Corner Transition Toolpath Generation Based on Velocity-Blending Algorithm for Glass Edge Grinding

Abstract

Sharp corners usually are used on glass contours to meet the highly increasing demand for personalized products, but they result in a broken wheel center toolpath in edge grinding. To ensure that the whole wheel center toolpath is of G1 continuity and that the grinding depth is controllable at the corners, a transition toolpath generation method based on a velocity-blending algorithm is proposed. Taking the grinding depth into consideration, the sharp-corner grinding process is planned, and a velocity-blending algorithm is introduced. With the constraints, such as traverse displacement and grinding depth, the sharp-corner transition toolpath is generated with a three-phase motion arrangement and with confirmations of the acceleration/deceleration positions. A piece of glass with three sharp corners is ground on a three-axis numerical-control glass grinding equipment. The experimental results demonstrate that the proposed algorithm can protect the sharp corners from breakage efficiently and achieve satisfactory shape accuracy. This research proposed a toolpath generation method based on a velocity-blending algorithm for the manufacturing of personalized glass products, which generates the transition toolpath as needed around a sharp corner in real time.

1 Introduction

Sharp corners usually are used on glass contours to highlight a unique, personalized appearance, but they create challenges in edge grinding. In Figure 1, the center toolpath of a diamond wheel is shown with black dashed lines that are parallel to the final contour but broken at the sharp corners. Transition toolpaths should be generated for wheel traverse. Moreover, arranging a proper transition velocity profile to protect fragile tips from possible breakage should be planned carefully.

Figure 1

Glass edge grinding

Velocity blending is one of the main techniques adopted by researchers for sharp-corner machining of metallic materials. The two data segments forming a corner are processed simultaneously by the controller to make the tool move smoothly. Accordingly, the machining efficiency is improved, but a certain contour accuracy is sacrificed. Shi et al. [1] proposed a velocity link algorithm to realize a smooth sharp-corner transition with such constraints as the corner angle, acceleration capacity, and maximum velocity limitation. Zhang et al. [2] and Tajima et al. [3] introduced the acceleration and contour error constraints to derive the velocity-blending control equations that had such parameters as motion time and transition velocity. Lee [4], Luo et al. [5], Wang [6] and Li et al. [7] adopted velocity look-ahead control strategies in kinematic planning for short linear segments to achieve maximum machining efficiency. Rewa et al. [8] proposed an asymmetrical acceleration and deceleration (acc/dec) algorithm for short-length segment machining with the acceleration limitation. With the S-type acc/dec algorithms and limitations of acceleration and jerk, Jahanpour et al. [9], Beudaert et al. [10] and Farouki et al. [11] used Non-Uniform Rational B-Splines (NURBS) or quintic Pythagorean hodograph (P-H) curves to correct the sharp corner toolpaths and achieve velocity planning along the whole toolpath.

The redundant material at a sharp corner is always greater than normal. Grinding force models [12, 13] for surface grinding of glass and other materials have been proposed. It is clear from the grinding force models that the total grinding force increases with the increase in cutting depth. Without the consideration of grinding depth control, direct application of a velocity-blending algorithm can further increase the grinding depth due to the sacrifice of contour accuracy, which results in easy corner breakage.

Corner rounding is another selection for sharp-corner machining and usually uses different kinds of parametric curves to correct the sharp corners and impose G1, G2, or higher-order continuity on the toolpaths. Zhao et al. [14] and Zhang et al. [15] used polynomial curves to correct the discrete toolpaths, which were smoothed under the constraints of maximum acceleration and velocity. Wu et al. [16], Zhao et al. [17], Pateloup et al. [18] and Lin et al. [19] pointed out the importance of toolpath planning at corners in high-speed cavity milling. Circular arcs and parametric curves were adopted to achieve tool smooth corner traverse. Duan et al. [20], Tulsyan et al. [21] and Zhou et al. [22] used NURBS curves and introduced contour error control equations to realize a shortest-time corner traverse. Ernesto et al. [23] and Bi et al. [24] used Bézier curves to interpolate the sharp corners with limitations of contour error and acceleration. Gassara et al. [25] presented a single circular arc transition method and built a velocity-planning model with a contour error constraint. Zhao et al. [26], Sencer et al. [27, 28], Beudaert et al. [29] and Pateloup et al. [30] used third- or higher-order B-spline curves to correct the sharp corners. In addition, different look-ahead control strategies with S-type acc/dec algorithms were proposed to achieve a smooth tool transition at the sharp corners.

Corner-rounding techniques can ensure that a tool traverses a sharp corner smoothly, but there is no interpolating point in the broken zones for direct curve fit, and it is time consuming for a controller to collect the proper interpolating points in real time.

To achieve sharp-corner grinding for personalized glass product manufacturing, a real-time toolpath generation method based on a velocity-blending algorithm is proposed. In Section 2, the desired sharp-corner transition planning is proposed, and a velocity-blending algorithm is presented. Section 3 introduces the constraints, such as traverse displacement and grinding depth, to generate the transition toolpath with a three-phase motion arrangement. In Section 4, a piece of glass with three sharp corners is ground to test the performance of the proposed algorithm. Section 5 concludes the work.

2 Toolpath-Planning Strategies

As shown in Figure 2, a typical sharp corner is formed by two linear segments, $\overline{SR}$ and $\overline{ST}$. S is the sharp-corner point and radius compensated to yield points A and C. Obviously, the wheel center toolpath is broken between the two radius-compensated linear segments, $\overline{{AQ_{0} }}$ and $\overline{{CQ_{2} }}$ which are extended and intersect at point Q. The corner-grinding process starts from point A and ends at point C.

Figure 2

Sharp-corner transition analyses

At the grinding zone, F is the total grinding force and composed of normal and tangential forces, F_n and F_t. According to the grinding-force model [19], F depends on four parameters: wheel speed, feeding velocity, grinding depth, and grit diameter. Usually, wheel speed is kept constant to protect the surface from possible scratches in grinding, whereas the feeding velocity and grinding depth can be adjusted as needed.

When a wheel penetrates into the sharp-corner zone from point A (see Figure 3), the glass becomes thinner, but the redundant material becomes greater, thereby increasing the total grinding force F. Especially, at point B, the maximum grinding depth corresponding to the maximum F is met, and, simultaneously, the wheel moving direction has a sudden change, resulting in an easily broken corner. Therefore, when a wheel traverses a fragile tip, feeding velocity and grinding depth should be arranged properly.

Figure 3

Corner geometrical analyses: a under the coordinate system X₁O₁Y₁, b under the coordinate system X₂AY₂

In conclusion, there are three requirements for transition toolpath planning.

• (1) To limit the total grinding force F, the grinding depth should be reduced gradually to be equal to or less than zero from point A to point B, whereas the grinding depth should increase gradually and returns to normal from point B to point C.

• (2) To avoid a sudden change, the moving direction at points A and C should be tangent to the radius-compensated contours, Q₀A and CQ₂, respectively.

• (3) Because of the maximum grinding depth at point B, a slower transition velocity should be planned.

When the grinding wheel arrives at point A with a velocity v_trans, a deceleration a₁ is imposed on AQ, and, meanwhile, an acceleration a₂ is imposed on QC (see Figure 2). With the coaction of a₁ and a₂, the wheel center moves to point C along the transitional toolpath denoted with a red dashed line. P_i, $i \ge 0$, is an arbitrary point on the transitional toolpath and expressed as

$$\varvec{P}_{\varvec{i}} = \varvec{A} + \left( {\varvec{v}_{trans} t_i { - }\frac{1}{2} \varvec{a}_{1} t_i^{2} } \right)\varvec{e}_{\varvec{s}} + \frac{1}{2} \varvec{a}_{2} t_i^{2} \varvec{e}_d,$$

(1)

where t_i is the elapsed blending time, e_s and e_d are unit vectors, and $\varvec{e}_{\varvec{s}} = \frac{{\varvec{AQ}}}{{\left\| {\varvec{AQ}} \right\|}},\varvec{e}_{\varvec{d}} = \frac{{\varvec{QC}}}{{\left\| {\varvec{QC}} \right\|}}$. Eq. (1) gives the following.

• (1) The arbitrary position P_i, i ≥ 0, is determined by a₁ and a₂, which can be adjusted to meet requirement 1 and is planned in Section 3.

• (2) The velocity is v_transe_s at point A and v_transe_d at point C, which guarantees that the generated transition toolpath is of G¹ continuity and meets requirement 2 naturally.

3 Toolpath Generation

3.1 Traverse Displacement Constraint

When the wheel center arrives at point C, command generations for two linear segments, $\overline{AQ}$ and $\overline{QC}$, are both finished. Along the linear segment $\overline{AQ}$,

$$\varvec{v}_{trans} = - \varvec{a}_{1} t_{m} ,$$

(2)

where t_m is the total time for velocity blending from point A to point C.

As shown in Figure 3(a), a local Cartesian coordinate system X₁O₁Y₁ is established. Let O₁ be the origin. In X₁O₁Y₁, the wheel center displacement D_p from point A to point C can be derived as

$$\varvec{D}_{\varvec{p}}=\left( {R_{T} {(1 + \cos}\beta), - R_{T} \sin\beta } \right).$$

(3)

According to the kinematic theory, D_p can also be derived as

$$\varvec{D}_{\varvec{p}} = \left( \begin{aligned} & \varvec{v}_{trans} t_{m} \sin \beta - \frac{1}{2}\varvec{a}_{1} \sin \beta t_{m}^{2} , \hfill \\ & \varvec{v}_{trans} t_{m} \cos \beta - \frac{1}{2}\left( {\varvec{a}_{1} \cos \beta + \varvec{a}_{2} } \right)t_{m}^{2} \hfill \\ \end{aligned} \right).$$

(4)

With Eqs. (3) and (4), a displacement equation can be derived as

$$\varvec{v}_{trans} t_{m} \cos \beta - \frac{1}{2}\left( {\varvec{a}_{1} \cos \beta + \varvec{a}_{2} } \right)t_{m}^{2} = - R_{T} \sin\beta .$$

(5)

3.2 Grinding Depth Constraint

Connecting point Q to point S, and when linear segment $\overline{QS}$ is the bisector of corner angle β, $0 \le \beta \le 180^\circ$. Around corner S, the maximum grinding depth d is

$$d = d_{n} /\sin \left( {\frac{\beta }{2}} \right),$$

(6)

where d_n is the normal grinding depth. According to the radius compensation algorithm,

$$\left\| {\varvec{SA}} \right\| = \left\| {\varvec{SC}} \right\| = R_{T} ,$$

(7)

where R_T is the wheel radius.

To simplify the calculations, as shown in Figure 3(b), a new Cartesian coordinate system X₂AY₂ is established, and let point A be the origin. In X₂AY₂, the corner point S = (R_T, 0), and an arbitrary point P_i on the transition toolpath can be expressed as

$$\varvec{P}_{i}=\left( {\frac{1}{2}\varvec{a}_{2} t_{i}^{2} \sin\beta , \, \varvec{v}_{trans} t_{i} - \frac{1}{2}\left( {\varvec{a}_{1} + \varvec{a}_{2} \cos\beta } \right)t_{i}^{2} } \right).$$

(8)

Let L(t) be the distance between two points, S and P_i,

$$\begin{aligned} L(t)^{2} & = \left( {\frac{1}{2}\varvec{a}_{2} t^{2} \sin\beta - R_{T} } \right)^{2} \\ & \quad + \left( {\varvec{v}_{trans} t - \frac{1}{2}\left( {\varvec{a}_{1} + \varvec{a}_{2} \cos\beta } \right)t^{2} } \right)^{2} . \end{aligned}$$

(9)

To satisfy the transition toolpath-planning requirement 1, a grinding depth constraint is given by

$$L(t)^{2} \ge R_{T}^{2} .$$

(10)

Let F(t) = L(t)² − R ² _T , and F(t) is a quartic function related to time t. In Figure 4, the derivatives of F(t) are shown with different colors. The expressions and solutions are shown in Table 1. Obviously, F(t) is a quartic curve and has the features:

Figure 4

Derivatives analyses of F(t)

Table 1 Derivatives of F(t)

• (1) If $0 \le t \le t_{{12}}$, F(t) is increasing monotonically due to $F^{\prime}(t) \ge 0$;

• (2) If $t_{{12}} < t \le t_{{13}}$, F(t) is decreasing monotonically due to $F^{\prime}(t) \le 0$.

Therefore, F(t), $0 \le t \le t_{{13}}$, meets requirement 1 and can be used to design the transition toolpath. Let the total blending time $t_{m} = t_{{13}}$,

$$F(t_{m}) = R_{T}^{2} .$$

(11)

For motion symmetry, letting t₁₂ = 0.5t_m, the maximum extreme value $F(t_{{12}} )$ should be limited by

$$F(t_{{12}} ) \ge \left( {R_{T} + d} \right)^{2} .$$

(12)

With Eqs. (2) and (7), constraint Eq. (12) is updated by

$$\frac{{d_{n} }}{{R_{T} }} < \left( {\sqrt {1 + \frac{{\sin^{2} \beta \cos\beta }}{{1 - \cos\beta }}} - 1} \right).$$

(13)

With Eqs. (5), (7), and (11), a₁ and a₂ can be derived as

$$\varvec{a}_{1} = \varvec{a}_{2} = \frac{{\varvec{v}_{trans}^{2} \left( {1 - \cos\beta } \right)}}{{2R_{T} \sin\beta }}.$$

(14)

The transition toolpath is determined by a₁ and a₂, which can be adjusted by control Eq. (14). With the constraints in Eqs. (11) and (12), the grinding depth reduces gradually to be equal to or less than zero from point A to point B, whereas the grinding depth increases gradually and returns to normal from point B to point C. Then, the transition requirements 1 and 2 are both satisfied.

3.3 Transition Velocity Planning

To meet transition requirement 3, motion through a sharp corner needs a deceleration and acceleration process. Hence, three motion phases are planned and depicted as follows:

• (1) A slowing-down portion, where the velocity is decelerating from v_nor to v_trans, and the deceleration distance is defined as L_d;

• (2) A uniform speed portion, where tools traverse the corner from point A to point C at a constant velocity v_trans;

• (3) A speed-up portion, where the velocity is accelerating from v_trans to v_nor, and the acceleration distance is defined as L_a.

A linear acc/dec algorithm is utilized; thereby, the required distances for deceleration and acceleration can be figured out by

$$L_{d} = L_{a} = \frac{{\varvec{v}_{nor}^{2} - \varvec{v}_{trans}^{2} }}{{2\sqrt {\varvec{a}_{x}^{2} + \varvec{a}_{y}^{2} } }},$$

(15)

where a_x and a_y are the normal accelerations of the X and Y axes, respectively. v_trans can be initialized by

$$\varvec{v}_{trans} = \frac{\beta }{180}\varvec{v}_{nor} .$$

(16)

A straight line, circular arc, and NURBS curve usually emerge before or after a corner. Ahead of the transition, a deceleration point P_d before point A and an acceleration point P_a after point C should be confirmed.

As shown in Figure 5(a), corner S is formed by two linear segments that are radius compensated to yield $\overline{{AA_{0} }}$ and $\overline{{CC_{0} }}$. With Eq. (15), lengths L_d and L_a can be figured out first, so

$$\varvec{P}_{d} = \varvec{A} + \frac{{\varvec{A}_{0} - \varvec{A}}}{{\left\| {\varvec{A}_{0} - \varvec{A}} \right\|}}L_{d} ,$$

(17)

and

Figure 5

Acc/dec point confirmations: a for a straight line, b for a circular arc, c for a NURBS curve

$$\varvec{P}_{a} = \varvec{C} + \frac{{\varvec{C}_{0} - \varvec{C}}}{{\left\| {\varvec{C}_{0} - \varvec{C}} \right\|}}L_{a} .$$

(18)

As shown in Figure 5(b), the corner S is formed by a circular arc and a straight line that are both radius compensated to yield ${{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{AA} }_0}$ and $\overline{{CC_{0} }}$. Supposing that the arc center is P_c,

$$\varvec{P}_{d} = \varvec{P}_{c} + \left[ {\begin{array}{*{20}c} {\cos \theta } & {\sin \theta } \\ { - \sin \theta } & {\cos \theta } \\ \end{array} } \right]\frac{{\left( {\varvec{A} - \varvec{P}_{c} } \right)^{\text{T}} }}{{\left\| {\varvec{A} - \varvec{P}_{c} } \right\|}},$$

(19)

where θ is decided by L_d. If ${{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{AA} }_0}$ is clockwise, $\theta = \frac{{L_{d} }}{{\left\| {\varvec{A} - \varvec{P}_{c} } \right\|}}$; otherwise, $\theta = - \frac{{L_{d} }}{{\left\| {\varvec{A} - \varvec{P}_{c} } \right\|}}$.

As shown in Figure 5(c), the corner is formed by a straight line and a NURBS curve, which are both radius compensated to yield $\widetilde{AA}_{0}$ and $\overline{{CC_{0} }}$. Suppose that curve $\widetilde{AA}_{0}$ is c(u), 0 ≤ u ≤ 1. Let u_i be an arbitrary knot value and Δu be the increasing step length. Set initial knot value u₀ = 0 and accumulating chord length L_c = 0. Let u_i = u_i−1 + Δu, i > 0. A point C_i on the NURBS curve can be generated corresponding to knot value u_i, and the chord length L_c is refreshed iteratively by

$$L_{c} = L_{c} + \left\| {\varvec{C}_{i} - \varvec{C}_{{i - 1}} } \right\|.$$

(20)

Once L_c is more than or equal to L_d or L_a, the current knot value u_i is recorded, and relevant point P_d or P_a can be confirmed.

3.4 Trajectory Generation Steps

Figure 6 shows the flowchart of the proposed algorithm application of which the steps are summarized briefly as follows.

Figure 6

Application of the proposed algorithm

3.4.1 Step 1

Basic parameters, such as wheel radius R_T, accelerations a_x and a_y, glass thickness T_g, and normal grinding depth d_n are input in the man–machine interface before machining starts.

The pattern file from a computer-aided design system is loaded into control system and then interpreted by the man–machine interface software, which is coded with Visual C++ 6.0. G-codes representing the contour are generated, and the sharp corners along the contour are detected and marked.

With Eq. (16), v_trans can be determined. Taking the known v_trans into Eq. (14), a₁ and a₂ are also determined. Eq. (15) is used to figure out both L_d and L_a. With Eqs. (17), (18), (19), and (20), points P_d and P_a are confirmed.

3.4.2 Step 2

The control framework with two rotary buffers is built. Each G-code segment is downloaded into the rotary buffer 1 with corresponding auxiliary machining information downloaded into the rotary buffer 2. The auxiliary machining information is stored in a data struct that has some member variables, such as normal velocity v_nor, transition velocity v_trans, corner angle β, accelerations a₁ and a₂, and lengths L_d and L_a, and is defined as:

In the INPUTPOINT struct, a Boolean variable sharpcorner is defined to mark a sharp corner, and another Boolean variable datatype is defined to mark the position of the data segment. If a sharp corner is detected, sharpcorner is true; otherwise, sharpcorner is false. If a data segment locates before the corner, datatype is false; otherwise, datatype is true.

3.4.3 Step 3

If sharpcorner is true, the desired transition toolpath can be generated with Eq. (1) and then inserted at the corner broken zone. After acc/dec planning is implemented, interpolation begins.

4 Experimental Results

The experiment was carried out with three-axis numerical-control glass grinding equipment, as shown in Figure 7(a). The X, Y and Z axes traveled together to span a 1500 × 2500 × 120 mm 3D space. The experimental parameters were set as: spindle speed S_p = 6000 r/min, normal accelerations of X and Y axes a_x = a_y = 20 mm/s², glass thickness T_g = 12 mm, maximum grinding velocity v_nor = 3600 mm/min, wheel radius R_T = 75 mm, normal grinding depth d_n = 0.13 mm, and increasing step length Δu = 0.05.

Figure 7

Experimental setup: a glass grinding center, b man-machine interface

As shown in Figure 7(b), a piece of glass balustrade with three sharp corners for the escalator is being ground. Three sharp corners are numbered sequentially. Figure 8 shows the real velocity and acceleration profiles of the X and Y axes, respectively. The real kinematic values at the corners are shown between two vertical straight lines of different color. In addition, v_x and v_y denote the velocities of the X and Y axes, respectively.

Figure 8

Kinematic profiles: a velocity profile of X axis, b velocity profile of Y axis, c acceleration profile of X axis, d acceleration profile of Y axis

Angle β of corner 3 is 60^o and taken as an example to depict how the transition toolpath is generated and implemented. The details are as follows.

• 1. With Eq. (16), the transition velocity v_trans = 1200 mm/min.

• 2. Taking β, v_trans, d_c, and R_T into Eq. (18) and checking whether the constraint is satisfied, one has t₁₂ = 6 s, and then F(t₁₂) = 8786.3 mm² and (R_T + d)² = 5643.0 mm²; thereby, constraint Eq. (13) is satisfied.

• 3. With Eq. (15), the required acc/dec lengths L_d = L_a = 56.57 mm. Thereby,

  $$\varvec{P}_{d} = \, \left( { 3 1 1. 9 3,{ 495}. 6 2} \right),$$
  $$\varvec{P}_{a} = \, \left( { 20 9. 4 2,{ 526}. 4 7} \right).$$

• 4. With Eq. (1), the transition toolpath is generated by

  $$\varvec{P}{\mathbf{(}}\varvec{t}{\mathbf{)}}= \varvec{A} + \left( {0.67}t^{2} ,20t - 1.15t^{2} \right),$$

  where 0 ≤ t ≤ 12.9.

The real acc/dec processes around the corners are explicitly shown in Figure 8 and in accordance with the planning. As shown in Figure 9, the wheel center trajectory is connected by the velocity-blending algorithm at each corner.

Figure 9

Generated transition toolpath at the sharp corners: a generated wheel center toolpath at the corners, b position profile of X axis, c position profile of Y axis

Figure 10 shows the distance change profiles between the wheel center and corner points when the diamond wheel traverses corners 1 and 3. The position of the X axis is relative to point A. On the left hand of the red dashed line, the increasing distance means the gradually decreasing grinding depth; otherwise, on the right, the decreasing distance means the gradually increasing grinding depth. The distance change profiles at two different corners are both in accordance with the design requirements.

Figure 10

Distance change profiles between the wheel center and the corner points when diamond wheel traverses corners 1 and 3: a corner 1; b corner 3

Figure 11 shows the final ground glass and its application. The proposed algorithm can protect a sharp corner from breakage efficiently, and the corner angle accuracy of ±0.1° fully meets the factory requirements.

Figure 11

Grinding results

5 Conclusions

A toolpath generation method based on a velocity-blending algorithm for the manufacturing of personalized glass products was proposed. The transition toolpath is generated based on the grinding depth and velocity control strategies, and this makes the wheel traverse a sharp corner smoothly.

Compared with the methods using velocity blending directly, the proposed algorithm uses constraints, such as traverse displacement and grinding depth, to derive an acceleration control equation that makes possible the adjustment of the transition toolpath as needed. Moreover, acc/dec distances and positions around a corner were confirmed, and they can be implemented by an interpolator easily.

Compared with other kinds of corner-rounding algorithm, the proposed algorithm generates the toolpath in real time under a control framework composed of two rotary buffers, and the main computational tasks are implemented by a powerful industrial computer, which alleviates the computational load of the control system and greatly improves the algorithm efficiency.

The experimental results show that the transition scheme proposed can achieve a personalized glass product with satisfactory corner shape accuracy and efficiently prevent fragile tips from breaking.