Tangential velocity tracking-based cross-coupled control method with variable feedrate for biaxial parametric curve following

Abstract

To address the inaccurate contour error calculation problem in traditional cross-coupled control (CCC) methods for biaxial motion systems, this paper presents a novel CCC method based on a recently developed tangential velocity tracking (TVT) strategy. It has the advantage that existing high-precision algorithms for searching the foot point can be directly integrated to obtain the excellent estimation accuracy of contour error. The cumbersome parameter tuning for position controllers of each axis is unnecessary. Moreover, a velocity interpolator for parametric curves is developed to extend the TVT strategy to the variable feedrate case. The stability of the TVT-based CCC system is proved using a quadratic Lyapunov function. Comparative experiments are carried out, and the results indicate that the estimation deviation of contour error in the TVT-based CCC method can be constrained within 1 μm. The maximum contour error is significantly reduced by 65.32% and 50.10% compared with the traditional CCC methods based on tangential and circular approximations, respectively.

Introduction

Contouring control is widely used in industrial applications, including computer numerical control (CNC) machine tools, laser cutting, robot manipulators, etc. In contouring control systems, each axis must follow a predetermined trajectory simultaneously. A critical measure of final product quality is the contour error,^1–3 which is geometrically the shortest distance between the desired trajectory and the actual position. In traditional contouring control methods, each axis is designed independently, and the control task is to reduce position tracking errors of each axis. Accordingly, the contour error can be indirectly reduced to some extent. Nonetheless, perfect position tracking control of each axis is hardly achieved. Moreover, purely reducing position tracking errors does not necessarily improve the contouring performance in theory. ⁴

To address the problem, Koren and Lo^5,6 pioneered the cross-coupled control (CCC) strategy for biaxial contouring control systems. In the CCC strategy, an additional contouring controller composed of a real-time contour error calculation algorithm and a control law is added to individual axial control loops. The basic idea of the CCC strategy is to generate contour error correction signals according to the estimated contour error and the control law, then distribute them into velocity control loops of each axis to reduce the contour error. Due to its simplicity, the CCC strategy receives more attention and has become popular in contouring control. ⁷ At present, different control laws, including fuzzy PID control ⁸ and deep reinforcement learning, ⁹ have been integrated into the CCC strategy to improve contouring performance under external disturbances. Meanwhile, advanced individual tracking control algorithms, such as sliding mode control, ¹⁰ model prediction control, ¹¹ and iterative learning control, ¹² have been combined with the CCC strategy to simultaneously minimize the contour error and the position track errors of each axis. Recently, the CCC strategy has been applied to the five-axis machine tools. ¹³

Note that the CCC is a feedback controller, and the contouring performance in CCC systems depends on the real-time estimation accuracy of contour error. Generally, the analytical solution of contour error for straight lines and circular trajectories can be directly obtained. Nevertheless, in reality most trajectories are represented in a complex form, and calculating the actual value of contour error is impractical. To address this problem, many researchers have proposed various contour error estimation methods (CEEMs) in the past decades, which can primarily be classified into two categories: the geometric approximation-based CEEM and the iterative search-based CEEM.

In the geometric approximation-based CEEM, the desired trajectory is approximated locally by the tangential line^14,15 or the osculating circle ¹⁶ at the current reference point. Then, the distance from the actual point to the tangential line or the osculating circle is taken as the estimation value of contour error. The geometric approximation-based CEEM is simple and easy to implement in practice. Note that the high estimation accuracy of contour error in the geometric approximation-based CEEM is achieved only when the actual point is close enough to the reference point. Although the CCC strategy has been combined with many advanced control algorithms for improving the position tracking performance of each axis, ¹⁷ significant position tracking errors of each axis may exist because of the uncertain external disturbances and the bandwidth limitations of motors and drivers.

As an alternative method, the iterative search-based CEEM aims to search the foot point, that is, the point closest to the current actual position in the desired trajectory. Geometrically, the vector from the foot point to the actual point is perpendicular to the desired trajectory. According to this property, the foot point can be searched accurately by tangential or circular backstepping methods.^18–20 Meanwhile, Newton’s method^21–23 and Brent’s method ²⁴ were developed to search the foot point in a numerical calculation way. Recently, Wang et al. ²⁵ realized a faster convergence rate based on Atiken acceleration for better real-time performance. The iterative search-based CEEM can achieve perfect estimation accuracy of contour error. However, it is necessary to point out that the stability analysis of CCC systems using the iterative search-based CEEM is still challenging.^26,27

Recently, a novel contouring control framework based on the tangential velocity tracking (TVT) strategy has been proposed to eliminate the adverse effect of position tracking errors on the estimation accuracy of contour error. ²⁸ Furthermore, the task coordinate frame method for biaxial motion control has been developed successfully based on the TVT strategy. However, it requires precision dynamics models and can only be used in the case of constant feedrate.

This paper will apply the TVT strategy to the CCC system to address the inaccurate contour error estimation problem. In the proposed TVT-based CCC method, the feed-direction control task is completed by tracking the desired tangential velocity command. The desired feedrate at the current foot point is calculated in real time. The error dynamics of the control system are derived, and the system stability is proved using a quadratic Lyapunov function.

The main contributions of this paper can be summarized as follows:

• (1) An innovative contouring control structure is proposed. It cancels the individual axial position tracking, avoiding the parameter tuning of position controllers that is particularly cumbersome. Different iterative search-based CEEMs for seeking the foot point can be directly integrated to achieve accurate contour error estimation. Moreover, the system stability is guaranteed.

• (2) Extend the TVT strategy to the variable feedrate case by developing a velocity interpolator for biaxial parametric curves.

The rest of this paper is organized as follows. Section 2 briefly discusses the traditional velocity-loop CCC method. In Section 3, we propose a novel TVT-based CCC method. The experiments are carried out and analyzed in Section 4. The conclusions are summarized in Section 5.

Traditional velocity-loop CCC method

Figure 1 shows the typical structure of the traditional velocity-loop CCC method for biaxial motion systems. ${\mathit{P}}_d=[P_{\mathrm{dx}}{P}_{\mathrm{dy}}{]}^T$ and ${\mathit{P}}_a=[P_{\mathrm{ax}}{P}_{\mathrm{ay}}{]}^T$ represent the reference point and the actual point, respectively. ( $e_{\mathrm{px}}$ , $e_{\mathrm{py}}$ ) are the position tracking errors of each axis. ( $K_{\mathrm{px}}$ , $K_{\mathrm{py}}$ ) are the proportional gains of position controllers. ${\mathit{V}}_f=[V_{\mathrm{fx}}{V}_{\mathrm{fy}}{]}^T$ and ${\mathit{V}}_{\epsilon }=[V_{\epsilon x}{V}_{\epsilon y}{]}^T$ denote the velocity commands generated by the individual axial position controllers and the CCC, respectively. ( $C_x$ , $C_y$ ) are the varying cross-coupled gains depending on the desired trajectory. ${\epsilon }_e$ denotes the estimated contour error. The generalized formula for ${\epsilon }_e$ is

$${\epsilon }_e=e_{\mathrm{px}}{C}_x+e_{\mathrm{py}}{C}_y$$

(1)

Figure 1.

Typical structure of the traditional velocity-loop CCC method for biaxial motion systems.

For example, the contour error for a linear trajectory can be determined by ⁶

$${\epsilon }_e=-e_{\mathrm{px}}\sin \theta +e_{\mathrm{py}}\cos \theta$$

(2)

where $\theta$ is the incline angle between the linear trajectory and the x-axis. In the CCC method, three issues should be considered.

The first one is the CEEM for the real-time calculation of ${\epsilon }_e$ . It can be seen from Figure 1 that the CCC is a feedback controller, which aims to eliminate the estimated contour error ${\epsilon }_e$ rather than the actual one. Thus, the contouring performance depends on the calculation accuracy of CEEM. Generally, the CCC method uses the geometric approximation-based CEEM because of the small computational load, such as the tangential line-based CEEM shown in Figure 2, where the estimated contour error ${\epsilon }_e$ is the distance from the actual point to the tangential line at the reference point. It is evident that the estimation accuracy of contour error depends on the magnitude of the position tracking error vector ${\mathit{e}}_p=[e_{\mathrm{px}}{e}_{\mathrm{py}}{]}^T$ . As a result, when significant position tracking errors exist because of the uncertain external disturbances and the bandwidth limitations of motors and drivers, the reliability of the tangential line-based CEEM deteriorates dramatically.

Figure 2.

Tangential line-based CEEM.

The second one is to distribute the control effort to minimize position tracking errors and the estimated contour error simultaneously. As shown in Figure 1, different position control objectives exist in the feed and contouring directions. The feed-direction control objective is to reduce the position tracking errors ( $e_{\mathrm{px}}$ , $e_{\mathrm{py}}$ ) through the velocity command ${\mathit{V}}_f=[V_{\mathrm{fx}}{V}_{\mathrm{fy}}{]}^T$ generated by individual axial position controllers, while the control objective in the contouring direction is to eliminate the estimated contour error ${\epsilon }_e$ through the velocity command ${\mathit{V}}_{\epsilon }=[V_{\epsilon x}{V}_{\epsilon y}{]}^T$ generated by the CCC. The velocity commands of each axis consist of ${\mathit{V}}_f$ and ${\mathit{V}}_{\epsilon }$ . Note that the velocity commands ${\mathit{V}}_f$ and ${\mathit{V}}_{\epsilon }$ are generally non-orthogonal and coupled. Thus, it is challenging to minimize both ( $e_{\mathrm{px}}$ , $e_{\mathrm{py}}$ ) and ${\epsilon }_e$ .

The last but not least one is the stability analysis. Because of the time-varying and coupled properties of the CCC system, it is difficult to analyze the system stability, especially when using the iteration search-based CEEM.^26,27

TVT-based CCC method

In this section, a novel TVT-based CCC approach is proposed. The contouring control framework is first introduced. Then, a velocity interpolator for parametric curves is developed to obtain the curve parameter-based tangential velocity commands. Finally, the error dynamics of the control system are derived, and the system stability is proved using a quadratic Lyapunov function.

Contouring control framework

In practical applications, the contouring-direction control task is to eliminate the contour error, while the feed-direction control task is to make the end-effector (e.g. tools) move along the desired trajectory with the desired feedrate. In traditional CCC methods, the feed-direction control is designed such that each axis must track the reference point according to the sampling time. As a result, the position tracking errors in the feed direction would degrade the estimation accuracy of contour error. Recently, a new contouring control framework based on the TVT strategy has been proposed. ²⁸ To eliminate the adverse effect of position tracking errors on the estimation accuracy of contour error, the feed-direction control task in the TVT strategy is completed by tracking the desired tangential velocity according to the estimated foot point. Note that the point closest to the current actual position in the desired trajectory is referred to as the foot point in this paper.

Accordingly, we propose a TVT-based CCC method for biaxial motion systems, as shown in Figure 3. In Figure 3(a), ${\mathit{P}}_r$ and ${\mathit{P}}_f$ denote the actual and estimated foot points, respectively. $\mathit{t}$ and $\mathit{n}$ denote the unit tangential and normal vectors at ${\mathit{P}}_f$ , respectively. $\theta$ is the incline angle between the vector $\mathit{t}$ and the x-axis. ${\epsilon }_a$ denotes the actual contour error. In Figure 3(b), ${•}_x$ and ${•}_y$ denote the components of • in the x-axis and y-axis, respectively. $u_f$ is the corresponding curve parameter of ${\mathit{P}}_f$ . The process in one sampling interval is summarized as follows.

• (1) Search for the foot point according to the current actual point ${\mathit{P}}_a$ and the desired trajectory.

• (2) Calculate the unit tangential vector $\mathit{t}$ at ${\mathit{P}}_f$ as follows:

$$\mathit{t}=[\cos \theta \sin \theta {]}^T$$

(3)

Figure 3.

TVT-based CCC method for biaxial motion systems: (a) schematic diagram and (b) control structure.

The unit normal vector $\mathit{n}$ is determined through rotating $\mathit{t}$ by 90° counterclockwise and is given by

$$\mathit{n}=[-\sin \theta \cos \theta {]}^T$$

(4)

• (3) The velocity interpolator determines the desired tangential velocity command ${\mathit{V}}_f$ according to the curve parameter $u_f$ as follows:

$${\mathit{V}}_f={\left[V_{\mathrm{fx}}{V}_{\mathrm{fy}}\right]}^T=V_d\left(u_f\right)\mathit{t}$$

(5)

where $V_d$ denotes the desired feedrate at ${\mathit{P}}_f$ . Thus, the desired tangential velocity command ${\mathit{V}}_f$ is always parallel to $\mathit{t}$ , and the magnitude of ${\mathit{V}}_f$ is equal to the desired feedrate at the real-time estimated foot point.

• (4) As shown in Figure 3(a), the contour error is estimated by the distance from ${\mathit{P}}_a$ to the tangential line at ${\mathit{P}}_f$ , which can be calculated by

$${\epsilon }_e=〈{\mathit{e}}_p·\mathit{n}〉=-e_{\mathrm{px}}\sin \theta +e_{\mathrm{py}}\cos \theta$$

(6)

where 〈·〉 denotes the inner product operator, and

$${\mathit{e}}_p={\mathit{P}}_f-{\mathit{P}}_a=\left[\begin{array}{c}{e}_{\mathrm{px}}\hfill \\ \multicolumn{1}{c}{e_{\mathrm{py}}}\end{array}\right]=\left[\begin{array}{c}{P}_{\mathrm{fx}}-P_{\mathrm{ax}}\hfill \\ \multicolumn{1}{c}{P_{\mathrm{fy}}-P_{\mathrm{ay}}}\end{array}\right]$$

(7)

Accordingly, the cross-coupled gains $C_x$ and $C_y$ shown in Figure 3(b) are represented by

$$C_x=-\sin \theta ,C_y=\cos \theta$$

(8)

• (5) As shown in Figure 3(b), the CCC generates the velocity command ${\mathit{V}}_{\epsilon }=[V_{\epsilon x}{V}_{\epsilon y}{]}^T$ according to ${\epsilon }_e$ and the control law. In this paper, the P-type control law is used. Thus, ${\mathit{V}}_{\epsilon }$ is calculated by

$${\mathit{V}}_{\epsilon }=\left[\begin{array}{c}{V}_{\epsilon x}\hfill \\ \multicolumn{1}{c}{V_{\epsilon y}}\end{array}\right]=K_c{\epsilon }_e\left[\begin{array}{c}{C}_x\hfill \\ \multicolumn{1}{c}{C_y}\end{array}\right]=K_c{\epsilon }_e\mathit{n}$$

(9)

where $K_c$ is the proportional gain of CCC, and $K_c>0$ .

• (6) Finally, the velocity commands of each axis shown in Figure 3(b) are given by

$$V_{\mathrm{cx}}=V_{\mathrm{fx}}+V_{\epsilon x}\mathrm{and}{V}_{\mathrm{cy}}=V_{\mathrm{fy}}+V_{\epsilon y}$$

(10)

• Remark 1: In the traditional CCC method with the geometric approximation-based CEEM, the poor position tracking performance in the feed direction would significantly deteriorate the estimation accuracy of contour error. For the proposed TVT-based CCC approach, the contour error is calculated by the distance from the actual point to the tangential line at the estimated foot point. At present, many high-precision algorithms have been proposed to search the foot point ${\mathit{P}}_f$ .^18–23 The error $e_{\mathrm{pt}}$ shown in Figure 3(a) is defined as follows:

$$e_{\mathrm{pt}}=〈{\mathit{e}}_p·\mathit{t}〉=〈({\mathit{P}}_f-{\mathit{P}}_a)·\mathit{t}〉$$

(11)

Note that $e_{\mathrm{pt}}$ is only determined by the calculation accuracy of foot point seeking algorithms and is usually very small. Consequently, it can be observed from Figure 3(a) that the actual contour error ${\epsilon }_a$ would be approximated accurately by ${\epsilon }_e$ .

• Remark 2: In the traditional CCC method described in Figure 1, each axis must track the desired position commands $P_{\mathrm{dx}}$ or $P_{\mathrm{dy}}$ . As shown in Figure 3(b), the feed-direction control task in the TVT-based CCC method is completed by tracking the desired tangential velocity command ${\mathit{V}}_f=[V_{\mathrm{fx}}{V}_{\mathrm{fy}}{]}^T$ . Thus, the control system cancels the individual axial position tracking, avoiding the cumbersome parameter tuning of position controllers.

• Remark 3: In the traditional CCC method described in Figure 1, the velocity commands ${\mathit{V}}_f$ and ${\mathit{V}}_{\epsilon }$ interfere with each other because of the non-orthogonal relationship. For the TVT-based CCC method, it can be seen from (5) to (9) that the velocity commands ${\mathit{V}}_f$ and ${\mathit{V}}_{\epsilon }$ are parallel to the vectors $\mathit{t}$ and $\mathit{n}$ , respectively. Hence, ${\mathit{V}}_f$ and ${\mathit{V}}_{\epsilon }$ always be orthogonal and decoupled. As a result, the control tasks in the feed and contouring directions can be realized together.

Curve parameter-based tangential velocity interpolator

The traditional interpolator parameterizes the desired trajectory as a function of time and calculates the reference point at each sampling instant. However, for the TVT-based CCC approach shown in Figure 3(b), the interpolator must determine the desired tangential velocity ${\mathit{V}}_f$ at every point of the desired trajectory.

Recall that the magnitude of ${\mathit{V}}_f$ is equal to the desired feedrate $V_d$ at ${\mathit{P}}_f$ . Thus, the first step is to obtain the desired feedrate for arbitrary curve parameters. After looking ahead and feedrate scheduling, the traditional interpolator for parametric curves is required to generate the curve parameter $u_d(k{T}_s)$ of the reference point ${\mathit{P}}_d(k{T}_s)$ at every sampling time ( $T_s$ is the sampling interval and $k=1,2,...$ ). ²⁹ The desired feedrate at the reference point can be determined by the forward difference formula, that is,

$$V_d(u_d(k{T}_s))=\left|\left|\frac{{\mathit{P}}_d((k+1)T_s)-{\mathit{P}}_d(k{T}_s)}{T_s}\right|\right|$$

(12)

where ||·|| denotes the Euclidean norm.

For the arbitrary curve parameter $u\in [u_d(k{T}_s),u_d((k+1)T_s)]$ , the desired feedrate can be approximated according to the linear relationship, that is,

$$\begin{array}{c}\multicolumn{1}{c}{V_d\left(u\right)=V_d\left(u_d\left(k{T}_s\right)\right)+\frac{u-u_d\left(k{T}_s\right)}{u_d\left((k+1)T_s\right)-u_d\left(k{T}_s\right)}·}\\ (V_d(u_d((k+1)T_s))-V_d(u_d(k{T}_s)))\hfill \end{array}$$

(13)

Note that ${\mathit{V}}_f$ is parallel to the unit tangential vector at the estimated foot point. The unit tangential vector $\mathit{t}$ for the arbitrary curve parameter $u$ can be obtained by:

$$\mathit{t}(u)=\frac{{\mathit{P}}_d\prime (u)}{||{\mathit{P}}_d\prime (u)||},{\mathit{P}}_d\prime (u)=\frac{d{\mathit{P}}_d(u)}{\mathrm{du}}$$

(14)

Thus, we have

$${\mathit{V}}_f(u)=[V_{\mathrm{fx}}(u)V_{\mathrm{fy}}(u){]}^T=V_d(u)\mathit{t}(u)$$

(15)

where $V_{\mathrm{fx}}(u)$ and $V_{\mathrm{fy}}(u)$ are the velocity commands of the x-axis and y-axis, respectively.

Error dynamics of the TVT-based CCC system

For most typical feed drive systems, the velocity closed-loop dynamics can be reasonably modeled as a first-order system,^30,31 which is described in the Laplace domain as follows:

$$\frac{V_{\mathrm{ai}}}{V_{\mathrm{ci}}}=\frac{K_{\mathrm{vi}}}{1+T_{\mathrm{ti}}s},i=x,y$$

(16)

where $T_{\mathrm{ti}}$ and $K_{\mathrm{vi}}$ represent the time constant and gain, respectively. Transforming (16) into the time domain, we obtain

$$T_{\mathrm{ti}}{\stackrel{·}{V}}_{\mathrm{ai}}+V_{\mathrm{ai}}=K_{\mathrm{vi}}{V}_{\mathrm{ci}}$$

(17)

In practical applications, it is essential to tune the velocity controller parameters to match axial dynamics. Hence, we assume that the mismatch of velocity closed-loop dynamics between the x-axis and y-axis can be negligible, that is, $T_t=T_{\mathrm{tx}}=T_{\mathrm{ty}}$ and $K_v=K_{\mathrm{vx}}=K_{\mathrm{vy}}$ . Accordingly, the dynamics of the biaxial feed drive system can be represented as follows:

$$T_t{\stackrel{·}{\mathit{V}}}_a+{\mathit{V}}_a=K_v{\mathit{V}}_c$$

(18)

$${\stackrel{·}{\mathit{P}}}_a={\mathit{V}}_a$$

(19)

where ${\mathit{V}}_a=[V_{\mathrm{ax}}{V}_{\mathrm{ay}}{]}^T$ , ${\mathit{V}}_c=[V_{\mathrm{cx}}{V}_{\mathrm{cy}}{]}^T$ , and ${\mathit{P}}_a=[P_{\mathrm{ax}}{P}_{\mathrm{ay}}{]}^T$ .

As seen from Figure 3(a), we can define a new local coordinate frame composed of the orthogonal unit vectors $\mathit{t}$ and $\mathit{n}$ at the estimated foot point ${\mathit{P}}_f$ . The coordinate transformation matrix from the world Cartesian coordinate frame to the local coordinate frame is

$${\mathit{X}}_F={\mathit{F}}^T\mathit{X}$$

(20)

where ${\mathit{X}}_F$ is the coordinate of $\mathit{X}$ in the local coordinate frame, and

$${\mathit{F}}^T=\left[\begin{array}{c}{\mathit{t}}^T\hfill \\ \multicolumn{1}{c}{{\mathit{n}}^T}\end{array}\right]$$

(21)

Note that ${\mathit{F}}^T={\mathit{F}}^{-1}$ because $\mathit{F}$ is unitary. Furthermore, the coordinate transformation matrix ${\mathit{F}}^T$ is time-varying when the local coordinate frame moves along the desired trajectory. Taking the time derivative of (21) and using (3) and (4), we can obtain

$${\stackrel{·}{\mathit{F}}}^T=\stackrel{·}{\theta }\left[\begin{array}{c}{\mathit{n}}^T\hfill \\ \multicolumn{1}{c}{-{\mathit{t}}^T}\end{array}\right]$$

(22)

where $\theta$ is the incline angle between the vector $\mathit{t}$ and the x-axis, as shown in Figure 3(a). Combining (6) and (7) with (11), the error $e_{\mathrm{pt}}$ and the estimated contour error ${\epsilon }_e$ can be expressed by

$$\left[\begin{array}{c}{e}_{\mathrm{pt}}\hfill \\ \multicolumn{1}{c}{{\epsilon }_e}\end{array}\right]={\mathit{F}}^T({\mathit{P}}_f-{\mathit{P}}_a)$$

(23)

By letting ${\mathit{e}}_F={[e_{pt}{\epsilon }_e]}^T$ , the time derivative of (23) is represented as

$${\stackrel{·}{\mathit{e}}}_F={\stackrel{·}{\mathit{F}}}^T\mathit{F}{\mathit{e}}_F-{\mathit{V}}_{\mathrm{aF}}+{\mathit{F}}^T{\stackrel{·}{\mathit{P}}}_f$$

(24)

where

$${\mathit{V}}_{\mathrm{aF}}={\mathit{F}}^T{\mathit{V}}_a=[V_{\mathrm{aFt}}{V}_{\mathrm{aFn}}{]}^T$$

(25)

In (25), $V_{\mathrm{aFt}}$ and $V_{aFn}$ are the actual velocities in the tangential and normal directions, respectively. Taking the time derivative of (25) and using (18), we can obtain

$${\stackrel{·}{\mathit{V}}}_{\mathrm{aF}}={\stackrel{·}{\mathit{F}}}^T\mathit{F}{\mathit{V}}_{\mathrm{aF}}+K_v{T_t}^{-1}{\mathit{F}}^T{\mathit{V}}_c-{T_t}^{-1}{\mathit{V}}_{\mathrm{aF}}$$

(26)

It can be seen from (10) that the desired velocity command ${\mathit{V}}_c$ consists of the velocity command ${\mathit{V}}_{\epsilon }$ generated by the CCC and the desired tangential velocity command ${\mathit{V}}_f$ , that is,

$${\mathit{V}}_c={\mathit{V}}_{\epsilon }+{\mathit{V}}_f$$

(27)

Accordingly, using (5) and (9), we can obtain

$${\mathit{F}}^T{\mathit{V}}_c=\left[\begin{array}{c}{\mathit{t}}^T\hfill \\ \multicolumn{1}{c}{{\mathit{n}}^T}\end{array}\right][K_c{\epsilon }_e\mathit{n}+V_d\mathit{t}]=\left[\begin{array}{c}{V}_d\hfill \\ \multicolumn{1}{c}{K_c{\epsilon }_e}\end{array}\right]$$

(28)

Substituting (21), (22), and (28) into (26) yields

$${\stackrel{·}{V}}_{\mathrm{aFt}}=\stackrel{·}{\theta }{V}_{\mathrm{aFn}}+K_v{T_t}^{-1}{V}_d-{T_t}^{-1}{V}_{\mathrm{aFt}}$$

(29)

$${\stackrel{·}{V}}_{\mathrm{aFn}}=-\stackrel{·}{\theta }{V}_{\mathrm{aFt}}+K_v{K}_c{T_t}^{-1}{\epsilon }_e-{T_t}^{-1}{V}_{\mathrm{aFn}}$$

(30)

Similarly, substituting (21) and (22) into (24) yields

$${\stackrel{·}{\epsilon }}_e=-\stackrel{·}{\theta }{e}_{\mathrm{pt}}-V_{\mathrm{aFn}}+{\mathit{n}}^T{\stackrel{·}{\mathit{P}}}_f$$

(31)

From (29) to (31), the resulting error dynamics of the TVT-based CCC system have the following form:

$$\stackrel{·}{\mathit{x}}=\mathit{A}(\stackrel{·}{\theta })\mathit{x}+\mathit{d}(V_d,{\stackrel{·}{\mathit{P}}}_f,\theta ,\stackrel{·}{\theta },e_{\mathrm{pt}})$$

(32)

where

$$\mathit{x}={\left[\begin{array}{ccc}{V}_{\mathrm{aFt}}\hfill & V_{\mathrm{aFn}}\hfill & {\epsilon }_e\hfill \end{array}\right]}^T$$

(33)

$$\mathit{A}(\stackrel{·}{\theta })=\left[\begin{array}{ccc}-{T_t}^{-1}\hfill & \stackrel{·}{\theta }\hfill & 0\hfill \\ \multicolumn{1}{c}{-\stackrel{·}{\theta }}& -{T_t}^{-1}\hfill & K_v{K}_c{T_t}^{-1}\hfill \\ \multicolumn{1}{c}{0}& -1\hfill & 0\hfill \end{array}\right]$$

(34)

$$\mathit{d}(t)={\left[\begin{array}{ccc}{K}_v{T_t}^{-1}{V}_d\hfill & 0\hfill & {\mathit{n}}^T{\stackrel{·}{\mathit{P}}}_f-\stackrel{·}{\theta }{e}_{\mathrm{pt}}\hfill \end{array}\right]}^T$$

(35)

Note that $\mathit{d}(V_d,{\stackrel{·}{\mathit{P}}}_f,\theta ,\stackrel{·}{\theta },e_{\mathrm{pt}})$ is represented as $\mathit{d}(t)$ hereafter for simplicity. In (32), $\mathit{x}$ is the state vector, $\mathit{A}(\stackrel{·}{\theta })$ is the linear system matrix, $\mathit{d}(t)$ is the disturbance input vector. Recall that $\theta$ denotes the incline angle between the x-axis and the unit tangential vector at the estimated foot point. For a linear trajectory, $\stackrel{·}{\theta }$ is always equal to zero. For complex trajectories, $\stackrel{·}{\theta }$ is trajectory-dependent and time-varying. Thus, $\mathit{A}(\stackrel{·}{\theta })$ is time-varying, linear, and parametrically varying.

Stability analysis

This subsection first gives a necessary and sufficient condition for positive definite matrices and a sufficient condition for quadratic stability. Then, the system (32) without the disturbance input is proved to be quadratically stable, which is used to further prove that the solution of the TVT-based CCC system is bounded.

• Theorem 1 ³² : Given the continuous, bounded matrix function $\mathit{G}:\mathbb{R}\to {\mathbb{R}}^{n\times n}$ , and a closed and bounded subset $\mathrm{\Psi }\subset \mathbb{R}$ , consider the time-varying, linear, and parametrically varying system

$$\stackrel{·}{\mathit{y}}(t)=\mathit{G}(\rho (t))\mathit{y}(t),\mathit{y}(t_0)={\mathit{y}}_0,t\ge t_0$$

(36)

where $\rho$ is a continuous function mapping $\mathbb{R}$ (time) into $\mathrm{\Psi }$ .

Define the scalar valued function $V:{\mathbb{R}}^n\to \mathbb{R}$ as

$$V(\mathit{y})={\mathit{y}}^T\mathit{Py}$$

(37)

where $\mathit{P}\in {\mathbb{R}}^{n\times n}$ , and $\mathit{P}$ is a symmetric and positive definite matrix. Along trajectories of (36), the time derivative of $V(\mathit{y})$ is expressed as

$$\stackrel{·}{V}(\mathit{y}(t))={\mathit{y}}^T(t)[{\mathit{G}}^T(\rho (t))\mathit{P}+\mathit{PG}(\rho (t))]\mathit{y}(t)$$

(38)

Let

$$\mathit{M}=-({\mathit{G}}^T(\rho )\mathit{P}+\mathit{PG}(\rho ))$$

(39)

If $\mathit{M}$ is positive definite, then the system (36) is quadratically stable over $\mathrm{\Psi }$ . Moreover, there exist finite positive constants ${\gamma }_1$ and ${\gamma }_2$ such that the state-transition matrix $\mathit{\Phi }(t,t_0)$ of the system (36) satisfies that

$$||\mathit{\Phi }(t,t_0)||\le {\gamma }_1{e}^{-{\gamma }_2(t-t_0)}$$

(40)

Note that ||·|| denotes the norm for vectors and the corresponding induced norm for matrices hereafter. When the condition (39) is satisfied, (37) is a quadratic Lyapunov function of the system (36).

• Theorem 2. (Sylvester’s Criterion) ³³ : Suppose $\mathit{M}\in {\mathbb{R}}^{n\times n}$ , and $\mathit{M}$ is a symmetric matrix with entries $m_{\mathrm{ij}}$ $(1\le i,j\le n)$ . The $i$ th leading principal minor is denoted as

$$det\mathit{M}(i)=det\left[\begin{array}{ccc}{m}_{11}\hfill & \cdots \hfill & m_{1i}\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{m_{i1}}& \cdots \hfill & m_{\mathrm{ii}}\hfill \end{array}\right]$$

(41)

Then, $\mathit{M}$ is positive definite if and only if all its leading principal minors are positive.

Let us consider the following system simplified from (32):

$$\stackrel{·}{\mathit{z}}=\mathit{A}(\stackrel{·}{\theta })\mathit{z}$$

(42)

• Theorem 3: The system (42) is quadratically stable.

Proof: Define a real and symmetric matrix $\mathit{K}$ as

$$\mathit{K}=\left[\begin{array}{ccc}r\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& r\hfill & -1\hfill \\ \multicolumn{1}{c}{0}& -1\hfill & {T_t}^{-1}(1+K_v{K}_{c}r)\hfill \end{array}\right]$$

(43)

where $r$ is a finite positive constant. All its leading principal minors are

$$\begin{array}{c}\multicolumn{1}{c}{det\mathit{K}\left(1\right)=r,det\mathit{K}\left(2\right)=r^2,det\mathit{K}(3)=r^2{T_t}^{-1}}\\ +K_v{K}_c{T_t}^{-1}r(r^2-T_t{K_v}^{-1}{K_c}^{-1})\hfill \end{array}$$

(44)

Note that $K_v$ , $K_c$ , and $T_t$ are finite positive constants. Thus, when $r>\sqrt{T_t{K_v}^{-1}{K_c}^{-1}}$ , all leading principal minors of $\mathit{K}$ are positive. Then $\mathit{K}$ is positive definite according to Theorem 2.

Let $\mathit{Q}=-({\mathit{A}}^T(\stackrel{·}{\theta })\mathit{K}+\mathit{KA}(\stackrel{·}{\theta }))$ . From (34) to (43), we obtain

$$\mathit{Q}=\left[\begin{array}{ccc}2{T_t}^{-1}r\hfill & 0\hfill & -\stackrel{·}{\theta }\hfill \\ \multicolumn{1}{c}{0}& 2{T_t}^{-1}r-2\hfill & 0\hfill \\ \multicolumn{1}{c}{-\stackrel{·}{\theta }}& 0\hfill & 2K_v{K}_c{T_t}^{-1}\hfill \end{array}\right]$$

(45)

Accordingly, all its leading principal minors are

$$\begin{array}{c}det\mathit{Q}(1)=2{T_t}^{-1}r,det\mathit{Q}(2)=4{T_t}^{-2}r(r-T_t),\hfill \\ det\mathit{Q}(3)=8K_v{K}_c{T_t}^{-3}(r-T_t)(r-\frac{{\stackrel{·}{\theta }}^2{T_t}^2}{4K_v{K}_c})\hfill \end{array}$$

(46)

For a regular desired trajectory, $\stackrel{·}{\theta }$ is bounded. Thus, there exists a finite positive constant ${\gamma }_3$ such that ${\gamma }_3>|\stackrel{·}{\theta }|$ for all time. It is evident from (46) that when $r$ satisfies that

$$r>max\{T_t,\frac{{{\gamma }_3}^2{T_t}^2}{4K_v{K}_c}\}$$

(47)

all leading principal minors of $\mathit{Q}$ are positive. Similarly, according to Theorem 2, the real and symmetric matrix $\mathit{Q}$ is positive definite.

To sum up, $r$ can be designed such that

$$r>max\{T_t,\frac{{{\gamma }_3}^2{T_t}^2}{4K_v{K}_c},\sqrt{T_t{K_v}^{-1}{K_c}^{-1}}\}$$

(48)

Then, both $\mathit{K}$ and $\mathit{Q}$ are positive definite. According to Theorem 1, the system (42) is quadratically stable. Thus, Theorem 3 is proved.

Based on Theorem 3, the stability of the TVT-based CCC system can be proved.

• Theorem 4: The solution of the TVT-based CCC system (32) is bounded.

Proof: Let $\mathit{\Phi }(t,t_0)$ be the state-transition matrix of the system (42). Then, the solution of the system (32) is ³⁴

$$\mathit{x}(t)=\mathit{\Phi }(t,t_0)\mathit{x}(t_0)+\underset{t_0}{\overset{t}{\int }}\mathit{\Phi }(t,\tau )\mathit{d}(\tau )d\tau ,t\ge t_0$$

(49)

Taking the norms of both sides of (49) yields the inequality

$$||\mathit{x}(t)||\le ||\mathit{\Phi }(t,t_0)||·||\mathit{x}(t_0)||+||\underset{t_0}{\overset{t}{\int }}\mathit{\Phi }(t,\tau )||·||\mathit{d}(\tau )||d\tau$$

(50)

For a regular desired trajectory, both $\stackrel{·}{\theta }$ and ${\stackrel{·}{\mathit{P}}}_f$ are bounded. Thus, it can be observed from (35) that all the elements of $\mathit{d}(t)$ are bounded. Hence, $\mathit{d}(t)$ is norm-bounded, that is, there exists a finite positive constant ${\gamma }_4$ such that

$$||\mathit{d}(t)||\le {\gamma }_4$$

(51)

Because the system (42) is quadratically stable according to Theorem 3, the state-transition matrix $\mathit{\Phi }(t,t_0)$ satisfies the condition (40) according to Theorem 1. Combining (40) and (51) with (50), we have

$$||\mathit{x}\left(t\right)||\le {\gamma }_1{e}^{-{\gamma }_2(t-t_0)}||\mathit{x}\left(t_0\right)||+\underset{t_0}{\overset{t}{\int }}{\gamma }_1{\gamma }_4{e}^{-{\gamma }_2(t-\tau )}d\tau$$

$$\le {\gamma }_1{e}^{-{\gamma }_2(t-t_0)}||\mathit{x}\left(t_0\right)||+\frac{{\gamma }_1{\gamma }_4}{{\gamma }_2}(1-e^{-{\gamma }_2(t-t_0)})$$

$$\le {\gamma }_1||\mathit{x}\left(t_0\right)||+\frac{{\gamma }_1{\gamma }_4}{{\gamma }_2}$$

(52)

which completes the proof.

• Theorem 5 ³⁵ : The internally connected systems are internally stable if the set of all input signals and output signals are bounded-input–bounded-output (BIBO) stable.

From Theorem 4 and (33), the actual velocities $V_{\mathrm{aFt}}$ and $V_{\mathrm{aFn}}$ in the tangential and normal directions, and the estimated contour error ${\epsilon }_e$ are bounded. Hence, for the proposed control structure depicted in Figure 3(b), the components $V_{\mathrm{ax}}$ and $V_{\mathrm{ay}}$ of the actual velocity vector in each axis are bounded. Because the distance between the actual point ${\mathit{P}}_a$ and the desired trajectory can be estimated accurately by the bounded contour error ${\epsilon }_e$ , the components $P_{\mathrm{ax}}$ and $P_{\mathrm{ay}}$ of the actual point in each axis are also bounded. It can be observed from (28) that the velocity commands of each axis are bounded. From Theorem 5, the proposed TVT-based CCC system is stable.

Experimental investigation

Experimental setup

Comparative experiments were carried out to verify the effectiveness of the proposed TVT-based CCC method. The three-axis motion control system shown in Figure 4 was used. The MATLAB/Simulink desktop real-time software (Version 10.4, R2021b), which can provide a real-time kernel for Windows, implements the control algorithms in the personal computer (PC) with a sampling interval of 1 ms. An ARM-based controller board converts the digital commands of the PC into the actual input voltage of the AC servo drivers over digital-to-analog converters (DACs). Meanwhile, the board collects position signals with a resolution of 0.5 μm by the encoder interfaces and feeds them back to the PC.

Figure 4.

Three-axis motion control system.

To improve the velocity tracking performance, the velocity controller of each axis consists of a PI-type feedback controller and a feedforward controller for the friction compensation, as depicted in Figure 5, where $V_c$ and $V_a$ are the desired and actual velocities, respectively. $K_{\mathrm{vp}}$ and $K_{\mathrm{vi}}$ are the proportional and integral gains of the feedback controller, respectively. The friction compensation strategy utilizes the Coulomb-plus-viscous friction model, ³⁶ that is,

$$u_{\mathrm{fc}}=B_m{V}_c+\mathrm{sgn}(V_c)C_m$$

(53)

where $C_m$ and $B_m$ are the Coulomb and viscous friction coefficients, respectively. $u_{\mathrm{fc}}$ is the output voltage of the feedforward controller. The values of $C_m$ and $B_m$ were identified by the half-period integral method. ³⁷ Table 1 lists the parameters of the velocity controllers.

Figure 5.

Block diagram of the velocity controller in each axis.

Table 1.

Parameters of the velocity controllers in each axis.

	x-axis	y-axis
$K_{\mathrm{vp}}$ (V · s/mm)	0.045	0.020
$K_{\mathrm{vi}}$ (V/mm)	0.9	0.4
$B_m$ (V · s/mm)	$9.6272\times {10}^{-4}$	$6.7144\times {10}^{-4}$
$C_m$ (V)	0.4530	0.3847

The butterfly-shaped curve shown in Figure 6(a) was applied in experiments. The parametric interpolator developed by Liu et al. ²⁹ was used to generate the time-stamped position commands offline, and the parameters used for the experiments are listed in Table 2. For the proposed TVT-based CCC method, the time-stamped position commands can be further converted into the curve parameter-based tangential velocity commands through the algorithm described in Subsection 3.2. Figure 6(b) shows the feedrate versus curve parameter when the maximum feedrate is set to 100 mm/s.

Figure 6.

Desired trajectory and the corresponding feedrate: (a) butterfly-shaped curve and (b) feedrate versus curve parameter with the maximum feedrate of 100 mm/s.

Table 2.

Parameters of the interpolator used for the experiments.

Parameters	Values
Arc-length tolerance	$1\times {10}^{-8}$ mm
Chord error tolerance	$1\times {10}^{-3}$ mm
Maximum centripetal acceleration	9800 mm/s2
Maximum tangential acceleration	9800 mm/s2
Maximum jerk	$1\times {10}^5$ mm/s 3

Three methods were conducted for comparison: the traditional velocity-loop CCC methods with the tangential line-based CEEM ¹⁴ (C1) and osculating circle-based CEEM ¹⁶ (C2), and the proposed TVT-based CCC method (C3). In C3, the foot point was searched by the tangential-error backstepping method ²⁰ with the maximum iteration number of 10 and the limited tangential error of 1 μm. The proportional gains of position controllers in the x-axis and y-axis were set to 40 s⁻¹ for C1 and C2, while there is no need for C3 to tune the parameters of position controllers. Note that ${\epsilon }_a$ and ${\epsilon }_e$ denote the actual and estimated contour errors, respectively. Moreover, we use ${\delta }_{\epsilon }$ to denote the estimation deviation of contour error in each sampling interval, that is, ${\delta }_{\epsilon }=|{\epsilon }_a-{\epsilon }_e|$ .

Experimental results

The estimation accuracy of contour error in the three methods was first verified. The experiments were carried out with the maximum feedrate of 100 mm/s, and the proportional gain $K_c$ of the CCC was set to 140 s⁻¹. Figure 7 shows the estimation deviation ${\delta }_{\epsilon }$ . It can be seen that C1 has a maximum estimation deviation of more than 1 mm, and C2 yields a better estimation accuracy than C1. For C3, the nearly perfect estimation performance is achieved, and the estimation deviation can be constrained within 1 μm. These experimental results verify that the proposed TVT-based CCC method could achieve much better estimation performance than the traditional CCC methods with the geometric approximation-based CEEMs.

Figure 7.

Estimation deviations of contour errors in the three methods when the proportional gain of the CCC was set to 140 s⁻¹.

The second experiment was conducted to test the contouring performance. The proportional gain $K_c$ of the CCC in the three methods was set from 20 to 180 s⁻¹ with an increment of 20 s⁻¹. Note that for C1 and C2 with $K_c=0$ , there is no cross-coupled effort, and each axis is controlled only by the individual position controller. Thus, we also set $K_c=0$ for C1 and C2 to investigate the contouring performance of the individual axial tracking control method. Experimental results in terms of ${\epsilon }_a$ and ${\epsilon }_e$ are shown in Figures 8 and 9, respectively.

Figure 8.

Performance indexes of estimated contour errors with different proportional gains of the CCC: (a) RMS values and (b) maximum values.

Figure 9.

Performance indexes of actual contour errors with different proportional gains of the CCC: (a) RMS values and (b) maximum values.

From Figure 9, when $K_c$ in C1 varies from 0 to 40 s⁻¹, the root-mean-square (RMS) value of ${\epsilon }_a$ is reduced from 209 to 80 μm (reduction rate of 61.72%). Similarly for C2, by increasing $K_c$ from 0 to 80 s⁻¹, the RMS value of ${\epsilon }_a$ is reduced to 103 μm (reduction rate of 50.72%). The results clearly indicate that the CCC method can significantly improve the contouring performance compared to the individual axial tracking control method by the cross-coupled effort.

It should be noted from Figure 8 that the RMS value of ${\epsilon }_e$ in C1 can decrease monotonically with respect to the proportional gain $K_c$ of the CCC as expected. But from Figure 9, the RMS value of ${\epsilon }_a$ increases when $K_c$ is set from 40 to 180. Similar results can also be observed for C2. The reason is that the CCC is a feedback controller, which aims to eliminate the estimation contour error rather than the actual one. For C1 and C2 with the large estimation deviation of contour error, the actual contour error does not necessarily decrease with the estimated one.

For C3 with the nearly perfect estimation performance, both the estimated and actual contour errors decrease monotonically with respect to the proportional gain $K_c$ of the CCC, as shown in Figures 8 and 9. Thus, C3 has the best contouring performance. The RMS value of ${\epsilon }_a$ is reduced by 53.75% and 64.08% compared to C1 and C2, respectively, and the maximum value of ${\epsilon }_a$ is reduced by 65.32% and 50.10% compared to C1 and C2, respectively. These results demonstrate that the proposed TVT-based CCC method can achieve outstanding contouring performance because of the accurate contour error estimation.

To investigate the dependence of contouring performance on the feedrate, the experiments were carried out for the maximum feedrate from 50 to 150 mm/s with an increment of 10 mm/s. The proportional gain of the CCC was set to 140 s⁻¹. Figure 10 illustrates the RMS values of ${\delta }_{\epsilon }$ and ${\epsilon }_a$ . From Figure 10(a), the RMS values of ${\delta }_{\epsilon }$ in C1 and C2 increase significantly when the maximum feedrate is set from 50 to 150 mm/s: from 66 to 248 μm for C1 (an increase of 276%) and from 32 to 161 μm for C2 (an increase of 403%). The reason is that both C1 and C2 use geometric approximation-based CEEMs, in which the estimation accuracy of contour error depends on the position tracking errors in the feed direction. For C3, the contour error is calculated by searching the foot point, and the estimation accuracy of contour error only depends on the iteration termination conditions (i.e. the maximum number of iterations and the limited tangential error). Thus, it can be seen from Figure 10(a) that the RMS value of ${\delta }_{\epsilon }$ in C3 can also be constrained within 1 μm when the feedrate varies.

Figure 10.

Dependence of contouring performance on the feedrate: (a) RMS values of estimation deviations of contour errors and (b) RMS values of actual contour errors.

From Figure 10(b), the RMS values of ${\epsilon }_a$ increase monotonically with the maximum feedrate in all three methods. However, the increase rates are quite different: 2.98 μm/(mm/s) for C1, 1.69 μm/(mm/s) for C2, and 0.53 μm/(mm/s) for C3. These results clearly indicate that the proposed TVT-based CCC method is less sensitive to the feedrate because of the accurate contour error estimation.

Table 3 lists the total travel time in the three methods when the proportional gain of the CCC was set to 140 s⁻¹. Although the total travel time $T_3$ of C3 is different from the total travel time $T_{1/2}$ of C1 and C2, the difference between $T_3$ and $T_{1/2}$ is less than 0.6% when the maximum feedrate increases from 50 to 150 mm/s. Hence, the proposed TVT-based CCC method has a similar machining efficiency to the traditional CCC methods.

Table 3.

Total travel time in the three methods when the proportional gain of the CCC was set to 140 s⁻¹.

	Maximum feedrate (mm/s)
	50	60	70	80	90	100	110	120	130	140	150
$T_{1/2}$ (ms)	7798	6562	5693	5051	4587	4201	3893	3645	3444	3277	3133
$T_3$ (ms)	7786	6544	5670	5024	4563	4182	3879	3637	3433	3267	3122
$|T_3-T_{1/2}|/T_{1/2}$	0.15%	0.28%	0.41%	0.54%	0.53%	0.45%	0.36%	0.22%	0.32%	0.31%	0.35%

Conclusion

The position tracking errors in the feed direction deteriorate the estimation accuracy of contour error in the traditional CCC method with geometric approximation-based CEEMs. This paper proposes a novel TVT-based CCC method. The feed-direction control task is completed by tracking the tangential velocity command, thus avoiding the cumbersome parameter tuning of individual axial position controllers. The contour error estimation is rather accurate by calculating the distance from the actual point to the tangential line at the foot point. Meanwhile, the system stability is guaranteed. Comparative experiments demonstrate that the proposed TVT-based CCC method can achieve outstanding contouring performance in high-speed and large-curvature contouring tasks because of the accurate contour error estimation. Note that only a simple P-type controller is used in the presented TVT-based CCC method. Future research will focus on designing advanced control laws to improve the contouring performance under external disturbances. Moreover, it will be interesting to apply the TVT strategy to three- and five-axis machine tools.