# Fractional-Order Active Disturbance Rejection Controller for Motion Control of a Novel 6-DOF Parallel Robot

Authors: Xinxin Shi; Jiacai Huang; Fangzheng Gao
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

A novel 6-degree-of-freedom (6-DOF) parallel robot driven by six novel linear motors is designed and controlled in this paper. Detailed structures of linear motors are illustrated. A control strategy based on kinematics of the 6-DOF parallel robot is used, and six linear motors are controlled to track their own desired trajectories under a designed fractional-order active disturbance rejection controller (FOADRC). Compared with the normal ADRC, two desired trajectories and three different working situations of a linear motor are simulated to show good performances of the FOADRC. Experimental results show that six linear motors can track their own desired trajectories accurately under payloads and disturbances, and the novel 6-DOF parallel robot can be controlled well.


---

## Body

## 1. Introduction

Industrial robots have two main types, i.e., serial and parallel robots. Each type of robot has its own advantages and disadvantages. Advantages of serial robots are simple structures, convenient control, and large workspace. Compared with serial counterparts, parallel robots have advantages of large loading capacity, high-speed and high-precision motions, and strong stiffness [1, 2]. With these superior characteristics, parallel robots have many industrial applications nowadays, such as pick-and-place [3], docking simulator [4], and medical surgery [5].Besides their own mechanical structures, the joint-driven ways are the very important aspects of parallel robots. There are different kinds of actuators to drive joints in parallel robots, such as cables [6], pneumatic or hydraulic cylinders [7], and rotary motors [8]. In this paper, a novel 6-DOF robot, which is actually a manipulator, was designed and driven by self-made linear motors. There is no need to use additional mechanisms such as ball screws to realize linear motions, and therefore, the whole system structure can be simplified.Control technologies of parallel robots are also very important, especially for the direct-drive actuators since uncertain system disturbances will act on them directly [9, 10]. Control methods of parallel robots can be classified based on kinematics and dynamics [11, 12]. Dynamics-based control methods require accurate dynamical models of parallel robots. In practice, there is inevitably an error between the established dynamical model and the real system. This error will lead to a decrease in the control performance [13, 14]. In addition, parameter values of dynamical models of parallel robots are not fixed, and they are constantly updated with movements of robots, which brings heavy calculations and higher requirements on the computing power of the controller hardware.Kinematics-based control methods assume that actuators of parallel robots are independent of each other [15]. By controlling each actuator to track their target trajectory, the desired trajectory and posture of the robot are achieved [16]. The kinematics-based control method is essentially the same as the design method of the single-axis motion controller. There is no need to consider the dynamical model of a parallel robot. It is simple to implement. However, in practical applications, actuators are not independent of each other. They are coupled with each other through the manipulator, and the moving manipulator acts on each actuator with constantly changing load forces. Therefore, performances of the kinematics-based control method depends heavily on the antidisturbance capability of the single-axis motion controller.To achieve high-performance motion control, a model predictive control was studied on a 5-DOF parallel robot in [17], and a modified robust control was designed in [18] for a 2-DOF parallel robot. Fractional robust control methods were proposed for a delta parallel robot in [19]. Disturbance observer (DOB) is a useful tool for motion control [20, 21]. An adaptive sliding-mode control based on a DOB was applied well on a manipulator in [22]. Parameter uncertainties can be seen as system disturbances, and a novel control strategy was designed to compensate nonlinear dynamics and model uncertainties of a long-stroke hydraulic robot [23].This paper adopts a control strategy based on kinematics and proposes the FOADRC, which effectively improves the disturbance rejection capability and trajectory tracking accuracy of each linear so that the novel 6-DOF parallel robot can achieve the given target motion better. Compared with other controllers such as fractional-order sliding-mode control [24], fuzzy and recurrent neural network [25], and dynamic terminal sliding-mode control [26], main differences of the proposed method are that accurate system models are not needed to obtain in advance, and it can be applied to practice easily. Modelling errors, parameter uncertainties, and external disturbances are estimated as the total disturbance which can be compensated online. However, control accuracy of the proposed method needs improvement.The proposed FOADRC is based on the conventional ADRC. In [27], details of the conventional ADRC are analyzed, including the stability. ADRC is a practical control strategy which has been used in many applications, such as coal-fired power plant [28], precise position tracking control [29], and gasoline engines [30]. In this proposed FOADRC, the component nonlinear PD is substituted by FOPD. FOPD is first validated stable and useful by experiments in [31]. Therefore, it can be concluded that the FOADRC is also stable. In addition, simulation and experimental results show that the proposed FOADRC is effective in practice.


### 1.1. Details of the Parallel Robot

The novel 6-DOF parallel robot is given in Figure1, which has components of six novel linear motors numbered 1 to 6, a motion plate and a base plate connected by rods and hinges, and an attitude sensor.Figure 1
Prototype of novel 6-DOF parallel robot.Linear motors are the main actuators of this 6-DOF parallel robot. Details of the linear motor are demonstrated in Figure2. The linear motor is modelled as the following equation:(1)Li˙=−Ri−Kex˙+uΣi,mx¨=−F+KfiΣx,F=Ff+Fr+Fd,where u, i, L, and R are the phase voltage, current, inductance, and resistance, x and m are the displacement and mass of the moving part, Ke and Kf are the back electromotive force coefficient and electromagnetic force coefficient, respectively, Ff is the friction, and Fd is the disturbances, including the interaction forces between linear motors and external disturbances.Figure 2
Details of the linear motor. (a) Appearance. (b) Internal structure.
(a)(b)


## 1.1. Details of the Parallel Robot

The novel 6-DOF parallel robot is given in Figure1, which has components of six novel linear motors numbered 1 to 6, a motion plate and a base plate connected by rods and hinges, and an attitude sensor.Figure 1
Prototype of novel 6-DOF parallel robot.Linear motors are the main actuators of this 6-DOF parallel robot. Details of the linear motor are demonstrated in Figure2. The linear motor is modelled as the following equation:(1)Li˙=−Ri−Kex˙+uΣi,mx¨=−F+KfiΣx,F=Ff+Fr+Fd,where u, i, L, and R are the phase voltage, current, inductance, and resistance, x and m are the displacement and mass of the moving part, Ke and Kf are the back electromotive force coefficient and electromagnetic force coefficient, respectively, Ff is the friction, and Fd is the disturbances, including the interaction forces between linear motors and external disturbances.Figure 2
Details of the linear motor. (a) Appearance. (b) Internal structure.
(a)(b)


## 2. FOADRC and Motion Control System

### 2.1. Motion Control System of Linear Motor

Here, a kinematics-based control strategy is adopted for the novel 6-DOF parallel robot, that is, six linear motors are controlled to track their own trajectories independently, and interactions among different linear motors are dealt with as disturbances. Therefore, a control system of one linear motor is constructed firstly based on the FOADRC, as illustrated in Figure3.Figure 3
Motion control of the linear motor based on FOADRC.


### 2.2. Details of FOADRC

There are three important components in the FOADRC, i.e., a tracking differentiator (TD), a fractional-order PD (FOPD), and an extended state observer (ESO). TD is expressed as the following equations in [27]:(2)x˙1=x2,x˙2=fhanx1−xd,x2,r,h0,where xd is the given position, x1 is the transition process of xd, and x2 is the velocity.(3)fhanx1−xd,x2,r,h0=−r⋅sgna,a>d−r⋅ad,a≤d.ESO is expressed as in the following equation:(4)e=z1−x,z˙1=z2−β01⋅e,z˙2=z3−β02⋅fale,0.5,δ+b⋅u,z˙3=−β03⋅fale,0.25,δ,where β01, β02, and β03 are ESO gains which can be selected as the following equation in [27]:(5)β01≈1h,β02≈11.6h1.5,β03≈18.6h2.2.fale,α,δ is given as the following equation:(6)fale,α,δ=e⋅δα−1,e≤δ,eα⋅sgne,e>δ.FOPD is expressed as the following equation:(7)Gs=Kp1+Kdsμ,where μ is a fraction between 0 and 1 and Kp and Kd are the controller gains.ESO_1 is given in the following equation:(8)e=z11−i,z˙11=z12−β11⋅e+b1⋅u,z˙12=−β12⋅fale,0.5,δ1,where β11、β12, and b1 are the parameters of the ESO_1.


## 2.1. Motion Control System of Linear Motor

Here, a kinematics-based control strategy is adopted for the novel 6-DOF parallel robot, that is, six linear motors are controlled to track their own trajectories independently, and interactions among different linear motors are dealt with as disturbances. Therefore, a control system of one linear motor is constructed firstly based on the FOADRC, as illustrated in Figure3.Figure 3
Motion control of the linear motor based on FOADRC.


## 2.2. Details of FOADRC

There are three important components in the FOADRC, i.e., a tracking differentiator (TD), a fractional-order PD (FOPD), and an extended state observer (ESO). TD is expressed as the following equations in [27]:(2)x˙1=x2,x˙2=fhanx1−xd,x2,r,h0,where xd is the given position, x1 is the transition process of xd, and x2 is the velocity.(3)fhanx1−xd,x2,r,h0=−r⋅sgna,a>d−r⋅ad,a≤d.ESO is expressed as in the following equation:(4)e=z1−x,z˙1=z2−β01⋅e,z˙2=z3−β02⋅fale,0.5,δ+b⋅u,z˙3=−β03⋅fale,0.25,δ,where β01, β02, and β03 are ESO gains which can be selected as the following equation in [27]:(5)β01≈1h,β02≈11.6h1.5,β03≈18.6h2.2.fale,α,δ is given as the following equation:(6)fale,α,δ=e⋅δα−1,e≤δ,eα⋅sgne,e>δ.FOPD is expressed as the following equation:(7)Gs=Kp1+Kdsμ,where μ is a fraction between 0 and 1 and Kp and Kd are the controller gains.ESO_1 is given in the following equation:(8)e=z11−i,z˙11=z12−β11⋅e+b1⋅u,z˙12=−β12⋅fale,0.5,δ1,where β11、β12, and b1 are the parameters of the ESO_1.


## 3. Comparative Simulations

### 3.1. Comparative Controller Details

Compared with the normal active disturbance rejection controller (ADRC), two designed trajectories and three different working situations of a linear motor are simulated to demonstrate good performances of the FOADRC in the MATLAB/Simulink software. Detailed parameters of the linear motor and the FOADRC are given in Tables1 and 2. Kp1 is the P controller gain in the current loop, and h and h1 are the sampling intervals of ESO and ESO_1, respectively.Table 1
Parameters of the linear motor.
ParametersR (Ω)L (mH)M (g)Kf/(N/A)Ke/(Vs/m)Values3.44.4225018.0118.01Table 2
Parameters of FOADRC.
ParametersValuesParametersValuesh00.000 2β015000h0.000 2β02220970h10.000 025β0315967450b1226δ0.000 2β1140000μ0.835β125000000Kp140 000δ10.000025B72Kp100 000Kd300Two desired trajectories are given in the following equation:(9)xd=5sin5t−0.5π+5mm,S1,20sin7t−0.5π+20mm,S2.


### 3.2. Simulation Results under Different Situations

Three different working situations are compared: without payloads, with parameter variations, and with external disturbances. Comparative results without payloads are illustrated in Figure4.Figure 4
Comparative results without payloads.To test the system robustness, a 7.75 kg mass is loaded and results are compared in Figure5. A disturbance force of 8 N acts on the system between 0.6 s and 1.2 s, and comparative results are illustrated in Figure 6.Figure 5
Comparative results with payloads.Figure 6
Comparative results with disturbances.Figures4–6 show that the FOADRC performs well under payloads and disturbances, and the control accuracy is higher than ADRC.


## 3.1. Comparative Controller Details

Compared with the normal active disturbance rejection controller (ADRC), two designed trajectories and three different working situations of a linear motor are simulated to demonstrate good performances of the FOADRC in the MATLAB/Simulink software. Detailed parameters of the linear motor and the FOADRC are given in Tables1 and 2. Kp1 is the P controller gain in the current loop, and h and h1 are the sampling intervals of ESO and ESO_1, respectively.Table 1
Parameters of the linear motor.
ParametersR (Ω)L (mH)M (g)Kf/(N/A)Ke/(Vs/m)Values3.44.4225018.0118.01Table 2
Parameters of FOADRC.
ParametersValuesParametersValuesh00.000 2β015000h0.000 2β02220970h10.000 025β0315967450b1226δ0.000 2β1140000μ0.835β125000000Kp140 000δ10.000025B72Kp100 000Kd300Two desired trajectories are given in the following equation:(9)xd=5sin5t−0.5π+5mm,S1,20sin7t−0.5π+20mm,S2.


## 3.2. Simulation Results under Different Situations

Three different working situations are compared: without payloads, with parameter variations, and with external disturbances. Comparative results without payloads are illustrated in Figure4.Figure 4
Comparative results without payloads.To test the system robustness, a 7.75 kg mass is loaded and results are compared in Figure5. A disturbance force of 8 N acts on the system between 0.6 s and 1.2 s, and comparative results are illustrated in Figure 6.Figure 5
Comparative results with payloads.Figure 6
Comparative results with disturbances.Figures4–6 show that the FOADRC performs well under payloads and disturbances, and the control accuracy is higher than ADRC.


## 4. Experimental Validations and Discussion

### 4.1. Experimental Prototype

An experimental setup of the novel 6-DOF parallel robot on the basis of a dSPACE real-time hardware-in-loop system was constructed as given in Figure7. The experimental setup is composed of dSPACE modular hardware boards (DS1005, DS4002, and DS2004), driving modules, current sensors, displacement sensors, attitude sensor, 6-DOF parallel robot, and personal computer (PC). The software running in the PC is used to realize the real-time simulation model of the control system, display the operating parameters of the system, and adjust the parameters of the controller in real time. DS1005 is the core of the real-time control system. DS4002 and DS2004 are connected to DS1005 through the external device high-speed bus peripheral high speed (PHS) for data communication.Figure 7
Experimental setup of 6-DOF parallel robot. (a) Actual. (b) Logical.
(a)(b)The double-loop series closed-loop hall current sensor TBC05SY and KTM series sliding variable resistance linear displacement sensor are selected. Both of the sensor linearity is 0.1%. Each linear motor is driven by its own driving module. The attitude sensor is used to measure the pose of the motion plate, including the rotation angle around thex-, y-, and z-axes. The measured data are sent to the MT Manager software for real-time display. Current sensors collect winding current signals of linear motors and send them to the analog-to-digital conversion board DS2004, and conversion results are sent to the main control board DS1005 to calculate control inputs. Pulse-width-modulation (PWM) signals are generated by the DS4002 board and the PWM frequency is 40 kHz.


## 4.1. Experimental Prototype

An experimental setup of the novel 6-DOF parallel robot on the basis of a dSPACE real-time hardware-in-loop system was constructed as given in Figure7. The experimental setup is composed of dSPACE modular hardware boards (DS1005, DS4002, and DS2004), driving modules, current sensors, displacement sensors, attitude sensor, 6-DOF parallel robot, and personal computer (PC). The software running in the PC is used to realize the real-time simulation model of the control system, display the operating parameters of the system, and adjust the parameters of the controller in real time. DS1005 is the core of the real-time control system. DS4002 and DS2004 are connected to DS1005 through the external device high-speed bus peripheral high speed (PHS) for data communication.Figure 7
Experimental setup of 6-DOF parallel robot. (a) Actual. (b) Logical.
(a)(b)The double-loop series closed-loop hall current sensor TBC05SY and KTM series sliding variable resistance linear displacement sensor are selected. Both of the sensor linearity is 0.1%. Each linear motor is driven by its own driving module. The attitude sensor is used to measure the pose of the motion plate, including the rotation angle around thex-, y-, and z-axes. The measured data are sent to the MT Manager software for real-time display. Current sensors collect winding current signals of linear motors and send them to the analog-to-digital conversion board DS2004, and conversion results are sent to the main control board DS1005 to calculate control inputs. Pulse-width-modulation (PWM) signals are generated by the DS4002 board and the PWM frequency is 40 kHz.


## 5. Results and Discussion

Suppose the desired motion of the robot is in the following equation:(10)θx=1.915sin3.749t+6.577cos3.749t+5.558°,θy=−5.024sin3.749t−0.965cos3.749t+0.313°,θz=3.449sin3.749t−0.418cos3.749t−3.202°,where θx, θy, and θz are the angles around three different axes, as shown in Figure 1. Desired trajectories of six linear motors are calculated as in the following equation:(11)xd1=24sin5t+24mm,xd2=18sin5t+18mm,xd3=6sin5t+6mm,xd4=12sin5t+12mm,xd5=24sin5t+24mm,xd6=18sin5t+18mm.Actual tracking results are given in Figures8 and 9. It can be seen from Figures 8 and 9 that six linear motors track their desired trajectories well, and the novel 6-DOF parallel robot is driven to reach its control goal. Effectiveness of the FOADRC is verified by experiments.Figure 8
Actual tracking results. (a) Linear motor nos. 1–3. (b) Linear motor nos. 4–6.
(a)(b)Figure 9
Motion performance of the robot.ESO is an important part of the FOADRC, which can be realized in the actual test. However, due to the limitation of the measurement accuracy of sensors, the observation effect of the ESO is not very good. The trajectory tracking accuracy in the experiment is not as good as that of the simulation. High-resolution current and displacement sensors can be chosen, such as optical encoders, to further improve the control performance.


## 6. Conclusions

A FOADRC is proposed in this paper, and it is implemented on a novel 6-DOF parallel robot for actual test. A kinematics-based control method is adopted to realize desired motions of this 6-DOF parallel robot. Six linear motors are controlled to track their own trajectories independently, and interactions among different linear motors are dealt with as disturbances. Different trajectories and working situations of a linear motor are simulated to demonstrate good performances of the FOADRC. Compared with the normal ADRC, the designed FOADRC can improve the system trajectory tracking performance effectively. Due to limitations of working space and singularity problem of the parallel robot, only sinusoidal trajectory tracking control is analyzed at present. In future work, we will further optimize the motion platform and conduct control research on various motion trajectories to promote the application of this 6-DOF parallel robot in industrial practice.



---

*Source: 3657848-2020-12-29.xml*