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Abstract

In view of the system uncertainty factors and bilateral drive synchronization problems of XY platform driven by stepping motor, on the basis of the mathematical model of single axis servo system, in order to improve traceability and disturbance rejection in X,Y direction, a double loop structure with PID controller as speed loop and active disturbance rejection as position loop is proposed to solve the contradiction between rapidity and overshoot, moreover, an ESO performance analysis method is proposed, and the parameter tuning rules of ADRC is summarized. On this basis, a deviation self-coupling compensation control strategy is proposed to solve the problem of bilateral drive synchronization in axis X direction, and the mechanism and parameter adjusting method of the compensator are studied. The comparative experiment between ADRC+PID+deviation self-coupling compensation and PID+PID+deviation self-coupling compensation, the comparative experiment between ADRC+PID+deviation self-coupling compensation and ADRC+PID are conducted, respectively. Finally, the experiment of synthetic trajectory of X and Y is carried out, the simulation and experimental results demonstrate that the response time of the control strategy proposed in this paper is 0.5 s, which is 0.2 s faster than PID’s, and the disturbance recovery time is 50% of PID control. Therefore, the proposed ADRC+PID+deviation self-coupling compensation control strategy can not only effectively improve the tracking performance of the XY platform system, but also enhance the robustness of the system.

Keywords

XY platform bilateral drive ADRC deviation self-coupling compensator (DSCC)synchronization

Introduction

XY platform is widely used in CNC machine tools, integrated circuit manufacturing, microelectronic packaging, and other fields. Usually, the XY platform adopts a unilateral drive. However, with the demand of the production process for long strokes, large driving force, and high load capacity, the uniaxial drive XY platform is limited. In literature,¹ Biaxial drive refers to the movement in a certain direction is driven by two motors, which has the advantages of the large driving force, strong load capacity, and large stroke, but also brings low synchronous control accuracy. In literature,² in order to improve the servo performance of the multi-axis linkage system, measures need to be taken in two aspects. One is to ensure the accuracy and dynamic performance of single-axis motion, and the other is to ensure the synchronous motion accuracy between multi-axis. Literature³ used the iterative learning strategy to achieve coordinated control, but the trajectory tracking accuracy is very low at the initial iteration, which is difficult to meet the high precision requirements. In literature,⁴ although Elman neural network complementary sliding mode control method improves the disturbance rejection of the single-axis system and improves the synchronization control performance of the biaxial system to a certain extent, but there is an overfitting phenomenon. Literature⁵ proposed a reduced order disturbance observer is proposed to improve the disturbance rejection of permanent magnet synchronous motor, but the observation accuracy depends on the model matching degree. In literature,⁶ in view of wheel permanent-magnet synchronous motor, a compound sliding mode disturbance observer is presented to compensate deadbeat prediction current and improve the disturbance rejection of system, literature⁷ proposed the fuzzy proportional integral sliding mode control method to deal with the unknown nonlinear system, literature⁸ presented hierarchical sliding mode control algorithm to realize the synchronous tracking performance of the control system, literature⁹ proposed the intelligent backstepping terminal sliding mode control method to improve the tracking accuracy of the control system, but four methods have an inevitable chattering phenomenon. Literature¹⁰ used a fuzzy adaptive control to improve the tracking synchronization performance, but the accuracy depends on the reference model. Literature¹¹ introduced a zero-phase control technology to reduce tracking error, but its ability to suppress disturbance is poor. Literature¹² used A $H_{\infty}$ control to reduce the synchronization error of multi axis drive, but it is necessary to reasonably select the sensitivity function and complementary sensitivity function. Literature¹³ used a cross coupled PID control to improve synchronous control performance, but PID control has the contradiction between overshoot and rapidity. Literature¹⁴ used a neural network PID is used to improve the deviation coupling synchronous control, but there are some problems such as local minimization and slow convergence speed.

In addition, the stepper motor is widely used in XY platforms because of its simple control, low cost, and easy maintenance, but there are also problems of step loss and oscillation, which will affect its application in precision control engineering. In literature,¹⁵ in order to solve the problem of step loss, the acceleration and deceleration curve method is used to reduce the step loss of the motor, but the error caused by the step loss cannot be completely eliminated. Literature¹⁶ proposed a fuzzy self-tuning PID control method to improve the position control accuracy of the stepper motor, but it is difficult to balance the contradiction between rapidity and overshoot. Literature¹⁷ used a parabolic acceleration and deceleration curve algorithm is used to optimize the open-loop control of the stepper motor, but the system accuracy is still low. In literature,¹⁸ in order to improve the tracking accuracy of the stepper motor drive system, the variable structure control method is adopted to improve the control accuracy of the system, but there is chattering. Literature¹⁹ proposed a multi-objective particle swarm optimization method to improve the control accuracy of the system, but the weighted objective function is complex. Literature²⁰ presented an additional iterative learning control method, but it is not suitable for non-repetitive tasks.

Aiming at the defects of conventional PID algorithm, Han proposed active disturbance rejection control,²¹ which is widely used in aerospace, electric power, precision machining and chemical systems. It has the advantages of high tracking accuracy and strong disturbance rejection.^21,22 XY platform is a kind of precision servo device. Its tracking, immunity and synchronization when driven by multiple motors are particularly important. Therefore, this paper adopts the double closed-loop control structure, proposes that the inner loop with PID control and the outer loop with active disturbance rejection control structure, and proposes a new ESO stability analysis method. PID is used as the inner loop to improve the speed tracking, which provides the basis for the application of second-order active disturbance rejection, and the active disturbance rejection is used as the outer loop to solve the contradiction between rapidity and overshoot, and improve the tracking and disturbance rejection. Then, to solve the synchronization problem of directional dual-axis drive, a cross-compensation control method, namely deviation self-coupling compensation (DSCC) is proposed to compensate for the asynchronous problem caused by the mismatch between the two axes. Finally, for verifying efficiency of the proposed method in improving the robustness and trajectory tracking accuracy of XY platform servo system, this paper carries out the experiment comparison between ADRC+PID+DSCC structure and ADRC+PID structure, the experiment comparison between ADRC+PID+DSCC structure and PID+PID+DSCC structure under the condition of different disturbance action points in the X direction of bilateral drive, as well as synthetic trajectory experimental of X,Y directions.

System structure and mathematical model

The X-direction of the platform is synchronously driven by two stepper motors, namely bilateral drive. The Y-direction of the platform is driven by a single stepper motor, namely unilateral drive. The transmission system of each axis is composed of driver, stepper motor, encoder, and leadscrews. The system structure is shown in Figure 1.

Figure 1.

Structure diagram of the system.

The stepper motor is the source of the driving force of the system. Its basic equations include voltage equation, torque equation, and motion equation. Under guaranteeing the premise of the control performance, the assumptions are as following: (1) the magnetic flux leakage of the permanent magnet loop, the stator pole and the end are not taken consideration; (2) ignoring the influence of saturation; (3) ignoring the influence of hysteresis and eddy current and the harmonic component of stator coil self-inductance.

Stepper motor a, b two-phase voltage equations can be expressed as

{\begin{matrix} u_{a} = L \frac{d i_{a}}{dt} + R i_{a} - k_{e} ω \sin (N_{r} θ) \\ u_{b} = L \frac{d i_{b}}{dt} + R i_{b} + k_{e} ω \cos (N_{r} θ) \end{matrix}

(1)

Where, $u_{a}$ , $u_{b}$ and $i_{a}$ , $i_{a}$ are voltage and current of a and b phase winding respectively, $R$ is the winding resistance, L is the equivalent inductance, $k_{e}$ is the torque coefficient, $θ$ is the angle of rotation, $ω$ is the revolution speed, $N_{r}$ is the number of rotor teeth.

The torque equation is written as

T_{e} = - k_{e} i_{a} \sin (Nr θ) + k_{e} i_{b} \cos (Nr θ)

(2)

The motion equation is

J \frac{d ω}{dt} + B ω + T_{L} = T_{e}

(3)

\frac{d θ}{dt} = ω

(4)

Where, $T_{e}$ is electromagnetic torque, $T_{L}$ is load torque, J is the moment of inertia, and B is the coefficient of viscosity.

According to equations (1) and (2), the two-phase hybrid stepper motor is highly nonlinear and coupling.

Since the motor speed has a large time constant relative to the electrical signal, the Counter EMF adjustment process is much larger than the current adjustment process. Therefore, according to equation (1), the Counter EMF part can be regarded as an uncertain term or ignored, and the transfer functions of a and b phase current and voltage can be obtained as

G_{i} (s) = \frac{1}{Ls + R} + Δ (s) or G_{i} (s) \approx \frac{1}{Ls + R}

(5)

In order to simplify the electromagnetic torque equation, the DQ transformation is introduced for linearization,²³ and the DQ transformation relationship is

[\begin{matrix} i_{d} \\ i_{q} \end{matrix}] = [\begin{matrix} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{matrix}] [\begin{matrix} i_{a} \\ i_{b} \end{matrix}]

(6)

Therefore, equation (2) electromagnetic torque equation is converted to

T_{e} = k'_{e} i_{q}

(7)

When $T_{L} = 0$ , it can be obtained from equations (3) and (7) as

G_{0} (s) = \frac{ω (s)}{i_{q} (s)} = \frac{{k'}_{e}}{Js + B}

(8)

The digital/pulse signal converter (D/F) and the driver are equivalent to a proportional element, and the transfer function is set to

G_{du} (s) = k_{1}

(9)

The transfer function between the running distance of the lead screw and the rotation angle of the motor is

G_{θ p} (s) = \frac{p (s)}{θ (s)} = k_{2}

(10)

The transfer control block diagram from the controller output to the screw slider position is shown in Figure 2.

Figure 2.

Block diagram of system transfer control.

Design of controller

The X-direction position control adopts the master-slave mode, that is the drive shaft of stepper motor 1 is active and the drive shaft of stepper motor 2 is slave driven. The double closed-loop structure is adopted to the driving shaft, the outer loop (position loop) adopts active disturbance rejection controller, and the inner loop (speed loop) adopts PID controller. Speed single closed loop structure is adopted to the slave shaft. In order to improve the synchronization of the two axes in the X-direction, the two-axis position deviation self-coupling compensation control is introduced. The control structure is shown in Figure 3. The Y-direction position adopts unilateral drive, and its control mode is the same as the active shaft control in the X-direction.

Figure 3.

Block diagram of X-direction control.

Design of inner-loop controller

The double closed-loop control structure of active shaft is shown in Figure 4. APR is the outer loop position controller and ASR is the inner loop controller.

Figure 4.

Block diagram of the double closed-loop structure.

In order to eliminate the adverse effect of large inertia link, and improve speed tracking, the speed loop controller adopts PD control, and its transfer function is written as

G_{ASR} (s) = {k_{s}}_{i} + K_{sd} s = Js + B

(11)

Then the closed-loop transfer function of the speed loop is

G_{φ s} (s) = \frac{k_{1} k_{e}^{'}}{Ls + R + k_{1} k_{e}^{'}}

(12)

Design of outer loop controller

When the speed loop is controlled by equation (11), the second-order object of the position loop-controlled object can be obtained as

Gp (s) = G_{φ s} (s) \frac{1}{s} k_{2} = \frac{k_{1} k_{2} k_{e}^{'}}{s (Ls + R + k_{1} k_{e}^{'})}

(13)

For the second-order controlled object, the second-order ADRC is adopted. Let the differential equation of the second-order system in equation (13) $\overset{\cdot\cdot}{y} = - a_{1} \overset{\cdot}{y} - a_{2} y + f (y, \overset{\cdot}{y}, w, t) + bu$ . $- a_{1} \overset{\cdot}{y} - a_{2} y$ is the known modeling dynamics of the system; $f_{0} (y, \overset{\cdot}{y}, w, t)$ is the sum of unknown modeling and external disturbance; b is the control gain. Let $x_{1} = y$ , $x_{2} = \overset{\cdot}{y}$ , $x_{3} = f (y, \overset{\cdot}{y}, w, t) = - a_{1} \overset{\cdot}{y} - a_{2} y + f_{0} (y, \overset{\cdot}{y}, w, t) + (b - b_{0)} u$ , the equation of state can be obtained as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = x_{3} + b_{0} u \\ {\overset{\cdot}{x}}_{3} = \overset{\cdot}{f} (y, \overset{\cdot}{y}, w, t) \end{matrix}

(14)

Where, $x_{1}$ , $x_{2}$ , and $x_{3}$ are system state variables, and $b_{0}$ is the approximate estimate of $b$ .

If $x_{3}$ can be obtained, then let $u = \frac{- x_{3} + u_{0}}{b_{0}}$ , the system can be transformed into an integral series type. To obtain $x_{3}$ , a nonlinear extended state observer (ESO)^21,22 is designed for equation (14) as following

{\begin{matrix} e = z_{1} - y \\ {\overset{\cdot}{z}}_{1} = z_{2} - β_{01} e \\ {\overset{\cdot}{z}}_{2} = z_{3} - β_{02} fal (e, α_{1}, δ) + b_{0} u \\ {\overset{\cdot}{z}}_{3} = - β_{03} fal (e, α_{2}, δ) \end{matrix}

(15)

Where, $fal (e, α, δ) = {\begin{matrix} \frac{e}{δ^{1 - α}}, | e | \leq δ \\ {| e |}^{α} sign (e), | e | > δ \end{matrix}$ , ${β_{0}}_{1}$ , $β_{02}$ , and $b_{0}$ are respectively output error correction gains, which determines the observation error and stability of the observer.

The purpose of error feedback control is to eliminate the error, but directly taking the error between the expected value and the actual value will lead to a large amount of initial output control and easy overshoot. The tracking differentiator proposed in Han²¹ and Gao²² is used to process the given signal, and the contradiction between rapidity and overshoot is solved by reasonably arranging the transition process and reasonably extracting the differential. Its equation is written as

{\begin{matrix} {\overset{\cdot}{v}}_{1} = v_{2} \\ {\overset{\cdot}{v}}_{2} = fst (v_{1} - v, v_{2}, r, h) \end{matrix}

(16)

Where, $v (t)$ is a given signal, $v_{1}$ and $v_{2}$ are the output of tracking differentiator, $h$ is the integration step, $r$ determines the speed of the tracking transition process, and fst is the optimal function in the form of $fst = {\begin{matrix} - rsign (a), | a | > δ \\ - r \frac{a}{δ}, | a | \leq δ \end{matrix}$ .

Since the nonlinear feedback control is easier to obtain better performance than the linear feedback control, the error nonlinear feedback control strategy is adopted for the system transformed into the integral series type,²⁴ and its expression is

{\begin{matrix} e_{1} = v_{1} - z_{1}, e_{2} = v_{2} - z_{2} \\ u_{0} = β_{1} fal (e_{1}, α_{1}, δ_{1}) + β_{2} fal (e_{2}, α_{2}, δ_{2}) \\ u = u_{0} - z_{3} / b_{0} \end{matrix}

(17)

Where, $β_{1}$ and $β_{2}$ are respectively the parameters of the nonlinear error feedback control law.

In summary, the system structure adopted by ADRC is shown in Figure 5.

Figure 5.

Block diagram of ADRC.

Performance analysis of ESO

Let the errors of the observer $e_{1} = z_{1} - x_{1}$ , $e_{2} = z_{2} - x_{2}$ , $e_{3} = z_{3} - x_{3}$ which are putting into equations (14) and (15), then

{\begin{matrix} {\overset{\cdot}{z}}_{1} - {\overset{\cdot}{x}}_{1} = z_{2} - β_{01} (z_{1} - x_{1}) - x_{2} = {\overset{\cdot}{e}}_{1} = e_{2} - β_{01} e_{1} \\ {\overset{\cdot}{z}}_{2} - {\overset{\cdot}{x}}_{2} = z_{3} - β_{02} fal (e_{1}) + b_{0} u - x_{3} - b_{0} u \\ = {\overset{\cdot}{e}}_{2} = e_{3} - β_{02} fal (e_{1}) \\ {\overset{\cdot}{z}}_{3} - {\overset{\cdot}{x}}_{3} = - w (t) - β_{03} fal (e_{1}) = {\overset{\cdot}{e}}_{3} = - w (t) - β_{03} fal (e_{1}) \end{matrix}

(18)

Where, $fal (\cdot)$ is a continuous function, it can be regarded as the connection of countless segments of linear functions. When only the equivalent function value is considered, the parameter time-invariant differential equation in equation (18) is equivalent to the parameter time-varying differential equation, which is presented as

{\begin{matrix} {\overset{\cdot}{e}}_{1} = e_{2} - β_{01} (t) e_{1} \\ {\overset{\cdot}{e}}_{2} = e_{3} - β_{02} (t) e_{1} \\ {\overset{\cdot}{e}}_{3} = - w (t) - β_{03} (t) e_{1} \end{matrix}

(19)

Let $β_{01} (t) = β_{01}$ , $β_{02} (t) = β'_{02}$ , $β_{03} (t) = β'_{03}$ , then

{\begin{matrix} ε_{1} = e_{1} = z_{1} - x_{1} \\ ε_{2} = e_{2} - β_{01} e_{1} = z_{2} - x_{2} - β_{01} e_{1} \\ ε_{3} = e_{3} - β_{02}^{'} e_{1} - β_{01} {\overset{\cdot}{e}}_{1} \end{matrix}

(20)

Taking the derivative of equation (20) to obtain

{\begin{matrix} {\overset{\cdot}{ε}}_{1} = {\overset{\cdot}{e}}_{1} = e_{2} - β_{01} e_{1} = ε_{2} \\ {\overset{\cdot}{ε}}_{2} = {\overset{\cdot}{e}}_{2} - β_{01} {\overset{\cdot}{e}}_{1} = e_{3} - β_{02}^{'} e_{1} - β_{01} {\overset{\cdot}{e}}_{1} = ε_{3} \\ {\overset{\cdot}{ε}}_{3} = {\overset{\cdot}{e}}_{3} - β_{02}^{'} {\overset{\cdot}{e}}_{1} - β_{01} {\overset{\cdot\cdot}{e}}_{1} = - β_{01} ε_{3} - β_{02}^{'} ε_{2} - β_{03}^{'} ε_{1} - w (t) \end{matrix}

(21)

If $w (t) = 0$ , the Laplace transform of the third line of equation (21) can be obtained as

s^{3} + β_{01} s^{2} + β'_{02} s + β'_{03} = 0

(22)

According to the Routh stability criterion, when $β_{01}$ , $β_{02}^{'}$ , and $β_{03}^{'}$ are greater than zero and $β_{01} \cdot β_{02}^{'} > β_{03}^{'}$ , it can be known that ESO is stable and the steady-state errors of observation $e_{1} (t)$ , $e_{2} (t)$ , $e_{3} (t)$ are all zero.

When w(t) ≠ 0, $w (t)$ is bounded, namely $| w (t) | < w_{0}$ , according to the final value theorem,²⁵ the steady-state error of each observation is

| e_{1} (t) | \leq \frac{w_{0}}{{β'}_{03}}

(23)

| e_{2} (t) | \leq \frac{β_{01} w_{0}}{{β'}_{03}}

(24)

| e_{3} (t) | \leq \frac{{β'}_{02} w_{0}}{{β'}_{03}}

(25)

According to equations (23)–(25), when $β_{01}$ and $β_{02}^{'}$ are small and $β_{03}^{'}$ is large, the steady-state error of each observation will be smaller, but the stability requires $β_{01} \cdot β_{02}^{'} > β_{03}^{'}$ . Therefore, the parameter tuning of ESO needs to comprehensively consider both stability and observation error.

The main parameter of the tracking differentiator is $r$ . the larger $r$ is, the faster the transition process is. When $r$ is greater than a certain value, it will not produce much effect. $h$ usually takes the execution cycle of the system. The parameter $β_{01}$ , $β_{02}$ , $β_{03}$ of the extended state observer is related to the fluctuation of system parameters. If the system parameters fluctuate greatly, the value of $β_{01}$ , $β_{02}$ , $β_{03}$ should be large. $β_{03}$ affects the speed of disturbance estimation. The larger the $β_{03}$ is, the faster the estimation is, but too large $β_{03}$ is easy to cause the oscillation of the estimation value; Excessive $β_{01}$ and $β_{02}$ will also cause oscillation of the estimated value, therefore $β_{01}$ , $β_{02}$ , and $β_{03}$ should be adjusted coordinately. The determination of $β_{1}$ and $β_{2}$ is related to the value of $b_{0}$ . $b_{0}$ is larger, and $β_{1}$ and $β_{2}$ is smaller; On the contrary, if $b_{0}$ is small, $β_{1}$ and $β_{2}$ are larger; Generally, when $b_{0}$ is small, the product of $β_{1}$ and $β_{2}$ can be 100, and when $b_{0}$ is large, the product of $β_{1}$ and $β_{2}$ can be 10–20.

Design of deviation self-coupling compensator

Although the two-axis drive in X-direction adopts two identical transmission mechanisms, the asymmetric movement in the Y-direction and various uncertainties in the machining process, the synchronization error between the two axes will occur, which will affect the tracking accuracy of the system. In this paper, from the perspective of compensation, cross compensation control is used to quickly overcome the asynchronous caused by disturbance. The cross-compensation control structure diagram is shown in Figure 6.

Figure 6.

Block diagram of cross-compensation control.

Let the second-order nonlinear system²⁶ be

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f (x_{1}, x_{2}, u) + d_{1} \\ y = x_{1} \end{matrix}

(26)

Uncertain dynamics and external disturbances are defined as total disturbances $d = f (x_{1}, x_{2}, u) + d_{1} - {b_{0}}_{1} u$ . The second-order nonlinear system is written as

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = d + b_{01} u \\ y = x_{1} \end{matrix}

(27)

According to definition of deviation compensation system, when $p_{1} (t) = r_{0}$ , $p_{2} (t) = y$ , then

e_{p 1} = r_{0} - y = p_{1} (t) - p_{2} (t)

(28)

Integral of defining error as

e_{0} = e_{p 0} = \int e_{p 1} dt

(29)

e_{p 2} = {\overset{\cdot}{e}}_{p 1} = {\overset{\cdot}{r}}_{0} - \overset{\cdot}{y} = {\overset{\cdot}{p}}_{1} - x_{2}

(30)

e_{p 2} = {\overset{\cdot\cdot}{e}}_{p 1} = {\overset{\cdot\cdot}{r}}_{0} - \overset{\cdot\cdot}{y} = {\overset{\cdot\cdot}{p}}_{1} - {\overset{\cdot}{x}}_{2} = {\overset{\cdot\cdot}{p}}_{1} - d - b_{01} u

(31)

The controlled error system is

{\begin{matrix} {\overset{\cdot}{e}}_{p 0} = e_{p 1} \\ {\overset{\cdot}{e}}_{p 1} = e_{p 2} \\ {\overset{\cdot}{e}}_{p 2} = {\overset{\cdot\cdot}{p}}_{1} - d - b_{01} u \end{matrix}

(32)

The self-coupling compensation control model defined by the controlled error system is shown in Figure 7.

Figure 7.

Deviation self-coupling compensation controller model.

In Figure 7, $i = 1, 2$ $r_{0}$ is the expected value of deviation, set $r_{0} = 0$ . The main innovation of this control strategy is that there is only one speed factor $z_{c}$ , Its main feature is that the physical links with three different attributes such as proportion, integral and differential are closely coupled together through $z_{c}$ to generate the control signal.²⁶ Therefore, the parameter setting rule of the self-coupling compensator is

k_{p} = 3 z_{c}^{2}, k_{i} = z_{c}^{3}, k_{d} = 3 z_{c}

(33)

According to $e_{0} (0) = 0$ , equation (32) can be transformed by Laplace transformation as

{\begin{matrix} s {e_{p}}_{0} (s) - e_{p 0} (0) = {e_{p}}_{1} (s) \\ s e_{p 1} (s) - e_{p 1} (0) = \frac{1}{s} [e_{2} (0) - d (s) - k_{i} e_{p 0} (s) - k_{p} e_{p 1} (s) \\ - k_{d} s e_{p 1} (s) + k_{d} e_{p 1} (0)] \\ s e_{p 2} (s) - e_{p 2} (0) = - d (s) - k_{i} e_{p 0} (s) - k_{p} e_{p 1} (s) \\ - k_{d} s e_{p 1} (s) + k_{d} {e_{p}}_{1} (0) \end{matrix}

(34)

Equation (34) is further derived as

\begin{matrix} e_{p 1} (s) = \frac{s \hat{d} (s) + s^{2} e_{p 1} (0) + s e_{p 2} (0) + k_{d} {e_{p}}_{1} (0)}{s^{3} + k_{d} s^{2} + k_{p} s + k_{i}} \\ = - \frac{sd (s)}{s^{3} + k_{d} s^{2} + k_{p} s + k_{i}} + \frac{s^{2} e_{1} (0) + s e_{2} (0) + k_{d} e_{1} (0)}{s^{3} + k_{d} s^{2} + k_{p} s + k_{i}} \end{matrix}

(35)

As can be seen from the characteristic equation of $e_{1} (s)$ , when $k_{p} = 3 z_{c}^{2}$ , $k_{i} = z_{c}^{3}$ , $k_{d} = 3 z_{c}$ , then the system has three negative real roots, all of which are $- z_{c}$ . The deviation compensation control system is stable, and the deviation can be eliminated. In addition, because $p_{1} (t)$ and $p_{2} (t)$ can be measured, it can be seen that the deviation compensation is simple, direct and easy to realize.

Research on simulation and experiment

In order to verify the control effect, digital simulation experiments and prototype experiments are carried out in this paper, respectively. Stepper motors’ parameters of the X-axis and Y-axis are all same and Controllers’ parameters are summarized in Table 1.

Table 1.

Parameters of stepping motor and controller.

$L$	$J$	$B$	Start-up frequency	Maximum frequency	Start load	Rotor rotary inertia
0.01005 H	1.05 $g \cdot cm \cdot s^{2}$	0.03 $N \cdot m \cdot s / rad$	0.112 kHz	8.73 kHz	1 N/m	260 $g \cdot c m^{2}$
Outer loop ADRC	$β_{1} = 200$	$β_{2} = 0.1$	${β_{0}}_{1} = 180$	${β_{0}}_{2} = 600$	${β_{0}}_{3} = 5$	$h = 0.01$
Inner loop PID	$k_{p} = 0.3$	$k_{i} = 0$	$k_{d} = 1.05$
Compensator	$z_{c} = 4$

Simulation and Experiment of X-direction synchronous control

Given a position signal $p^{*} = 10$ mm, and add the disturbance signal $d (t) = 4 (t - 5) - 8 (t - 10)$ at point $d_{1}$ and point $d_{2}$ shown in Figure 4 respectively. When a disturbance is added at point $d_{1}$ , the comparison curves of the position response by adopting cross compensation combined ADRC strategy and PID double closed-loop strategy are shown in Figure 8; when the disturbance is added at point $d_{2}$ , the same comparison strategies as the point $d_{1}$ is used, the comparison curves of the position response by adopting cross compensation combined ADRC strategy and PID double closed-loop strategy is obtained in Figure 9. The same procedure can be obtained, Figures 10 and 11 show the comparison curves of the position errors of the two-axis in x-direction controlled by the feedback compensator and the non-feedback compensator when the outer loop is ADRC and the inner loop is the PID control, respectively. Where: n is non-feedback compensator and h is feedback compensator.

Figure 8.

Position response curves in X direction when disturbance is added at point $d_{1}$ .

Figure 9.

Position response curves in X direction when disturbance is added at point $d_{2}$ .

Figure 10.

Position error curves n X direction when disturbance is added at point $d_{1}$ .

Figure 11.

Position error curves n X direction when disturbance is added at point $d_{2}$ .

It can be seen from Figures 8 and 9 that the steady-state time of the strategy proposed in this paper is about 0.5 s, while the steady-state time of PID control is about 0.7 s, and the recovery time is 50% of that of PID control. It can be seen from Figures 10 and 11 that the asynchronous deviation caused by disturbance can be eliminated within 0.3 s by using the deviation self-coupling compensator, while there is always synchronous deviation without compensator, and the deviation is related to the position of the action point of disturbance signal.

Simulation and Experiment of X, Y-direction contour trajectory

In X-direction, given position signal $p_{x}^{*} = 10 \sin 2 t$ mm, In Y-direction, given position signal $p_{y}^{*} = 10 \sin (2 t - 90)$ mm. When the disturbance signal $d (t) = 4 (t - 2) - 8 (t - 3)$ is added at the point $d_{1}$ of one axis in the X-direction, the response trajectory curves with and without deviation feedback compensators under the condition of the outer loop uses ADRC are shown in Figure 12. Then, X-direction given position signal $p_{x}^{*} = 10 (\sin 2 t)^{3}$ mm, Y-direction given position signal $p_{y}^{*} = 10 (\sin (2 t - 90))^{3}$ mm. When the disturbance signal $d (t) = 4 (t - 2) - 8 (t - 3)$ is added at the point $d_{1}$ of one axis in the x-direction, the response trajectory curves with and without deviation feedback compensators under the condition of the outer loop uses ADRC are shown in Figure 13. It can be seen from Figures 12 and 13 that, compared with the control without cross compensation, the cross compensation combined with ADRC strategy has the higher contour trajectory tracking accuracy, the smaller disturbance fluctuation, and the shorter recovery time.

Figure 12.

Circular trajectory.

Figure 13.

Curved quadrilateral trajectory.

Prototype experiment

According to the theoretical analysis, the XY experimental platform of the stepper motor control system is built. The experimental control system is composed of control unit STM32F103RCT6, stepper motor, driver, photoelectric encoder, and fixture. The XY experimental platform of the stepper motor control system is shown in Figure 14.

Figure 14.

Hardware XY experimental platform.

In order to verify the actual control effect, the device is used to draw the circular and curved quadrilaterals with radius and edge of 10 mm. Firstly, the conventional PID control experiment is carried out. The actual drawing curves of the experiments are shown in Figures 15 and 16, respectively. And then, the control strategy experiment proposed by paper is conducted, and the results are shown in Figures 17 and 18, respectively.

Figure 15.

Circular response curve of PID control strategy.

Figure 16.

Curved quadrilateral response curve of PID control strategy.

Figure 17.

Circular response curve of proposed control strategy.

Figure 18.

Curved quadrilateral response curve of proposed control strategy.

The experimental results show that the PID control has a long transition time, and even if it can eliminate the disturbance, it will produce contour tracking error. The control method proposed in this paper has fast response and strong robustness.

Conclusions

In this paper, in order to solve the uncertain factors in the step servo system of XY platform, an active disturbance rejection+PID closed-loop control structure is proposed; In order to improve the synchronization of bilateral drive, a deviation self-coupling compensation strategy is proposed. The experimental results between ADRC+PID+DSCC and ADRC+PID, the experimental results between ADRC+PID+DSCC and PID+PID+DSCC are compared, respectively.

The following conclusions are obtained through simulation and experiments:

Aiming at the problems of nonlinearity, strong coupling and uncertainty of the model, the active disturbance rejection strategy can improve the tracking accuracy and disturbance rejection. An ESO stability analysis method is proposed, and the tunning rules of ADRC parameters are summarized.

For the bilateral drive system, by introducing the deviation self-coupling compensator into the feedback loop, the asynchronous phenomenon can be effectively suppressed and the synchronization performance of bilateral drive can be improved. The parameter setting of the deviation self-coupling compensation controller is simple.

The control strategy proposed in this paper explores a new way for developing high-power and high-precision automatic machining servo equipment.

The strategy proposed in this paper has a good effect on dual motor synchronous drive, but when the number of motors is more than 3, the compensation algorithm is complex, and the fixed speed factor may have the problem of integral saturation and cause compensation oscillation.

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 Talent Development Fund Projects of Jilin Province (Grant 201728), and Science and Technology Research Project of the Education Department of Jilin Province (Grant JJKH20210042KJ, Grant 2015148) and Natural Sciences Transverse Projects of Beihua University (BHKJ211008-1).

ORCID iDs

Dejun Liu

Yanming Cheng

Data availability

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

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An active disturbance rejection synchronous control strategy based on cross compensation for stepper motor XY platform

Abstract

Keywords

Introduction

System structure and mathematical model

Design of controller

Design of inner-loop controller

Design of outer loop controller

Performance analysis of ESO

Design of deviation self-coupling compensator

Research on simulation and experiment

Simulation and Experiment of X-direction synchronous control

Simulation and Experiment of X, Y-direction contour trajectory

Prototype experiment

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

Data availability

References