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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.

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 [

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 [

A simplified model of the 2-DOF manipulator is depicted in Figure

A simplified model of the 2-DOF manipulator.

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 as

Technical parameters of the BLDC reduction motor.

0.7 | 0.4 | 0.032 | 0.032 | 90 | 0.15 |

Since the inertial matrix

Considering the influence of friction, parameter perturbation, and external disturbances, we add these uncertainties to equation (

Equation (

According to the application of LADRC in MIMO systems [

We can observe from equation (

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.

LESO: the third-order LESO is used to dynamically estimate the total disturbances

where

Disturbance compensation:

where

State error feedback control: the PD controller is designed to control the double integrator, and its control algorithm is as follows:

where

Reference angular acceleration feedforward: in the case that the reference trajectory

It can be seen from equation (

Since the output torque of the BLDC reduction motor is proportional to the current and

The first-order LADRC is designed 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

Trajectory tracking control structure based on the MLADRC of joint 1.

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.

Considering the joint 1 system of

Meanwhile, equation (

Let

Assuming

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

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 (

Let

Since the estimation error of LESO has been proved to be bounded, there exists

Trajectory tracking control structure based on PD of joint 1.

In Matlab/Simulink, PD, LADRC, and MLADRC are applied for controlling plant (

Parameter values of the control system based on PD.

Parameter | Value (joint 1) | Value (joint 2) |

150 | 60 | |

15 | 5 | |

300 | 300 | |

50 | 50 |

Parameter values of the control system based on MLADRC.

Parameter | Value (joint 1) | Value (joint 2) |

240 | 300 | |

100 | 80 | |

200 | 200 | |

800 | 800 | |

2.5 | 2.5 |

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

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

MLADRC: MLADRC, i.e., “modified LADRC.” The structure of the trajectory tracking control system composed of MLADRC is plotted in Figure

where

In MLADRC,

Step 1: first, determine the initial value of

Step 2: keep

Step 3: keep

Step 4: weigh the stability, transient performance, anti-interference, and noise suppression of the system, and determine the optimal values of

After multiple tests and adjustments, the ideal parameter values are obtained as shown in Table

LADRC: LADRC, i.e., “regular LADRC.” The LADRC-based trajectory tracking control system is obtained by removing the reference angular acceleration feedforward from Figure

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

Comparison of the tracking performance among PD, LADRC, and MLADRC with the friction. (a) Trajectory tracking of joint 1. (b) Tracking errors of joint 1. (c) Trajectory tracking of joint 2. (d) Tracking errors of joint 2.

As is seen from Figure

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

Comparison of the tracking errors among PD, LADRC, and MLADRC with the parameter perturbation and friction. (a) Tracking errors of joint 1. (b) Tracking errors of joint 2.

From Figure

High-performance controllers must be able to reject external disturbances. To test the disturbance rejection property of MLADRC, a disturbance of

Comparison of the tracking errors among PD, LADRC, and MLADRC with the total disturbances. (a) Tracking errors of joint 1. (b) Tracking errors of joint 2.

Observations of the total disturbances of the MLADRC. (a) The total disturbances of joint 1. (b) The total disturbances of joint 2.

It can be observed from Figure

Control torques of the two joints based on the MLADRC. (a) Control torque of joint 1. (b) Control torque of joint 2.

According to Figures

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

2-DOF manipulator control experiment platform.

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.

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 as

From equation (

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 (

Step 1: give time

Step 2: solve equation (

Step 3: substitute

Step 4: send

Step 5: go back to the beginning of the loop

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

Control program flowchart of the real-time motion control system.

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

The tracking responses of the two joints. (a) Trajectory tracking of joint 1. (b) Tracking error of joint 1. (c) Trajectory tracking of joint 2. (d) Tracking error of joint 2.

The trajectory tracking curves of the end tip. (a) Trajectory tracking without external disturbances. (b) Trajectory tracking with the external disturbance.

From Figure

As can be seen from Figure

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.

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.

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

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

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).