#### Abstract

To solve the problems of model uncertainties, dynamic coupling, and external disturbances, a modified linear active disturbance rejection controller (MLADRC) is proposed for the trajectory tracking control of robot manipulators. In the computer simulation, MLADRC is compared to the proportional-derivative (PD) controller and the regular linear active disturbance rejection controller (LADRC) for performance tests. Multiple uncertain factors such as friction, parameter perturbation, and external disturbance are sequentially added to the system to simulate an actual robot manipulator system. Besides, a two-degree-of-freedom (2-DOF) manipulator is constructed to verify the control performance of the MLADRC. Compared with the regular LADRC, MLADRC is significantly characterized by the addition of feedforward control of reference angular acceleration, which helps robot manipulators keep up with target trajectories more accurately. The simulation and experimental results demonstrate the superiority of the MLADRC over the regular LADRC for the trajectory tracking control.

#### 1. Introduction

Robot manipulators are well-known mechanical systems with controllable trajectories, which are widely used in modern industry and other fields. Trajectory tracking control of robot manipulators requires that the end-effector can move precisely along the given trajectories. However, the nonlinearity, strong coupling, and uncertainty of the system make the trajectory tracking very complicated and difficult, so it has always been a hot spot for researchers.

Trajectory tracking control methods of robot manipulators can be divided into “motion control” and “dynamic control.” The motion control only carries out negative feedback control through the deviation between target trajectories and actual trajectories. Therefore, such methods cannot guarantee control performance. The dynamic control is designed according to the dynamic characteristics of robot manipulators, so it can make the control quality of the system better [1, 2]. At present, the commonly used dynamic control methods mainly include intelligent PID control [3–5], iterative learning control [6–9], adaptive neural network control [10–13], sliding mode control [14–16], and active disturbance rejection control [17, 18]. Aiming at *n*-degree-of-freedom rigid robots, Hernández-Guzmán and Orrante-Sakanassi [4] proposed a control scheme for direct-drive brushless direct-current (BLDC) motors, which solved the position control problem of *n* direct-drive BLDC with complex, nonlinear, and highly coupled mechanical loads. Bouakrif and Zasadzinski [7] designed a high-order iterative learning controller for the trajectory tracking of robot manipulators subject to external disturbances and performing repetitive tasks. A dual-link manipulator was taken as the research object to prove that the closed-loop system was asymptotically stable in the finite time interval when the number of iterations tended to infinity. Liu et al. [11] proposed the adaptive neural network control with the optimal number of hidden layer nodes. The method could approach the uncertainty of the robot manipulator to ensure high-precision trajectory tracking. Baek and Kwon [14] proposed a strong and stable adaptive sliding-mode control method by designing two adaptive laws related to switching gain. It enhanced the robustness of the robot manipulator system and achieved accurate tracking performance. Chen et al. [17] proposed a robust active disturbance rejection controller (ADRC) based on sliding mode control technology. The uncertainties in control gains and disturbance estimation errors were considered in the design process of the controller, which improved tracking accuracy and minimized link vibrations.

The most important feature of ADRC is the ability to estimate and compensate for system uncertainties. Hence, it is very suitable for the application in multiple-input and multiple-output (MIMO) systems such as robot manipulators [19]. However, ADRC has many control parameters, and the physical meaning of some parameters is not very clear. So, the parameter tuning is difficult. To simplify the tuning process, Gao [20] proposed a linear ADRC, i.e., LADRC, and replaced nonlinear ESO with a linear ESO (LESO). In this paper, we make some improvements to LADRC to make it have higher control precision and better dynamic performance compared with the regular LADRC.

#### 2. Modeling of a 2-DOF Manipulator System

##### 2.1. A Dynamic Model of the 2-DOF Manipulator

A simplified model of the 2-DOF manipulator is depicted in Figure 1, which has two rotary joints. Ignoring gravity, friction, and external disturbances, the robot manipulator can be modeled aswhere is the joint angle vector, is the joint control torque vector, is the inertia matrix, and is the Coriolis matrix. The expressions of and are as follows:wherewhere and represent the lengths of the two rods and and represent the masses of the two rods. The values of , and are

##### 2.2. A Mathematical Model of the BLDC Reduction Motor

The robot manipulator is driven by the BLDC reduction motor on each joint. According to the working principle of BLDC reduction motors, the mathematical model of the motor can be described aswhere , , , and represent the voltage, current, resistance, and inductance of the motor, respectively. and are the rotation angle and output torque of the motor. and are the torque coefficient and the back electromotive force (EMF) coefficient. represents the reduction ratio. The parameter values of the BLDC reduction motor are listed in Table 1.

#### 3. Design of the Trajectory Tracking Control System

##### 3.1. Modified LADRC Design

Since the inertial matrix is symmetric and positive definite, equation (1) can be converted into

Considering the influence of friction, parameter perturbation, and external disturbances, we add these uncertainties to equation (6) and take them together with the coupling term as total disturbances applied to the joint. The total disturbances can be expressed aswhere is the disturbance torque vector, is the friction torque vector, is treated as the total disturbances of joint 1, and is the total disturbances of joint 2. Furthermore, assuming

Equation (6) can be rewritten as

According to the application of LADRC in MIMO systems [21–23], a “virtual control torque” vector is introduced to the control system. Letand equation (9) can be represented by

We can observe from equation (11) that the two joints of the 2-DOF manipulator are decoupled; each joint becomes an independent second-order system with the total disturbances. So MLADRC can be designed to control them separately.

For the sake of simplicity, only the MLADRC algorithm for controlling joint 1 is presented. MLADRC is mainly composed of a third-order LESO, disturbance compensation, a PD controller, and reference angular acceleration feedforward. The controller design of joint 2 is the same as that of joint 1.(1)LESO: the third-order LESO is used to dynamically estimate the total disturbances , which is characterized as where and are the estimates of and , respectively, and is the estimate of , i.e., the extended state. , , and are the control gains.(2)Disturbance compensation: is dynamically compensated by where is an intermediate control quantity. Ignoring the estimation error of to , joint 1 is reduced to a unit-gain double integrator:(3)State error feedback control: the PD controller is designed to control the double integrator, and its control algorithm is as follows: where is the reference input trajectory of joint 1 and and are the PD controller parameters.(4)Reference angular acceleration feedforward: in the case that the reference trajectory is known, the reference angular acceleration, i.e., the second derivative of , can be solved first. The calculation result is then assigned to ; that is,

It can be seen from equation (16) that the joint trajectory can track the reference input trajectory very well. Therefore, we call the LADRC that incorporates the reference angular acceleration feedforward control as “modified linear active disturbance rejection control,” i.e., MLADRC.

##### 3.2. BLDC Motor Control

Since the output torque of the BLDC reduction motor is proportional to the current and is relatively easy to be collected in experiments, a first-order LADRC is designed to control the current [24]. Considering the design process of LADRC, equation (5) is simplified aswherewhere is regarded as the total disturbances of the motor and is the control input gain.

The first-order LADRC is designed aswhere is the estimate of and is the estimate of , i.e., the extended state; and are the second-order LESO control gains; is the P controller parameter; is the compensation coefficient, and its value is selected as

Based on the above design and analysis of the controller, we adopt a double closed-loop control structure for the trajectory tracking system of the 2-DOF manipulator, as shown in Figure 2. The position loop controls the joint angle and the current loop controls motor torque. In the current loop, the proportional (P1) controller acts as the state error feedback control law of the first-order LADRC. The controller structure of joint 2 is consistent with that of joint 1.

##### 3.3. Stability Analysis of the Control System

LESO is a key component of MLADRC. Whether the total disturbances and other system states can be accurately observed by LESO will directly affect the dynamic performance and quality of the entire control system. Therefore, we should first analyze the estimation ability of LESO in the control system, and then the stability of the manipulator trajectory tracking system is analyzed and verified.

###### 3.3.1. LESO Observation Performance Analysis

Considering the joint 1 system ofassuming , the state equation of plant (21) can be expressed aswhere , and are system state variables and . Let , , and ; we have

Meanwhile, equation (12) can also be expressed in matrix form aswhere , , and then subtracting equation (24) from equation (23), we obtain

Let and , and it follows that

Assumingwhere , , and are LESO coefficients, and , the stability condition of LESO can be expressed as

Thus, as long as we can select the appropriate gains, the LESO estimation error will be bounded; that is, there exists constant , such that .

###### 3.3.2. Closed-Loop System Stability Analysis

Assuming that the reference input trajectory is bounded and according to the state estimates of LESO, the system error feedback control law of joint 1 can be described as

Equation (21) is then rewritten as

Let , , , and ; we havewhere is the estimation error of LESO and . Let , , and ; the matrix form of equation (31) can then be expressed as

Since the estimation error of LESO has been proved to be bounded, there exists to make the tracking error of system (32) bounded [25, 26]. Thus, for bounded input, the plant (21) output is bounded; that is, the system is bounded-input bounded-output (BIBO) stable.

#### 4. Simulation Research

In Matlab/Simulink, PD, LADRC, and MLADRC are applied for controlling plant (11). By comparing the simulation results of the three, the rationality of MLADRC design is verified. The dynamic control performance of MLADRC is analyzed according to the simulation results under different disturbances.

##### 4.1. Controller Parameters’ Tuning

(1)PD controller: because PD is simple and easy to be realized in practical engineering, it is often used in robot manipulator control. The trajectory tracking system based on PD of joint 1 is plotted in Figure 3. The current loop adopts a proportional-integral (PI) controller. The controller parameters of the PD and PI are tuned based on the Ziegler–Nichols method, and the final tuning results are listed in Table 2. and are the proportional coefficient and differential coefficient of PD, and and are the proportional coefficient and integral coefficient of PI.(2)MLADRC: MLADRC, i.e., “modified LADRC.” The structure of the trajectory tracking control system composed of MLADRC is plotted in Figure 2. According to the method of parameter bandwidthization proposed in [20], the MLADRC parameters are configured as where is denoted as the third-order LESO bandwidth, represents the bandwidth of the PD controller in MLADRC, and is the second-order LESO bandwidth. As a result, the parameters that need to be tuned are greatly reduced and the physical meaning of each parameter is very clear. In MLADRC, , , , and determine the various performances of the system, such as stability, transient performance, anti-interference, and noise suppression. Each performance needs to be weighed when adjusting the controller parameters. For the controlled plant (11) and plant (17), is known, and , , and need to be tuned. The specific configuration process is as follows [27]: Step 1: first, determine the initial value of according to the adjustment time of the system, and then let . Step 2: keep constant, and adjust in a small range until the system is stable. If adjustment of fails to stabilize the system, decrease or increase both and in the same proportion, and then adjust individually to make the system stable. Step 3: keep constant, and gradually decrease . On the premise of ensuring the stability of the system, weigh the transient performance and noise suppression ability, and select the appropriate value for . Step 4: weigh the stability, transient performance, anti-interference, and noise suppression of the system, and determine the optimal values of and . After multiple tests and adjustments, the ideal parameter values are obtained as shown in Table 3.(3)LADRC: LADRC, i.e., “regular LADRC.” The LADRC-based trajectory tracking control system is obtained by removing the reference angular acceleration feedforward from Figure 2. All parameter values of LADRC are in line with those of MLADRC. Since the derivative of cosine is continuous, it is often used as the reference input trajectory in simulation research. So the target trajectories of the two joints of the robot manipulator are set as

##### 4.2. Simulation Results

###### 4.2.1. Research on Joint Friction

In working processes, the robot manipulator will be hindered by the friction at the joint, so the influence of friction cannot be ignored when designing the controller. Assuming the frictional force at each joint isthe tracking curves and tracking errors among PD, LADRC, and MLADRC are compared in Figure 4.

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As is seen from Figure 4, the trajectory tracking accuracy of MLADRC is the highest. The maximum tracking error of joint 1 is limited to and that of joint 2 is limited to . Joint friction in actual systems is very complex; especially at the moment when the velocity direction changes, there will be a spiking error, as shown in Figures 4(b) and 4(d). The friction needs to be observed by LESO before it is compensated, which leads to a lag in compensation.

###### 4.2.2. Research on System Robustness

In addition to the friction, the preidentified parameter values of the system model will change with the variety of working states. For example, in actual work, the end of robot manipulators will clamp different loads. Assuming the model perturbation caused by varying loads is , the tracking errors with the friction and parameter perturbation are depicted in Figure 5.

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From Figure 5, we can observe that the tracking error of MLADRC is still the smallest. Comparing Figures 4 and 5, the error curves are very close. So it illustrates that MLADRC is strongly robust against the system parameter perturbation.

###### 4.2.3. Research on Disturbance Rejection

High-performance controllers must be able to reject external disturbances. To test the disturbance rejection property of MLADRC, a disturbance of is applied to joint 2 between 4.5 and 5.5 seconds. The tracking errors of the two joints are shown in Figure 6, and the total disturbances (the friction, parameter perturbation, external disturbances, and dynamic coupling) and their estimates are described in Figure 7.

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It can be observed from Figure 6 that MLADRC can respond quickly and make timely adjustments when joint 2 is subjected to the external disturbance. The convergence time of the tracking error caused by the external disturbance is very short, which is mainly thanks to LESO’s excellent disturbance observation ability (see Figure 7). Figure 8 shows the input torques of the two joints, where the increased torque in joint 2 between 4.5 and 5.5 seconds is used to compensate for the external disturbance. The torque curves are relatively smooth and there is no violent chattering.

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According to Figures 4–8, we can conclude that the trajectory tracking system based on MLADRC can keep up with the target trajectory rapidly and has a strong robustness to the total disturbances. Under the same conditions, the tracking accuracy of MLADRC is enhanced compared with the regular LADRC.

#### 5. Experiment

##### 5.1. Experimental Platform

A self-developed horizontal 2-DOF manipulator is used as the controlled object to conduct experimental researches on PD, LADRC, and MLADRC. The experimental platform includes a horizontal 2-DOF manipulator, an STM32 microcontroller, a host computer, two Hall current sensors, two DC motor drivers, and a switching power supply, which is shown in Figure 9. An angle sensor with dual-channel pulse output is mounted at the tail of the BLDC reduction motor to measure the rotation angle and angular velocity of the joint. The linearity of the current sensor is 0.1% and the resolution of the angle sensor is 1024p/r. A smart power chip of BTS7960 is adopted to drive the BLDC motor.

STM32 receives the position and control instructions from the host computer and collects the feedback signals such as joint angles and motor currents. The PD, LADRC, and MLADRC control algorithms run in STM32 to complete the calculation and output of the control quantity. The current sensor sends the current signal to the A/D conversion module in the STM32, and then the conversion result is sent to the current loop. The host control system is developed through Microsoft Foundation Classes (MFC) in Visual Studio 2015 and is responsible for such tasks as kinematics calculation, trajectory planning, data processing, and human-computer interaction.

##### 5.2. Trajectory Planning and Motion Control

The experiment requires the end tip of the 2-DOF manipulator to track a circular trajectory with a diameter of 0.17 m. The circular trajectory is preset in the host control system, and its mathematical equation is expressed aswhere and are the rectangular coordinates of the end tip. According to the structure and coordinate definition of the 2-DOF manipulator, the kinematic relationship between the end tip and the joints is described as

From equation (37), the desired motion equations of the two joints can be obtained as(1)Joint trajectory planning: when the robot manipulator is running, the target trajectories of the two joints are calculated and generated in real time by equations (36) and (38). To explain how the trajectories are produced, a detailed trajectory planning procedure is given as follows: Step 1: give time an increment: Step 2: solve equation (36) to get the position coordinate Step 3: substitute and into equation (38) to solve for and Step 4: send and to STM32 for tracking control Step 5: go back to the beginning of the loop(2)Motion control: control algorithms are loaded into STM32 before the system is powered on. The sampling period is set to 2 ms. When receiving the running command sent by the host computer, the end tip of the robot manipulator will be driven to move along target trajectories. The lower computer control system is designed by the modularization method, which includes function modules such as system initialization, data acquisition, algorithm design, and serial communication. The control program flow is shown in Figure 10.

##### 5.3. Experimental Results and Discussion

According to the above experimental design process, a comparative experiment is carried out on LADRC and MLADRC. Two kinds of experimental results are given: undisturbed experiment and disturbed experiment. During the operation of the robot manipulator, joint angles are read in real time by STM32 and then transmitted to the host computer. Without external disturbances, the tracking performances of the two joints are shown in Figure 11. Figure 12(a) describes the tracking curves of the end tip without disturbance. A step disturbance is applied to joint 1 at 4 seconds, and then the tracking curves are depicted in Figure 12(b).

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From Figure 11, we observed that joints 1 and 2 can track target trajectories very well, but the tracking error of MLADRC is less than that of LADRC. When the robot manipulator is just started, the system will be disturbed by sudden coupling and friction. The initial tracking error of each joint is large because of the lag of dynamic coupling and friction compensation.

As can be seen from Figure 12, the motion trajectory of the end tip is very close to the preset circular trajectory. When the robot manipulator is just started or is subject to the step disturbance, both MLADRC and LADRC can quickly adjust the motion state of the system. The tracking accuracy of MLADRC is higher than that of LADRC, and the MLADRC system responds faster.

In the experiment, the measurement accuracy of the sensor will affect the observation performance of LESO, so the tracking accuracy is not as high as that in the simulation. In future research, high-precision current sensors and angle sensors can be selected to further enhance the trajectory tracking accuracy.

#### 6. Conclusions

Aiming at the trajectory tracking control problem of robot manipulators, a more in-depth study is carried out based on the regular LADRC. The control quality of the system is improved by adding the reference angular acceleration feedforward control, and the stability of the proposed MLADRC closed-loop system is analyzed. In the control system design, LESO is used to estimate and compensate for the total disturbances composed of internal uncertainties, external disturbances, and dynamic coupling. The feedforward control is used to improve the trajectory tracking accuracy. The double closed-loop control structure enhances the robustness of the system. In addition, according to the proposed control method, the error convergence, robustness, and external disturbance suppression of the system are studied, respectively. The comparative simulations and experiments verify the excellent control performance of MLADRC.

#### Data Availability

The data used to support the ﬁndings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conﬂicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was supported by the Major Scientific Research Project Cultivation Plan Fund of Ningde Normal University (no. 2017ZDK20) and the Natural Science Foundation of Fujian Province, China (no. 2018J01556).