Collision detection between toolholder and workpiece on ball nut grinding
György Hegedűs
University of Miskolc, Department of Machine Tools
Hungary
hegedus.gyorgy@unimiskolc.hu

Introduction


This application presents numerical methods for the determination of collision detection of toolholder (quill) and workpiece on ball nut grinding. Beside the collision detection the method is capable of the determination of proper grinding angle with the prescribed safety gap between the toolholder and workpiece. The applied NewtonRaphson and Broyden numerical algorithms were executed and compared to each other.
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Overview of the problem
Gothicarc profile ballscrew motion transforming mechanisms are widely used in machine tools and the demand for highlead ballscrews is increasing due to the highspeed manufacturing. The gothic arc is a symmetrical combined curve of two arcs with equal radius and distance between their centres. These types of ballscrews are manufactured by form grinding, where the grinding tool has corresponding profile [6], ultraprecision ballscrews sometimes lapped after grinding process [4], [5]. In case of long and high lead threaded ball nut the grinding wheel is not tilted at the lead angle of the thread to avoid the collision between the quill and workpiece. Fig. 1 shows the real manufacturing process on conventional thread grinding machine.
Fig. 1. Ball nut thread grinding on conventional machine
Due to these conditions the profile obtained is not gothicarc, because the grinding wheel tends to overcut the thread surface. The problem is well known in gear and worm manufacturing, different methods had been worked out to solve this task [3], [8], [2]. In case of long threaded ball nut the setting of optimum tilt angle is not possible due to the collision of quill and workpiece. This angle parameter has to be determined for the real manufacturing process. In this paper a numerical method will be presented for the determination of grinding wheel tilt angle on cylindrical and conical toolholders.


Collision detection between quill and ball nut


Collision detection between cylindrical bodies is widely used in three dimensional mechanical systems, for example machine tools, robots, different mechanisms. There are different software package for collision detection, for example ICollide, VCollide, Rapid, Solid. In this work collision detection is only determined between cylindricalcylindrical and conicalcylindrical bodies without using of third party developed software.
In case of the ball nut and the quill the determination of collision is equivalent with a minimum distance computation between cylinders or conical and cylindrical surfaces. Detecting of collision between cylindrical rigid bodies were developed using line geometry by Ketchel and Larochelle [7]. Distance computation between cylinders has four different types according to their three dimensional positions in space. Fast and accurate computation method was developed by Vranek [11]. To determine the maximum tilt angle for the grinding the minimum distance determination is required between the tilted quill axis and the edge of the ball nut represented as a circle (see Fig. 2., where is tilt angle, ϕ lead angle – optimal tilt angle – of the ball nut, length of the ball nut, pitch of the ball nut).
Fig. 2. Schematic figure of collision between cylindrical quill and ball nut
Collision detection between cylindrical quill and ball nut
Determination of minimum distance between the quill and the ball nut is equivalent with the computation of the distance between the quill axis and the circular edge of the ball nut.
Fig. 3. Spatial position of toolworkpiece
Applying notations of Fig. 3. the circle equation described by
>


 (2.1) 
where 2[0, 2π], is point of the circle, C is centre of the circle, is diameter of the quill and u and v are unit vectors in the plane containing the circle. The minimum distance between the quill axis and the circular edge is
>


 (2.2) 
where is a point on the quill axis and Q is the projection of on the circle plane. Applying the expressions from [10] a nonlinear equation system can be formulated for the unknown parameters. The equation for the minimum distance between the quill and the edge of the ball nut using the notations of Fig. 3. is written by
>


 (2.3) 
where is a safety gap between the quill and the ball nut, and is the tilt angle from the vk direction vector of the quill axis. Minimizing (2.2) a quartic equation can be formulated [10]
>


 (2.4) 
where
>


 (2.5) 
The roots of the nonlinear equation system from (2.3) and (2.4) are found by root finder algorithm.


Numerical algorithms for nonlinear equation systems


In this work NewtonRaphson and Broyden numerical algorithms [12] are used for the solving of the nonlinear equation system and the results are compared. Initial values are required for the two unknown parameters on both methods for correct solutions.

NewtonRaphson method for nonlinear systems


Let N functional relations to be zeroed, involving variables , i = 1, 2, . . .,n, thus
=1,2,...,n.
x the vector of values and F denotes the vector of functions . The expanded functions in Taylor series in the neighbourhood of x
The matrix of partial derivatives appearing in the above equation is the Jacobian matrix J, where
.
In matrix notation
By neglecting terms of order and higher and by setting F(x + δx) = 0, we obtain a set of linear equations for the corrections δx that move each function closer to zero simultaneously, namely
Jδx = F.
The above matrix equation can be solved by LU decomposition. The corrections are then added to the solution vector,
and the process is iterated to convergence. In general it is a good idea to check the degree to which both functions and variables have converged. Once either reaches machine accuracy, the other won’t change.


Broyden method for nonlinear systems


Newton’s method as showed perviously above is quite powerful, but it still has several disadvantages. One drawback is that the Jacobian matrix is needed. In many problems analytic derivatives are unavailable. If function evaluation is expensive, then the cost of finitedifference determination of the Jacobian can be prohibitive. Just as the quasiNewton methods provide cheap approximations for the Hessian matrix in minimization algorithms, there are quasiNewton methods that provide cheap approximations to the Jacobian for zero finding. These methods are often called secant methods, since they reduce to the secant method in one dimension. The best of these methods still seems to be the first one introduced, Broyden’s method [11]. Let us denote the approximate Jacobian by B. Then the ith quasiNewton step is the solution of
where . The quasiNewton or secant condition is that statisfy
where . This is the generalization of the onedimensional secant approximation to the derivative, δF/δx. However, does not determine uniquely in more than one dimension. The bestperforming algorithm to pin down in practice results from Broyden’s formula. This formula is based on the idea of getting by making the least change to consistent with the secant . Broyden showed that the resulting formula is
Early implementations of Broyden’s method used the ShermanMorrison formula to invert the above equation analytically. Then instead of solving equation by e.g., LU decomposition, one determined
by matrix multiplication in operations. The disadvantage of this method is that it cannot easily be embedded in a globally convergent strategy, for which the gradient of requires B, not ,



Determining of the initial values


Previous analyses pointed out that correct initial guesses are needed for real solutions. In this section the determination of initial guesses described in detail.
Initial values on cylindrical quill and ball nut
Fig. 4. Quill and workpiece position for initial value determination on cylindrical quill
Assuming that the quill and the ball nut axes are in the same plane. The initial tilt angle and the initial quill axis parameter are determined by
where
>


 (4.1) 
the rotation matrix for x axis is
>


 (4.2) 
and the rotated points are
>


 (4.3) 
the direction vector
>


 (4.4) 
the equation for the unknown and parameters is
>


 (4.5) 
applying the Fig. 4. notations. Solving equation (4.5) and simplifying the results the required initial values for the iterative algorithms are
>


 (4.6) 
and
>


 (4.7) 
Initial values on conical quill and ball nut
In the previous section the equation system was formulated for the determination of the tilt angle of the quill, where the quill was cylindrical. In certain cases conical quills are used in grinding process. The determination of the conical quill tilt angle is similar to the cylindrical case, but equation (3) has to be modified according to the taper angle of the quill (see Fig. 5.).
Fig. 5. Quill and workpiece position for initial value determination on conical quill
The modified expression is
 (4.8) 
where is the smaller diameter of the quill, is the grinding wheel width and is the width of additional parts. Solving equation (4.5) similarly the initial values are
>


 (4.9) 
where
>


 (4.10) 
the parameter is
>


 (4.11) 
and the larger diameter of the conical quill is
without additional parts
>


 (4.12) 
and
>


 (4.13) 
and the runout of the tool is
>


 (4.14) 
(as seen on Fig.5.).


Parameters and result on different ball nuts and grinding tools


In this section the parameters of the different ball nuts and tools are collected and showed which have been analysed in this work. Table 1. shows the dimensions of the ball nuts, Table 2. shows the dimensions of grinding tools and Table 3. shows the results of different applied procedures.
Table 1: Dimensions of different ball nuts 







32x25

34

60

32.71

2.68

0.25

5

40x20

43.5

90

41.69

3.77

0.28

7.144

40x30

42

100

39.47

3.38

0.255

6.35

50x30

54

133

51.69

4.22

0.264

8


Table 2: Dimensions of different tools 







32x25

7

26.5

12

12

12



40x20

10

37

12

14

16.5

100

40x30

10

31.5

12

14

16.5

120

50x30

12

42

12

20

20




Table 3: Results of different methods 

Newton method I.

Newton method II.

Broyden method

Maple's fsolve()

Number of iterations


Number of iterations


Number of iterations


Number of iterations


32x25

5


25


78


5


40x20

5


23


71


5


40x30

5


33


101


5


50x30

5


22


101


5





References



[1] Broyden, C.G. (1965), Mathematics of Computation, Vol. 19, pp. 577–593.


[2] Dudás, I. (2004), The Theory and Practice of Worm Gear Drives, pp. 320, ISBN 9781903996614, ButterworthHeinemann


[3] Dudás, L. (2010), New way for the innovation of gear types, Engineering the Future, Rijeka, Dudás, L. (Ed.),pp.111140, ISBN 9789533072104


[4] Guevarra, D. S.; Kyusojin, A.&Isobe, H.& Kaneko, Y. (2001), Development of a new lapping method for high precision ball screw (1st report) – feasibility study of a prototyped lapping tool for automatic lapping process, Precision Engineering (25), pp.6369, ISSN 01416359


[5] Guevarra, D. S.; Kyusojin, A. &Isobe, H. & Kaneko, Y. (2002), Development of a new lapping method for high precision ball screw (2nd report) Design and experimental study of an automatic lapping machine with in–process torque monitoring system, Precision Engineering (26), pp. 389–395,ISSN 01416359


[6] Harada, H.; Kagiwada, T. (2004), Grinding of highlead and gothicarc profile ballnuts with free quillinclination, Precision Engineering (28), pp. 143151, ISSN 01416359


[7] Ketchel, J.;Larochelle, P. (2005), Collision Detection of Cylindrical Rigid Bodies Using Line Geometry, Proceedings of the 2005 ASME International Design Engineering Technical Conferences, pp. 313,ISBN: 0791847446


[8] Litvin, F. L.;Fuentes, A. (2004), Gear Geometry and Applied Theory – Second Edition, pp. 801, ISBN 9780521815178, Cambridge University Press


[9] Mihálykó Cs., Virágh J., (2011), Közelítő és szimbolikus számítások feladatgyűjtemény, Typotex, 2011.


[10] Schneider, P. J.;Eberly, D. H. (2003).Geometric Tools for Computer Graphics,pp. 1056, ISBN 1558605940, Morgan Kaufmann Publishers, San Fransisco


[11] Vranek, D (2002). Fast and accurate circle–circle and circle–line 3D distance computation, Journal of Graphics Tools, Vol. 7(1), pp. 23–32, ISSN 10867651


[12] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, (1992), Numerical Recipes in C, The Art of Scientific Computing, Second Edition, Cambridge University Press


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