 Original Article
 Open Access
 Published:
Corner Transition Toolpath Generation Based on VelocityBlending Algorithm for Glass Edge Grinding
Chinese Journal of Mechanical Engineering volumeÂ 32, ArticleÂ number:Â 87 (2019)
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 velocityblending algorithm is proposed. Taking the grinding depth into consideration, the sharpcorner grinding process is planned, and a velocityblending algorithm is introduced. With the constraints, such as traverse displacement and grinding depth, the sharpcorner transition toolpath is generated with a threephase motion arrangement and with confirmations of the acceleration/deceleration positions. A piece of glass with three sharp corners is ground on a threeaxis numericalcontrol 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 velocityblending 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 FigureÂ 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.
Velocity blending is one of the main techniques adopted by researchers for sharpcorner 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 sharpcorner 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 velocityblending 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 lookahead 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 shortlength segment machining with the acceleration limitation. With the Stype acc/dec algorithms and limitations of acceleration and jerk, Jahanpour et al. [9], Beudaert et al. [10] and Farouki et al. [11] used NonUniform Rational BSplines (NURBS) or quintic Pythagorean hodograph (PH) 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 velocityblending 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 sharpcorner machining and usually uses different kinds of parametric curves to correct the sharp corners and impose G1, G2, or higherorder 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 highspeed 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 shortesttime corner traverse. Ernesto et al. [23] and Bi et al. [24] used BÃ©zier 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 velocityplanning 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 higherorder Bspline curves to correct the sharp corners. In addition, different lookahead control strategies with Stype acc/dec algorithms were proposed to achieve a smooth tool transition at the sharp corners.
Cornerrounding 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 sharpcorner grinding for personalized glass product manufacturing, a realtime toolpath generation method based on a velocityblending algorithm is proposed. In SectionÂ 2, the desired sharpcorner transition planning is proposed, and a velocityblending algorithm is presented. SectionÂ 3 introduces the constraints, such as traverse displacement and grinding depth, to generate the transition toolpath with a threephase motion arrangement. In SectionÂ 4, a piece of glass with three sharp corners is ground to test the performance of the proposed algorithm. SectionÂ 5 concludes the work.
2 ToolpathPlanning Strategies
As shown in FigureÂ 2, a typical sharp corner is formed by two linear segments, \(\overline{SR}\) and \(\overline{ST}\). S is the sharpcorner point and radius compensated to yield points A and C. Obviously, the wheel center toolpath is broken between the two radiuscompensated linear segments, \(\overline{{AQ_{0} }}\) and \(\overline{{CQ_{2} }}\) which are extended and intersect at point Q. The cornergrinding process starts from point A and ends at point C.
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 grindingforce 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 sharpcorner zone from point A (see FigureÂ 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.
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 radiuscompensated 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.
When the grinding wheel arrives at point A with a velocity v_{trans}, a deceleration a_{1} is imposed on AQ, and, meanwhile, an acceleration a_{2} is imposed on QC (see FigureÂ 2). With the coaction of a_{1} and a_{2}, 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
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}, iÂ â‰¥Â 0, is determined by a_{1} and a_{2}, which can be adjusted to meet requirement 1 and is planned in SectionÂ 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
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 \(\overline{AQ}\),
where t_{m} is the total time for velocity blending from point A to point C.
As shown in FigureÂ 3(a), a local Cartesian coordinate system X_{1}O_{1}Y_{1} is established. Let O_{1} be the origin. In X_{1}O_{1}Y_{1}, the wheel center displacement D_{p} from point A to point C can be derived as
According to the kinematic theory, D_{p} can also be derived as
With Eqs. (3) and (4), a displacement equation can be derived as
3.2 Grinding Depth Constraint
Connecting point Q to point S, and when linear segment \(\overline{QS}\) is the bisector of corner angle Î², \(0 \le \beta \le 180^\circ\). Around corner S, the maximum grinding depth d is
where d_{n} is the normal grinding depth. According to the radius compensation algorithm,
where R_{T} is the wheel radius.
To simplify the calculations, as shown in FigureÂ 3(b), a new Cartesian coordinate system X_{2}AY_{2} is established, and let point A be the origin. In X_{2}AY_{2}, the corner point SÂ =Â (R_{T}, 0), and an arbitrary point P_{i} on the transition toolpath can be expressed as
Let L(t) be the distance between two points, S and P_{i},
To satisfy the transition toolpathplanning requirement 1, a grinding depth constraint is given by
Let F(t)Â =Â L(t)^{2}Â âˆ’Â R ^{2}_{ T} , and F(t) is a quartic function related to time t. In FigureÂ 4, the derivatives of F(t) are shown with different colors. The expressions and solutions are shown in TableÂ 1. Obviously, F(t) is a quartic curve and has the features:

(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 \(t_{m} = t_{{13}}\),
For motion symmetry, letting t_{12}Â =Â 0.5t_{m}, the maximum extreme value \(F(t_{{12}} )\) should be limited by
With Eqs. (2) and (7), constraint Eq. (12) is updated by
With Eqs. (5), (7), and (11), a_{1} and a_{2} can be derived as
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
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 slowingdown 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 speedup 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
where a_{x} and a_{y} are the normal accelerations of the X and Y axes, respectively. v_{trans} can be initialized by
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 FigureÂ 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
and
As shown in FigureÂ 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},
where Î¸ 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 FigureÂ 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), 0Â â‰¤Â uÂ â‰¤Â 1. Let u_{i} be an arbitrary knot value and Î”u be the increasing step length. Set initial knot value u_{0}Â =Â 0 and accumulating chord length L_{c}Â =Â 0. Let u_{i}Â =Â u_{iâˆ’1}Â +Â Î”u, iÂ >Â 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
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
FigureÂ 6 shows the flowchart of the proposed algorithm application of which the steps are summarized briefly as follows.
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 manâ€“machine interface before machining starts.
The pattern file from a computeraided design system is loaded into control system and then interpreted by the manâ€“machine interface software, which is coded with Visual C++ 6.0. Gcodes 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
The control framework with two rotary buffers is built. Each Gcode 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 Î², accelerations a_{1} and a_{2}, 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 threeaxis numericalcontrol glass grinding equipment, as shown in FigureÂ 7(a). The X, Y and Z axes traveled together to span a 1500Â Ã—Â 2500Â Ã—Â 120 mm 3D space. The experimental parameters were set as: spindle speed S_{p}Â =Â 6000 r/min, normal accelerations of X and Y axes a_{x}Â =Â a_{y}Â =Â 20Â mm/s^{2}, glass thickness T_{g}Â =Â 12 mm, maximum grinding velocity v_{nor}Â =Â 3600 mm/min, wheel radius R_{T}Â =Â 75 mm, normal grinding depth d_{n}Â =Â 0.13 mm, and increasing step length Î”uÂ =Â 0.05.
As shown in FigureÂ 7(b), a piece of glass balustrade with three sharp corners for the escalator is being ground. Three sharp corners are numbered sequentially. FigureÂ 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.
Angle Î² 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 Î², v_{trans}, d_{c}, and R_{T} into Eq. (18) and checking whether the constraint is satisfied, one has t_{12}Â =Â 6 s, and then F(t_{12})Â =Â 8786.3 mm^{2} and (R_{T}Â +Â d)^{2}Â =Â 5643.0 mm^{2}; thereby, constraint Eq. (13) is satisfied.

3.
With Eq. (15), the required acc/dec lengths L_{d}Â =Â L_{a}Â =Â 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 0Â â‰¤Â tÂ â‰¤Â 12.9.
The real acc/dec processes around the corners are explicitly shown in FigureÂ 8 and in accordance with the planning. As shown in FigureÂ 9, the wheel center trajectory is connected by the velocityblending algorithm at each corner.
FigureÂ 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.
FigureÂ 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 Â±0.1Â° fully meets the factory requirements.
5 Conclusions
A toolpath generation method based on a velocityblending 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 cornerrounding 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.
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Authorsâ€™ Contributions
KR carried out the toolpathplanning 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.
Authorsâ€™ Information
Kun Ren, born in 1979, is currently an associate professor at College of Mechanical Engineering and Automation, Zhejiang SciTech 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 SciTech University, China.
Danyan Jiang, born in 1993, is currently a master candidate at College of Mechanical Engineering and Automation, Zhejiang SciTech University, China.
Jun Pan, born in 1974, is currently a professor at College of Mechanical Engineering and Automation, Zhejiang SciTech University, China.
Wenhua Chen, born in 1963, is currently a professor at College of Mechanical Engineering and Automation, Zhejiang SciTech University, China.
Xuxiao Hu, born in 1965, is currently a professor at the College of Mechanical Engineering and Automation, Zhejiang SciTech 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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Ren, K., Pan, Y., Jiang, D. et al. Corner Transition Toolpath Generation Based on VelocityBlending Algorithm for Glass Edge Grinding. Chin. J. Mech. Eng. 32, 87 (2019). https://doi.org/10.1186/s1003301903987
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DOI: https://doi.org/10.1186/s1003301903987
Keywords
 Glass edge grinding
 Toolpath planning
 Velocity blending
 Grinding depth control