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Edited by: Yongping Pan, National University of Singapore, Singapore

Reviewed by: Jie Ling, Wuhan University, China; Xinan Pan, Shenyang Institute of Automation (CAS), China

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The robot arm with flexible joint has good environmental adaptability and human robot interaction ability. However, the controller for such robot mostly relies on data acquisition of multiple sensors, which is greatly disturbed by external factors, resulting in a decrease in control precision. Aiming at the control problem of the robot arm with flexible joint under the condition of incomplete state feedback, this paper proposes a control method based on closed-loop PD (Proportional-Derivative) controller and EKF (Extended Kalman Filter) state observer. Firstly, the state equation of the control system is established according to the non-linear dynamic model of the robot system. Then, a state prediction observer based on EKF is designed. The state of the motor is used to estimate the output state, and this method reduces the number of sensors and external interference. The Lyapunov method is used to analyze the stability of the system. Finally, the proposed control algorithm is applied to the trajectory control of the flexible robot according to the stability conditions, and compared with the PD control algorithm based on sensor data acquisition under the same experimental conditions, and the PD controller based on sensor data acquisition under the same test conditions. The experimental data of comparison experiments show that the proposed control algorithm is effective and has excellent trajectory tracking performance.

With the increasing use of robots in the fields of industry, the rehabilitation, aviation, and marine exploration, the demand for robots that can adapt to complex environments and enable human-robot interaction is increasing, which introduces flexible structure onto robotic joints (Schiavi et al.,

In recent years, research on the control of flexible-joint robots has become more and more attractive. A spring-damping mode was first proposed to simplify the flexible-joint, and at the same time, the flexible-joint is divided into two subsystems for control by integral flow and perturbation theory (Spong,

In order to solve the problem of jitter and friction in flexible robot tracking control, the adaptive CFBC (command-filtered backstepping control) was proposed to improve tracking accuracy (Pan et al.,

Throughout the above control methods, the classic PD controller, with its “natural” anti-interference and model-independence, is widely used in the control of series elastic actuators by matching feedforward control (Zhu et al.,

In order to solve the above problems, this paper first proposes the state estimation of non-linear stiffness-driven flexible robot with EKF, as it has good convergence and low computational complexity, and can handle system uncertainty and external disturbances in real time (Reif et al.,

This paper is organized as follows. The model, principle and dynamics analysis of the flexible-joint robot are introduced in section Robotic Prototype and its Dynamic Model. Subsequently, the design of the closed-loop PD controller and EKF state observer is introduced in section EKF Based Controller. The stability analysis based on Lyapunov method is presented in section Lyapunov Stability Analysis. The experimental results are presented in section Experimental Results. Finally, a conclusion is provided in section Conclusion.

In order to demonstrate our method, a 3-DOF robot with flexible joint is introduced in this section to verify the algorithm. As shown in

Robotic prototype.

Since the control target of this paper is the trajectory output of the tip, based on the configuration of the robot, the variables of the three joints are respectively calculated by the inverse kinematics according to the desired trajectory output, and the three joints are respectively controlled according to the timing.

The structure of the three actuators is basically the same. In this paper, the first joint actuator is taken as an example to introduce the structure. As shown in

Actuator structure

The non-linear elastic structure is shown in

In addition, the change of the deflection angle also affects the vertical component of the contact point position. Therefore, the relationship between the positional change in the vertical direction of the contact point and the pressing force is no longer a simple proportional relationship, but the combination of the deflection and the deflection angle. In short, the contour curve of the contact surface determines the relationship between the pressing force and displacement of the contact point, that is, the stiffness variation curve. The specific design scheme and the non-linear mechanism have been deeply studied by the researcher group (Lan and Song,

Non-linear stiffness actuator can be divided into power systems, transmission systems, elastic structures, and external loads. The power system is the motor combination, which mainly includes the motor rotor and the gear reducer. The equivalent moment of inertia of the motor combination can be obtained from the dynamics model of the motor combination. The dynamic equation of the rotor of the motor is:

where _{r} and _{r} are the moment of inertia and damping of the rotor of the motor respectively; _{m} is the torque generated by the rotor of the motor; τ_{r} is the torque output by the rotor of the motor.

The dynamic equation of the motor reducer is:

where _{g} and _{g} are the moment of inertia and damping of the motor reducer respectively; _{1} is the reduction ratio; τ_{g} is the torque output by the motor reducer.

Since the motor rotor and the gear reducer are rigidly connected, the following relationship is used:

where θ_{r} and θ_{g} are the motor rotor angle and the gear reducer angle respectively. Combined with Equations (1–4) can be obtained:

The schematic diagram of the actuator from the motor combination to the output is shown in

Schematic diagram of the actuator.

The dynamic equation of the outer cylinder section is:

where: _{w} and _{w} are the moment of inertia and damping of the outer drum, respectively; _{2} is the reduction ratio of the wire drive, and the relationship between the angular velocity and the angular velocity of the outer cylinder is:

where θ_{w} is the angle of rotation of the outer cylinder for the non-linear stiffness drive, simultaneous Equations (4–6) can obtain:

Then the equivalent dynamic equation of the motor assembly to the elastic part is:

where

The dynamic equation of the outer cylinder part is:

where: _{e} and _{e} are the moment of inertia and damping of the external load, respectively; τ_{e} is the output torque of the drive;

It can be seen from the above formula that the dynamic equation from the motor to the output shaft without considering the external torque input is:

The EKF based PD controller is shown in

EKF based PD controller.

In the control system, τ_{m} can be considered to consist of two parts:

where τ_{dy} is the part consumed by the equilibrium dynamics, and its expression is:

τ_{d} is the torque required for the end output, which is output by the PD controller. The design expression is as follows:

where _{p} is the proportional stiffness coefficient and _{d} is the differential damping coefficient, substituting (11–13) into (10), we can get:

It can be simplified to:

where _{0} = 0;

The formula is a typical input-output equation with a derivative term, so the state variables are chosen as follows:

where _{1} = _{1} − _{1}_{0}, then the equation of state of the system is:

where _{2} = (_{2} − _{2}_{0}) − _{1}_{1}, rewritten into a matrix form:

According to the control frame we designed in the previous section, we can see that the PD position controller based on EKF requires the output shaft angle and angular velocity to be the feedback amount. In order to solve the sensor's measurement interference, cost, and structural design issues, we use the EKF state observer to predict the angle of the output shaft and the angular velocity. The inputs are only the angle and angular velocity of the motor. The angular acceleration is obtained from the first derivative of the angular velocity and filtered by a low-pass filter to eliminate high-frequency interference.

According to [29], the relationship between the output torque of the elastic component and the rotor angle of the motor and the output angle of the shutdown section is as follows:

Then the overall dynamic equation can be written as:

According to the formula, it can be seen that the experimental platform of this paper is a typical non-linear system. According to the EKF observation method in document [27], combined with the control objectives of this paper, the state variables are defined as:

Deriving it to time

where

where

Define the observation vector as:

Then the state observation matrix is:

The EKF iteration formula is as follows:

where

According to the research of extended Kalman filter, the stability of the control system of the flexible joint robot proposed in this paper is that the overall PD control system is stable, and the EKF observer is stable. The system stability analysis in this paper is divided into the following two steps:

Step 1: Proof of PD controller stability.

The Lyapunov method is used to prove the stability of the controller. Therefore, the previous system state Equation (18) can be written as:

where

Define the Lyapunov equation as:

where

It is a positive definite matrix. Then the derivative of the Lyapunov equation is:

Through the adjustment of the PD parameters, it can make

Step 2: Proof of EKF observer stability

Defining observation error:

Expand

where

Define the Lyapunov equation as:

where Π = ^{−1}, then:

Assumption: _{α}, _{β},

Lemma: According to the assumptions, there are ε > 0 and κ > 0, then:

for any ‖

Proof: According to ^{−1} and assumption, it can be obtained that:

Using triangular inequalities, ^{T}^{−1}, Π = ^{−1}, and ‖

Inequality can be obtained as follows according to assumption:

According to the lemma and ^{τ}^{−1}

where

By using separation variable method, we can obtain that

so Ẇ(

That is, the EKF state observer is exponentially stable.

In summary, step 1 and step 2 respectively prove that the PD control system is stable and the EKF state observer is stable. Therefore, the stability of the incomplete state feedback control system of the entire flexible joint robot is proved.

In this part, the proposed control algorithm is applied to the prototype to prove the feasibility and stability of the control algorithm to compare with the sensor-based PD trajectory controller under the same experimental conditions.

A 64-bit-Windows-8.1-based host computer with an Intel Core i7 processor @2.40 GHz and 8-GB RAM is used to run the Kalman estimation and calculate the input torque to the motor. The control algorithm is able to operate on an execution rate of 1 kHz using Visual C++ 2010, which is enough for real-time applications. The DSP board is used to read, process, and calculate the signal from the motor encoder and transmit it to the computer, that is, obtain the real-time position information of the motor through the QEP module of the DSP chip. Then the results calculated by the host computer are sent to the DSP board through a RS232 serial port. The DSP board then converts the input torque command into a PWM wave signal to drive the motor, and the A-D electromagnetic tracking system (trakSTAR, produced by NDI) is used to measure the position of the robot end. Through the USB cable, the location data is sent to the host for comparative verification of the experimental results. The entire experimental platform is shown in

The entire experimental platform.

For the fairness of the experiment and the validity of the comparison verification, the experiment was carried out in the same environment using the same machine, and the same PD controller parameters were used. The desired trajectory is a closed circular trajectory. The trajectory tracking results are shown in

Experimental results

To further analyze the experimental data, define the error mean square error:

where ξ_{e}(_{exp}(

Through the experimental results shown in

Trajectory tracking variance mean square.

Comparison of tracking error of each joint.

The EKF observer in the control system can handle external disturbances in real time. In order to prove its ability to handle real-time interference, a force of sinusoidal variation along the direction of the guide rail is applied at the end of the robot. The force changes are shown in

Disturbance Force.

Experimental results with disturbance force

Trajectory tracking variance mean square with disturbance force.

Comparison of tracking error with disturbance force of each joint.

From

In this paper, the trajectory tracking control problem of flexible joint robot is discussed. Aiming at the problem that sensor data acquisition is susceptible to interference, an EKF-based PD controller is proposed. The EKF state observer is designed for the control target to observe the output position, and only the position and speed feedback amount of the motor rotor is needed. And the stability analysis of the designed control system is given according to the Lyapunov method. Finally, the effectiveness and superiority of the proposed control algorithm are verified by experiments.

All datasets generated for this study are included in the manuscript and/or the supplementary files.

TM: theoretical analysis. ZS: guide control plan. ZX: guide doing experiment. JD: guide writing paper.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was supported by Tianjin Municipal Science and Technology Department Program (Grant No. 17JCZDJC30300), the Natural Science Foundation of China (Project No. 51475322), and the Programme of Introducing Talents of Discipline to Universities (111 Program) under Grant No. B16034.