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

Liu, Dawei; Gao, Qinhe; Chen, Zhixiang; Liu, Zhihao

doi:https://doi.org/10.1155/2020/6969207

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2020 | Article ID 6969207 | https://doi.org/10.1155/2020/6969207

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

Dawei Liu,¹Qinhe Gao,¹Zhixiang Chen,¹and Zhihao Liu¹

Academic Editor: Jean Jacques Loiseau

Received12 Jan 2020

Accepted27 Feb 2020

Published06 May 2020

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.

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 tune

The 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. Section 3 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 Figure 1.

The dynamic model can be derived using the Euler–Lagrange dynamic equation, which comprises two coupled nonlinear second-order differential equations. Define , and the model can be represented aswherewhere and represents the length of the 2 rods, are the joint angles shown in Figure 1, represent the mass of the 2 rods, are the control laws of rod 1 and rod 2, respectively, and 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 is 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 point to another point the displacement in the x and y directions can be written as

From point A to point B, the motion in x 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 aswhere where is the time at the initial point A, is time to reach the point B, and represent the positions of the end effector in x and y directions at the initial point, respectively, and and 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 asand 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

From (7), the corresponding Jacobian matrix J can be calculated asand let ; we havewhere

From (9), the desired velocity signal of the 2-DOF manipulator can be written asand the desired angle signal can be written as

Considering the case where J is irreversible, assumingwe can obtain , when J is irreversible. This situation must be avoided. Hence, when letting , 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

To avoid any calculation error, (11) is modified aswhere is error gain.

The desired signal generation principle of the 2-DOF manipulator is depicted in Figure 2. The desired trajectory planned in this study is a regular octagonal.

3. Design of LADRC

3.1. Problem Description

Before designing an LADRC, we describe its mathematical model in (1). Letting (1) becomes

From (16), the descriptions (i)-(ii) are made:(i)The condition can be verified according to the definition of the parameters in (1). The control input and both have an effect on and . 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, and can be controlled by and , respectively.

3.2. Controller Design

From (16), we have

Assuming the total disturbances of rod 1 and rod 2 as(17) can be rewritten as(19) is then rewritten aswhere is the sum of and the coupling control , is the sum of and the coupling control , and and are the nominal values of the control gain of rod 1 and rod 2, respectively. Assuming are differentiable and that (20) can be written in an augmented state-space form:where are the states of the system, and are the input and output of the system, respectively.

LADRC is designed for (21) and the LESO is given aswhere and are selected such that the characteristic polynomial is Hurwitz.

Then, the ADRC control law is given aswhere is the desired signal of , is the desired signal of ς, and the values are assigned and bounded. Let

are also bounded and are the controller gains selected to make and Hurwitz.

4. Stability Analysis

4.1. Stability Analysis of LESO

Let From (21) and (22), the observer estimation error can be written as

Now, let and (25) can be rewritten aswhere .

Theorem 1. Assuming and is bounded, there exists constant and a finite , such that and Furthermore, and for some positive integer k.

Proof. Solving (26), it follows thatand letas and be bounded, i.e., , , where is a positive constant. For we haveFor and defined in (26)where As is Hurwitz, there exists a finite time such thatfor all , = Hence,for all , =
Let , , and we have for all , = , where .
From (29), (30), and (33), we obtainFrom (29), (30), and (33), we obtainfor all , = .
Let and and it follows that for all , = .
From (27), one hasLet and According to , and (34)–(37), we havefor all = 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 havewhere are the controller gains selected to make and Hurwitz. The closed-loop system becomes

Considerwhere and is a matrix.

Lemma 1. If N is Hurwitz and , then ; Zheng et al. [32] provided detailed proof for the lemma.

Theorem 2. Assuming and is bounded, there exist and a finite time such that Furthermore, and and and

Proof. From (23), we haveand it follows thatLet and ; then, wheresince are selected such that the characteristic polynomials and are Hurwitz and and are Hurwitz. For tunning simplicity, we let and where , [32]. This implies that and , which are controller bandwidths of rod 1 and rod 2, respectively, the only tunning parameters to be adjusted for the controller.
Solving the equation in (44), we obtainAccording to the equation and Theorem 1, we havewhere
Defining and letting it follows that Since is Hurwitz, there exists a finite time such thatfor all
Let It follows thatfor all andfor all From (48) and (51), we obtainfor all Let It follows thatfor all From (46), we haveAccording to (47) and (52)–(54), we havefor all whereSimilarly, we can obtainwhereAccording to (55) and (57), we can observe that the error of the desired signal and the system output and the error of desired signal and the system output are bounded, and the upper bound monotonously decreases with the two controllers’ bandwidth.

Corollary 1. Assuming is globally Lipschitz with respect to and is globally Lipschitz with respect to , there exist constants , such that the closed-loop system (40) is asymptotically stable.

Proof. From (44), we knowAccording to Theorem 1 postulated by [27], and , if is globally Lipschitz with respect to and is globally Lipschitz with respect to . Furthermore, according to Lemma 1, since and is Hurwitz and such that 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 Figure 3. 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.

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.

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 Figure 5. 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.

(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 Figure 6, 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.

(a)

(b)

(c)

(d)

(e)

(f)

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 the x-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)

5. Experiment

Here, the proposed LADRC is further examined through an experiment, which is conducted on a 2-DOF manipulator, as shown in Figure 8. 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.

The sampling period in the experiment is set as 0.001 s and the solver in Matlab/Simulink environment is set as ode l (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 Table 3.

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 Figure 9. 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.

(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 Figures 10 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.

(a)

(b)

(c)

(d)

(e)

(f)

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 the x-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)

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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundations of China (1905541) and Young Talent fund of University Association for Science and Technology in Shannxi, China (20190412).

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Copyright © 2020 Dawei Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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