Iterative learning contouring controller based on trajectory generation with linearly interpolated contour error estimation and Bézier reposition trajectory for computerized numerical control machine tool feed drive systems

Abstract

In industrial applications, highly accurate mechanical components are generally required to produce advanced mechanical and mechatronic systems. In machining mechanical components, contour error represents the product shape quality directly, and therefore it must be considered in controller design. Although most existing contouring controllers are based on feedback control and estimated contour error, it is generally difficult to replace the feedback controller in commercial computerized numerical control machines. This article proposes an embedded iterative learning contouring controller by considering the linearly interpolated contour error compensation and Bézier reposition trajectory, which can be applied in computerized numerical control machines currently in use without any modification of their original feedback controllers. While the linearly interpolated contour error compensation enhances tracking performance by compensating the reference input with an actual value, the Bézier reposition trajectory enables smooth velocity transitions between discrete points in the reference trajectory. For performance analysis, the proposed controller was implemented in a commercial three-axis computerized numerical control machine and several experiments were conducted based on typical three-dimensional sharp-corner and half-circular trajectories. Experimental results showed that the proposed controller could reduce the maximum and mean contour errors by 45.11% and 54.48% on average, compared to embedded iterative learning contouring controller with estimated contour error. By comparing to embedded iterative learning contouring controller with linearly interpolated contour error compensation, the maximum and mean contour errors are reduced to 20.54% and 26.92%, respectively.

Keywords

Actual contour error Bézier curve contouring control feed drive iterative learning control machine tool

Introduction

Highly accurate mechanical components with high-curvature surface are widely required in the fields of aerospace, energy, power, medical, and automobile, such as knee joints,¹ propellers,² impellers with small splitter blades,^3–5 gears,⁶ and ship structural components.⁷ To produce such components, many engineers choose the computerized numerical control (CNC) machining process, which has advantages of high production efficiency, high machining precision, and stable product quality. Five-axis CNC machining further improves the flexibility, precision, and efficiency.^8,9 Due to the above objective, two or more axes on these machines have to simultaneously move and their motions must be controlled accurately in order to produce precise machined parts.¹⁰ Achieving high precision in machining highly depends on control performance of each individual feed drive axis.¹¹ Under independent drive axial control, load disturbance or performance variance of either axis causes contour error.¹² Efforts to reduce or eliminate contour errors have been made through either the design of advanced feedback controllers (FBCs) for each feed drive or modifying the reference axial positions.¹⁰ This issue has attracted many researchers in the machine tool community because it is directly related to the product quality.

A cross-coupled control (CCC) system was proposed for biaxial manufacturing systems to improve the contour accuracy by coupling the individual axial errors.¹³ A new structure of cross-coupling controller with a simpler design process for precise tracking in motion control was proposed,¹⁴ which can be implemented easily on most motion systems via reprogramming the reference position command subroutine. Many cross-coupled methods enhanced the contouring performance by coupling axial errors under high feed rates such as adaptive feed rate and cross-coupled iterative learning control (CCILC).^15–19 The CCILC improved the motion control performance by combining individual axial iterative learning control (ILC) and CCILC into a single control input. Applying the techniques of ILC to CCC enables learning of the cross-coupled error which leads to a modified control signal and subsequent improvements in the contour trajectory tracking performance.¹⁹ However, there are three major limitations. The CCILC iteratively modified the control input signal which requires full access to the existing controller for redesigning or reprogramming, so it will be difficult to implement to commercial CNC machines in which access to the controller is generally not allowed. Second, the CCILC requires more iterations to reduce contour error in higher curvature trajectory because the simple contour error estimation does not provide sufficient estimation performance of the actual contour error. Third, commercial CNC machines generally equip already well-tuned FBCs, and hence they should be maintained in particular for machines currently in use.

In ILC, a machining error is reduced iteratively through modification of the control input or the reference trajectory. In linear-type ILCs, the convergence of the output error is guaranteed under a certain condition even when system parameters are not known exactly and periodic disturbances exist.^20,21 Besides, ILCs with a contouring controller or an iterative contouring controller minimize the contour error through an iterative estimation of the instantaneous curvature of the reference trajectory and coordinate transformation.²² Furthermore, the contour error can be significantly reduced by an ILC which considers both tracking and contour errors.²³ The contour error is approximated as the distance from the actual motion position to the tangential line of the desired contour.¹³ The contour error for a biaxial free-form curve is estimated by approximating the desired curved path with a straight line that passed through the desired position, and the slope of the approximated line was determined by the average vector direction of the desired and the actual feed speed vectors.^15,24 In addition, the contour error is estimated by calculating the deviation between the actual current position and the secant line of the desired path, and the secant line was obtained by passing through the command point of a certain time period before the current desired point.²⁵ Using circular approximation, the contour error estimation method is simplified by two-dimensional (2D) circular contours, which could be used for free-form curved paths.²⁶ A similar second-order method was also proposed by approximating the desired contour with an osculating circle. Then, the contour error is estimated by computing the distance from the actual position to the approximated circle.²⁷ Parameter-based contour error estimation algorithms were also proposed. However, the parameter equations of the desired tool path must be known, and the computational burden is also increased.²⁸ An embedded iterative learning contouring controller (EILCC) with contour error approximation by coordinate transformation was proposed in Hendrawan et al.,^29,30 which could be applied to a commercial machine without any modification to the existing controller.

All of the methods mentioned above estimate 2D contour error and use approximated contour errors instead of the actual ones to design contouring controllers or modify reference trajectories.^22,31,32 Most of the error approximation methods do not represent the contour error accurately, especially for a high-curvature surface application. However, contouring control with actual contour errors is difficult to be implemented in real-time control systems because the actual contour error cannot be obtained immediately for complex contour profiles. Therefore, it requires contour error estimation to obtain approximated values. On the contrary, with an embedded iterative control system, it is possible to obtain actual contour errors by offline calculation. However, this method generates nonuniform trajectory points along the reference path as explained in section “BRT,” which may produce nonsmooth velocity profiles. Therefore, the smoothing algorithm is needed to obtain smooth velocity profiles. Parametric splines, particularly Bézier curves (BCs), have been widely used for trajectory smoothing by interpolating linear trajectory points for both mobile robots and feed drive systems.^33,34 Thus, they can be applied for the aforementioned scenario as well.

This study proposes an EILCC with linearly interpolated contour error compensation (LICEC) and Bézier reposition trajectory (BRT) to generate uniform trajectory points as shown in Figure 1. By calculating the minimum distance between the actual position and the reference trajectory, the LICEC is used to achieve the desired contour precisely³⁵ in the three-dimensional (3D) space. The EILCC modifies the original trajectory (NC program) with LICEC as an iterative learning controller. The modified reference trajectory is smoothed by quintic BCs through interpolation of the reference trajectory with uniformly distanced reference points to obtain a BRT. The BRT is executed by a CNC machine to reduce the contour error, iteratively. The proposed method is proven to provide better performance than the EILCC either with or without LICEC. The proposed method could be applied to commercial CNC machine tools without changing the existing controller. Generally, a commercial CNC machine has an existing well-tuned controller, which has good stability and disturbance rejection properties. The feedforward control is rather expected to improve the tracking performance by optimizing learning gain only with lower computational effort. The proposed method is suitable for mass production application because it may be desirable to provide dummy material in the beginning of the production to reduce the contour error in the EILCC. However, there is also a study on feed drive control, which mentions that feed drive motion accuracy mostly affects the precision of products because typical industrial machines have sufficient stiffness to the disturbance such as cutting force. Therefore, once the reference trajectory for the feed drive motion is optimized, it may still be effective for the finishing cutting process.³⁶ In typical heavy cutting cases, after rough cutting, finishing cutting is conducted. In that case, rough cutting does not need accuracy. Once the final reference trajectory is optimized by the proposed ILC in the air cutting process, the optimized trajectory can be applied for the finishing cutting process, which enables to produce high-precision products.

Figure 1.

EILCC block diagram.

The rest of this article is organized as follows: section “EILCC with LICEC and BRT for three-axis CNC machine tools” gives a description of the dynamic model of three-axis machine tools, definition of the contour error, LICEC, BRT, controller design, and convergence analysis. Experimental results, which compare EILCC, EILCC with LICEC, and EILCC with LICEC and BRT in 3D sharp-corner and 3D half-circular trajectories, are presented in section “Experiment,” followed by concluding remarks in section “Conclusion.”

EILCC with LICEC and BRT for three-axis CNC machine tools

The dynamic model of three-axis machine tools

The dynamics of three-axis machine tools is represented as follows

\begin{matrix} M \overset{\cdot\cdot}{q} + C \overset{\cdot}{q} = f \\ M = diag {m_{i}}, C = diag {c_{i}}, i = x, y, z \\ f = {[f_{x}, f_{y}, f_{z}]}^{T} \end{matrix}

(1)

where $m_{i}, c_{i}, f_{i}, and q_{i}$ are the mass matrix, viscous friction coefficient, angular position, driving force, and actual position for the axis $i$ , respectively.

Definition of contour error

The contour error is the error components orthogonal to the desired contour curves³⁷ or it can be defined as the shortest distance between the actual and desired contours. The difference between tracking error and contour error in machine tools is shown in Figure 2. The tracking error in each axis is the difference between the reference and actual positions. The reference position from the starting position of the machine tool system at time $t$ in the coordinate frame $Σ_{w}$ is denoted by $r = [\begin{matrix} r_{x} & r_{y} & r_{z} \end{matrix}]^{T}$ . The tracking error in each axis is defined as

e_{w} = {[\begin{matrix} e_{x} & e_{y} & e_{z} \end{matrix}]}^{T} = r - q

(2)

e = ‖ e_{w} ‖

(3)

Figure 2.

3D tracking and contour errors.³⁵

The coordinate frame $Σ_{l}$ is attached at $r$ and its axial directional vectors are $I$ , $N$ , and $B$ , respectively. The axis $I$ is in the tangential direction of the reference trajectory at $r$ , the direction of $N$ is perpendicular to $I$ , and the $B - axis$ is the binormal component to $I$ and $N$ . The axial directions of frame $Σ_{l}$ are calculated as follows

I = {[\begin{matrix} t_{x} & t_{y} & t_{z} \end{matrix}]}^{T} = \frac{\overset{\cdot}{r}}{‖ \overset{\cdot}{r} ‖}

(4)

N = {[\begin{matrix} n_{x} & n_{y} & n_{z} \end{matrix}]}^{T} = \frac{\overset{\cdot}{I}}{‖ \overset{\cdot}{I} ‖}

(5)

B = {[\begin{matrix} b_{x} & b_{y} & b_{z} \end{matrix}]}^{T} = I \times N

(6)

The tracking error vector $e_{l}$ with respect to $Σ_{l}$ can be expressed as

e_{l} = [\begin{matrix} e_{t} & e_{n} & e_{b} \end{matrix}] = L^{T} e_{w}

(7)

L = {[\begin{matrix} t_{x} & n_{x} & b_{x} \\ t_{y} & n_{y} & b_{y} \\ t_{z} & n_{z} & b_{z} \end{matrix}]}^{T}

(8)

The closest position of the desired contour to $q$ is denoted by $r_{c} = [\begin{matrix} r_{cx} & r_{cy} & r_{cz} \end{matrix}]^{T}$ . The actual contour error, which is the shortest distance between the actual position $q$ and the reference trajectory is calculated as follows

e_{c} = {[\begin{matrix} e_{cx} & e_{cy} & e_{cz} \end{matrix}]}^{T} = r_{c} - q

(9)

\overset{‵}{e} = ‖ e_{c} ‖

(10)

According to the contour error definition, it may define multiple closest positions and contour errors in a trajectory with an intersection part, for example, as shown in Figure 3 ( $r_{c 1}, r_{c 2}, e_{c 1}$ , and $e_{c 2}$ ). In this case, conventional contour error estimation may estimate a wrong value of the contour error. The LICEC algorithm is also designed to solve this problem.

Figure 3.

Sample trajectory with intersection part.

LICEC

LICEC is a direct method to compensate for the actual contour error by finding the closest position between the actual position and the reference trajectory as shown in Figure 2. Here we assume that the reference trajectory is represented by discrete points as ${\overset{‵}{r}}_{c}, {\overset{‵}{r}}_{c + 1}, \dots$ in the figure and $r_{n}$ denotes the current reference position corresponding to the position $q$ . $\overset{‵}{q}$ is a candidate position, which is defined in the same manner with contour error estimation by a rotation matrix to estimate projected contour error in the tangential direction on coordinate frame $Σ_{l}$ as in Hendrawan et al.³⁰ and is represented as follows

\overset{‵}{q} = q + L^{- 1} [\begin{matrix} 0 \\ e_{n} \\ e_{b} \end{matrix}]

(11)

The shortest distance $\overset{‵}{d}$ between $\overset{‵}{q}$ and the reference trajectory is calculated as follows

\overset{‵}{d} = \min_{0 < m < n} ‖ r_{n - m} - \overset{‵}{q} ‖

(12)

where $r_{n - m}$ is the closest candidate reference position in the $(n - m) th$ point and $m$ is determined to find shortest distance $\overset{‵}{d}$ . The approximated closest position of the desired contour to $q$ is denoted by ${\overset{‵}{r}}_{c}$ , which is obtained by finding the minimum distance ${\overset{‵}{d}}_{c}$ between the actual position $q$ and a part of reference trajectory from the $(n - m - k) th$ to the $(n - m + k) th$ discrete position, where $(n - m)$ is calculated by equation (12) and $k$ is determined to avoid high computation. This step guarantees that the LICEC estimates the true contour error $e_{c 1}$ in Figure 3 because a candidate’s closest position is calculated by considering the axial directions of local frame $Σ_{l}$ . ${\overset{‵}{r}}_{c}$ is calculated as follows

{\overset{‵}{r}}_{c} = q + e_{c}

(13)

{\overset{‵}{d}}_{c} = \min_{- k < h < k} ‖ e_{c} ‖

(14)

where

e_{c} = r_{n - m + h} - q

In machining application, because the reference trajectory (G-code) consists of many discrete points which are connected by a linear short-line segment as shown in Figure 4, and hence linear interpolation is chosen for contour error estimation. The actual position $r_{c}$ is defined by linear interpolation of the neighborhood points as shown in Figure 4 with the equations below

r_{c} = {\overset{‵}{r}}_{c} + \frac{(r_{c + 1} - {\overset{‵}{r}}_{c}) D_{1}}{D_{2}}

(15)

D_{1} = \frac{{\overset{`}{d}}_{c}^{2} - {\overset{`}{d}}_{c + 1}^{2} + {‖ r_{c + 1} - {\overset{`}{r}}_{c} ‖}^{2}}{2 ‖ r_{c + 1} - {\overset{`}{r}}_{c} ‖}

(16)

D_{2} = ‖ r_{c + 1} - {\overset{‵}{r}}_{c} ‖ - D_{1}

(17)

where ${\overset{‵}{r}}_{c + 1}, {\overset{‵}{d}}_{c + 1}, D_{1}$ , and $D_{2}$ are the discrete position next to the approximated discrete position ${\overset{‵}{r}}_{c}$ , the shortest distance from ${\overset{‵}{r}}_{c + 1}$ to the actual position $q$ , the distance between ${\overset{‵}{r}}_{c}$ and $r_{c}$ , and the distance from $r_{c}$ to ${\overset{‵}{r}}_{c + 1}$ , respectively.

Figure 4.

Contour error interpolation.

BRT

Herein, BRT refers to the modified reference trajectory that has been smoothed by quintic BCs through interpolation of the EILCC-modified reference trajectory with uniformly distanced discrete points as shown in Figure 5, where $r_{n}, q_{n}, l_{n}, e_{cn}, {\overset{‵}{r}}_{n}, {\overset{‵}{l}}_{n}, and {\overset{‵}{u}}_{cn}$ are the reference position, the actual position, the length between two points in the reference trajectory, the contour error, the reference position in the modified trajectory, the length between two points in the modified trajectory, and the contouring compensation for the $n th$ trajectory segment, respectively.

Figure 5.

Bézier reposition trajectory concept.

BC is a parametric curve defined by several control points depending on its order, and it always passes through the starting and ending control points.³³ A BC of order $w$ can be represented mathematically as follows³⁸

B (ψ) = \sum_{v = 0}^{w} b_{v}^{w} (ψ) P_{v}, 0 \leq ψ \leq 1

(18)

where $b_{v}^{w}, P_{v}, v, and ψ$ are the $w th - order$ Bernstein polynomial, the control points, control point number, and the Bézier parameterization variable, respectively. The Bernstein polynomial is represented as follows

b_{v}^{w} (ψ) = \frac{w!}{v! (w - v)!} ψ^{v} {(1 - ψ)}^{w - v}, v = 0, 1, 2, 3, 4, 5

(19)

The BRT has two main parts: BC generation and reposition of the trajectory points. BC generation considers parameterization of a linear segment by a BC and corner smoothing by an inserted Bézier curve (IBC) as detailed in Simba et al.³⁴

From equations (18) and (19), the BC and the IBC with the fifth-order Bernstein polynomials for $n th$ segments are represented as follows

B_{n} (ψ) = {(1 - ψ)}^{5} P_{0 n} + 5 ψ {(1 - ψ)}^{4} P_{1 n} + 10 ψ^{2} {(1 - ψ)}^{3} P_{2 n}

(20)

and

{\overset{‵}{B}}_{n} (ψ) = {(1 - ψ)}^{5}' P_{0 n} + 5 ψ {(1 - ψ)}^{4}' P_{1 n} + 10 ψ^{2} {(1 - ψ)}^{3}' P_{2 n}

(21)

where ${\overset{‵}{B}}_{n} and {\overset{‵}{P}}_{in}, i = 0, \dots, 5$ are the IBC and its control points, respectively. The control points for the BC and the IBC are defined by the following equations

\begin{matrix} P_{0 n} = r_{n - 1} + l_{n - 1} μ_{n} \\ P_{1 n} = r_{n - 1} + l_{n - 1} μ_{n} (2 - α_{n}) \\ P_{2 n} = r_{n - 1} + l_{n - 1} μ_{n} (β_{n} - 4 α_{n} + 4) \\ P_{3 n} = r_{n} - l_{n} μ_{n} (β_{n} - 4 α_{n} + 4) \\ P_{4 n} = r_{n} - l_{n} μ_{n} (2 - α_{n}) \\ P_{5 n} = r_{n} - l_{n} μ_{n} \end{matrix}

(22)

and

\begin{matrix} {\overset{‵}{P}}_{0 n} = r_{n} - l_{n} μ_{n} \\ {\overset{‵}{P}}_{1 n} = r_{n} - l_{n} α_{n} μ_{n} \\ {\overset{‵}{P}}_{2 n} = r_{n} - l_{n} β_{n} μ_{n} \\ {\overset{‵}{P}}_{3 n} = r_{n} + l_{n} β_{n} μ_{n + 1} \\ {\overset{‵}{P}}_{4 n} = r_{n} + l_{n} α_{n} μ_{n + 1} \\ {\overset{‵}{P}}_{5 n} = r_{n} + l_{n} μ_{n + 1} \end{matrix}

(23)

where $l_{n}, μ_{n}, α_{n}, and β_{n}$ are the length, tangential vector, the one-third and the two-third fractions of the length $l_{n}$ for the $n th$ trajectory segment, respectively. Because a geometrical error $γ$ is induced at the corners by the smoothing process, the smoothing algorithm must guarantee that $γ$ is within the predefined tolerance of the contour error.³⁹ The maximum geometry error occurs at the middle point of the IBC, that is, at $ψ = 0.5$ as follows

γ = ‖ r_{n} - {\overset{‵}{B}}_{n} (0.5) ‖

(24)

From equations (21) and (23), the point $\overset{‵}{B} (0.5)$ is found by

{\overset{‵}{B}}_{n} (0.5) = r_{n} + \frac{l_{n}}{32} (1 + 5 α_{n} + 10 β_{n}) (μ_{n + 1} - μ_{n})

(25)

From equations (24) and (25), the length $l_{n}$ is calculated as

l_{n} = \frac{32 γ}{(1 + 5 α_{n} + 10 β_{n}) ‖ μ_{n} - μ_{n + 1} ‖}

(26)

Because $μ_{n}$ and $μ_{n + 1}$ are unit vectors, equation (24) can be written as

l_{n} = \frac{32 γ}{(1 + 5 α_{n} + 10 β_{n}) \sqrt{2 - 2 \cos θ_{n}}}

(27)

where $θ_{n}$ is the inclination between the corresponding linear segments.

The generated BRT has a higher number of discrete points than the original reference trajectory, which directly affect the velocity profile. Therefore, the number of points is re-calculated based on the sampling time and the desired velocity as follows

n_{s} = \frac{\sum_{k = 1}^{n} ‖ B_{k} (ψ) - B_{k - 1} (ψ) ‖}{l_{s}}

(28)

l_{s} = t_{s} f

(29)

where $n_{s}, l_{s}, t_{s}, and f$ are the re-calculated number of discrete points, length between the points, sampling time, and velocity, respectively. The BRT is re-defined by considering the following condition

r_{B_{n}} = B_{n} (ψ), if \sum_{k = 1}^{n} ‖ B_{k} (ψ) - B_{k - 1} (ψ) ‖ = l_{s} n, 1 \leq n \leq n_{s}

(30)

Controller design

The proposed controller for the three-axis machine tool consists of an FBC and an EILCC with LICEC and BRT. The input of the FBC is the modified reference trajectory BRT generated from the ILC with LICEC as shown in Figure 6, where $mem u_{c}, q, and r$ are the memories of learning compensation value, actual position, and reference trajectory for the $x, y,$ and $z$ axes, respectively. The transfer function of the plant including the FBC is represented as follows

G_{i} (s) = \frac{M_{i} (s) K_{i} (s)}{1 + M_{i} (s) K_{i} (s)}

(31)

where $G_{i}, M_{i}, K_{i}, and s$ are the transfer function from the reference command to the actual position, the machine plant, the feedback compensator, and the variable of the Laplace transform for the $i th$ axis, respectively. The FBC compensator $K_{i} (s)$ is defined as a proportional–integral–derivative (PID) compensator as follows

K_{i} (s) = K_{Pi} (s) + \frac{1}{s} K_{Ii} (s) + s K_{Di} (s)

(32)

where $K_{Pi}, K_{Ii}, and K_{Di}$ are the proportional, integral, and derivative gains for the $i th$ axis, respectively. The BRT for the $(j + 1) th$ iteration is represented as follows

r_{B_{ij + 1}} (t) = B (ψ) r_{i_{j + 1}}

(33)

where $B, t, ψ, r_{B_{ij + 1}}, and r_{ij + 1}$ are the 3D BC, the sampling instant, the Bézier parameterization value, Bézier modified trajectory, and the reference signal for the $(j + 1) th$ and $i th$ axes, respectively. Reference signal $r_{ij + 1} (t)$ is compensated by an iterative learning compensator $K_{l}$ with LICEC approximation which is represented as follows

r_{ij + 1} (t) = r_{ij} (t) + u_{cij} (t)

(34)

with

\begin{matrix} u_{cij} (t) = e_{cij} K_{li} (s) \\ K_{li} (s) = K_{Pli} (s) + \frac{1}{s} K_{Ili} (s) + s K_{Dli} (s) \end{matrix}

where $r_{ij}, u_{cij}, e_{cij}, K_{Pli}, K_{Ili}, and K_{Dli}$ are the reference signal, contouring compensated input, contour error for the $j th$ iteration, and the proportional, integral, and derivative learning compensator gain $K_{l}$ for the $i th$ axis, respectively. Therefore

u_{cij} (t) = K_{Pli} e_{cij} (t) + K_{Ili} \int_{0}^{t} e_{cij} (τ) d τ + K_{Dli} {\overset{\cdot}{e}}_{cij} (t)

(35)

Substituting equation (34) into equation (33) leads to

r_{B_{ij + 1}} (t) = B (ψ) (r_{i} (t) + \sum_{n = 0}^{j} u_{cij - n} (t))

(36)

Figure 6.

The proposed controller design.

Convergence analysis

Performance of the EILCC with LICEC and BRT can be guaranteed through convergence analysis to ensure that the contour error is iteratively reduced. The convergence condition of ILC is represented as

ε_{i} (t) = ‖ \frac{e_{cij + 1} (t)}{e_{cij} (t)} ‖ < 1

(37)

where $ε_{i} (t)$ is the convergence factor for the $i th$ axis. Based on the cascade ILC,⁴⁰ the convergence can be guaranteed if the error magnitude at time $t$ in iteration $j + 1$ is smaller than that in iteration $j$ . From equations (9) and (14) and the assumption that $r_{B_{ij + 1}}$ is equal to $r_{ij + 1}$ , the convergence factor is derived as follows

\begin{matrix} e_{cij + 1} & = \min (r_{i} - G_{i} (s) r_{B_{ij + 1}}), r_{B_{ij + 1}} \approx r_{ij + 1} \\ = \min (r_{i} - G_{i} (s) r_{ij + 1}) \\ = \min (r_{i} - G_{i} (s) (r_{ij} + u_{cij})) \\ = \min (r_{i} - G_{i} (s) r_{ij} - G_{i} (s) e_{cij} K_{li} (s)) \\ = \min (r_{i} - G_{i} (s) r_{ij}) - G_{i} (s) e_{cij} K_{li} (s) \\ = e_{cij} - G_{i} (s) e_{cij} K_{li} (s) \\ = (1 - G_{i} (s) K_{li} (s)) e_{cij} \\ \frac{e_{cij + 1}}{e_{cij}} & = 1 - \frac{M_{i} (s) K_{i} (s) K_{li} (s)}{1 + M_{i} (s) K_{i} (s)} \end{matrix}

(38)

Substituting equation (38) into equation (37) leads to

ε_{i} (s) = ‖ 1 - \frac{M_{i} (s) K_{i} (s) K_{li} (s)}{1 + M_{i} (s) K_{i} (s)} ‖ < 1

(39)

Considering the discrete-time form for the implementation, $M_{i} (s)$ is represented as follows

M_{i} (z^{- 1}) = \frac{1}{m_{i} {(\frac{z^{- 1} - 1}{t_{s} z^{- 1}})}^{2} + c_{i} \frac{z^{- 1} - 1}{t_{s} z^{- 1}}}, z = e^{j ω t_{s}}

(40)

where $ω$ and $z$ are the angular frequency and the z-domain variable, respectively. The feedback and learning compensators in the z domain are represented as

K_{i} (z^{- 1}) = K_{Pi} + K_{Ii} (\frac{t_{s} z^{- 1}}{z^{- 1} - 1}) + K_{Di} (\frac{z^{- 1} - 1}{t_{s} z^{- 1}})

(41)

K_{li} (z^{- 1}) = K_{Pli} + K_{Ili} (\frac{t_{s} z^{- 1}}{z^{- 1} - 1}) + K_{Dli} (\frac{z^{- 1} - 1}{t_{s} z^{- 1}})

(42)

The convergence speed depends on the parameter $ε$ , which should be kept as minimum as possible for fast convergence with a delay factor $η$ due to the system delay. Considering equation (39) and the delay, the following objective function for minimization is considered

J_{i} = \min_{K_{Pli}, K_{Ili}, K_{Dli}} ‖ 1 - \frac{z^{η} M_{i} (z^{- 1}) K_{i} (z^{- 1}) K_{li} (z^{- 1})}{1 + M_{i} (z^{- 1}) K_{i} (z^{- 1})} ‖_{\infty}, \forall ω \in Ω

(43)

where $z^{- 1} and Ω$ are the delay operator and the considered domain of the operational frequency. Equation (43) is solved by “fmincon” function in MATLAB to find learning compensator gains.

Experiment

Experimental configuration

A three-axis commercial CNC machine (Figure 7) with a ball screw mechanism, which is attached on a table and three servo motors. It is controlled by a Mitsubishi M70 controller which was used in the experiment to verify the effectiveness of the proposed controller. The actual position of the X–Y–Z table was measured by linear encoders with a resolution of $5 μ m$ and a microcontroller ATMega16 attached to each axis for interface. The EILCC was programmed in Microsoft Visual Basic 6 in a separate personal computer (Windows OS) and embedded into the machine by a direct numerical control (DNC) system with an RJ45 connection. The experimental interface design and system parameters are shown in Figure 8 and Table 1, respectively.

Figure 7.

Three-axis CNC machine: (a) CNC machine and (b) table mechanism.

Table 1.

Plant and control parameters.

Parameter	X-axis	Y-axis	Z-axis
m_i (×10 kg)	9.62	13.99	11.3
c_i (kg/s)	0.24	0.24	0.24

Figure 8.

Experimental interface design.

The original FBC gain was originally assigned gains of the existing commercial CNC machine as $[K_{P}, K_{I}, K_{D}] = [0.96 V / mm, 0.01 V / smm, 8 Vs / mm]$ . The optimal learning compensator gains were calculated with equation (43) by “fmincon” function in MATLAB as $[K_{Pl}, K_{Il}, K_{Dl}] = [0.468, 0.044, 0.443]$ . Based on the defined feedback and optimum learning controller gains, the convergence factor $ε_{i} = 0.6025$ was obtained for the considered operational frequency $0 < ω < 100 Hz$ with a discretization of $1 Hz$ and the sampling rate of $5 ms$ .

The following three types of controllers were considered for the experiment:

EILCC (with estimated contour error);

EILCC with LICEC;

EILCC with LICEC and BRT.

For each controller, two types of reference trajectories were executed, 3D sharp-corner trajectory and 3D half-circular trajectory, which were implemented in G-code form.

Experimental results for the 3D sharp-corner trajectory

The 3D sharp-corner trajectory consists of two linear segments, and sharp corners are generally difficult to be tracked by feed drive systems because of infinite acceleration due to a trajectory discontinuity at each corner. All of the drive axes move in the same direction before the sharp corner. After the sharp corner, the drive axis X continues with the same motion direction, whereas the drive axes Y and Z reverse their directions. This requires instant deceleration and acceleration in the drive axes Y and Z. If the typical PID control is used, an overshoot occurs at the corner.^41,42 Because the proposed method can reduce contour error for the difficult trajectory such as a sharp-corner trajectory, it may work also for other shapes of trajectories properly.

The 3D sharp-corner trajectory is represented in G-code form as shown in Figure 9. For each controller, the five-time experiment was conducted until the minimum contour error was reached as shown in Figures 10 –13. Figures 10(a), 11(a), and 12(a) and Figures 10(b), 11(b), and 12(b) show the trajectory tracking profiles for the 3D sharp-corner trajectory and the contour errors based on the EILCC, EILCC with LICEC, and EILCC with LICEC and BRT, respectively. In each case, results of the first iteration refer to the FBC only. Under EILCC, the initial largest contour error was $0.1172 mm$ on average, which was reduced to $0.0266 mm$ in the fifth iteration as shown in Figure 10(b). On the contrary, under EILCC with LICEC, the final average maximum contour error was $0.018 mm$ in the fourth iteration, which is equivalent to the reduction of $32.33 %$ of the maximum contour error from that of EILCC. In the last controller, EILCC with LICEC and BRT, the smallest maximum contour error of $0.0146 mm$ on average was achieved as shown in Figure 12(b). This is equivalent to the reduction of $45.11 %$ and $18.89 %$ of the maximum contour error compared to the EILCC and the EILCC with LICEC, respectively.

Figure 9.

3D sharp-corner trajectory G-code.³⁵

Figure 10.

Experimental results for the 3D sharp-corner trajectory by EILCC: (a) trajectory tracking profiles and (b) contour error.

Figure 11.

Experimental results for the 3D sharp-corner trajectory by EILCC with LICEC: (a) trajectory tracking profiles and (b) contour error.

Figure 12.

Experimental results for the 3D sharp-corner trajectory by EILCC with LICEC and BRT: (a) trajectory tracking profiles and (b) contour error.

Figure 13.

Contour error experimental results for the 3D sharp-corner trajectory: (a) maximum contour error and (b) mean contour error.

Figure 13(a) and (b), respectively, shows the comparisons of the maximum and mean contour errors for the five-time experiments of the three controllers. It can be noted that the proposed controller, EILCC with LICEC and BRT, yielded the smallest values of the maximum and mean contour errors. Details of the experimental results are summarized in Table 2.

Table 2.

Summary of experimental results.

3D sharp-corner trajectory
Experiment no.	$e_{c}$ max (mm)				$e_{c}$ mean (mm)
	FBC	EILCC	EILCC + LICEC	EILCC + LICEC + BRT	FBC	EILCC	EILCC + LICEC	EILCC + LICEC + BRT
1	0.093	0.008	0.008	0.006	0.025	0.009	0.007	0.004
2	0.111	0.029	0.019	0.016	0.035	0.013	0.011	0.009
3	0.148	0.045	0.023	0.02	0.017	0.013	0.013	0.012
4	0.132	0.032	0.028	0.022	0.037	0.011	0.006	0.005
5	0.102	0.019	0.012	0.009	0.049	0.024	0.015	0.008
Mean	0.1172	0.0266	0.018	0.0146	0.0326	0.014	0.0104	0.0076
Reduction ratio^a (%)	0	77.3	84.64	87.54	0	57.06	68.10	76.69
3D half-circular trajectory
Experiment no.	$e_{c}$ max (mm)				$e_{c}$ mean (mm)
	FBC	EILCC	EILCC + LICEC	EILCC + LICEC + BRT	FBC	EILCC	EILCC + LICEC	EILCC + LICEC + BRT
1	0.181	0.045	0.025	0.007	0.031	0.018	0.015	0.014
2	0.207	0.052	0.041	0.04	0.039	0.033	0.011	0.009
3	0.181	0.062	0.052	0.033	0.029	0.026	0.016	0.013
4	0.178	0.067	0.063	0.051	0.034	0.027	0.018	0.015
5	0.16	0.083	0.077	0.074	0.044	0.03	0.017	0.01
Mean	0.1814	0.0618	0.0516	0.041	0.0354	0.0268	0.0154	0.0122
Reduction ratio^a (%)	0	65.93	71.55	77.40	0	24.29	56.5	65.54

FBC: feedback controller; EILCC: embedded iterative learning contouring controller; LICEC: linearly interpolated contour error compensation; BRT: Bézier reposition trajectory.

Compared to FBC.

3D half-circular trajectory experimental results

The 3D half-circular trajectory in G-code form was generated by the following equation

q_{x} = 5 \cos (\frac{π}{12} t) mm, q_{y} = q_{z} = 5 \sin (\frac{π}{12} t) mm

(44)

Similar to the 3D sharp-corner trajectory, the five-time experiment was conducted for each controller until the minimum contour error was reached as shown in Figures 14 –17. Figures 14(a), 15(a), and 16(a) show the trajectory tracking profiles, while Figures 14(b), 15(b), and 16(b) show the contour errors based on the EILCC, EILCC with LICEC, and EILCC with LICEC and BRT, respectively. Here also, the results of the first iteration refer to the FBC only. Under EILCC, the initial maximum contour error was $0.1814 mm$ on average and was reduced to $0.0618 mm$ in the fifth iteration as shown in Figure 14(b). Using EILCC with LICEC, the final average maximum contour error was $0.0516 mm$ in the fourth iteration, which is equivalent to the reduction of $16.5 %$ of the maximum contour error compared to that of EILCC. On the contrary, using EILCC with LICEC and BRT, the smallest maximum contour error of $0.041 mm$ was obtained on average as shown in Figure 16(b). This is equivalent to the reduction of the maximum contour error by $33.66 %$ and $20.54 %$ compared to EILCC and EILCC with LICEC, respectively. All of the experiments show that all of the EILCC generates oscillated contour error profile and reaches zero contour error slowly after contour error peak appeared as shown in Figure 16, because EILCC does not modify control input voltage directly but enforces the reference trajectory only. Hence, the oscillated profile may increase the mean of contour error although the maximum contour error is more reduced.

Figure 14.

Experimental results for the 3D half-circular trajectory by EILCC: (a) trajectory tracking profiles and (b) contour error.

Figure 15.

Experimental results for the 3D half-circular trajectory by EILCC with LICEC: (a) trajectory tracking profiles and (b) contour error.

Figure 16.

Experimental results for the 3D half-circular trajectory by EILCC with LICEC and BRT: (a) trajectory tracking profiles and (b) contour error.

Figure 17.

Contour error experimental results for the 3D half-circular trajectory: (a) maximum contour error and (b) mean contour error.

Comparison of the maximum and mean contour errors of the three controllers in the five-time experiments is shown in Figure 17(a) and (b), respectively. The average mean and maximum contour errors based on EILCC with actual contour error compensation (ACEC) and BRT were, respectively, reduced by $65.54 %$ and $77.4 %$ compared to FBC. The rest of the experimental results are summarized in Table 2, where the tendency of error reduction is similar in both the 3D sharp-corner and half-circular trajectories.

Conclusion

This study proposed a method to enhance the tracking performance of machine tool feed drive systems using EILCC with LICEC and BRT. The LICEC can estimate precise contour error for a higher curvature trajectory. Because the existing methods require access to the embedded controller for modifying the control input, it cannot be applied in commercial CNC machines. The proposed method is designed by a cascade scheme which enables it to be implemented in commercial CNC machines currently in use. An FBC of industrial machines is generally well tuned already. Although the ILC based on feedback control requires changing the well-tuned controller, the proposed approach is based on feedforward control, which is expected to improve the tracking performance, by iterative NC program modification. Its performance was verified experimentally with the 3D sharp-corner and half-circular trajectories in a commercial three-axis CNC machine. The experimental results of the proposed controller were compared to those of the traditional FBC and EILCC with estimated contour error, where the proposed controller achieved that the maximum and mean contour errors were reduced by 45.11% and 54.48% and 20.54% and 26.92% on average compared to EILCC with approximated contour error and EILCC with LICEC, respectively.

In the future, the EILCC could be extended to reduce energy consumption by energy analysis as well as convergence because industrial CNC machines consume a lot of energy. Furthermore, because the machining process is nonlinear, the EILCC performance can be improved by considering this non-linearity in the convergence analysis.

Footnotes

Handling Editor: James Baldwin

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 Ministry of Research, Technology and Higher Education of the Republic of Indonesia, which we would like to thank.

ORCID iD

Yogi Muldani Hendrawan

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