Modified Linear Active Disturbance Rejection Control for Uncertain Robot Manipulator Trajectory Tracking

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. *e simulation and experimental results demonstrate the superiority of the MLADRC over the regular LADRC for the trajectory tracking control.


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." e motion control only carries out negative feedback control through the deviation between target trajectories and actual trajectories. erefore, such methods cannot guarantee control performance. e 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][4][5], iterative learning control [6][7][8][9], adaptive neural network control [10][11][12][13], sliding mode control [14][15][16], and active disturbance rejection control [17,18]. Aiming at ndegree-of-freedom rigid robots, Hernández-Guzmán and Orrante-Sakanassi [4] proposed a control scheme for directdrive 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. e 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. e 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. e 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.

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 as where q � q 1 q 2 T is the joint angle vector, τ � τ 1 τ 2 T is the joint control torque vector, D(q) is the inertia matrix, and C(q, _ q) is the Coriolis matrix. e expressions of D(q) and C(q, _ q) are as follows: where where l 1 and l 2 represent the lengths of the two rods and m 1 and m 2 represent the masses of the two rods. e values of l 1 , l 2 , m 1 , and m 2 are

A Mathematical Model of the BLDC Reduction
Motor. e 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 as where u, i, R, and L represent the voltage, current, resistance, and inductance of the motor, respectively. q m and τ m are the rotation angle and output torque of the motor. k t and k e are the torque coefficient and the back electromotive force (EMF) coefficient. α represents the reduction ratio. e parameter values of the BLDC reduction motor are listed in Table 1.

Design of the Trajectory Tracking
Control System 3.1. Modified LADRC Design. Since the inertial matrix D(q) 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. e total disturbances can be expressed as where τ d � τ d1 τ d2 T is the disturbance torque vector, T is the friction torque vector, f 1 (·) is treated as the total disturbances of joint 1, and f 2 (·) 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][22][23], a "virtual control torque" vector T is introduced to the control system. Let and 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. e 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 f 1 (·), which is characterized as where z 1r and z 2r are the estimates of q 1 and _ q 1 , respectively, and z 3r is the estimate of f 1 (·), i.e., the extended state. β 1r , β 2r , and β 3r are the control gains.
(2) Disturbance compensation: z 3r is dynamically compensated by where T 01 is an intermediate control quantity. Ignoring the estimation error of z 3r to f 1 (·), 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 q d1 is the reference input trajectory of joint 1 and k pr and k dr are the PD controller parameters. (4) Reference angular acceleration feedforward: in the case that the reference trajectory q d1 is known, the reference angular acceleration, i.e., the second derivative of q d1 , can be solved first. e calculation result is then assigned to T 01 ; that is, It can be seen from equation (16) that the joint trajectory can track the reference input trajectory very well. erefore, we call the LADRC that incorporates the reference angular acceleration feedforward control as "modified linear active disturbance rejection control," i.e., MLADRC.

BLDC Motor Control.
Since the output torque of the BLDC reduction motor is proportional to the current and i 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 as where f m (i, _ q m ) is regarded as the total disturbances of the motor and b m is the control input gain. e first-order LADRC is designed as where z 1m is the estimate of i and z 2m is the estimate of f m (i, _ q m ), i.e., the extended state; β 1m and β 2m are the second-order LESO control gains; k pm is the P controller parameter; b 0 is the compensation coefficient, and its value is selected as Table 1: Technical parameters of the BLDC reduction motor.
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. e 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. e controller structure of joint 2 is consistent with that of joint 1.

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. erefore, 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.

LESO Observation Performance
Analysis. Considering the joint 1 system of assuming _ f 1 (·) � h(·), the state equation of plant (21) can be expressed as where x 1 , x 2 , and x 3 are system state variables and Meanwhile, equation (12) can also be expressed in matrix form as and then subtracting equation (24) from equation (23), we obtain Let e � X − Z r and A 2 � A − β r , and it follows that Assuming where c ∈ R + , a 1 , a 2 , and a 3 are LESO coefficients, and a 1 , a 2 , a 3 ∈ R + , the stability condition of LESO can be expressed as us, as long as we can select the appropriate gains, the LESO estimation error will be bounded; that is, there exists constant M > 0, such that ‖e‖ ≤ M.

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 where e i is the estimation error of LESO 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 k pr > 0, k dr > 0 to make the tracking error of system (32) bounded [25,26]. us, for bounded input, the plant (21) output is bounded; that is, the system is bounded-input bounded-output (BIBO) stable.

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. e dynamic control performance of MLADRC is analyzed according to the simulation results under different disturbances.

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. e trajectory tracking system based on PD of joint 1 is plotted in Figure 3. e current loop adopts a proportional-integral (PI) controller. e 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. k p and k d are the proportional coefficient and differential coefficient of PD, and k p ′ and k i are the proportional coefficient and integral coefficient of PI.
According to the method of parameter bandwidthization proposed in [20], the MLADRC parameters are configured as where ω 0r is denoted as the third-order LESO bandwidth, ω cr represents the bandwidth of the PD controller in MLADRC, and ω 0m is the secondorder 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, b 0 , ω 0r , ω cr , and ω 0m 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), b 0 is known, and ω 0r , ω cr , and ω 0m need to be tuned. e specific configuration process is as follows [27]: Step 1: first, determine the initial value of ω cr according to the adjustment time of the system, and then let ω 0r � (2 ∼ 10)ω cr .
Step 2: keep ω 0r constant, and adjust ω cr in a small range until the system is stable. If adjustment of ω cr fails to stabilize the system, decrease or increase both ω cr and ω 0r in the same proportion, and then adjust ω cr individually to make the system stable.

Mathematical Problems in Engineering
Step 3: keep ω cr constant, and gradually decrease ω 0r . On the premise of ensuring the stability of the system, weigh the transient performance and noise suppression ability, and select the appropriate value for ω 0r .
Step 4: weigh the stability, transient performance, anti-interference, and noise suppression of the system, and determine the optimal values of ω 0r and ω cr .

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 is the tracking curves and tracking errors among PD, LADRC, and MLADRC are compared in Figure 4. As is seen from Figure 4, the trajectory tracking accuracy of MLADRC is the highest. e maximum tracking error of joint 1 is limited to ±5 × 10 − 4 rad and that of joint 2 is limited to ±2 × 10 − 4 rad. 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). e friction needs to be observed by LESO before it is compensated, which leads to a lag in compensation.

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 Δp 1 � 30%p 1 , Δp 2 � 20%p 2 , Δp 3 � 20%p 3 , the tracking errors with the friction and parameter perturbation are depicted in Figure 5.
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 τ d2 � 0.2 is applied to joint 2 between 4.5 and 5.5 seconds. e 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.
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. e 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. e torque curves are relatively smooth and there is no violent chattering.
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.

Experimental Platform.
A self-developed horizontal 2-DOF manipulator is used as the controlled object to conduct experimental researches on PD, LADRC, and MLADRC. e 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. e 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. e PD, LADRC, and MLADRC control algorithms run in STM32 to complete the calculation and output of the control quantity. e 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. e 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.

Trajectory Planning and Motion Control.
e experiment requires the end tip of the 2-DOF manipulator to track a circular trajectory with a diameter of 0.17 m. e circular trajectory is preset in the host control system, and its mathematical equation is expressed as x d � 0.285 + 0.085 sin(0.25π(t + 2))(m), where x d and y d 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 x d � l 1 cos q d1 + l 2 cos q d1 + q d2 , y d � l 1 sin q d1 + l 2 sin q d1 + q d2 . (37) 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 t an increment: t � t + Δt Step 2: solve equation (36) to get the position coordinate (x d , y d ) Step 3: substitute x d and y d into equation (38) to solve for q d1 and q d2 Step 4: send q d1 and q d2 to STM32 for tracking control   Mathematical Problems in Engineering 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. e 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. e 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. e control program flow is shown in Figure 10.

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).
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. e 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. e 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.

Conclusions
Aiming at the trajectory tracking control problem of robot manipulators, a more in-depth study is carried out based on the regular LADRC. e 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. e feedforward control is used to improve the trajectory tracking accuracy. e 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. e comparative simulations and experiments verify the excellent control performance of MLADRC.

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

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.