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Abstract

This paper proposes an adaptive PID controller based on linear extended state observer (LESO) for the two-degree-of-freedom joint servomechanism of industrial robot with time-varying load, uncertainties of parameters and disturbance. The third-order extended state space equations of the system approximate model is established to obtain LESO which is applied to estimate the state values and the total disturbance. The model reference adaptive algorithm is used to estimate the variable moment of inertia to design the controller parameter for control law which is designed with disturbance compensation. By appropriately selecting the adaptive gain coefficient of model reference adaptive algorithm and the bandwidth of LESO, the influences of parameter uncertainty, unknown dynamics and disturbances are effectively attenuated. Simulation and experimental results show that the proposed method achieves both satisfactory disturbance rejection and tracking performances of the two-degree-of-freedom joint servomechanism.

Keywords

servomechanism linear extended state observer disturbance compensation inertia identification industrial robot

Introduction

Servomechanism is widely used in high-precision operations such as guidance of missiles and spacecrafts, antenna position control and precision manufacturing, higher requirements have put forward for the control performance of servo system, especially with the widespread use of industrial robots.¹ Servomechanism is a key technology of robotics with uncertainties including nonlinear time-varying load, external disturbance and parameter perturbations. It is important to research on the anti-disturbance control strategy of servomechanism to suppress the influence of the system uncertainties for improving the tracking accuracy and dynamic performance of servo system.^2,3

For the above problems, numerous advanced control techniques have been proposed, such as internal model control,⁴ adaptive control,^5,6 sliding mode control (SMC)⁷ and active disturbance rejection control (ADRC). Among these control techniques, ADRC has received substantial interest owing to its remarkable robustness against uncertainties and disturbances.⁸

Active disturbance rejection control was proposed by Jingqing Han. The key part of ADRC is the extended state observer (ESO) which can estimate and compensate the total disturbance of the system including the internal uncertainties and the unknown external disturbances.^9,10 In order to simplify parameters design, the linear active disturbance rejection controller (LADRC) with linear extended state observer (LESO) was proposed which adjust the bandwidth of the observer. The error feedback control law was replaced by common proportional-derivative control.¹¹ In the frequency domain analysis, LADRC was regarded as the feedback compensator.¹² A two degree-of-freedom PID controller was derived from LADRC which made the proposed controller more powerful and easy to tune the parameters in engineering practice.¹³ The estimation accuracy of ESO and the performance of ADRC were systematically analyzed. The relationship between the system dynamic characteristics and the controller parameters was discussed, and the engineering method of the control parameters was proposed.¹⁴ The adaptive ESO was presented which could automatically reduce the estimation error of the states and total disturbances in real time.¹⁵ For the filtering problem of general nonlinear uncertain systems, an extended state filter (ESF) with real-time estimation of system uncertainty was proposed.¹⁶ More improved approaches with ESO were presented by which the stability of the system were proved with total disturbance and uncertainties compensated.^17–20

PID controller has been widely used in industry which is attributed primarily to its simplicity and performance characteristics. However, the limitations of conventional PID control rapidly become evident, because the performance is often not satisfied in nonlinear systems such as a high-DOF robot. In order to overcome the limitations of traditional PID controller, an adaptive PID control strategy is necessary. Moreover, LESO has shown advantages in disturbance estimation and compensation with convenient parameter tuning.¹¹ Thus, in this paper, by borrowing ideas from the LESO and the inertia estimation approach, an adaptive PID controller based on LESO is proposed for high-accuracy control of the two-degree-of-freedom joint servo mechanism. The proposed approach presents a prescribed output tracking performance in the presence of parameter uncertainty, unknown dynamics and disturbances.

This paper is organized as follows. LESO is designed by established mathematical model to estimate and compensate disturbance. The inertia of the system is estimated with the method of parameter identification by which the observer parameter is adjusted to improve the estimation precision of the LESO. The PID controller with variable gain is designed for high accuracy control of the system. Simulations and experiments results are obtained to verify the high-performance feature of the proposed control strategies compared with traditional PID control.

Modeling

The dynamic equation of the robot with n degrees of freedom can be described in matrix form as

H (q) \ddot{q} + C (q, \dot{q}) + G (q) = τ = [\begin{matrix} τ_{1} \\ ⋮ \\ τ_{n} \end{matrix}]

(1)where

q, \dot{q}, \ddot{q} \in R^{n \times 1}

are the position, velocity and acceleration vectors of the joint, respectively, and

H (q) \in R^{n \times n}, C (q, \dot{q}) \in R^{n \times 1}

represent the inertial matrix and Coriolis force and centrifugal force matrix respectively;

G (q) \in R^{n \times 1}

is the gravitational term;

τ \in R^{n \times 1}

is the joint drive torque.²¹

This paper takes a two-degree-of-freedom robot as an example, as shown in Figure 1. The Cartesian coordinate system is established, the turning angles of joint one and joint two are θ₁、θ₂, the corresponding moments are τ ₁、 τ ₂, the masses are m₁、m₂, the rod lengths are l₁、l₂, and the distances of the center of mass C₁、C₂ from each joint connection are p₁、p₂.

Figure 1.

Diagram of 2-DOF joint robot.

The equation of motion of the two joints can be described as

{\begin{cases} τ_{1} = D_{11} {\ddot{θ}}_{1} + D_{12} {\ddot{θ}}_{2} + D_{112} {\dot{θ}}_{1} {\dot{θ}}_{2} + D_{122} {\dot{θ}}_{2}^{2} + D_{1} \\ τ_{2} = D_{21} {\ddot{θ}}_{1} + D_{22} {\ddot{θ}}_{2} + D_{212} {\dot{θ}}_{1} {\dot{θ}}_{2} + D_{222} {\dot{θ}}_{1}^{2} + D_{2} \end{cases}

(2)where

{\begin{cases} D_{11} = m_{1} p_{1}^{2} + m_{2} p_{2}^{2} + m_{2} l_{1}^{2} + 2 m_{2} l_{1} p_{2} \cos θ_{2} \\ D_{12} = m_{2} p_{2}^{2} + m_{2} l_{1} p_{2} \cos θ_{2} \\ D_{112} = - 2 m_{2} l_{1} p_{2} \sin θ_{2} \\ D_{122} = - m_{2} l_{1} p_{2} \sin θ_{2} \\ D_{1} = (m_{1} p_{1} + m_{2} l_{1}) g \sin θ_{1} + m_{2} p_{2} g \sin (θ_{1} + θ_{2}) \\ D_{21} = m_{2} p_{2}^{2} + m_{2} l_{1} p_{2} \cos θ_{2} \\ D_{22} = m_{2} p_{2}^{2} \\ D_{212} = - m_{2} l_{1} p_{2} \sin θ_{2} + m_{2} l_{1} p_{2} \sin θ_{2} \\ D_{222} = m_{2} l_{1} p_{2} \sin θ_{2} \\ D_{2} = m_{2} p_{2} g \sin (θ_{1} + θ_{2}) \end{cases}

(3)

Noting (2), variable matrices in (1) can be written as

{\begin{cases} H (q) = [\begin{matrix} D_{11} & D_{12} \\ D_{21} & D_{22} \end{matrix}] \\ C (q, \dot{q}) = [\begin{matrix} D_{112} {\dot{θ}}_{1} {\dot{θ}}_{2} + D_{122} {\dot{θ}}_{2}^{2} \\ D_{212} {\dot{θ}}_{1} {\dot{θ}}_{2} + D_{222} {\dot{θ}}_{1}^{2} \end{matrix}] \\ G (q) = [\begin{array}{l} D_{1} \\ D_{2} \end{array}] \end{cases}

(4)

When the load is added to the end of the joint, the driving torque can be expressed as

τ_{l o a d} = P^{T} (q) F

(5)Where F represents the force or moment of the load;

P^{T} (q)

is the Jacobian matrix, which linearly maps the generalized external force at the end to the corresponding joint driving torque. At this time, the torque of the loaded joint is

τ_{a l l} = τ + τ_{l o a d}

(6)

From (3), (4) and (6), we can obtain that the load torque of the joint is similar to feature of sine wave.²²

The servomechanism of the robot joint is driven by a surface-mounted permanent magnet synchronous motor. To design the controller, the motor is usually analyzed in the d-q synchronous rotating coordinate system and its voltage equation²³ is

{\begin{array}{l} u_{d} = R_{s} i_{d} + L_{d} \frac{d i_{d}}{d t} - ω_{m} P_{n} L_{q} i_{q} \\ u_{q} = R_{s} i_{q} + L_{q} \frac{d i_{q}}{d t} + C_{e} ω_{m} \end{array}

(7)where u_d 、 u_q are motor d and q axis stator voltage; R_s is armature resistance; L_d and L_q are motor d and q axis inductance; i_d and i_q are motor d and q axis stator current; C_e is back electromotive force coefficient; p_n is the number of pole pairs of the motor; ω_m is the angular speed of the motor.

The dynamic equations of the system²³ can be described as follows

{\begin{cases} J \frac{d ω_{m}}{d t} = T_{e} - T_{L} - B ω_{m} \\ T_{e} = K_{t} i_{q} \\ ω_{m} = \frac{d θ_{m}}{d t} \end{cases}

(8)where T_e is the motor electromagnetic torque; T_L is the load torque; J is the inertia of the system in the motor side; B is the viscous friction coefficient; K_t is the torque constant; θ_m is the angular displacement of the motor.

The servo system parameters used in this article are shown in Table 1.

Table 1.

Parameter list of the servomechanism.

Parameter	Units	Value
Rated speed n	r/min	3000
Rated power p	W	200
Rated torque Te	N·m	0.64
Peak torque Tp	N·m	1.91
Back EMF coefficient Ce	V/rad/s	0.0764
Torque coefficient K_t	N·m/A	0.112
Motor inertia J	kg·m²	0.175×10⁻⁴
Inductance factor Lq	H	1.01×10⁻³
Motor resistance Rs	Ω	0.15
Number of pole pairs pn	Pairs	4
Reduction ratio i	—	100:1
Speed feedback factor Kω	V·s/rad	9.55
PWM power amplification factor Ku	—	8.32

From (7), (8), the structure of the joint servo system is established. Ignoring the inductance with current and velocity loop to be open loop, the block diagram of the model is shown in Figure 2, which is used in the controller design.

Figure 2.

Joint servo system model of industrial robot.

In the Figure 2, θ_ref is the reference position signal; u(t) is the control input; K_u is the amplifier coefficient; i is the reducer speed ratio.

Considering the system model in Figure 2, dynamic equation of the servomechanism is given by

\ddot{θ} = - (\frac{B}{J} + \frac{K_{t} C_{e}}{J R_{s}}) \dot{θ} + \frac{K_{t} K_{u}}{J i R_{s}} u (t) - \frac{1}{J i} T_{L}

(9)where θ is the position angle of the load side;

\dot{θ}

is the angular velocity. Let

θ = x_{1}

\dot{θ} = x_{2}

, we can rewrite the system (9) in a state-space form as follows

\begin{array}{l} [\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} 0 & 1 \\ 0 & - \frac{B R_{s} + K_{t} C_{e}}{J R_{s}} \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] \\ + [\begin{matrix} 0 \\ \frac{K_{t} K_{u}}{J i R_{s}} \end{matrix}] u (t) + [\begin{matrix} 0 \\ - \frac{1}{J i} \end{matrix}] T_{L} \end{array}

(10)where x₁ is the position angle of the load side; x₂ is the angular velocity.

From (10), the extended state space form can be obtained as

{\begin{cases} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} + b u + a_{1} x_{2} \\ {\dot{x}}_{3} = g (x, T_{L}) \\ x_{1} = θ \end{cases}

(11)where

{\begin{cases} a_{1} = - \frac{B R_{s} + K_{t} C_{e}}{J R_{s}} \\ b = \frac{K_{t} K_{u}}{J i R_{s}} \end{cases}

(12)x₃ is an augmented state, and g(x,T_L) is unknown disturbance.

The three-loop model of the joint servomechanism is shown in Figure 3. In the figure, $G_{P} (s) 、 G_{V} (s) 、 G_{i} (s)$ are the controller transfer functions of the position loop, speed loop, and current loop, respectively, and $K_{ω}$ is the speed feedback coefficient.

Figure 3.

Three-loop model of the joint servomechanism.

Controller design

Active disturbance rejection control technology is utilized in servomechanism of industrial robot. The system structure is shown in the Figure 4 which is composed of tracking differentiator (TD), linear extended state observer (LESO) and PID controller.

Figure 4.

Block diagram of adaptive disturbance rejection control.

As shown in Figure 4, ${\hat{x}}_{1}$ is the estimated value of x₁; ${\hat{x}}_{2}$ is the estimated value of the speed; ${\hat{x}}_{3}$ is the estimated value of the unknown disturbance; $\hat{b}$ is the estimated value of b which is varied by the change of the rotational inertia.

Tracking Differentiator

Tracking differentiator²⁴ in discrete form is described by equation (13)

{\begin{cases} x_{r e f 1} (k + 1) = x_{r e f 1} (k) + h x_{r e f 2} (k) \\ x_{r e f 2} (k + 1) = x_{r e f 2} (k) + h f s t (\cdot) \end{cases}

(13)where h is the sampling period, fst (·) is given by

\begin{matrix} f s t (\cdot) = f s t (x_{r e f 1} (k) - θ_{r e f} (k), x_{r e f 2} (k), δ, h) \\ = {\begin{cases} - δ \frac{a}{d}, | a | \leq d \\ - δ sgn (a), | a | > d \end{cases} \end{matrix}

(14)

Let, then

a = {\begin{cases} r {}_{2}+ \frac{(a_{0} - d)}{2} sgn (y), | y | > d_{0} \\ r_{2} + \frac{y}{h}, | y | \leq d_{0} \end{cases}

(15)where

a_{0} = \sqrt{d^{2} + 8 δ | y |}, d = h δ, y = r_{1} + h r_{2}, d_{0} = h d

; θ_ref(k) is the position reference signal at k-th moment. δ and h are the two adjustable parameters in equation (14). δ is the parameter relating to the speed of tracking the signal, h plays a role in filtering noise, called filter factor. The output of TD are x_ref1 and x_ref2 which are approximated to

θ_{ref}

and

{\dot{θ}}_{ref}

respectively.

PID controller with variable gain based on linear extended state observer

From (11), LESO is designed as follows ¹¹

{\begin{cases} {\dot{\hat{x}}}_{1} = {\hat{x}}_{2} + 3 ω_{o} (x_{1} - {\hat{x}}_{1}) \\ {\dot{\hat{x}}}_{2} = {\hat{x}}_{3} + 3 ω_{o}^{2} (x_{1} - {\hat{x}}_{1}) + a_{1} {\hat{x}}_{2} + \hat{b} u \\ {\dot{\hat{x}}}_{3} = ω_{o}^{3} (x_{1} - {\hat{x}}_{1}) \\ x_{1} - {\hat{x}}_{1} = y - {\hat{x}}_{1} \end{cases}

(16)where ω_o is the bandwidth of the observer which is the parameter to be selected. The convergence of the LESO has been proved in [11].

Noting (11) and according to the theory of ADRC, the control law is given as

u = u_{0} - \frac{{\hat{x}}_{3} + a_{1} {\hat{x}}_{2}}{\hat{b}}

(17)

$u_{0}$ is expressed as

u_{0} = k_{p} (1 + \frac{1}{T_{i} s} + T_{D} s)

(18)

k_{p}

is the proportional coefficient;

T_{i}

is the integral time constant;

T_{D}

is the derivative time constant. If the estimation error of

x_{3}

is slight, the system (11) can be approximately written as the integral series system by the control law (17).

The joint servomechanism adopts three-loop PID control as shown in Figure 3. The position-loop PID controller is implemented in the real-time control system, and its gains tuned carefully via an error-and-try method are k_pp=5, k_pi=0.03 and k_pd=0, which denote the proportional gain, integral gain, and derivative gain respectively. Similarly, PID controller is used in the current loop and speed loop with the controller parameters k_ip=1, k_ii=0.01, k_id=0, k_vp=0.8, k_vi=0.8, k_vd=0.

The final PID parameters are shown in Table 2.

Table 2.

Parameters of PID controller.

	Proportion	Integral
position loop	5	0.03
Speed loop	0.8	0.8
Current loop	1	0.01

Inertia identification

Substituting (12) into (16) and noting (17), it can be obtained that the parameter b is affected by rotational inertia which is time-varying changed when the system is in motion and the control law resulting the dynamic performance of the system is related to $\hat{b}$ . Therefore, it is necessary to identify the variable inertia to make the system adaptive. ²⁵

In this paper, the rotational inertia is identified by Model Reference Adaptive System (MRAS). The reference model is the servomechanism and the adjustable model is derived from its mechanical equations. Denoting T_s is the sampling period of the system, system (8) is discretized. The discrete forms of system (8) at $k - 1$ , $k - 2$ moments are

{\begin{cases} ω_{m} (k) = ω_{m} (k - 1) + \frac{T_{s}}{J} [T_{e} (k - 1) - T_{L} (k - 1)] \\ ω_{m} (k - 1) = ω_{m} (k - 2) + \frac{T_{s}}{J} [T_{e} (k - 2) - T_{L} (k - 2)] \end{cases}

(19)

The rotational inertia identification algorithm is with much higher frequency than the load torque fluctuation. Therefore, the change of load torque can be ignored during the identification, i.e. $T_{L} (k - 1) = T_{L} (k - 2)$ . From (19), the reference model can be obtained as

ω_{m} (k) = 2 ω_{m} (k - 1) - ω_{m} (k - 2) + σ (k) U (k - 1)

(20)

Let $U (k - 1) = T_{e} (k - 1) - T_{e} (k - 2), σ (k) = \frac{T_{s}}{J}$ The adjustable model is expressed as

{\hat{ω}}_{m} (k) = 2 ω_{m} (k - 1) - ω_{m} (k - 2) + \hat{σ} (k - 1) U (k - 1)

(21)where

{\hat{ω}}_{m} (k)

is the estimated velocity;

\hat{σ}

is the estimated value of

σ

. Based on Popov hyperstability theory, the adaptive law of rotational inertia identification²⁵ can be obtained as

\hat{σ} (k) = \hat{σ} (k - 1) + β \frac{U (k - 1)}{1 + β U^{2} (k - 1)} ε (k)

(22)where

ε (k) = ω_{m} (k) - {\hat{ω}}_{m} (k)

; β is adaptive gain factor. Since T_s is known, the identification value of rotational inertia that can be found as

\hat{J} = T_{s} / \hat{σ}

At this point

\hat{b} = \frac{K_{t} K_{u}}{\hat{J} i R_{s}} = \frac{K_{t} K_{u} \hat{σ}}{T_{s} i R_{s}}

(23)

Simulation

In this section, the servomechanism under different operating conditions such as normal load condition, sudden disturbance at start time, and sudden change of load at steady state is constructed in Matlab/Simulink with variable load. Simulation parameters are set as follows: simulation step is 10 $μ s$ ; h=0.00,001, δ=15,000; $a_{1} = - 3276, ω_{o} = 1500$ ; the parameter b is variable with the change of rotational inertia;

The following three performance indexes will be used to measure the quality of each control algorithm, i.e. the maximum (M_e), average (μ), and standard deviation (σ) of the tracking errors.²⁶

The desired trajectory in the simulation is given as θ_ref=sin(t) rad with sinusoidal rotational inertia of the system $J = 0.1 + 0.03 \times s i n (3.14 t) k g \cdot m$ ; $T_{s} = 10 μ s$

The simulation results are shown in Figure 5 in which the convergence speed of the identification of inertia is related to the adaptive parameter β. In contrast, when the adaptive parameter β=3, the identification convergence speed is faster. The estimated error of J will be raised with β increasing. From equations (17) and (23), it can be seen that the identification value of J will affect the value of $\hat{b}$ , and the value of $\hat{b}$ is related to the control law, so the identification accuracy of J is quite critical. Here select β=3 as the system parameter value.

Figure 5.

Rotational inertia identification.

Step response of the system with proposed controller compared to PID controller are shown in Figure 6. As displayed, the position response under the proposed controller has almost no overshoot, while the overshoots under PID controller are relatively larger, which is 0.6%. Therefore, the proposed controller can solve the problem of position overshoot.

Figure 6.

System step response.

The desired position signal is θ_ref=sin (2t) rad, the simulation results are shown in Figures 7 to 9. In addition, the performance indexes of sine position tracking error are shown in Table 3. As shown, the maximum position error with PID control is 0.0189 rad, while the maximum position tracking error of the proposed controller is 0.0012 rad. The tracking accuracy of the proposed controller is further improved, and the position tracking error is reduced by 93.3%. Therefore, the system with proposed controller has higher performance.

Figure 7.

Position tracking.

Figure 8.

Tracking error.

Figure 9.

Controller output.

Table 3.

Performance indexes of sinusoidal position tracking error in simulation.

Indexes	M _e	μ	σ
PID	0.0189	0.00085	0.0135
PID+LESO	0.0012	0.00008	0.0009

In order to illustrate the system performance with added variable load, the position reference signal is $θ_{r e f} = 1 r a d$ , the external disturbance $T_{L} = 3 s i n (2 t) N \cdot m$ . Figure 10 show that the steady state error of the system using PID control is 0.002 rad with the regulation time of 0.5s compared to the proposed controller with the error of 0.0001 rad.

Figure 10.

Step response with variable load.

The step load applied to the system is shown in Figure 11. The position response under different algorithms are shown in Figure 12. Figure 12 obviously show that LESO can effectively estimate and compensate the disturbance. The anti-disturbance performance of the proposed controller is improved by 84.6% compared with PID control and the system has higher accuracy with more rapid response.

Figure 11.

Step load.

Figure 12.

Position response curve of sudden load.

Comparative experimental results

Experiment setup

To verify the effectiveness of the proposed controller, the verification platform for the joint servomechanism of the industrial robot has been set up in Figure 13. The platform is used to simulate the load converted to a single joint of the industrial robot which is obtained by mathematical modeling. The platform consists of motor, reducer, position sensor, torque sensor, slider bar and control system. The device can change the inertia of the slider bar when the slider moves, changing the distance between the slider and the center of rotation. The permanent magnet synchronous motor is the actuator of the servo system. The IR2136 driver and the DSP-TMS320f28335 are used to control the servomechanism with control software. The torque sensor is used for data acquisition of the real torque value. Figure 14 show that the load torque of the platform presents the similar change of sinusoidal torque when the system is in motion.

Figure 13.

Experimental platform.

Figure 14.

Torque curve of the experimental platform.

According to the simulation results, further adjust the experimental parameters, and finally select h=0.00,001, δ=15,000 $a_{1} = - 3276, ω_{o} = 1500$ , β=3.

Experimental results

The position reference signal is set as $θ_{r e f} = 180 °$ . The experimental results of position tracking and tracking error with the proposed controller compared to PID control are shown in Figure 15 and Figure 16. And the experimental results in terms of performance indexes after reaching the steady state are given in Table 4. As shown in the table, the proposed controller performs better performance in all indexes, as compared with PID. It is worth noting that, from the standard deviation performance index σ given in Table 4, the smoother tracking performance is achieved by the proposed controller. Moreover, The steady state error of the proposed controller is reduced by about 70.4%, which can indicate that the proposed controller has better tracking performance.

Figure 15.

Step response.

Figure 16.

Step signal position tracking error.

Table 4.

Performance indexes of step response.

Indexes	M _e	μ	σ
PID	0.2966	0.0275	0.1826
PID+LESO	0.0879	0.0193	0.0418

To verify the performance with the controller under large rotational inertia, an additional weight of 1 kg is attached to the end of the slider bar at the beginning. The desired position is $θ_{r e f} = 180 °$ . The experimental results of position tracking with proposed controller and PID control are shown in Figure 17. The performance indexes with this case are collected in Table 5. From these results, again, it can be observed that the proposed controller has the excellent tracking performance, with almost 77% performance improvement compared to the PID controller. It is worth noting that the change of standard deviation performance index σ given in Table 5 and Table 4 is not obvious, which can indicate that LESO can efficiently compensate the disturbance.

Figure 17.

Step response with additional loading.

Table 5.

Performance indexes of step response with additional weight.

Indexes	M _e	μ	σ
PID	0.4065	0.0196	0.2592
PID+LESO	0.0934	0.0183	0.0453

Weight of 2 kg is added to the end of the slider bar in steady state to observe the performance of the system. Figure 18 show that under the sudden increase of load, the position error by disturbance for PID control is 2.8°, with the regulation time of 2s and the steady-state error of 0.1°. The position error by disturbance for the proposed controller is 0.4°, with regulation time of 0.3s and the steady-state error of 0.05°. The results show that the proposed controller is better than PID control for disturbance rejection, and its anti-disturbance performance is improved by 85.7%.

Figure 18.

Position tracking under sudden step load.

Setting the position reference signal as $θ_{r e f} = 180 * s i n (t / 3) °$ . The position tracking curve and tracking error are shown in Figure 19, Figure 20 and Figure 21. And Figure 22 is the control output. The performance indexes of sine position tracking error are shown in Table 6. As shown, the proposed controller has better starting characteristics. From Figure 21, we can obtain that when the angle reaches the position of 180°, the motor output torque changes. At this time, the maximum position tracking error with PID control is 3.62°, compared to the maximum position tracking error is 2.15° with the proposed controller. The standard deviation of the tracking error of the proposed controller is significantly smaller than that of PID, which is reduced by 43.6%. The experimental results illustrate that the system with the proposed controller has higher dynamic performance and strong ability of disturbance rejection.

Figure 19.

Position tracking curve.

Figure 20.

Partial position tracking.

Figure 21.

Position tracking error.

Figure 22.

Controller output.

Table 6.

Performance indexes of sinusoidal position tracking error.

Indexes	M _e	μ	σ
PID	10.21	0.338	4.827
PID+LESO	9.175	0.179	2.724

Conclusion

In this paper, the method of PID with variable gain based on LESO is proposed for servomechanism of industrial robot. LESO with adaptive parameter b is designed for variable inertia load to estimate the states values more accurately. The PID controller and the proposed controller are compared under different conditions of inertia load, including normal variable load, additional load and step load in steady state. The proposed controller takes into account the effect of model uncertainties with the inertia load, modeled and unmodeled disturbances. LESO with variable parameter can estimate and compensate the disturbance effectively to improve the tracking precision and system performance. The comparative experimental results show that the proposed controller significantly improves tracking performance under different load conditions compared with PID, and the error of disturbance rejection is reduced by 85.7%, which proves the effectiveness of the proposed algorithms in servomechanism of industrial robot.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Zhejiang Province under Grant No (LQ19E050009).

ORCID iD

Dafang Chen

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Active disturbance rejection control based on inertia estimation and variable gain for servomechanism of industrial robot

Abstract

Keywords

Introduction

Modeling

Controller design

Tracking Differentiator

PID controller with variable gain based on linear extended state observer

Inertia identification

Simulation

Comparative experimental results

Experiment setup

Experimental results

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References