2. Design of Delta High-Speed Parallel Robots

The controller structure of the LADRC strategy is shown in Figure5.Figure 5
Structure diagram of LADRC for the second-order system.As shown in Figure5, the LADRC strategy only needs to be set by setting up stability conditions of LESO which are β1>0, β2>0, and β3>0 and meet β1β2>β3.As depicted in Figure2, Delta high-speed parallel robot is composed of three inputs X1s, X2s, and X3s and three outputs Y1s, Y2s, and Y3s, which are coupled with each other and can be constructed as shown in the following formula:(23)Y1s=G11sX1s+G12sX2s+G13sX3s,Y2s=G21sX1s+G22sX2s+G23sX3s,Y3s=G31sX1s+G32sX2s+G33sX3s.According to formulas (6) and (23), the system equation of Delta high-speed parallel robot can be obtained as expressed in(24)θ¨1=f1θ1,θ˙1,μ1t,t+k1μ1,θ¨2=f2θ2,θ˙2,μ2t,t+k2μ2,θ¨3=f1θ3,θ˙3,μ3t,t+k3μ3,where f1θ1,θ˙1,μ1t,t, f2θ2,θ˙2,μ2t,t, and f3θ3,θ˙3,μ3t,t are equivalent comprehensive disturbance which are the comprehensive functions of the coupling term between the three axes, the uncertainty term in high-speed motion, and various disturbance terms in the field. The control structure block diagram is shown in Figure 6.Figure 6
Simulation structure block diagram.


## 4. Simulation and Results Analysis

In this section, to verify the performance of the controller based on the proposed LADRC strategy, the primary parameters of Delta high-speed parallel robot are summarized in Table1.Table 1
Primary parameters of Delta high-speed parallel robot for simulation.
ParametersValueRated output power of servo motor750WRated speed of servo motor3000rpmRated torque of servo motor2.38M⋅mMoment of inertia of servo motor1.59×10−4kg/m2Reduction ratio20:1Mass of active arm2.35kgMass of driven arm0.9kgLength of active arm400mmLength of driven arm1000mmAn 8-shaped trajectory that is a representative space helical variance trajectory is taken as inputs of the system, which is expressed as(25)x1=t,x2=800sin0.3t+1360∗π,x3=800sin0.6t∗π.The pictures of different views corresponding to formula (25) are depicted in Figure 7.Figure 7
8-shaped trajectory. (a) The space trajectory in 3D coordination system. (b) The projection on thex-y plane. (c) The projection on the y-z plane. (d) The projection on the x-z plane.
(a)(b)(c)(d)For obtaining trajectories under some conditions, the first step is to verify the performance of the system when the disturbance is not introduced and then verify the performance of the system when the disturbance is introduced. To better estimate, the control effect, the angle limit, time (speed) limit, and space limit of the robot are removed.Case 1.
The case of 8-shaped trajectory input without disturbance.
According tozii=1,2,3 in formula fd9(9), stability conditions of LESO must meetβ1>0, β2>0, β3>0, and β1β2>β3. On this basis, it is concluded that coefficientsβ1=1, β2=65000, and β3=600 through a large number of simulation experiments. The trajectory of Delta high-speed parallel robot is set as the 8-shaped trajectory. The given trajectories of joint 1, joint 2, and joint 3 are set according to formula (25). According to Figure 6, the simulation curve of trajectory tracking control is obtained in Figure 8, the blue circle denotes the given circular trajectory, and the red curve denotes the trajectory controlled by the LADRC strategy.
In order to verify the robustness of the controller against disturbance, a triangular wave external disturbance with an amplitude of 50 mm and a period of 0.2 Hz is added after the Delta high-speed parallel robot runs to 5 s. The disturbance signal is shown in Figure9.
Figure10 shows the tracking trajectories of three joints in the scenario of only PID control without the LADRC strategy.
The tracking trajectory of only using PID control under the condition of without the LADRC strategy is obtained in Figure11.
From Figures10 and 11, it is very obvious that only using PID control completely cannot control the tracking trajectory. Therefore, it is very necessary to introduce the LADRC strategy to resolve the antidisturbance of PID control. Figures 12 and 13 show the effects of tracking trajectory control when the LADRC strategy is used.
In Figure13, the fluctuation of the three-dimensional trajectory is not obvious, and the 8-shaped trajectory is slightly vibrated. It is verified that the LADRC strategy can realize the decoupling control of three joints and has strong robustness under the disturbance of the triangular wave for a representative space helical variance trajectory of 8-shaped trajectory input.
Simulation results show that the LADRC strategy can realize the decoupling control of three joints and realize the quick and precise real-time tracking of the given trajectory and has good robustness.Figure 8
Trajectories without disturbance.Figure 9
Disturbance signal.Figure 10
Trajectories of 3 joints without the LADRC strategy. (a) Trajectories of joint 1. (b) Trajectories of joint 2. (c) Trajectories of joint 3.
(a)(b)(c)Figure 11
The tracking trajectory of without the LADRC strategy.Figure 12
Trajectories of 3 joints with the LADRC strategy. (a) Trajectories of joint 1. (b) Trajectories of joint 2. (c) Trajectories of joint 3.
(a)(b)(c)Figure 13
The tracking trajectory with the LADRC strategy.


## 5. Conclusion

Aiming at the motion requirements, nonlinear characteristics, and the omission of the relevant factors for modeling Delta high-speed parallel robot, this paper proposes an LADRC strategy to be applied to the dynamic control of Delta high-speed parallel robot. Through the stability analysis, the LADRC strategy can realize the dynamic control of the trajectory tracking of Delta high-speed parallel robot, and when the robot is running at high speed, the stability of the model will not be affected by the uncertain factors of the system.In order to verify the proposed LADRC strategy, the circular trajectory, a representative 8-shaped space helical variance trajectory, and a triangular wave external disturbance with large amplitude are input, respectively. The PID method and the linear ADRC control strategy are used to simulate, respectively. The simulation and comparative analysis show that the LADRC strategy is applied to the Delta high-speed parallel robot control which has a good quick and precise real-time trajectory tracking and strong robustness.



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