Online Compensation Method for Contour Error in the Parallel Module of a Hybrid Gantry Machining Center

Abstract

Contour error is the deviation between the actual displacement and reference trajectory, which is directly related to the machining accuracy. Contour error compensation poses substantial challenges because of the time-varying, nonlinear, and strongly coupled characteristics of parallel machining modules. In addition, the time delay in the system reduces the timeliness of the feedback data, thereby making online contour error calculations and compensation particularly difficult. To solve this problem, the generation mechanism of the time delay of the feedback data and contour error is revealed, and a systematic method for the identification of the time-delay parameter based on Beckhoff's tracking error calculation mechanism is proposed. The temporal alignment between the position commands and feedback data enables the online calculation of the contour error. On this basis, the tracking error of the drive axes (an important factor resulting in end-effector contour errors) is used for the contour error calculation. Considering the ambiguous parameter-setting logic of the servo drive, the servo parameter is calculated in reverse using the steady-state error to obtain the tracking error model of the drive axes. Furthermore, combined with the system time-delay model, an online correction method for the tracking error estimation model is established. To achieve an accurate mapping of the drive-axis tracking error and end-effector contour error, a bounded iterative search method for the nearest contour point and online calculation model for the contour error are respectively established. Finally, an online compensation controller for contour error is designed. Its effectiveness is verified by a machining experiment on a frame workpiece. The machining results show that the contour error reduces from 68 μm to 45 μm, and the finish machining accuracy increases by 34%. This study provides a feasible method for online compensation of contour error in a system with time delay.

Introduction

Complex structural parts with large aspect ratios [1] are the core components of aerospace equipment [2, 3]. Such parts usually have difficult-to-machine local features such as variable curvature, large corners, and large open/close angles. Their high-efficiency and high-precision machining requires equipment that display a remarkable machining performance.

During the machining process, the errors in the servo drive system are kinematically reflected at the tool tip. The deviation between the actual displacement and reference trajectory is referred to as the contour error [4], which is directly related to the machining accuracy. The online control of contour errors is a feasible and reasonable means to address these problems. However, a crucial issue remains unresolved: the time delay in the system. Time delay is common in control systems [5, 6]. This reduces the gain margin of the system and can even cause an overshoot. The Siemens control system is one of the most widely used systems. The position loop can be deployed in a numerical control (NC) system to create a flexible online control structure. Dynamic servo-control technology has been developed for high-speed motion. Both this and the position loop can be deployed only in the driver (rather than in the NC system) to improve the gain of the position loop for a high dynamic response. Time delay impacts online control. Moreover, the calculation of the contour error involves feedback data from multiple drive axes, and the control algorithm can be deployed only in an NC system. Therefore, realizing the online control of contour errors by considering the time delay is challenging. The sources of the time delay mainly include the response delay of the actuator and transmission delay of the feedback data. Thus, the development of equipment with a high dynamic response capability and the establishment of a control model that considers time delay are significant.

At present, complex structural parts with large aspect ratios are mainly processed using gantry multi-axis CNC equipment [7]. As high-tech equipment with a wide range of applications, it can satisfy the machining workspace requirements of various parts. The growing manufacturing demands continue to challenge the precision and efficiency of existing machining equipment. However, conventional machining equipment has certain limitations. In the machining of local complex features, the significant difference in load inertia among the drive axes limits the machining efficiency and dynamic accuracy [8]. In addition, the synthetic motion of the A/C swing angle head may cause reductions in efficiency and accuracy in open-and-close angle conversion feature machining. In response to these problems, a gantry machining center (X-RIM) with hybrid kinematics [9] was designed based on the five degree-of-freedom (5-DoF) parallel machining module (PMM) [10, 11] (Figure 1). The moving gantry realizes the large-stroke positioning of the PMM, and the PMM realizes the machining of local complex features. The 5-DoF PMM exhibits the characteristics of attitude coupling motion and a small load difference between the drive axes. Its high attitude adjustment efficiency and dynamic response capability make it suitable for machining complex features. Although the PMM has advantages in terms of response capability, the nonlinear and strongly coupled characteristic of parallel mechanisms also introduces new challenges to the online control of contour errors.

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Figure 1

Hybrid gantry machining center (X-RIM)

To realize contour error control, many studies have focused on the tracking error control of drive axes. This method indirectly improves the end-trajectory accuracy by improving the single-axis tracking performance, e.g., feedforward control algorithms [12, 13], trajectory constraint methods [14, 15], and predictive compensation methods [16, 17]. Because the tracking error of the drive axes cannot be eliminated completely [18], the improvement in end-effector tracking performance by this method is limited. Moreover, an improvement in single-axis tracking performance cannot completely ensure an improvement in the end-effector trajectory accuracy [19]. Therefore, coordinating the error between the drive axes is necessary to ensure that the actual end-effector trajectory matches the reference trajectory to the extent feasible, thereby improving the machining accuracy.

Unlike tracking error control, the contour error control method directly modifies the position command to improve the trajectory accuracy. Cross-coupled control (CCC) is a typical contour error control method [20]. The contour error is decoupled from the error component of the drive axes through CCC [21], which is used as an additional control command to compensate for the contour error. Therefore, the calculation and compensation methods for contour errors are crucial for CCC. The nonlinear and strong coupling of the five-axis PMM complicates the design of the contour error controller.

The five-axis contour error control mainly includes the feedback-based online control method [22] and model-based compensation method [23]. Lo [24] and Yang et al. [25] performed online calculations of contour errors by tracking error decomposition and iterative search, respectively. In the feedback-based online control method, omitting the impact of the feedback delay reduces the contour error estimation accuracy and limits the gain margin of the feedback. This is unfavorable for the control accuracy and stability of the system. To mitigate the impact of time delays [26, 27], scholars have developed various model-based contour error compensation methods. Liu et al. [28] analytically represented the contour error as the control point of a spline path and compensated for it by adjusting the control point. Khoshdarregi et al. [29] established a contour error estimation model by identifying the error transfer function for each drive axis. The position command was then modified according to the estimation results. Notably, in these methods, the compensation trajectory is generated offline according to the tracking error model obtained from the identified drive model. The accuracy of drive model identification and the time-varying characteristics of the equipment directly affect the compensation.

According to the above analysis, the impact of the time delay is non-negligible in the feedback-based online control method, and contour error estimation relies on an accurate drive model in the model-based compensation method. Because the feedback data can reflect the actual operating state, using it for online correction of the contour error calculation model is feasible. The main challenge is that the time delay results in the asynchronization of the position commands and feedback data, which reduces the real-time performance of the control system. Therefore, this study proposed an online compensation method for contour errors. First, a time-delay model was developed based on Beckhoff's tracking error calculation mechanism, and an identification experiment was conducted. Subsequently, by comparing the theoretical and actual values of the tracking error, a correction factor was introduced to achieve online correction of the drive-axis tracking error estimation model. Based on this, an online contour error compensation controller was designed. Finally, experiments were performed to verify the effectiveness of the proposed model. This method involves only basic functions of the control system, including tracking error calculations and feedback data collection. The method is generic and is applicable to commercial control systems.

The remainder of this paper is organized as follows: In Section 2, the development of the time-delay model is described, and an identification experiment is presented. In Section 3, online correction of a single-axis tracking error estimation model is proposed. Furthermore, its experimental validation by establishing a single-axis control model and identifying its servo parameters is described. In Section 4, a bounded iterative search method is proposed for the design of an online contour error compensation controller, and the validation experiments are described. Finally, Section 5 concludes the study.

Time-delay Parameter Identification

The current control system utilizes an NC system for computation, with position commands executed by the servo system and communication facilitated by EtherCat. Although a servo drive can achieve closed-loop control of a single motor, in complex control systems, coordination between multiple motors based on the feedback data of the drive is essential. This complex algorithm can be implemented only within the NC system. Therefore, the impact of time delay on the drive-feedback data should be considered. Time delay is common in control systems because of the time-consuming characteristic of sampling and holding, power conversion, and digital calculations. It results in the asynchronization of the position commands and feedback data, thereby posing a challenge for real-time control.

The control system of the hybrid gantry machining center (Figure 1) utilizes a Beckhoff C6920 as the central controller. The position command and feedback data can be collected. A conceptual diagram of the tracking error calculation of Beckhoff C6920 is shown in Figure 2. The tracking error calculation logic of Beckhoff system is as follows. At time t, it calculates the difference between the current position command and current feedback position, which is then considered as the tracking error (TE_Beckhoff):

1

$$\begin{array}{cc}\hfill T{E}_{\mathit{Beckhoff}}=& r(t)-y(t)\hfill \\ \hfill =& r(t)-g_c·r(t-t_1-t_2)\hfill \\ \hfill =& [r(t)-g_c·r(t)]+g_c·\mathrm{\Delta }r(t_1+t_2)\hfill \\ \hfill =& [r(t)-g_c·r(t)]+\mathrm{\Delta }y(t_{\mathit{delay}})\hfill \\ \hfill =& T{E}_{\mathit{act}}+Er{r}_{\mathit{delay}},\hfill \\ \hfill \end{array}$$where $e^{-t_{1}s}$ and $e^{-t_{2}s}$ are the time delays of the system, $r(t)$ is the reference trajectory, $R(s)$ is its Laplace transform, $y(t)$ is the actual trajectory, and $Y(s)$ is its Laplace transform. Err_delay is the variation in $y(t)$ over the time interval t_delay (t_delay = t₁ + t₂). $G_c(s)$ is the transfer function of the servo system, and its time-domain representation is $g_c$. TE_act is the actual tracking error of the drive axes and is the feedback value of the encoder. In the Beckhoff system, position commands are issued periodically, and the speed within each cycle can be considered uniform. Therefore, Err_delay can also be expressed as

2

$$Er{r}_{\mathit{delay}}=\sum _{t=0}^{t=t_{\mathit{delay}}}{v}_t·t,$$where v_t is the actual velocity of the motor per servo cycle, which can also be obtained from the drives. Owing to the presence of a time delay, the tracking error directly calculated by the Beckhoff system consists of two components: the actual tracking error ($T{E}_{\mathit{act}}$) and time-delay error ($Er{r}_{\mathit{delay}}$). TE_act represents feedback data obtained from the drive. $Er{r}_{\mathit{delay}}$ is related to the time delay and v_t. Complex control algorithms can be implemented only in a Beckhoff controller. Therefore, the impact of the time delay should be considered. Otherwise, it would severely affect the stability of the system. Thus, identify time-delay parameters is essential.

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Figure 2

Block diagram of the Beckhoff control system

According to Eqs. (1) and (2), the time-delay calculation equation can be expressed as

3

$$T{E}_{\mathit{Beckhoff}}-T{E}_{\mathit{act}}=\sum _{t=0}^{t=t_{\mathit{delay}}}{v}_t·t.$$

According to the time-delay calculation model derived above, a time-delay parameter identification experiment was conducted on the PMM [11], as shown in Figure 3. The feedback data for the five drive axes were collected simultaneously. The five motors are completely consistent in the control structure. Motors 2 and 3 and motors 4 and 5 are mechanically symmetrical. Although identification can be completed using the feedback data of one motor, identification experiments were conducted on motors 1, 2, and 4 to verify the consistency of this method. The time delay has a certain value and is independent of the motion trajectory. Therefore, the PMM executes the simple trajectory shown in Figure 3 while simultaneously collecting feedback data from the five motors. The detailed data of motor 1 are used as an example to show the time-delay identification process data. The tracking error calculated by the Beckhoff controller (TE_Beckhoff), tracking error feedback from the driver (TE_act), and drive axis velocity (v_t) are shown in Figure 4(a), 4(b), and 4(c), respectively. Err_delay under the current micro displacement conditions can be obtained by aligning the specified positions of the Beckhoff and drive-side commands, as shown in Figure 4(d). By substituting these data into Eq. (3), the time delay (t_delay) can be calculated, as shown in Figure 4(e).

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Figure 3

Five-axis PMM of the hybrid gantry machining center

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Figure 4

Time-delay parameter identification data

This method was also used to process the feedback data of motors 2 and 4, and the time delay was obtained (Figure 4(f)). The time delay of the system has a certain value, and the sharp points in Figure 4(e) and 4(f) are the error values generated by friction variations and the mechanical backlash during motion reversal, respectively. After removing the error points in Figure 4(e) and 4(f), the mean value of the time delay is approximately 8.75 ms. The identification of the system delay parameter enables the synchronization of position command and feedback data and the deployment of real-time control algorithms based on feedback data within the NC system. Additionally, it lays a foundation for the online calculation of tracking and contour errors.

Single-axis Tracking Error Estimation

Single-axis Tracking Error Model

The single-axis tracking error is ultimately reflected as the contour error of the end effector through forward kinematic mapping. To realize the online calculation and compensation of contour errors, analyzing the tracking error generation mechanism and establishing a single-axis tracking error model is necessary. The current control system uses Kollmorgen commercial drives, and its control block is shown in Figure 5.

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Figure 5

Single-axis control model

The main loop of the single-axis control system uses a conventional three-loop PID controller. The current loop is equivalent to the proportional gain $K_t$, while the velocity loop uses a proportional–integral controller $K_{\mathit{vp}}\left(1+K_{\mathit{vi}}\left./\phantom{K_{\mathit{vi}}s}\right)s\right)$. The position loop also uses a proportional–integral controller $K_{\mathit{pp}}\left(1+K_{\mathit{pi}}\left./\phantom{K_{\mathit{pi}}s}\right)s\right)$. The velocity loop includes a velocity feedforward term, $K_{\mathit{ff}}s$, which is related to the desired velocity. Similarly, the current loop includes an acceleration feedforward term, $T_{\mathit{ff}}{s}^2$, which is related to the desired acceleration. $R(s)$ and $Y(s)$ are the input and output of the single-axis control system, respectively. The closed-loop transfer function can be expressed as

4

$$G_c(s)=\frac{K_1{s}^4+K_2{s}^3+K_3{s}^2+K_{4}s+K_5}{J{s}^4+K_6{s}^3+K_7{s}^2+K_{8}s+K_9},$$where $K_1=K_t{T}_{\mathit{ff}}$, $K_3=K_t{K}_{\mathit{vp}}\left(K_{\mathit{pp}}+K_{\mathit{ff}}{K}_{\mathit{vi}}\right)$, $K_2=K_{\mathit{ff}}{K}_t{K}_{\mathit{vp}}$, $K_4=K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}\left(K_{\mathit{pi}}+K_{\mathit{vi}}\right)$, $K_8=K_4$,

$K_5=K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}{K}_{\mathit{pi}}{K}_{\mathit{vi}}$, $K_6=(B+K_t{K}_{\mathit{vp}}),$ $K_7=K_t{K}_{\mathit{vp}}\left(K_{\mathit{pp}}+K_{\mathit{vi}}\right)$, and $K_9=K_5$.

The error transfer function of the drive axes is established based on the closed-loop transfer function as follows:

5

$$E\left(s\right)=R\left(s\right)-Y\left(s\right)=\left(1-G_c(s)\right)R\left(s\right)=\delta \left(s\right)R\left(s\right).$$

$E\left(s\right)$ represents the tracking error caused by the closed-loop feedback characteristics. It is an inherent property of the closed-loop system and is related to specific inputs such as position, velocity, and acceleration. $\delta \left(s\right)$ denotes the error transfer function associated with the input signal. By expanding $\delta \left(s\right)$ at point $s=0$ into a Taylor series expression, the approximate expression for the steady-state error transfer function of the system can be obtained as

6

$$\delta \left(s\right)=\delta \left(0\right)+\dot{\delta }\left(0\right)s+0.5\ddot{\delta }\left(0\right)s^2+o\left(s^3\right).$$

In this case, Eq. (6) can be substituted into Eq. (5), and a Laplace inverse transform is performed to obtain the time-domain expression for the steady-state error of the single-axis system:

7

$$\stackrel{⌢}{e}\approx \left(\frac{1-K_{\mathit{ff}}}{K_{\mathit{pi}}{K}_{\mathit{pp}}}\right)\ddot{r}.$$

It can be observed that the single-axis tracking error is related to the current acceleration ($\ddot{r}$), and its scale coefficient is related only to the position loop proportional–integral link and the velocity feedforward. However, this theoretical model cannot be used directly for tracking error estimation. Because the underlying operating logic of the drive is unknown, the servo parameters given in the PID tuning may be inconsistent with the actual servo parameters. If these servo parameters are used directly, the tracking error estimation model may be incorrect. Therefore, the servo parameters involved in the tracking-error estimation model must be identified.

Servo-System Parameter Identification

The steady-state error of the equivalent control system differs under the action of different input signals. Moreover, its numerical value is related to the static error coefficient, which corresponds to the parameters of the PID controller in the underlying layer of the system. The common input signals used for identification include the step, ramp, and acceleration signals. The time domain expressions for these signals are as follows:

Step signal:

8

$$y(t)=r_0·1(t).$$

Ramp signal:

9

$$y(t)=r_{0}t·1(t).$$

Acceleration signal:

10

$$y(t)=\frac{1}{2}{r}_0{t}^2·1(t).$$

The steady-state error of a Type 0 system under a step input is given by

11

$$e_{\mathit{ss}}=\underset{s\to 0}{lim}E(s)=\frac{r_0}{1+K}.$$

The steady-state error of a Type I system under a ramp input is given by

12

$$e_{\mathit{ss}}=\underset{s\to 0}{lim}sE(s)=\frac{r_0}{K}.$$

The steady-state error of a Type II system under the influence of an acceleration input is given by

13

$$e_{\mathit{ss}}=\underset{s\to 0}{lim}{s}^{2}E(s)=\frac{r_0}{K}.$$

An identification experiment was also performed on the PMM, as shown in Figure 3. In practice, to ensure the control stability of the PMM under complex force interactions, all the feedforward parameters are set to zero. Therefore, only the positional loop parameters were identified. The servo parameter identification experiment was performed using a step-by-step identification method. The specific experimental process and data were as follows:

1) Identification of position loop proportional parameters

By setting the position loop integral parameter to zero, the control system becomes equivalent to a Type-I system. Its equivalent system control block diagram is shown in Figure 6.

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Figure 6

Type-I equivalent system control block diagram

Its open-loop transfer function is

14

$$G_1\left(s\right)=\frac{K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}s+K_{\mathit{pp}}{K}_t{K}_{\mathit{vi}}{K}_{\mathit{vp}}}{J{s}^3+(b+K_t{K}_{\mathit{vp}})s^2+K_t{K}_{\mathit{vp}}{K}_{\mathit{vi}}s}.$$

The steady-state error of this Type I system under the ramp input signal is given by

15

$$e_{ss1}=r_0\left./\phantom{r_0{K}_{\mathit{pp}}}\right)K_{\mathit{pp}}.$$

The experimental parameter settings and statistical means of the steady-state errors are presented in Figure 7 and Table 1, respectively. The number of encoder bits is 20. Thus, the equation for the actual calculation of the position-loop proportional parameter is

16

$$e_{ss1}=r_0\left./\phantom{r_0{K}_{\mathit{pp}}}\right)K_{\mathit{pp}}=T{E}_{\mathit{Kpp}}\left./\phantom{T{E}_{\mathit{Kpp}}{2}^{20}}\right)2^{20},$$where $T{E}_{\mathit{Kpp}}$ is the tracking error of the drive axes under a given input signal, and its unit is count. The experimental results show that the coefficient factor of the proportionality parameter of the positional loop is approximately one.

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Figure 7

Identification of position loop proportional parameter

Table 1. Identification of position loop proportional parameters: Experimental data

Group	r0 (r/s)	Kpp set value (rad/s)	Tracking error (count)	Kpp actual value (rad/s)	Coefficient
1	0.1	30	3494.8	30.00389	1
2	0.05	30	1747.2	30.00733	1
3	0.05	40	1310.2	40.01588	1
4	0.025	40	654.9	40.02809	1

2) Identification of position loop integral parameter

When both proportional and integral parameters of the position loop exist, the control system can be equivalently represented as a Type-II system. Its equivalent system control block diagram is shown in Figure 8.

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Figure 8

Type-II equivalent system control block diagram

Its open-loop transfer function is

17

$$G_2\left(s\right)=\frac{K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}{s}^2+K_{9}s+K_t{K}_{\mathit{vp}}{K}_{\mathit{vi}}{K}_{\mathit{pp}}{K}_{\mathit{pi}}}{J{s}^4+(b+K_t{K}_{\mathit{vp}})s^3+K_t{K}_{\mathit{vp}}{K}_{\mathit{vi}}{s}^2},$$where $K_9=K_{\mathit{pi}}{K}_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}+K_{\mathit{pp}}{K}_t{K}_{\mathit{vi}}{K}_{\mathit{vp}}$.

The steady-state error of the Type-II system under the acceleration input signal is given by

18

$$e_{ss2}=r_0\left./\phantom{r_0{K}_{\mathit{pp}}{K}_{\mathit{pi}}}\right)K_{\mathit{pp}}{K}_{\mathit{pi}}=T{E}_{\mathit{Kpi}}\left./\phantom{T{E}_{\mathit{Kpi}}{2}^{20}}\right)2^{20},$$where $T{E}_{\mathit{Kpi}}$is the tracking error of the drive axes under the acceleration input signal, and its unit is count.

The experimental parameter settings and statistical means of the steady-state errors are presented in Figure 9 and Table 2, respectively. In the experiment, K_pp was set to 40 rad/s for all the cases. The experimental results show that the coefficient factor of the position loop integral parameter is approximately 6.34.

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Figure 9

Identification of position loop integral parameter

Table 2. Identification of position loop integral parameters

Group	r0 (r/s2)	Kpi set value (rad/s2)	Tracking error (count)	Kpi actual value (rad/s2)	Coefficient
1	0.05	10	20.4	64.2509	6.42509
2	0.1	10	41.3	63.4721	6.34721
3	0.05	2	103.9	12.6152	6.30760
4	0.1	2	208.1	12.5970	6.29850

The error equation group was formed by combining Eqs. (15) and (18). The results of the position-loop parameter identification are summarized in Table 3.

Table 3. Position-loop parameter identification results

Servo parameter	Set value	Actual value	Coefficient
Kpp	30/40 rad/s	30/40 rad/s	1
Kpi	10/2 rad/s2	63.4/12.7 rad/s2	6.34

From the table, it is evident that there exists a proportional relationship between the set and actual parameters. The coefficients of the position-loop proportional and integral parameters are 1 and 6.34, respectively. These coefficients were multiplied by the tuned PID parameters to obtain the actual servo parameters, and a tracking error estimation model was established.

Online Correction of Single-axis Tracking Error Estimation Model

Single-axis tracking error modeling makes it feasible to precompensate for tracking errors. In complex trajectories, the effect of the tracking error precompensation may be suboptimal. In addition, owing to the presence of a time delay, even if the driver can provide feedback on the actual tracking error, it cannot be aligned with the previously estimated compensation value. However, after the system time delay is accurately identified, the situation transitions. Because the time-delay parameter and system servo parameters have been identified, this study proposes an online correction method for a single-axis tracking error estimation model. In this method, a correction factor k is introduced. Its calculation involves the estimated and actual tracking errors, thereby realizing an online correction of the tracking error estimation model based on the previous operational state.

The reference trajectory set in the experiment is

19

$$y(t)=2cos(t)-0.5,$$where t is the system time and y(t) is the system output. According to the tracking error estimation model established in Section 3.2, y(t) can be precompensated. The system output y₁(t) is

20

$$y_1(t)=y(t)+\left(\frac{1}{K_{\mathit{pi}}{K}_{\mathit{pp}}}\right)\ddot{r}.$$

Furthermore, owing to the time-varying and nonlinear characteristic of the external load, the estimated tracking error may differ from the actual feedback tracking error, and the identification of the system delay parameter enables the synchronization of the position command and feedback data. Therefore, a correction parameter k is introduced, which is calculated by comparing the estimated tracking error and the actual feedback tracking error during the operation. It is multiplied by the tracking error estimated by the model to achieve online correction. The system output y₂(t) is expressed as follows:

21

$$y_2(t)=y(t)+k\left(\frac{1}{K_{\mathit{pi}}{K}_{\mathit{pp}}}\right)\ddot{r},$$

22

$$k=\frac{T{E}_{\mathit{est}}+T{E}_{\mathit{act}}}{T{E}_{\mathit{est}}},$$where TE_est is the estimated value of the tracking error, and TE_act is the feedback value of the actual tracking error. The introduction of k enables online correction of the tracking error estimation model. Its control block is illustrated in Figure 10, and its closed-loop transfer function is expressed by Eq. (23).

23

$$\frac{\left[2(K_{\mathit{pi}}+K_{\mathit{pp}})+s^2\right]G_c(s)e^{-(t_1+t_2)}}{(K_{\mathit{pi}}+K_{\mathit{pp}})\left[1+G_c(s)\right]},$$where $G_c(s)$ is the closed-loop transfer function of the single-axis control model, as expressed in Eq. (4). The values of each parameter in the transfer function are listed in Table 4. $f(l,p,q)$ is a function with three inputs and one output, and its expression is shown in Figure 10. The time-delay term is approximated using the first-order Pade approximation expression [30], which is represented by Eq. (24):

24

$$e^{-\tau s}\approx \frac{N_r(\tau s)}{D_r(\tau s)},$$where $N_r(\tau s)={\sum }_{k=0}^r\frac{(2r-k)!}{k!(r-k)!}{(-\tau s)}^k,$ $D_r(\tau s)={\sum }_{k=0}^r\frac{(2r-k)!}{k!(r-k)!}{(\tau s)}^k$.

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Figure 10

Online correction method for single-axis tracking error estimation model

Table 4. Main parameters of servo drive system

Servo parameter	Value	Unit
Kpp	95.1	rad/s
Kpi	50	rad/s2
Kvp	1000	Arms/(rad/s)
Kvi	10	Arms/(rad/s2)
Kt	100	V/A
J	546.63	kg·cm2
b	0.1	N·m·s/rad
t1+t2	8.75	ms

Substituting Eqs. (4) and (24) into Eq. (23), the closed-loop characteristic equation can be expressed as Eq. (25). The characteristic roots obtained by solving this equation are listed in Table 5.

25

$$M_1{s}^5+M_2{s}^4+M_3{s}^3+M_4{s}^2+M_{5}s+M_6=0,$$where $M_1=J(t_1+t_2),$$M_2=2J+(b+K_t{K}_{\mathit{vp}})(t_1+t_2),$$M_3=2(b+K_t{K}_{\mathit{vp}})+(2K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}+K_{\mathit{vi}}{K}_t{K}_{\mathit{vp}})(t_1+t_2),$

Table 5. Characteristic roots

Roots	Value
s1	−500
s2	−500
s3	−132.951
s4	−10.006
s5	−19.991+112.595i
s6	−19.991−112.595i

$M_4=2K_t{K}_{\mathit{vp}}(2K_{\mathit{pp}}+K_{\mathit{vi}})+2K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}(K_{\mathit{vi}}+K_{\mathit{pi}})(t_1+t_2),$$M_5=4K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}(K_{\mathit{vi}}+K_{\mathit{pi}})+2K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}{K}_{\mathit{vi}}{K}_{\mathit{pi}}(t_1+t_2),$$M_6=4K_{\mathit{pp}}{K}_t{K}_{\mathit{vp}}{K}_{\mathit{vi}}{K}_{\mathit{pi}}$.

From Table 5, it is evident that the system is stable because all the real parts of the characteristic roots are less than zero. The effectiveness of this model is demonstrated by the following single-axis trajectory tracking experiment: In the cosine trajectories given in Eq. (19), the tracking error precompensation and online correction function blocks are activated in that order. Simultaneously, the position, velocity, and tracking error information are recorded to compare the effects of "precompensation only" and "precompensation + online correction" on tracking error control. The experimental platform and results are presented in Figures 3 and 11, respectively.

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Figure 11

Position, velocity, and tracking error of single-axis cosine trajectory tracking experiment

In the experiment of cosine trajectory tracking, the maximum tracking error of single-axis was approximately 20 μm without compensation. After the precompensation control method was applied, the maximum tracking error reduced to 14 μm. Moreover, after the online correction block was active, the maximum tracking error of single-axis further reduced to 5 μm. The experimental results indicate that in cases where the effect of precompensation for the tracking error is suboptimal, the proposed method can further improve the tracking accuracy of a single axis. More importantly, the method provides more accurate tracking error estimation results than precompensation alone. This lays the foundation for further implementation of online calculation and compensation of contour error.

Online Compensation of Contour Error

An improvement in single-axis tracking performance cannot completely ensure an improvement in end-effector trajectory accuracy [21]. Therefore, coordinating the error between the drive axes is necessary to modify the position command directly and improve the trajectory accuracy. The basic logic of the proposed online compensation method for the contour error is to calculate the contour error of the end effector based on the single-axis tracking error online estimation model established in Section 3.3. The single-axis tracking error model is corrected online using the parameter k, which is calculated considering the previous operational state. This study focused on the impact of motor drive accuracy on end-effector tracking performance. It omitted the errors caused by backlash, elastic deformation, and other factors. Thus, the estimated position of the end effector can be obtained through forward kinematic mapping. To evaluate the deviation between the estimated position and the reference trajectory, this study proposed a bounded iterative search method for the nearest contour point, which is combined with the linear estimation method to calculate the contour error and compensation vector. Finally, the position command is modified, and the online calculation and compensation of the contour error are realized.

Kinematic Analysis of PMM

A kinematic schematic of the PMM is shown in Figure 12. The fixed reference coordinate system $O-xyz$ and mobile coordinate system are located at the center of the frame and tool center point, respectively. The position and orientation of the end-effector are described as $\mathit{pos}={[x,y,z,\phi ,\theta ]}^{\text{T}}$, where ${[x,y,z]}^{\text{T}}$ is the position of the tool center point and ${[\phi ,\theta ]}^{\text{T}}$ is the orientation of the tool axis vector expressed by the tilt-and-torsion (T&T) angles [12]. According to the T&T angles, the transformation matrix from $O^{\prime }-x^{\prime }{y}^{\prime }{z}^{\prime }$ to $O-xyz$ can be expressed as follows:

26

$$\mathit{R}\mathbf{=}\left[\mathit{n},\mathit{o},\mathit{q}\right],$$where $\mathit{q}={[\text{cos}\phi \text{sin}\theta ,\text{sin}\phi \text{sin}\theta ,\text{cos}\theta ]}^{\text{T}}$. The kinematic constraint equation of the parallel machining module is

27

$${\mathit{B}}_1{\mathit{O}}^{\mathbf{\prime }}·\mathit{n}=0.$$

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Figure 12

Kinematic scheme of parallel machining module

Therefore, $\mathit{n}={\mathit{B}}_1{\mathit{O}}^{\mathbf{\prime }}\times \mathit{q}$, and $\mathit{o}=\mathit{q}\times \mathit{n}$. Let $L_i=∥{\mathit{B}}_i{\mathit{P}}_i∥$, $\mathit{O}{\mathit{O}}^{\mathbf{\prime }}=\mathit{t}$, ${\mathit{OB}}_i={\mathit{b}}_i$, ${{\mathit{O}}^{\mathbf{\prime }}\mathit{P}}_i={\mathit{p}}_i$, and $i=1,2,...,5$. The length of each driving limb $L_i$ can be derived as follows:

28

$$L_i=∥{\mathit{B}}_i{\mathit{P}}_i∥=∥\mathit{O}{\mathit{O}}^{\mathbf{\prime }}+{{\mathit{O}}^{\mathbf{\prime }}\mathit{P}}_i-{\mathit{OB}}_i∥=∥\mathit{t}+{\mathit{Rp}}_i-{\mathit{b}}_i∥.$$

The iterative solution of the forward kinematics is achieved using the Newton iteration method, and its iterative equation can be expressed as

29

$$\begin{array}{c}f(\mathit{pos})=∥\mathit{t}+{\mathit{Rp}}_i-{\mathit{b}}_i∥-L_i,\\ {\mathit{pos}}_{k+1}={\mathit{pos}}_k-{[g({\mathit{pos}}_k)]}^{-1}f({\mathit{pos}}_k),\\ \end{array}$$where $g({\mathit{pos}}_k)=[\text{d}[f({\mathit{pos}}_k)]/\text{d}x,\text{d}[f({\mathit{pos}}_k)]/\text{d}y,...,$$\text{d}[f({\mathit{pos}}_k)]/\text{d}z,\text{d}[f({\mathit{pos}}_k)]/\text{d}\phi ,\text{d}[f({\mathit{pos}}_k)]{/\text{d}\theta ]}^{\text{T}}$.

Contour Error Calculation

The single-axis tracking error eventually results in both end-effector tracking error and end-effector contour error, which are different approaches to describing the deviation of the actual end-effector trajectory from the reference trajectory (Figure 13).

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Figure 13

Actual position point and the reference trajectory

Here, $B_{\text{s}}$ represents the theoretical position point, $B_{\text{r}}$ denotes the actual position point, $B_{\mathit{ij}}$ represents the ideal trajectory interpolation point, and $B_{\text{c}}$ represents the nearest contour point. The tracking error is the distance between the actual displacement of the end effector and reference displacement (|B_rB_s| in Figure 13), whereas the contour error is the minimum distance between the actual displacement of the end effector and reference trajectory. Although these two are interrelated, these do not provide a quantitative description. Considering the actual position point B_r as the center of the sphere, a spherical region is drawn with the tracking error |B_rB_s| as the radius. It can be observed that the interpolation point closest to the actual position point satisfies two constraints simultaneously: it lies inside (or on) the spherical region and on the trajectory. Therefore, a qualitative relationship exists between |B_rB_s| and |B_rB_c|, as shown in Eq. (30):

30

$$\left|B_{\text{r}}{B}_{\text{c}}\right|\le \left|B_{\text{r}}{B}_{\text{s}}\right|\text{(Tracking error)}.$$

Based on this relationship, the tracking error is used as a boundary to iteratively search for the trajectory interpolation point with the minimum distance from the actual position point. In the iterative search, starting from $B_{\text{s}}$, the search proceeds simultaneously in both the directions along the reference trajectory until the constraint condition shown in Eq. (27) is no longer satisfied. The iteration is bounded and requires only simple calculations, thereby enabling it to be completed within a servo cycle. According to Eq. (30), the single-step calculation of the iterative search involves simple mathematical operations, and its computational complexity is O(1). If the search process requires n iterations, the complexity of the entire iterative calculation is O(n). In actual machining, the curvature radius of the actual trajectory is at the millimeter level, whereas the end-effector tracking error is at the ten-micron level, with a difference of two orders of magnitude between these. Therefore, in a circle with a radius of the tracking error, the actual trajectory can be approximated as a straight line. In the worst case, the trajectory of the iterative search is the longest chord of the circle, i.e., the diameter of the circle. Additionally, the interpolation step length of the NC system is approximately 1 µm. Therefore, the number of iterative searches is at most 200 in the worst case. The PMM controller is the Beckhoff C6920 equipped with an i7 processor with a computing frequency of 2.2 GHz. Therefore, the iterative operation does not affect the real-time performance of the system. The bounded iterative search method is combined with the linear estimation method, where a straight line is constructed by considering the velocity vector direction at $B_{\text{c}}$ as the direction vector and $B_{c^{\prime }}$ as the point on the line (Figure 14).

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Figure 14

Linear estimation method

In contour error calculation, the contour error $\left|B_{\text{r}}{B}_{c^{\prime }}\right|$ can be expressed as

31

$$\left|B_{\text{r}}{B}_{c^{\prime }}\right|=\left|\overrightarrow{B_{\text{r}}{B}_{\text{c}}}\times \overrightarrow{B_{\mathit{ij}}{B}_{\text{c}}}\right|/\left|\overrightarrow{B_{\mathit{ij}}{B}_{\text{c}}}\right|.$$

The coordinates of the foot point $B_{c^{\prime }}$ can be expressed as

32

$$B_{\text{c}}^{\prime }={\left[X_{\text{c}}(t),Y_{\text{c}}(t),Z(t)\right]}^{\text{T}}.$$

Therefore, the end-effector contour error vector can be expressed as

33

$${\mathit{B}}_{\mathbf{r}}{\mathit{B}}_{{\mathbf{c}}^{\prime }}={\left[\mathrm{\Delta }{X}_{\text{c}}(t),\mathrm{\Delta }{Y}_{\text{c}}(t),\mathrm{\Delta }{Z}_{\text{c}}(t)\right]}^{\text{T}}.$$

According to the above analysis, the contour error and compensation vectors are calculated by analyzing the positional relationship between the actual position point and reference trajectory.

Online Compensation Controller for Contour Error

The control system of the PMM can be established by integrating the single-axis control model, online estimation and compensation model of tracking errors, and forward and inverse kinematics models of the PMM. Its control model is shown in Figure 15. The specific control strategy is as follows: After the servo system executes the position command, the actual tracking error is fed back to the online compensation controller. In addition, the reference trajectory is input into the controller as prior information. Its output is the trajectory compensation value, which is then fed back to the position input to achieve the online compensation of the end-effector displacement. The core of the online compensation controller consists of a tracking error estimation model and contour error calculation model. This is further expanded to illustrate the control logic (Figure 16).

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Figure 15

Control system of the PMM

[See PDF for image]

Figure 16

Online compensation controller for contour error

After the actual tracking error is fed back, the factor k is adjusted by comparing the actual tracking error with the theoretical tracking error. This adjustment realizes online correction of the tracking error estimation model by considering the previous operational state of the drive axes. Furthermore, the estimated position of the drive axes is mapped using forward kinematics to obtain the estimated position of the end effector. Based on the contour error calculation model, the contour error compensation vector is calculated and used to modify the reference trajectory. In this manner, online contour error compensation is realized. The results of the experimental testing show that the computation time of the entire algorithm is less than 2 ms, which makes it fully deployable in a real-time control system with a 2 ms computation cycle. The application effects are discussed in detail in the following sections.

Online Compensation Experiment for Contour Error

Based on the above analysis and experiments, the simulation control model and the actual control system of the PMM were established. The proposed method was first verified in a simulation system and then in an actual system (Figure 17). A rabbit-shaped trajectory with variable curvature and large corner features was designed, and its position–velocity curve is shown in Figure 18. The experimental feed speed was set to 3000 mm/min. The simulation results were compared with actual experimental results to verify the consistency between the simulation and actual systems, and the effectiveness of the proposed method in improving the contour error accuracy was evaluated further. The maximum values of the experimental and simulation data are listed in Table 6, and the specific data are shown in Figure 19.

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Figure 17

Rabbit-shaped trajectory experiment

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Figure 18

Position and velocity curve of rabbit-shaped trajectory

Table 6. Maximum contour error data

Experiment setting	Simulation max. (μm)	Experiment max. (μm)
Before compensation	42	44
After compensation	25.3	25.47

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Figure 19

Contour error data of rabbit-shaped trajectory

The end-effector position was calculated based on the encoder data through the forward kinematic model and was compared with the reference trajectory to obtain the contour error. According to the data shown in Figure 19 (a) and (b) and Table 6, the maximum contour errors of the rabbit trajectory in the simulation and actual system before compensation were 42 μm and 44 μm respectively. After compensation, the maximum contour error of the rabbit trajectory in the simulation and actual system were 25.3 μm and 25.47 μm, respectively. Interferences (such as joint damping and a nonlinear force) and the operational state of the actual system are complex. These factors are excluded from the simulation model. Moreover, the motor model was simplified. These factors prevent the simulation model from being consistent with the actual system. In this comparative experiment, the errors between the simulation and actual system were 4.5% and 0.6%, respectively. This indicates that the trajectory tracking performance of the simulated system is consistent with that of the actual system.

A comparison of the contour error compensation data in both simulation and actual systems is shown in Figure 19(c) and (d) and Table 6. In the simulation system, the maximum contour error of the rabbit trajectory after compensation reduced from 42 μm to 25.3 μm. In the actual system, the maximum contour error of the rabbit trajectory after compensation reduced from 44 μm to 25.47 μm. The contour accuracy improvement in both simulation and actual system was approximately 42%. It can be concluded that the proposed method can effectively reduce contour errors.

The above experimental data analysis is based on encoder feedback data, and the contour accuracy improvement effect of the end effector is an indirect calculation result. To directly verify the effectiveness of the proposed method, experiments were conducted to verify the accuracy of the end-effector through a rotated tool center-point (RTCP) motion. The dynamic accuracy of the end effector was measured using a rotary-axis analyzer at various swing angles and motion velocities. The experimental settings and data are shown in Figures 20, 21, and Table 7.

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Figure 20

RTCP experiment

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Figure 21

RTCP experiment data (1500°/min, angle = 10°)

Table 7. Maximum contour error in RTCP experiment

Group	Velocity (°/min)	C angle (°)	Before compensation (μm)	After compensation (μm)
1	F800	10	49.0	35.2
2	F1200	10	67.8	40.7
3	F1500	10	79.0	46.8
4	F1500	15	106.0	61.3

In the RTCP experiment, four comparison experiments were conducted to verify the effectiveness of the proposed method at different feed velocities and maximum swing angles. The end-effector posture represented by the T-T angle was set with a C angle of 10° or 15°, and the A-axis velocity was set between 800°/min and 1500°/min.

The third group of experiments was considered as an example to analyze the accuracy improvement of the end effector in three directions, as shown in Figure 21. X_Ori, Y_Ori, and Z_Ori represent the errors of the end effector in the three directions before comparison. X_Com, Y_Com, and Z_Com represent the errors of the end effector in the three directions after the comparison. In the third experiment where the feed speed was set to F1500 and the maximum swing angle was 10°, the maximum errors of the three directions reduced from 54.8 μm, 48.7 μm, and 41.3 μm to 30.7 μm, 35.1 μm, and 24.5 μm, respectively. The maximum contour error reduced from 79.0 μm to 46.8 μm. The contour accuracy improved by approximately 40.76%. Comparing the RTCP experimental data from the four comparison experiments, it can be observed that the contour accuracy improvement in the end effector was consistently higher than 28%. This demonstrates that the proposed method is effective at improving the accuracy of the end-effector contour. Moreover, in the comparison experiments shown in Table 7, the A-axis velocity increased from 800°/min to 1500°/min, and the C-angle was set to 10° or 15°. The contour accuracy improvements were 28.17%, 39.97%, 40.76%, and 42.17%, respectively. The experiment with a high feed speed and large swing angle showed better results in terms of contour error accuracy improvement. According to Eq. (4), the tracking error of the drive axis is proportional to the acceleration. With an increase in the feed rate and swing angle, the acceleration and deceleration of the drive axes became more frequent, which resulted in a higher proportion of servo system-induced errors in the end-effector error. The proposed method compensates for the errors caused by the servo system, which makes it more effective in high-speed motion scenarios.

Machining Experiment Verification

Furthermore, the proposed method was applied to frame-workpiece machining. An area with varying curvature characteristics was selected for experimental verification. The comparison experiment was set as follows. The upper layer added the compensation algorithm, whereas the lower layer did not. The experimental setup is illustrated in Figure 22.

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Figure 22

Machining comparative experiment

The machining results were obtained using a three-coordinate machine, and the distributions of the measurement points and contour errors are illustrated in Figures 23 and 24. The error statistics show that the mean contour error was approximately 68 μm without compensation. After compensation, the mean contour error was approximately 45 μm, with an improvement of approximately 34%. Note that the volume accuracy of the PMM was approximately 43 μm after kinematic calibration. It can be concluded that the proposed online contour error compensation method effectively improves the final machining accuracy of the frame workpiece and promotes the full utilization of the performance of PMMs.

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Figure 23

Distribution and error of measured points

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Figure 24

Machining error of measured points

Conclusions

To address the challenge posed by the system time delay in online control of contour error, an online compensation method for contour error in PMM is presented in this paper. The conclusions of this study are as follows:

1. Based on the tracking error calculation mechanism with a time delay in Beckhoff, an identification method for the time delay was proposed. The identification experiment revealed that the system time-delay parameter is approximately 8.75 ms. The proposed method enables the alignment of the command position and feedback data of the servo system, and the online calculation of the contour error.

2. A precompensation plus online correction method for single-axis tracking error estimation was established by combining a time-delay model and single-axis tracking error estimation model. In the comparison experiment of the cosine trajectory, the proposed method reduced the tracking error from 14 μm to 5 μm. This indicates that this model provides more accurate tracking error estimation results than precompensation alone.

3. Based on the qualitative relationship between the contour error and tracking error of the end-effector, a bounded iterative search method for the nearest contour error point was proposed. It was combined with a linear estimation method to establish a contour error calculation model, thereby enabling the online calculation of the contour error compensation vector. In this manner, the position command was modified, and online compensation of the contour error was realized.

4. The proposed method was applied to the PMM of a hybrid gantry machining center for machining frame workpieces with varying curvature characteristics. The three coordinate measurement results show that the mean machining error reduced from 68 μm to 45 μm. Thus, the effectiveness of the proposed method was verified.

Acknowledgements

Not applicable.

Authors' Contributions

Jie Wen was in charge of the whole trial and wrote the manuscript; Fugui Xie, Zenghui Xie and Xinjun Liu supported with their extensive experience and gave advices on the manuscript. All authors read and approved the final manuscript.

Funding

Supported by National Natural Science Foundation of China (Grant Nos. 52375018, 92148301).

Data availability

The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing financial interests.

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