Corner Transition Toolpath Generation Based on Velocity-Blending Algorithm for Glass Edge Grinding
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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.
Keywords
Glass edge grinding Toolpath planning Velocity blending Grinding depth control1 Introduction
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
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.
- (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, Q_{0}A and CQ_{2}, respectively.
- (3)
Because of the maximum grinding depth at point B, a slower transition velocity should be planned.
- (1)
The arbitrary position P_{i}, i ≥ 0, is determined by a_{1} and a_{2}, which can be adjusted to meet requirement 1 and is planned in Section 3.
- (2)
The velocity is v_{trans}e_{s} at point A and v_{trans}e_{d} at point C, which guarantees that the generated transition toolpath is of G^{1} continuity and meets requirement 2 naturally.
3 Toolpath Generation
3.1 Traverse Displacement Constraint
3.2 Grinding Depth Constraint
Derivatives of F(t)
Derivatives | Expressions | Solutions |
---|---|---|
F^{(4)}(t) | \(6\left( {{\varvec{a}_1}^2 + {\varvec{a}_2}^2 + 2{\varvec{a}_1}{\varvec{a}_2}\cos \beta } \right)\) | F^{(4)}(t) is a constant and greater than zero |
F^{(3)}(t) | \(F^{{(4)}} (t)t - 6\varvec{v}_{trans} \left( {\varvec{a}_{1} + \varvec{a}_{2} \cos\beta } \right)\) | F^{(3)}(t) has only one solution named t_{31} |
\(F^{\prime\prime}\left( t \right)\) | \(- 0.5F^{{(4)}} (t)t^{2} + F^{{(3)}} (t)t + 2\left( {\varvec{v}_{trans}^{2} - \varvec{a}_{2} R\sin\beta } \right)\) | \(F^{\prime\prime}\left( t \right)\) has two solutions named t_{21} and t_{22} |
\(F^{\prime}\left( t \right)\) | \(\frac{1}{3}F^{{(4)}} (t)t^{3} + \frac{1}{6}F^{{(3)}} (t)t^{2} - \frac{2}{3}F^{\prime\prime}(t)t + \frac{{10}}{3}\left( {\varvec{v}_{trans}^{2} - \varvec{a}_{2} R\sin\beta } \right)t\) | \(F^{\prime}\left( t \right)\) has three solutions named t_{11}, t_{12} and t_{13} |
- (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 \(F^{\prime}(t) \le 0\).
The transition toolpath is determined by a_{1} and a_{2}, 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
- (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 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.
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
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), a_{1} and a_{2} 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
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
- 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 t_{12} = 6 s, and then F(t_{12}) = 8786.3 mm^{2} and (R_{T} + d)^{2} = 5643.0 mm^{2}; 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
where 0 ≤ t ≤ 12.9.$$\varvec{P}{\mathbf{(}}\varvec{t}{\mathbf{)}}= \varvec{A} + \left( {0.67}t^{2} ,20t - 1.15t^{2} \right),$$
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.
Notes
Authors’ Contributions
KR carried out the toolpath-planning studies, derived the main equations and constraints, designed the experimental steps, and drafted the manuscript. YP derived some equations and collected the experimental data. DJ participated in the toolpath planning and derived some equations. JP analyzed the geometrical features of a sharp corner, proposed the transition strategies, and drafted the manuscript. WC participated in the mechanical design of the experimental device and helped to draft the manuscript. XH designed the control system of the experimental device. All authors read and approved the final manuscript.
Authors’ Information
Kun Ren, born in 1979, is currently an associate professor at College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, China. He received his PhD degree from Zhejiang University, China, in 2008. His research interests include numerical control technology and automatic equipment development.
Yujia Pan, born in 1995, is currently a master candidate at College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, China.
Danyan Jiang, born in 1993, is currently a master candidate at College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, China.
Jun Pan, born in 1974, is currently a professor at College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, China.
Wenhua Chen, born in 1963, is currently a professor at College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, China.
Xuxiao Hu, born in 1965, is currently a professor at the College of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, China.
Competing Interests
The authors declare that they have no competing interests.
Funding
Supported by National Key R&D Program of China (Grant No. 2017YFB0309800), and National Natural Science Foundation of China (Grant No. 51405445).
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