Dual-Bézier path smoothing and interpolation for five-axis linear tool path in workpiece coordinate system

Abstract

The tool path composed of consecutive short linear segments (G01 blocks) is still the widespread tool path representation form in five-axis machining. The inherent shortcoming of linear tool path is first-order discontinuity at the corner, which is the bottleneck to achieve high-speed and high-accuracy machining. In this article, a dual-Bézier path smoothing algorithm for five-axis linear tool path in workpiece coordinate system is proposed. There are three steps involved in our method. First, the corner error distribution model is introduced to assign the given tolerance to the smoothing approximation error constraint and the chord error constraint to ensure the interpolation trajectory error within the given tolerance. Second, segment junctions of the linear tool path in workpiece coordinate system are smoothed by double G² continuous cubic Bézier curves. One cubic Bézier curve is used to round the corner of the tool tip point path, and the other Bézier curve is used to round the corner of the tool axis point path. This algorithm takes the conditions of approximation error constraint, the parameterized synchronization constraint, and continuous curvature constraint into consideration. Hence, the tangency and curvature continuities are both guaranteed in the new path. Third, an adaptive feedrate scheduling method is introduced to interpolate the new path. Simulation and experiment are performed to verify the effectiveness of the proposed method in five-axis tool path smoothing, speed smoothing, and trajectory accuracy controlling.

Introduction

Five-axis computer numerical control (CNC) machining is widely used to machine complex curved surface parts, such as impeller, blade, and molds. Compared with five-axis machining in machine coordinate system (MCS), five-axis machining in workpiece coordinate system (WCS) has many virtues. The speed of cutting tool center point can be directly controlled, and the nonlinear error is reduced. Especially, in side milling, it can effectively avoid the overcut and undercut and improve the machining accuracy.

Linear tool path (G01 blocks) is a widespread format for CNC machining due to its high efficiency in approximating complex curvilinear tool paths under predefined tolerance. ¹ As shown in Figure 1, commercial CAM software (e.g. CATIA, UG, Cimatron, Master CAM) generates a series of discrete reference points in WCS. Each discrete reference point contains position and direction information of the tool. The position vector describes the tip point trajectory, and direction vector describes the tool orientation. In post-processing software, the cutter locations in WCS are transferred into the NC code in MCS. Then, NC instructions will be sent to CNC system, and CNC system will real-time plan out the position, velocity, and acceleration of each axis under the kinematical constraints of the machine tool. In the linear tool path, at every segment junction, tangency of the tool path is discontinuous. The discontinuity often leads to speed fluctuations and acceleration oscillations. The fluctuations will affect the machine tool movement stability, and be harmful to the surface quality, and decrease the machining efficiency. Although the look-ahead schemes were proposed to alleviate feedrate and acceleration fluctuations,^2–4 the discontinuities are still the most important source of feedrate and acceleration fluctuations. Previous research works also show that smoothing consecutive short linear segments with smooth parametric curves is an important method to improve the machining efficiency and quality, such as parametric curves,^5,6 Bézier,^7–9 B-spline, ¹⁰ and non-uniform rational basis spline (NURBS).^11,12

Figure 1.

Five-axis linear tool path.

Five-axis tool path smoothing mainly includes two categories. One is performed in WCS, in which the continuous motion of the tool is defined by two splines describing the trajectories of the tool tip, and the second point on the tool axis. Siemens840D NC system adopts double NURBS curve for five-axis linkage NC machining instruction format, which can improve the interpolation speed and precision. ¹³ Langeron et al. ¹⁴ utilized dual NURBS tool path with unequal distance (DNTPUD) to describe the five-axis linkage cutting tool path and demonstrated the C² continuous advantages on improving machining efficiency in comparison with the linear tool path. Bi et al. ¹⁵ extended this idea by giving a method to obtain dual NURBS tool path with equal distance (DNTPED). Compared with DNTPUD, DNTPED is more suitable for CNC interpolation calculation, because DNTPED interpolation calculation is more simple and efficient. Fleisig and Spence ¹⁶ proposed a method using three C² continuous splines to describe the tool path. The position spline is a near arc-length parameterized quintic polynomial spline. The orientation spline is a near arc-length parameterized quintic spherical Bézier spline. The third spline was used to synchronize the parameters of the two splines. Koch ¹⁷ adopts two B-spline curves to guide tool path for five-axis machining. The tool is guided to move along a first guide path while a further point on the axis of the tool, which is located at a distance from the work portion, is guided along a prescribed second path. The above methods did not consider the effect of the fitting error on tool posture. The implementations of them are so time-consuming that they are difficultly implemented in real-time environment. Beudaert et al. ¹⁸ proposed an interesting five-axis local corner rounding method to smooth the tool path in the WCS. The corner rounding method was implemented by blending the junctions in two trajectories that represented the traces of tip center point and a point on the tool axis in the WCS. The curvature continuity was realized by inserting a transition spline curve, and the approximation error was also guaranteed. The orientation parametrization was optimized with the third parametrization spline to smooth the rotational speed. Sneha ¹⁹ proposed to insert quintic and septic micro-splines for the tool tip and tool orientation, respectively, at the adjacent linear tool path segments. However, it is mathematically complex to control the synchronization of the position and orientation splines. Shi ¹ proposed an on-line path smoothing algorithm using dual Pythagorean Hodograph (PH) curves to blend the corner of the linear tool path. A simple linear synchronization of the position and orientation trajectories is introduced. Although the approximation error of this method can be guaranteed according to the given tolerance, the trajectory error cannot be controlled within the given tolerance because approximation error and chord error cannot be restricted simultaneously.

Another smoothing method is performed in MCS, in which the five movements of the machine tools are divided into translational movement and rotational movement. Both Fanuc ²⁰ and Heidenhain ²¹ NC systems introduce the spline interpolation function using five parameter curves to describe the trajectories of five axes in MCS. It proves that smoothing the linear path in MCS is also an effective means to improve the machining quality and machining efficiency. Using the quaternion interpolation scheme, Ho et al. ²² smoothed the tool orientation to reduce the cutting error. Tutunea-Fatan and Feng ²³ proposed a dual-spline tool path smoothing algorithm in the MCS. The method fitted a three-dimensional (3D) B-spline for the three translational joints and fitted a two-dimensional (2D) B-spline for two rotational joints. The synchronization of the five machine joints was achieved using the same knot sequence in the two fitting processes. Yuen et al. ²⁴ proposed a smooth spline interpolation for five-axis machining, in which the smoothed positional and rotational movements are represented by quintic splines. The nonlinear relationship between spline parameters and displacements along the path is approximated with ninth-order and seventh-order feed correction splines for position and orientation, respectively. This method decreases tracking error for each axis in machine tools. Beudaert et al. ²⁵ proposed a five-axis tool path smoothing method by taking the drive constraints, such as maximum speed, acceleration, and jerk, into consideration. It can increase the average feedrate effectively. Bi et al. ²⁶ presented a G² dual-Bézier corner transition algorithm. One Bézier curve is used to blend the segment junction of the translational path, and another Bézier curve smoothes the segment junction of the rotational path. The linear synchronous mechanism is used to guarantee the parameterized synchronization of the translational axes’ path and the rotational axes’ path. In the above methods, the trajectory error cannot be controlled within the permissible maximum error range because approximation error and chord error cannot be restricted simultaneously.

To satisfy the requirements of high-speed and high-accuracy five-axis machining, this article proposes a G² dual-Bézier path smoothing algorithm based on the corner error distribution model. To guarantee the interpolation trajectory error within the given tolerance, the corner error distribution model is introduced to assign the given tolerance to the smoothing approximation error constraint and the chord error constraint. Then, segment junctions of the linear tool path in WCS are rounded by two G² continuous cubic Bézier curves. One cubic Bézier curve is used to round the corner of the tool tip point path, and the other Bézier curve is used to round the corner of the tool axis point path. This algorithm takes the conditions of approximation error constraint, the parameterized synchronization constraint, and continuous curvature constraint into consideration. Hence, the tangency and curvature continuities and interpolation trajectory error within the given tolerance are guaranteed in the new path. The comparison of our method with those available in the literatures is presented in Table 1.

Table 1.

Comparison of path smoothing methods.

Articles	G2 continuity	Approximation error restriction	Trajectory error restriction	Parameterized synchronization	Coordinate system	Real-time application
J Shi et al. 1	Yes	Yes	No	Yes	WCS	Yes
Langeron et al. 14	Yes	No	No	No	WCS	No
QZ Bi et al. 15	Yes	No	No	No	WCS	No
RV Fleisig and AD Spence 16	Yes	No	No	Yes	WCS	No
OR Tutunea-Fatan and HY Feng. 23	Yes	Yes	No	Yes	MCS	No
A Yuen et al. 24	Yes	Yes	No	No	MCS	No
QZ Bi et al. 26	Yes	Yes	No	Yes	MCS	Yes
This article	Yes	Yes	Yes	Yes	WCS	Yes

WCS: workpiece coordinate system; MCS: machine coordinate system.

The remainder of this article is organized as follows: section “Definition of the corner error distribution model” describes the corner error distribution model in WCS. Dual-Bézier path smoothing algorithm is given in section “Dual-Bézier path smoothing for five-axis linear path in WCS.” In section “Interpolation method for five-axis continuous tool path in WCS,” an adaptive feedrate scheduling method is introduced to interpolate the new path. Simulation and experiment are presented in section “Simulation and experiment.” Section “Summary and conclusion” concludes this article.

Definition of the corner error distribution model

When path smoothing algorithm and parametric interpolation algorithm are utilized to interpolate the linear tool path, approximation error and chord error may occur. They are the main error sources causing trajectory error. The given tolerance ${\epsilon }_{max}$ is predefined in CNC system to restrict the approximation error and chord error. The given tolerance ${\epsilon }_{max}$ is set according to the precision of the machine tool and the requirement of the machining accuracy of parts. First, the given tolerance should be set smaller than the precision of the machine tool, so the CNC system can utilize the ability of the machine tool better. Second, the given tolerance should satisfy the requirement of the machining accuracy of parts.

In the existing studies,^{1,5–12,14–25} approximation error and chord error are calculated, respectively, and they are all restricted with the given tolerance value ${\epsilon }_{max}$ usually. As shown in Figure 2, in fact, the maximal actual trajectory error may exceed the given tolerance, when the transition curve is interpolated. In order to guarantee precision and gain the maximum machining efficiency, an optimal error assignment algorithm is deduced to distribute the given tolerance value into approximation error and chord error.

Figure 2.

Definitions of approximation error and chord error.

Definition of three-axis corner error distribution model

As shown in Figure 2, two line segments are defined by three points P_{i− 1}, P_i , and P_{i +1}. Assuming that segment junction is rounded with a curve C(t), the maximum approximation error is $\Vert P_i{D}_i\Vert ={\epsilon }_{\mathrm{ap}}$ . When the transition curve C(t) is being interpolated, the chord error may occur. $C(t_j)$ and $C(t_{j+1})$ are two contiguous interpolation points. The chord error is defined as $\Vert D_{i}N\Vert ={\epsilon }_{\mathrm{ch}}$ , and the trajectory error is defined as $\Vert P_{i}N\Vert =\epsilon$ . The maximum trajectory error must be within the given tolerance ${\epsilon }_{max}$

$${\epsilon }_{\mathrm{ap}}+{\epsilon }_{\mathrm{ch}}\le {\epsilon }_{max}$$

(1)

Assuming that the maximum curvature of the transition curve C(t) is ${\kappa }_{max}$ and the interpolation period of CNC system is T_s , the maximum transitional velocity is restricted by the maximum normal acceleration $A_{max}$ and the chord error constraint

$$\begin{array}{cc}\multicolumn{1}{c}{f_{\mathrm{Tr}}}& =min(\frac{2}{T_s}\sqrt{{\left(\frac{1}{{\kappa }_{max}}\right)}^2-{(\frac{1}{{\kappa }_{max}}-{\epsilon }_{\mathrm{ch}})}^2},\sqrt{\frac{A_{max}}{{\kappa }_{max}}})\hfill \\ \multicolumn{1}{c}{}& \approx min(\frac{1}{T_s}\sqrt{\frac{8{\epsilon }_{\mathrm{ch}}}{{\kappa }_{max}}},\sqrt{\frac{A_{max}}{{\kappa }_{max}}})\hfill \end{array}$$

(2)

According to Bi et al., ⁸ ${\kappa }_{max}$ lies on ${\epsilon }_{\mathrm{ap}}$ ; moreover, ${\epsilon }_{\mathrm{ch}}$ restricts feedrate. Setting optimal ${\epsilon }_{\mathrm{ap}}$ and ${\epsilon }_{\mathrm{ch}}$ could help gain bigger transitional velocity. If ${\epsilon }_{\mathrm{ch}}=A_{max}{T}_s^2/8$ , $\sqrt{A_{max}/{\kappa }_{max}}=\sqrt{8{\epsilon }_{\mathrm{ch}}/{\kappa }_{max}}/T_s$ . Optimal ${\epsilon }_{\mathrm{ap}}$ and ${\epsilon }_{\mathrm{ch}}$ can be calculated as follows

$$\{\begin{array}{cc}\hfill {\epsilon }_{\mathrm{ch}}\hfill & \hfill =\{\begin{array}{cc}\hfill A_{max}{T}_s^2/8\hfill & \hfill \mathrm{if}{\epsilon }_{max}\ge A_{max}{T}_s^2/4\hfill \\ \multicolumn{1}{c}{{\epsilon }_{max}/2}& \hfill \mathrm{other}\hfill \end{array}\hfill \\ \multicolumn{1}{c}{{\epsilon }_{\mathrm{ap}}}& \hfill ={\epsilon }_{max}-{\epsilon }_{\mathrm{ch}}\hfill \end{array}$$

(3)

Definition of five-axis corner error distribution model

As shown in Figure 3, the discrete cutter posture in WCS is defined $(P_i,Q_i)$ ; $P_i$ is the tool tip point and $Q_i$ is the second point on the tool axis. Two line segments in WCS are defined by three points $(P_{i-1},Q_{i-1})$ , $(P_i,Q_i)$ , and $(P_{i+1},Q_{i+1})$ . Assuming that segment junction of tool tip point is rounded with a curve $C_P(t)$ and segment junction of tool axis point is rounded with a curve $C_Q(t)$ , the maximum approximation errors are $\Vert P_i{E}_P\Vert ={\epsilon }_{\mathrm{ap}1}$ and $\Vert Q_i{E}_Q\Vert ={\epsilon }_{\mathrm{ap}2}$ , respectively. The chord errors of $C_P(t)$ and $C_Q(t)$ are $\Vert E_P{W}_P\Vert ={\epsilon }_{\mathrm{ch}1}$ and $\Vert E_Q{W}_Q\Vert ={\epsilon }_{\mathrm{ch}2}$ , respectively. The trajectory error of the tool axis is expressed as the deviation from $W_P{W}_Q$ to $P_i{Q}_i$

$$\epsilon =(1-u)\Vert W_P{P}_i\Vert +u\Vert W_Q{Q}_i\Vert (0\le u\le 1)$$

(4)

Figure 3.

Five-axis linear tool path and corner rounding curve.

To guarantee the trajectory error of the tool axis, the errors of the tool tip point, the tool axis point, and the tool axis should be restricted as follows

$${\epsilon }_{\mathrm{ap}1}+{\epsilon }_{\mathrm{ch}1}\le {\epsilon }_{max}$$

(5)

$${\epsilon }_{\mathrm{ap}2}+{\epsilon }_{\mathrm{ch}2}\le {\epsilon }_{max}$$

(6)

$$\begin{array}{cc}\multicolumn{1}{c}{}& \epsilon \le (1-u)||W_p{P}_i||+u||W_Q{Q}_i||\hfill \\ \multicolumn{1}{c}{}& \le (1-u)({\epsilon }_{\mathrm{ap}1}+{\epsilon }_{\mathrm{ch}1})+u({\epsilon }_{\mathrm{ap}2}+{\epsilon }_{\mathrm{ch}2}),(0\le u\le 1)\hfill \\ \multicolumn{1}{c}{}& \le max({\epsilon }_{\mathrm{ap}1}+{\epsilon }_{\mathrm{ch}1},{\epsilon }_{\mathrm{ap}2}+{\epsilon }_{\mathrm{ch}2})\hfill \\ \multicolumn{1}{c}{}& \le {\epsilon }_{max}\hfill & \hfill \end{array},(0\le u\le 1)$$

(7)

The maximum transitional velocity of the tool tip point is expressed as

$$f_{\mathrm{Tr},P}\approx min(\frac{1}{T_s}\sqrt{\frac{8{\epsilon }_{\mathrm{ch}1}}{{\kappa }_{max,P}}},\sqrt{\frac{A_{max}}{{\kappa }_{max,P}}})$$

(8)

The maximum transitional velocity of the tool axis point is expressed as

$$f_{\mathrm{Tr},Q}\approx min(\frac{1}{T_s}\sqrt{\frac{8{\epsilon }_{\mathrm{ch}2}}{{\kappa }_{max,Q}}},\sqrt{\frac{A_{max}}{{\kappa }_{max,Q}}})$$

(9)

In order to obtain maximum transitional velocity, the maximum approximation errors and the chord errors should be calculated as

$$\{\begin{array}{cc}\hfill {\epsilon }_{\mathrm{ch}1}\hfill & \hfill ={\epsilon }_{\mathrm{ch}2}=\{\begin{array}{cc}\hfill A_{max}{T}_s^2/8\hfill & \hfill \mathrm{if}{\epsilon }_{max}\ge A_{max}{T}_s^2/4\hfill \\ \multicolumn{1}{c}{{\epsilon }_{max}/2}& \hfill \mathrm{other}\hfill \end{array}\hfill \\ \multicolumn{1}{c}{{\epsilon }_{\mathrm{ap}1}}& \hfill ={\epsilon }_{\mathrm{ap}2}={\epsilon }_{max}-{\epsilon }_{\mathrm{ch}1}\hfill \end{array}$$

(10)

The approximation errors ${\epsilon }_{\mathrm{ap}1}$ and ${\epsilon }_{\mathrm{ap}2}$ are used to restrict the transition curve generation in section “Dual-Bézier path smoothing for five-axis linear path in WCS.”

Dual-Bézier path smoothing for five-axis linear path in WCS

In our previous works, ⁸ a corner smoothing method for three-axis linear tool path based on cubic Bézier curve was introduced, which can restrict the approximation error exactly, as shown in equation (11). In this section, G² continuous dual-Bézier transition algorithm is proposed for five-axis linear tool path in WCS. One Bézier curve rounds the corner of the tool tip point path, and another Bézier curve rounds the corner of the tool axis point path. Parameter synchronization between the tip point trajectory and tool axis point trajectory, approximation error constraints, and transition length optimization is taken into consideration

$$\frac{d}{4}\sin \frac{\theta }{2}\le {\epsilon }_{\mathrm{ap}}$$

(11)

where the transition length d is defined as the length of the linear segment that is replaced by the transition curves and $\theta$ is the angle between two adjacent segments.

Definition of five-axis dual-Bézier curve

A shown in Figure 3, five-axis linear tool path is defined by two linear trajectories. The tip point trajectory with dots sequence is defined as $\{P_i=(x_i,y_i,z_i),i=1,\dots ,n\}$ , and the tool axis point path is defined by the points’ sequence $\{Q_i=(x_i,y_{i,}{z}_i),i=1,\dots ,n\}$ . The corresponding points $(P_i,Q_i)$ describe the discrete cutter posture. The Bézier curve $B_{\mathrm{Pi}}(t)$ is used for smoothing the tool tip point trajectory, and another Bézier curve $B_{\mathrm{Qi}}(t)$ is used for smoothing the tool axis point path. The dual-Bézier transition curves are defined as follows

$$\begin{array}{c}\multicolumn{1}{c}{B_{\mathrm{Pi}}(t)=\sum _{j=0}^3\left(\begin{array}{c}\hfill 3\hfill \\ \multicolumn{1}{c}{j}\end{array}\right)B_{\mathrm{Pij}}{(1-t)}^{3-j}{t}^j}\\ \multicolumn{1}{c}{B_{\mathrm{Qi}}(t)=\sum _{j=0}^3\left(\begin{array}{c}\hfill 3\hfill \\ \multicolumn{1}{c}{j}\end{array}\right)B_{\mathrm{Qij}}{(1-t)}^{3-j}{t}^j}\end{array}$$

(12)

where t is the Bézier parameter; $0\le t\le 1$ ; $B_{\mathrm{Pij}}$ and $B_{\mathrm{Qij}}$ are the control points of Bézier curves $B_{\mathrm{Pi}}(t)$ and $B_{\mathrm{Qi}}(t)$ . The derivatives of the dual-Bézier curves at both ends are

$$\begin{array}{c}\multicolumn{1}{c}{B{\prime }_{\mathrm{Pi}}(0)=3(B_{\mathrm{Pi}1}-B_{\mathrm{Pi}0})B{\prime }_{\mathrm{Pi}}(1)=3(B_{\mathrm{Pi}3}-B_{\mathrm{Pi}2})}\\ \multicolumn{1}{c}{B{\prime }_{\mathrm{Qi}}(0)=3(B_{\mathrm{Qi}1}-B_{\mathrm{Qi}0})B{\prime }_{\mathrm{Qi}}(1)=3(B_{\mathrm{Qi}3}-B_{\mathrm{Qi}2})}\end{array}$$

(13)

$$\begin{array}{cc}\multicolumn{1}{c}{}& B{″}_{\mathrm{Pi}}(0)=6(B_{\mathrm{Pi}0}-2B_{\mathrm{Pi}1}+B_{\mathrm{Pi}2})\hfill \\ \multicolumn{1}{c}{}& B{″}_{\mathrm{Pi}}(1)=6(B_{\mathrm{Pi}1}-2B_{\mathrm{Pi}2}+B_{\mathrm{Pi}3})\hfill \\ \multicolumn{1}{c}{}& B{″}_{\mathrm{Qi}}(0)=6(B_{\mathrm{Qi}0}-2B_{\mathrm{Qi}1}+B_{\mathrm{Qi}2})\hfill \\ \multicolumn{1}{c}{}& B{″}_{\mathrm{Qi}}(1)=6(B_{\mathrm{Qi}1}-2B_{\mathrm{Qi}2}+B_{\mathrm{Qi}3})\hfill \end{array}$$

(14)

The synchronization of parametrization of two smoothed curves is crucial in the smoothing of five-axis path. If the linear interpolation method is used, the two continuous trajectories between two reference points are defined as ¹

$$\begin{array}{c}\hfill L_{\mathrm{Pi}}(t)=(1-t)P_i+{\mathrm{tP}}_{i+1}\hfill \\ \multicolumn{1}{c}{L_{\mathrm{Qi}}(t)=(1-t)Q_i+{\mathrm{tQ}}_{i+1}}\end{array},0\le t\le 1$$

(15)

The continuous movement trajectory of tool axis along the linear trajectories $L_{\mathrm{Pi}}(t)$ and $L_{\mathrm{Qi}}(t)$ is expressed

$$T_i(u)=(1-u)L_{\mathrm{Pi}}(t)+{\mathrm{uL}}_{Q_i}(t),0\le u\le 1$$

(16)

The continuous movement trajectory of tool axis along the dual-Bézier transition curves is expressed

$$T_i(u)=(1-u)B_{\mathrm{Pi}}(t)+{\mathrm{uB}}_{Q_i}(t)0\le u\le 1$$

(17)

According to equations (15) and (17), continuous path of tool axis is a ruled surface. The deviation between smoothing tool path ruled surface and the original linear path ruled surface is the maximum approximation errors of corner transition.

Synchronization of parameterization

The synchronization of parameterization of two smoothed curves is crucial in making sure that the variation in the tool axis direction is continuous and is the most important constraint in the tool path smoothing. As shown in Figure 3, the corner of the segments $(P_{i-1},Q_{i-1})-(P_i,Q_i)-(P_{i+1},Q_{i+1})$ is blended with dual-Bézier transition curves. $d_{\mathrm{Pi}1}$ , $d_{\mathrm{Pi}2}$ , $d_{\mathrm{Qi}1}$ , and $d_{\mathrm{Qi}2}$ are, respectively, the transition lengths on the segments $P_{i-1}{P}_i$ , $P_i{P}_{i+1}$ , $Q_{i-1}{Q}_i$ , and $Q_i{Q}_{i+1}$ . To guarantee the tool tip point and tool axis point synchronous, the transition lengths must meet the following conditions

$$\begin{array}{cc}\multicolumn{1}{c}{}& \frac{{\mathrm{dQ}}_{i1}}{\Vert Q_i-Q_{i-1}\Vert }=\frac{{\mathrm{dP}}_{i1}}{\Vert P_i-P_{i-1}\Vert }=t_{i_1}\hfill \\ \multicolumn{1}{c}{}& \frac{{\mathrm{dQ}}_{i2}}{\Vert Q_{i+1}-Q_i\Vert }=\frac{{\mathrm{dP}}_{i2}}{\Vert P_{i+1}-P_i\Vert }=t_{i_2}\hfill \end{array}$$

(18)

Equation (18) can be represented as

$$\begin{array}{cc}\multicolumn{1}{c}{}& \frac{{\mathrm{dQ}}_{i1}}{l_{\mathrm{Qi}-1}}=\frac{{\mathrm{dP}}_{i1}}{l_{\mathrm{Pi}-1}}=t_{i_1}\hfill \\ \multicolumn{1}{c}{}& \frac{{\mathrm{dQ}}_{i2}}{l_{\mathrm{Qi}}}=\frac{{\mathrm{dP}}_{i2}}{l_{\mathrm{Pi}}}=t_{i_2}\hfill \end{array}$$

(19)

Approximation error constraint

According to equation (11), the approximation error of the tool tip point curve $B_{\mathrm{Pi}}(t)$ is expressed as

$${\epsilon }_{\mathrm{ap}1}\le \frac{d_{\mathrm{Pi}}}{4}\sin \frac{{\theta }_{\mathrm{Pi}}}{2}$$

(20)

where $d_{\mathrm{Pi}}=max(d_{\mathrm{Pi}1},d_{\mathrm{Pi}2})$ ; ${\theta }_{\mathrm{Pi}}$ is the angle between $\overrightarrow{P_i{P}_{i+1}}$ and $\overrightarrow{P_{i-1}{P}_i}$ .

The approximation error of the tool axis point curve $B_{\mathrm{Qi}}(t)$ is expressed as

$${\epsilon }_{\mathrm{ap}2}\le \frac{d_{\mathrm{Qi}}}{4}\sin \frac{{\theta }_{\mathrm{Qi}}}{2}$$

(21)

where $d_{\mathrm{Qi}}=max(d_{\mathrm{Qi}1},d_{\mathrm{Qi}2})$ ; ${\theta }_{\mathrm{Qi}}$ is the angle between $\overrightarrow{Q_i{Q}_{i+1}}$ and $\overrightarrow{Q_{i-1}{Q}_i}$ .

Real-time G2 path smoothing

The smoothing methods for five-axis linear tool path linear need to take curvature-continuous constraint, the approximation error constraint, and the synchronization of parameterization into consideration. According to equations (19)–(21), approximation error and synchronization constraints can be accomplished by designing the transition lengths. As shown in Figure 4, the transition lengths of two adjacent segments also need to meet the constraints of the corresponding line segment lengths

$$\begin{array}{c}\multicolumn{1}{c}{d_{Q(i-1)2}+d_{\mathrm{Qi}1}\le \Vert Q_{i-1}{Q}_i\Vert \le l_{\mathrm{Qi}-1}}\\ \multicolumn{1}{c}{d_{P(i-1)2}+d_{\mathrm{Pi}1}\le \Vert P_{i-1}{P}_i\Vert \le l_{\mathrm{Pi}-1}}\\ \multicolumn{1}{c}{d_{\mathrm{Qi}2}+d_{Q(i+1)1}\le \Vert Q_i{Q}_{i+1}\Vert \le l_{\mathrm{Qi}}}\\ \multicolumn{1}{c}{d_{\mathrm{Pi}2}+d_{P(i+1)1}\le \Vert P_i{P}_{i+1}\Vert \le l_{\mathrm{Pi}}}\end{array},(3\le i\le n-2)$$

(22)

Figure 4.

Corner rounding for the point on tool axis.

The transition length of the first segment meets $d_{Q21}\le \Vert Q_1{Q}_2\Vert =l_{Q1}$ . The transition length of the last segment meets $d_{Q(n-1)2}\le \Vert Q_{n-1}{Q}_n\Vert =l_{Q(n-1)}$ .

According to equation (19), to meet the tool tip point and the tool axis point synchronous constraint, the transition lengths need to satisfy the following relations

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathrm{dQ}}_{i1}}& =t_{i1}{l}_{Q(i-1)}{\mathrm{dQ}}_{i2}=t_{i2}{l}_{\mathrm{Qi}}\hfill \\ \multicolumn{1}{c}{{\mathrm{dP}}_{i1}}& =t_{i1}{l}_{P(i-1)}{\mathrm{dP}}_{i2}=t_{i2}{l}_{\mathrm{Pi}}\hfill \end{array}$$

(23)

According to equations (20) and (21), in order to meet the approximation error constraints ${\epsilon }_{\mathrm{ap}1}$ and ${\epsilon }_{\mathrm{ap}2}$ assigned by equation (10), the transition lengths are calculated by

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathrm{dQ}}_{i1}\le 4{\epsilon }_{\mathrm{ap}2}\csc \left(\frac{{\theta }_{\mathrm{Qi}}}{2}\right){\mathrm{dQ}}_{i2}\le 4{\epsilon }_{\mathrm{ap}2}\csc \left(\frac{{\theta }_{\mathrm{Qi}}}{2}\right)\hfill \\ \multicolumn{1}{c}{}& {\mathrm{dP}}_{i1}\le 4{\epsilon }_{\mathrm{ap}1}\csc \left(\frac{{\theta }_{\mathrm{Pi}}}{2}\right){\mathrm{dP}}_{i2}\le 4{\epsilon }_{\mathrm{ap}1}\csc \left(\frac{{\theta }_{\mathrm{Pi}}}{2}\right)\hfill \end{array}$$

(24)

Equation (22) is simplified, and the transition length is less than half the length of the corresponding line. The key parameters of the transition curves can be gained

$$\begin{array}{c}\multicolumn{1}{c}{t_{i1}=min(\frac{4{\epsilon }_{\mathrm{ap}2}\csc (0.5{\theta }_{\mathrm{Qi}})}{l_{Q(i-1)}},\frac{4{\epsilon }_{\mathrm{ap}1}\csc (0.5{\theta }_{\mathrm{Pi}})}{l_{P(i-1)}},0.5)}\\ \multicolumn{1}{c}{t_{i2}=min(\frac{4{\epsilon }_{\mathrm{ap}2}\csc (0.5{\theta }_{\mathrm{Qi}})}{l_{\mathrm{Qi}}},\frac{4{\epsilon }_{\mathrm{ap}1}\csc (0.5{\theta }_{\mathrm{Pi}})}{l_{\mathrm{Pi}}},0.5)}\end{array}$$

(25)

Substituting $t_{i1}$ and $t_{i2}$ into equation (23), the transition lengths are obtained.

The control points for Bézier curve $B_{\mathrm{Pi}}(t)$ and Bézier curve $B_{\mathrm{Qi}}(t)$ are calculated by

$$\begin{array}{cc}\multicolumn{1}{c}{}& B_{\mathrm{Pi}0}=t_{i1}{P}_{i-1}+(1-t_{i1})P_i\hfill \\ \multicolumn{1}{c}{}& B_{\mathrm{Pi}1}=B_{\mathrm{Pi}2}=P_i\hfill \\ \multicolumn{1}{c}{}& B_{\mathrm{Pi}3}=(1-t_{i2})P_i+t_{i2}{P}_{i+1}\hfill \end{array}$$

(26)

$$\begin{array}{cc}\multicolumn{1}{c}{}& B_{\mathrm{Qi}0}=t_{i1}{Q}_{i-1}+(1-t_{i1})Q_i\hfill \\ \multicolumn{1}{c}{}& B_{\mathrm{Qi}1}=B_{\mathrm{Pi}2}=Q_i\hfill \\ \multicolumn{1}{c}{}& B_{\mathrm{Qi}3}=(1-t_{i2})Q_i+t_{i2}{Q}_{i+1}\hfill \end{array}$$

(27)

Substituting equations (26) and (27) into equation (12), Bézier curve $B_{\mathrm{Pi}}(t)$ and Bézier curve $B_{\mathrm{Qi}}(t)$ are obtained.

Substituting equations (26) and (27) into equation (13), it is proved that the tangency of the smoothed path is continuous

$$\begin{array}{cc}\multicolumn{1}{c}{}& B{\prime }_{\mathrm{Pi}}(0)=3t_{i1}(P_i-P_{i-1})B{\prime }_{\mathrm{Pi}}(1)=3t_{i2}(P_{i+1}-P_i)\hfill \\ \multicolumn{1}{c}{}& B{\prime }_{\mathrm{Qi}}(0)=3t_{i1}(Q_i-Q_{i-1})B{\prime }_{\mathrm{Qi}}(1)=3t_{t2}(Q_{i+1}-Q_i)\hfill \end{array}$$

(28)

Substituting equations (26) and (27) into equation (14), the second derivatives of the curves $B_{\mathrm{Pi}}(t)$ and $B_{\mathrm{Qi}}(t)$ at t = 0 and t = 1 are obtained

$$\begin{array}{cc}\multicolumn{1}{c}{}& B{″}_{\mathrm{Pi}}(0)=6t_{i1}(P_{i-1}-P_i)B{″}_{\mathrm{Pi}}(1)=6t_{i2}(P_{i+1}-P_i)\hfill \\ \multicolumn{1}{c}{}& B{″}_{\mathrm{Qi}}(0)=6t_{i1}(Q_{i-1}-Q_i)B{″}_{\mathrm{Qi}}(1)=6t_{t2}(Q_{i+1}-Q_i)\hfill \end{array}$$

(29)

Then, the curvatures of the curves $B_{\mathrm{Pi}}(t)$ and $B_{\mathrm{Qi}}(t)$ at t = 0 and t = 1 are obtained

$$\begin{array}{c}\multicolumn{1}{c}{{\kappa }_{\mathrm{Pi}}(0)=\frac{\Vert B{\prime }_{\mathrm{Pi}}(0)\times B{″}_{\mathrm{Pi}}(0)\Vert }{\Vert B{\prime }_{\mathrm{Pi}}(0){\Vert }^3}=0}\\ \multicolumn{1}{c}{{\kappa }_{\mathrm{Pi}}(1)=\frac{\Vert B{\prime }_{\mathrm{Pi}}(1)\times B{″}_{\mathrm{Pi}}(1)\Vert }{\Vert B{\prime }_{\mathrm{Pi}}(1){\Vert }^3}=0}\\ \multicolumn{1}{c}{{\kappa }_{\mathrm{Qi}}(0)=\frac{\Vert B{\prime }_{\mathrm{Qi}}(0)\times B{″}_{\mathrm{Qi}}(0)\Vert }{\Vert B{\prime }_{\mathrm{Qi}}(0){\Vert }^3}=0}\\ \multicolumn{1}{c}{{\kappa }_{\mathrm{Qi}}(1)=\frac{\Vert B{\prime }_{\mathrm{Qi}}(1)\times B{″}_{\mathrm{Qi}}(1)\Vert }{\Vert B{\prime }_{\mathrm{Qi}}(1){\Vert }^3}=0}\end{array}$$

(30)

According to equation (30), it is proved that the curvature of the smoothed path is continuous. So the smoothed path is G² continuity.

Interpolation method for five-axis continuous tool path in WCS

For five-axis dual-spline interpolation, a real-time adaptive feedrate scheduling method considering the comprehensive constraints is proposed. First, the comprehensive constraints (such as the chord error constraint, the maximum normal acceleration, the servo capacity of each axis) are deduced to obtain the maximum feed and feasible acceleration range. On this basis, the bi-directional scanned feed scheduling method is given.

Comprehensive constraints’ computation

Constraints of chord error and the maximum normal acceleration

Based on the corner error distribution model, the constraints of chord error and the maximum normal acceleration are calculated by

$$f_{\mathrm{Tr},P}(u)=min(\sqrt{A_{max}{\rho }_P(u)},2\sqrt{2{\epsilon }_{\mathrm{ch}1}{\rho }_P(u)-{\epsilon }_{\mathrm{ch}1}^2}/T_s)$$

(31)

$$f_{\mathrm{Tr},Q}\left(u\right)=min(\sqrt{A_{max}{\rho }_Q\left(u\right)},2\sqrt{2{\epsilon }_{\mathrm{ch}1}{\rho }_Q\left(u\right)-{\epsilon }_{\mathrm{ch}1}^2/T_s})$$

(32)

where $f_{\mathrm{Tr},P}(u)$ and $f_{\mathrm{Tr},Q}(u)$ are the feedrate of the tool tip point and tool axis point restricted by the interpolation error and maximum normal acceleration, respectively. ${\rho }_P(u)$ and ${\rho }_Q(u)$ are the radii of curvature of the tool tip point and tool axis point, respectively.

Constraints of the servo capacity

In interpolation process, when the input command of the axis exceeds its servo capability, the tracking error will increase. Therefore, the servo ability of each axis should also be considered when the feed scheduling is implemented.

However, in CNC system, the constraints of the servo capacity of each axis are defined in MCS. The key point of five-axis interpolation in WCS is to transfer the constraints of the servo capacity to the constraints for motion of tool in WCS. These mapping relations are deduced as follows:

1. The feed and acceleration of machine tool

$P(t)$ and $Q(t)$ are the paths for tool tip point and tool axis point, respectively. The tool axis direction is expressed as $O(t)=(Q(t)-P(t))/\Vert Q(t)-P(t)\Vert$ , and the tool pose in WCS is expressed as $W(t)$

$$W(t)=\left[\begin{array}{c}\hfill P(t)\hfill \\ \multicolumn{1}{c}{O(t)}\end{array}\right]$$

(33)

The coordinate of machine tool is $\mathbf{M}(t)$

$$M(t)={[M_x(t),M_y(t),M_z(t),M_{\alpha }(t),M_{\beta }(t)]}^T$$

Assuming that a given kinematics inverse transformation for five-axis machine tool is G: R ⁶→R ⁵, then $M(t)=G(W(t))$ .

The speed of the axis $\Lambda$ ( $\Lambda =x,y,z,\alpha ,\beta$ ) is given as

$$v_{\Lambda }(t)=\frac{M{\prime }_{\Lambda }(t)}{\sigma (t)}{v}_f$$

(34)

where $\sigma$ is the arc-length of tool tip point curve $B_{\mathrm{Pi}}(t)$ at parameter t, $\sigma (t)=\Vert B{\prime }_{\mathrm{Pi}}(t)\Vert$ , $v_f$ is the feed of the tool tip point, and $k_{\Lambda }=M{\prime }_{\Lambda }(t)/\sigma (t)$ .

The acceleration of the axis $\Lambda$ ( $\Lambda =x,y,z,\alpha ,\beta$ ) is given as

$$a_{\Lambda }(t)=k_{\mathrm{aa}\Lambda }(t)a_f+k_{\mathrm{av}\Lambda }{v}_f^2$$

(35)

where $a_f$ is the acceleration of the tool tip point, $k_{\mathrm{aa}\Lambda }=M{\prime }_{\Lambda }(t)/\sigma (t)$ , and $k_{\mathrm{av}\Lambda }=(\sigma (t)M{″}_{\Lambda }(t)-\sigma \prime (t)M{\prime }_{\Lambda }(t))/\sigma {(t)}^3$ .

2. The maximum feed constraint of machine tool

The maximum feed of axis Λ is V _Λm . The following equation is obtained

$$|k_{v\Lambda }(t)v_f|\le V_{\Lambda m}$$

(36)

The feedrate of the tool tip is restricted by $f_{\mathrm{vm}}(t)$

$$v_f\le f_{\mathrm{vm}}(t)=\underset{\Lambda =x,y,z,\alpha ,\beta }{min}\left\{\frac{V_{\Lambda m}}{|k_{v\Lambda }(t)|}\right\}$$

(37)

3. The maximum acceleration constraint of machine tool

Assuming that acceleration limit ranges for axis Λ are [−a _Λm , a _Λm ], velocity constraint polygon is surrounded by groups of five pair parallel lines, green area in Figure 5. Restricted by the maximum acceleration of each axis, the feedrate constraint $f_{\mathrm{am}}(t)$ and acceleration $a_f(t)$ of the tool tip point are as follows

$$\begin{array}{cc}\multicolumn{1}{c}{v_f}& \le f_{\mathrm{am}}(t)\hfill \\ \multicolumn{1}{c}{}& =\underset{i\ne j\ne k}{\overset{i,j,k\in \{x,y,z,\alpha ,\beta \}}{max}}\{f(i,j)|-a_{\mathrm{km}}\le k_{\mathrm{aak}}{a}_f(i,j)+k_{\mathrm{avk}}{f}^2(i,j)\le a_{\mathrm{km}}\}\hfill \end{array}$$

(38)

$$a_{fmin}(t,v_f)\le a_f\le a_{fmax}(t,v_f)$$

(39)

where

$$a_{fmax}(t,v_f)=min\left\{\begin{array}{c}\hfill \underset{\Lambda =x,y,z,\alpha ,\beta }{min}\left\{\frac{a_{\Lambda m}-k_{\mathrm{av}\Lambda }{v}_f^2}{k_{\mathrm{aa}\Lambda }(t)}|k_{\mathrm{av}\Lambda }>0\right\}\hfill \\ \multicolumn{1}{c}{\underset{\Lambda =x,y,z,\alpha ,\beta }{min}\left\{\frac{-a_{\Lambda m}-k_{\mathrm{av}\Lambda }{v}_f^2}{k_{\mathrm{aa}\Lambda }(t)}|k_{\mathrm{av}\Lambda }<0\right\}}\end{array}\right\}$$

(40)

$$a_{fmin}(t,v_f)=min\left\{\begin{array}{c}\hfill \underset{\Lambda =x,y,z,\alpha ,\beta }{max}\left\{\frac{-a_{\Lambda m}-k_{\mathrm{av}\Lambda }{v}_f^2}{k_{\mathrm{aa}\Lambda }(t)}|k_{\mathrm{av}\Lambda }>0\right\}\hfill \\ \multicolumn{1}{c}{\underset{\Lambda =x,y,z,\alpha ,\beta }{max}\left\{\frac{a_{\Lambda m}-k_{\mathrm{av}\Lambda }{v}_f^2}{k_{\mathrm{aa}\Lambda }(t)}|k_{\mathrm{av}\Lambda }<0\right\}}\end{array}\right\}$$

(41)

Figure 5.

Feasible polygon determined by axis acceleration constraints.

4. The maximum feed and acceleration restricted by comprehensive constraints

The maximum feed $f_{lim}(t)$ and feasible acceleration range considering all the above constraints comprehensively are obtained

$$v_f\le f_{lim}(t)=min\{f_{\mathrm{Tr}}(t),f_{\mathrm{vm}}(t),f_{\mathrm{am}}(t),F_{\mathrm{cmd}}\}$$

(42)

The feasible acceleration of the tip point is expressed as equation (40) or (41).

Adaptive feedrate scheduling

In five-axis parametric curve interpolation in WCS, tool tip point is usually selected to implement feedrate planning. The optimal feedrate of tool tip point is gained by considering various constraints and adopting the bi-directional scanned feed scheduling method. The increment of arc-length of the tool tip point curve is calculated and the parameter is estimated. Then the coordinate of the tool tip point is obtained. According to the parameters of synchronous mechanism, the tool axis direction is also obtained. Finally, using kinematics inverse transformation, the command position of each axis in MCS is gained.

As shown in Figure 6, the bi-directional scanned feed scheduling method based on the look-ahead window consists of two functions: the reverse scanning to generate limit feed curve and the forward scanning to obtain adaptive feedrate.

Figure 6.

Adaptive feed scheduling with a look-ahead window.

Reverse scanning

A look-ahead window is employed here. The window covers a set of linear segments, the number of which is dependent on the CPU’s power. At the beginning of the task, N linear segments are pushed into the look-ahead first-in first-out (FIFO) buffer from the decoding shared memory.

First, using the five-axis corner error distribution model, the optimization error constraint values are obtained, ${\epsilon }_{\mathrm{ap}1}$ , ${\epsilon }_{\mathrm{ch}1}$ , ${\epsilon }_{\mathrm{ap}2}$ , and ${\epsilon }_{\mathrm{ch}2}$ . After path smoothing, N linear segments are transformed into G² continuous smoothing path, which consists of 2N − 1 curves (lines or Bézier curves). There are N lines and N − 1 Bézier curves. Reverse scanning starts from Nth curve to the first curve. Various constraints including the chord error, the maximum normal acceleration, the maximum tangential acceleration/jerk, the axes’ capabilities, and the command feedrate are taken into account.

When the ith segment curve is being scanned, the accumulative arc-length, the parameter, the feed limitation, and the acceleration limitation of jth sampling period are defined as $l_j,u_i^j,f_{lim,i}^j,\mathrm{and}{a}_{lim,i}^j$ , respectively. Let them be $(i,l_j,u_i^j,f_{lim,i}^j,a_{lim,i}^j)$ .

To solve the problem of cross section of deceleration, the cumulative arc-length criterion is defined as $l_c[\mathrm{index}]$

$$l_c[1]=0,l_c[2]=l_c[1]+l_1,\dots ,l_c[i]=l_c[i-1]+l_{i-1},\dots$$

Reverse scanning process is as follows:

1. Let $i=2N-1$ , $j=0$ , $f_{lim,i}^0=0$ , $a_{lim,i}^0=0$ , $l_i=l_c[2N-1]+l_{2N-1}$ , $u_i^0=1$ , $j=j+1$ .

2. If $l_j\le 0$ , go to step (3); otherwise go to step (2a).

3. Let $i=i-1$ . If $i>0$ , go to step (2); otherwise, it is over.

Forward scanning

Forward scanning is performed from the start of the look-ahead window to the end of the look-ahead window. Forward scanning generates optimal feedrate profile using jerk-limited acceleration method and generates the interpolating point.

For the kth sampling period, the accumulative arc-length, the sequence number of curve, the parameter, the feedrate, and the acceleration are denoted as (l_k , i, $u_i^k$ , f_k , a_k ). Forward scanning is described in detail as follows:

Step 1. The feedrate is initialized by setting $f_0=0$ at the start of the whole tool path.

Step 2. Jerk-limited acceleration method is utilized to evaluate the feed of kth ( $k>0$ ) sampling period.

The acceleration of kth sampling period is calculated using equation (40)

$$a_k=min\{a_{k-1}+J_m{T}_s,a_{fmax}(u_i^{k-1},f_{k-1})\}$$

(47)

Then, the feed of kth sampling period is evaluated as

$$f_k=f_{k-1}+a_k{T}_s$$

(48)

Step 3. The parameter $u_k$ is calculated with the first-order Taylor’s expansion

$$u_i^k=u_i^{k-1}+\frac{f_k{T}_s}{\sigma (u_i^{k-1})}$$

(49)

If $u_i^k\ge 1$ , let $i=i+1$ , and $u_i^k$ is calculated again

$$u_i^k=0+\frac{f_k{T}_s-(l_c[i]-l_{k-1})}{\sigma (0)}$$

(50)

Step 4. After obtaining $u_i^k$ , the feed $f_k$ is compared with the reverse limit feed profile.

First, two adjacent reverse feed points $(l_j,u_i^j,f_{lim,i}^j,a_{lim,i}^j)$ and $(l_j,u_i^{j-1},f_{lim,i}^{j-1},a_{lim,i}^{j-1})$ ( $u_i^j\le u_i^k\le u_i^{j-1}$ ) are picked out. Then, feed limitation of reverse feed profile at $u_i^k$ is evaluated as

$$f_{lim}(u_i^k)=\frac{u_i^k(f_{lim,i}^{j-1}-f_{lim,i}^j)+u_i^{j-1}{f}_{lim,i}^j-u_i^j{f}_{lim,i}^{j-1}}{u_i^{j-1}-u_i^j}$$

(51)

If $f_k<f_{lim}(u_i^k)$ , the optimal feed is obtained; otherwise, $f_k$ is adjusted as shown in Figure 7.

$$u_i^k=\frac{T_s(u_i^j{f}_{lim,i}^{j-1}-u_i^{j-1}{f}_{lim,i}^j)-u_i^{k-1}\sigma (u_{k-1})(u_i^j-u_i^{j-1})}{T_s(f_{lim,i}^{j-1}-f_{lim,i}^j)-\sigma (u_i^{k-1})(u_i^j-u_i^{j-1})}$$

(52)

$$f_k=\frac{\sigma \left(u_i^{k-1}\right)(u_i^k-u_i^{k-1})}{T_s}$$

(53)

Step 5. After obtaining the feedrate $f_k$ and the parameter $u_i^k$ , the interpolating point $[P(u_i^k),Q(u_i^k{)]}^T$ is calculated and the tool axis direction is calculated

$$O(u_i^k)=\frac{(Q(u_i^k)-P(u_i^k))}{\Vert Q(u_i^k)-P(u_i^k)\Vert }$$

(54)

Figure 7.

Adjust the optimal feedrate within reverse feed limit.

The tool pose in WCS is expressed as

$$W(u_i^k)=\left[\begin{array}{c}\hfill P(u_i^k)\hfill \\ \multicolumn{1}{c}{Q(u_i^k)}\end{array}\right]$$

(55)

The coordinate of each axis in MCS is obtained

$$M(u_i^k)=G(W(u_i^k))$$

(56)

Finally, the motion command for each axis is updated in programmable motion controller.

Simulation and experiment

In this section, simulation and experiment are performed on a five-axis machining center with dual rotary tables, as shown in Figure 8. The in-house developed research CNC, which is PC-based, has an open architecture and allows rapid implementation of the proposed interpolation algorithm. The kinematics inverse transformation G is expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{}& A=\arccos (O_z)\hfill \\ \multicolumn{1}{c}{}& C=\arctan \left(\frac{O_x}{O_y}\right)\hfill \\ \multicolumn{1}{c}{}& X=\cos (C)P_x-\sin (C)P_y\hfill \\ \multicolumn{1}{c}{}& Y=\cos (A)\sin (C)P_x\hfill \\ \multicolumn{1}{c}{}& +\cos (A)\cos (C)P_y-\sin (A)(P_z-z_c)\hfill \\ \multicolumn{1}{c}{}& Z=\sin (A)\sin (C)P_x\hfill \\ \multicolumn{1}{c}{}& +\sin (A)\cos (C)P_y+\cos (A)(P_z-z_c)+z_c\hfill \end{array}$$

where $\mathbf{M}={(X,Y,Z,A,C)}^T$ is the coordinate in MCS and $\mathbf{R}={(P_x,P_y,P_z,O_x,O_y,O_z)}^T$ is the coordinate in WCS. $(0,0,z_c)$ is the rotary center. $z_c=28.251\mathrm{mm}$ .

Figure 8.

VMC-C30 five-axis machining center.

This experiment demonstrates the advantages of the proposed algorithm by machining a polyhedron workpiece, as shown in Figure 9. In the experiment, the drive constraints of each axis are listed in Table 2. The maximum jerk is set as 8.1 × 10⁸ mm/min³. The maximum normal acceleration is set as 1.08 × 10⁶ mm/min². The command feedrate is set to 1500 mm/min and the sampling period T_s is 0.004 s. The maximum look-ahead N of blocks is set as 50, and the predefined tolerance error is set as 0.02 mm.

Figure 9.

Polyhedron workpiece.

Table 2.

Machine tool driving constraints.

Axis	V max	A max
X	4000 mm/min	2.16 × 106 mm/min2
Y	4000 mm/min	2.16 × 106 mm/min2
Z	4000 mm/min	2.16 × 106 mm/min2
A	1500°/min	3.6 × 105°/min2
C	2000°/min	3.6 × 105°/min2

First, the approximation error constraint and the chord error constraint are calculated:

$\because A_{\max }*T_s^2/4=0.0012<0.02\mathrm{mm}$ ;

$\therefore {\epsilon }_{\mathrm{ch}}=A_{\max }*T_s^2/8=0.0006\mathrm{mm},{\epsilon }_{\mathrm{ap}}={\epsilon }_{max}-{\epsilon }_{\mathrm{ch}}=0.0194$ .

A side milling tool path for polyhedron is taken for example to demonstrate dual-Bézier path smoothing algorithm presented in section “Dual-Bézier path smoothing for five-axis linear path in WCS.” The approximation error constraint value is set to 0.0194 mm. The generated G² continuous tool path is shown in Figure 10.

Figure 10.

Dual-Bézier tool path for the side milling of the polyhedron.

According to Figure 11, in the new smoothing path, the trajectory of the tool tip point is curvature-continuous. There are three peaks of the curvature, corresponding tags in Figure 11, which are in conformity with the locations on the polyhedron parts’ computer-aided design (CAD) model. The points marked with ① and ② are where the surface curvature change is bigger. The point marked with ③ is retract of the side milling.

Figure 11.

Relationship of curvature and arc-length for the new tool tip point path.

Figure 12 shows three types of error profiles for the trajectory generated by the proposed method. The trajectory error, the chord error, and the approximation error are all less than or equal to the predefined tolerance 0.02 mm.

Figure 12.

Errors of the tool tip and tool axis point based on error distribution model: (a) three types of error of the tool tip point and (b) three types of error of the tool axis point.

In the existing studies,^{5–12,14–25} approximation error and chord error are calculated, respectively, and they are all restricted with the predefined tolerance usually. Take Shi’s ¹ method, for example, as shown in Figure 13, the chord error and the approximation error of the trajectory satisfy the predefined tolerance constraint, but the trajectory error exceeds the predefined error. The maximum trajectory error of tool tip point reaches 0.02048 mm, and the maximum trajectory error of tool axis point is 0.02062 mm. They are all bigger than the predefined tolerance of 0.02 mm. According to Figure 13, we find that the point whose trajectory error exceeds the predefined error is the point with bigger curvature. Thus, the proposed algorithm is useful to control the trajectory error at the corner of the linear path.

Figure 13.

Errors of the tool tip and tool axis point without error distribution model: (a) three types of error of the tool tip and (b) three types of error of the tool axis point.

As shown in Figure 14(a) and (b), the feedrate and acceleration of tool tip have few fluctuations, which just appear at the corners with big angles of the linear tool path. The feedrate profile obtained with the proposed feedrate scheduling method is smooth, and the feedrate is close to the command feedrate 1500 mm/min. Because the speed of C-axis reaches the limitation 2000°/min at the middle, as shown in Figure 15(a), the feedrate of the tool tip point slows down at the middle part.

Figure 14.

Speed and acceleration of the tool tip point: (a) tool tip feedrate and (b) tool tip acceleration.

Figure 15.

Speed and acceleration of each axis: (a) speed of each axis and (b) acceleration of each axis.

From the analysis presented above, it can be found that the new tool path generated by the proposed path smoothing algorithm for five-axis linear tool path is curvature-continuous. In addition, the trajectory error, the chord error, and the approximation error are all guaranteed within the predefined tolerance. Based on smoothing the tool path, considering the comprehensive constraints and utilizing jerk-limited acceleration method, the feedrate and acceleration fluctuations are alleviated.

Summary and conclusion

The most widespread tool paths are represented linear segments (G01 blocks) to approximate the complicated surface in machining. The discontinuities at the corners are the main obstacle to perform high-speed and high-accuracy machining.

In this work, a dual-Bézier path smoothing algorithm for five-axis linear tool path in WCS is proposed. First, the corner error distribution model is presented to guarantee the trajectory error, the chord error, and the approximation error within the predefined tolerance. Second, every junction of the five-axis tool path is blended by inserting two cubic Bézier curves to generate a curvature-continuous tool path; moreover, the tool tip point curve and tool axis point curve are parameterized synchronous. Third, a real-time adaptive feedrate scheduling method considering the comprehensive constraints is presented. Compared with the previous methods, the novel path smoothing algorithm has the following advantages: (1) the tool tip point curve and tool axis point curve are parameterized synchronous, (2) the smoothed tool path is everywhere curvature-continuous, (3) the trajectory error is guaranteed within the predefined tolerance, and (4) the feedrate and acceleration fluctuations are alleviated.

The proposed dual-Bézier transition algorithm has been integrated into an open five-axis CNC system. The experiments are carried out to test its practicability and effectiveness. The interpolator integrated with the dual-Bézier corner transition algorithm guarantees the trajectory accuracy and generates smooth speed and acceleration curves.