Optimal circular arc interpolation for NC tool path ... - Science Direct

pp.

PII: soolo-4485(97)ooo19-5

Computer-Aided Design, Vol. 29, No. 11, 751-760, 1997 6 1997 Elsevier Science Ltd. Ail rights resewed Printed in Great Britain 001 O-4465/97/$1 7.00+0.00

ELSEVIER

Optimal circular arc interpolation for NC tool path generation in curve contour manufacturing Hua Qiu*, Kai Chengt

and Yan Li*

required in NC apparatus, tool paths for a contour curve and its offset profile curves to be separately generated in semi-finish and final finishing. When linear interpolation is used, the finished profile of a work piece is a polygon which is not smooth enough for some applicationsES9. Using circular arc interpolation is a better way because the finished profile is smooth. The biarc method’ is currently widely used in many CAD/CAM systems. In order to make the connection between adjacent circular arcs look more smooth and natural, many optimal biarc algorithms have been developed4*5*7,11.16,20.On the other hand, several algorithms have been proposed in which parameters of circular arcs are sequentially determined by a numerical technique so as to restrict the interpolation error less than the desired value lo317.However, these methods are not minimax approximations15. The number of circular arcs and the amount of NC data are large relevant to a desired interpolation accuracy. The problem becomes in particular obvious for a contour curve with steep change in local radius of curvature 12. Marciniak and F’utz have proposed an algorithm of minimax approximation with circular arcs for a spiral segment of planar curve, whose curvature is of one sign and is monotonic increasing or monotic decreasing as the curve is traversed6. Marciniak and Putz have proved that the minimax approximation generates the lowest number of the circular arcs for a spiral so as to satisfy a desired interpolation accuracy. However, the algorithm is not sufficient for real engineering applications and the problem of how to smoothly connect the circular arcs at the joining of two spiral segments has not been solved. In addition, the golden section technique is used in optimizing computing and it may result in a lower computation efficiency. This paper proposes a simple and convenient algorithm for optimal circular arc interpolation. The algorithm is based on the principle of minimax approximation for accurate manufacturing of planar curve contours. A basic method of how to sequentially determine the parameters of every interpolation circular arc is described for a spiral contour. An improved algorithm is proposed on perfectly solving the problems of the sequential algorithm for practical application. The resulted interpolation error is minimax and the number of circular arcs is fewest with reference to a desired interpolation accuracy under the condition that the circular arcs are tangential to the contour at the start and the end points. In this algorithm, the Newton-Ralphson method is used for resolving non-linear equations so as to obtain a high

Circular arc interpolation is essential in manufacturing of curve contours. However, the problems of how to determine the parameters of the circular arcs and how to minimize the number of arcs according to desired interpolation accuracy have not been completely solved. This paper presents a new algorithm of circular arc interpolation for planar curves. It is based on the principle of minimax approximation. The algorithm has three features which include: (1) the interpolation error is minimax in any spiral segment of the curve, tangent-continuous at the joints of two spiral segments and satisfies the desired accuracy; (2) the parameters of the circular arcs depend on the desired accuracy and the number of arcs is fewest; (3) the circular arcs are compatible with the curve in concavity and convexity, and have almost the same radius of curvature, in other words, the interpolation looks very smooth and natural. Experimental trials show the algorithm is simple and robust for the application to NC contour manufacturing. The trials are carried out on producing the contour of a planar cam and a quadratic NURBS curve. 0 1997 Elsevier Science Ltd Keywords: numerical control, tool path generation, circular arc interpolation

optimal

INTRODUCTION When we accurately finish curve contours, such as a planar cam with machining center, it is necessary to generate a tool path by some interpolation method. At the present time, the function of the linear or the circular arc interpolation in NC apparatus is generally used for this purpose. There are several methods ‘J’J~ developed to reduce the amount of NC data using B-splines or clothoids as interpolation curves. However, there are some limitations on their practical use in industry. For example, special function generators are ‘Department of Mechanical Engineering, Faculty of Engineering, Kyushu Sangyo University, 2-3-l Matsukadai, Higashi-ku, Fukuoka City, Fukuoka 813, Japan. Tel: 81-92-673-5619, Fax:Sl-92-673-5699, E-mail:chiu@ip. Kyusan-u.ac.jp TDepartment of Engineering, Glasgow Caledonian University, Cowcadden Road, Glasgow, UK *Department of Mechanical Engineering, Sichuan Union University, 610065 Chengdu, Sichuan, People’s Republic of China Paper Received: 18 March 1996. Revised: 18 March 1997

751

Optimal circular arc interpolation:

Hua Qiu

et al.

computation efficiency, and the initial values of unknown variables are formulated in a simple and universal form. The interpolation result of using this algorithm for producing a planar cam profile is compared with that of the measured profile of an actual cam manufactured by a machining center. The authors also compare both interpolation results for a quadratic NURBS curve, obtained with the present algorithm and with a biarc algorithm proposed by Meek and Walton’. These results confirm the validity of the algorithm developed by the authors.

OPTIMAL CIRCULAR ARC INTERPOLATION FOR PLANAR CURVE CONTOURS Considering a parametric planar curve (.x(d), ~(4)) with parameter 4. whose radius of curvature is monotonic for $ and is of one sign in [+o. $J, i.e. it is a spiral curve segment. If the spiral lies on the left side of its tangent line relative to the increasing direction of 4, the radius of curvature is defined as positive, otherwise it is defined as negative. Circular arc interpolation error is measured along the normal line of the spiral. If a circular arc is on the right side of the spiral, the interpolation error is positive, otherwise it is negative, relative to the increasing direction of 4 as shown in Figure 3.

Sequential curve

circular arc interpolation

for a spiral

Using a circular arc through three points on a spiral curve to approximate the spiral, the error curve of the minimax approximation can be illustrated in Figure I and it is unique6. Based on this result, the error curve of the minimax interpolation with a desired accuracy E can be illustrated in Figure 2, where several circular arcs are used to interpolate a spiral curve and the connection between two circular arcs is tangent-continuous. In the light of Chebyshev theory on function approximations, Marciniak and Putz have proved that, in a minimax interpolation, the number of the interpolation circular arcs is fewest with reference to the interpolation accuracy E and the obtained parameters of the circular arcs are unique, except for the last one whose shape can be selected6. It should be noted that we have used

=E/2

=-E/2

Figure 2

The error of an optimal interpolation

a different technique from that developed by Marciniak and Putz to deal with the last circular arc in Figure 2 in order to simplify the calculation. In our proposed method, the interpolation error extremum of the last circular arc, if it exists. is equal to that of other circular arcs. This extremum may be less than those of other circular arcs in Marciniak and Putz’s method. But the results on the number of circular arcs and the interpolation error are completely the same as those achieved by Marciniak and Putz. This technique will be described in detail in the section of “determination of the following circular arcs”. In Figure 2, the features of the minimax interpolation error are clearly illustrated.

There is one relative minimum E, and one relative maximum Ei’ in each circular arc segment respectively, except the last segment. Their absolute values are equal to each other. The connection of two adjacent circular arcs is tangentcontinuous. The absolute value of the error &a at start point ($ = $o) of the spiral is equal to E,+. In the part of the curve interpolated by the last circular arc, two or one or no extremum exists according to its length, except the end point of the spiral. If the extremum exists, its absolute value is equal to cif. The error aN at end point (4 = $N) of the spiral is /F/J 5 a+. Hence, if we determine parameters of every circular arcs so that Equation (1) and the above relations hold, the circular arc interpolation is a minimax approximation and the number of the circular arcs is fewest, at the same time, the desired interpolation accuracy E is guaranteed. e, t = -E,

~

&+

0 E-

(b) Figure 1 The error of the minimax approximation: (a) the case of a curve with monotonic increasing radius of curvature. (b) the case of a curve with monotonic decreasing radius of curvature.

752

E = -

2

Determination

(4

with the desired value E.

(1)

of the first circular arc

As shown in Figure 3 (considering i = I), it is assumed that the interpolation errors at points A, and B, on spiral curve are error extremum ~1 and .a:, and their coordinates are ]~.,i(&r), v,i(&i)] and {Xbl(+blh ybl(+bl)}T respectively. At an error extremum point, the tangent lines on both the spiral and the circular arc are parallel to each other. Thus the normal lines of the spiral through the points A, and B, are the ones of the first circular arc simultaneously. That is. the intersection point of the both is the center point 0, of the first circular arc. Equations of the normal lines can be expressed as follows:

Hua Qiu et a/.

Optimal circular arc interpolation:

where

the

symbol

““’

indicates a differential on 4, kz, = -x’(&)/Y’(&) and k,, = -x’(&)~~‘(&). To simplify the expression of equations, ~~~(4~~) is written as x,1* Y,~(#Q) as ~~1, ... m, and so on, in the following. The coordinates (xol, y,i) of the point Oi can be calculated using Equation (3) as follows: ‘%Pb, - kz,x,, +Y~I - Ybl hl

- ka,

kalkb, (Xb, -X,1)

+ bYa,

- kalYb,

kb, - ka, For the desired interpolation accuracy 2 E/2, the distances L,, and &,i from 0, to A, and B,, respectively, can be obtained with Equation (4).

(4) cxol

-Ybd2=p,

-xbd2+(Yol

T E

d

2

(Xo,i+l

where p 1is the radius of the first circular arc. The upper sign is for positive radius of curvature and the lower sign is for negative radius of curvature in compound signs as the same as the following equations of this paper. From Equation (4) La,

+Lbl

=2&

&,-Lb,=

1

>

(5)

-CEj

have been obtained. We assume that the interpolation errors at point Ai+, and point Bi+i on spiral curve are error extremum ei;i and cizi respectively. The coordinates of the points Ai + 1 and Bi + 1 are (Xa,i+ 12Ya,i+ I) and (Xb,i+lT Yb,i+lh and the curve parameters are +a,i+i and $b,i+i respectively. It is obvious that the relation between the points A,+, and Bi+i and the center point Oi+i of the (i + 1)th circular arc is all just the same as that between the points A 1,B 1 and the point 0, relative to the first circul.ar arc. Therefore, Equations (2)-(5) also hold to the (i + 1)th circular arc, only where the suffix “1” is substituted by “i + 1”. That is, all of the radius pi+ 1 and the coordinates x0,;+,, Y~,~+, of Oi+ 1 are the functions on +,i+i and +b,i+i for the (i + 1)the circular arc. On the other hand, we assume that the (i+ 1)th circular arc is tangent to the ith circular arc at point Ci. The straight line connecting the center points Oi+i and Oi of the both circular cucs passes through the point Cf. The distances from Ci to Oi and to Oi+i are equal to the radii pi and pi+, of the both circular arcs, respectively. Thus

(X01

-Yd2=P1

-%>2+(Yol

mim;?E - 2

(6)

If the spiral has a positive radius of curvature then the value of ml is defined as 1, otherwise it is - 1. If the radius of curvature of the spiral is monotonic increasing then the value of m2 is defined as 1, otherwise it is - 1. We can solve Fi and F2 of Equation (7) for the values of $,, and 4 . The position of center point and the radius of the first circular arc can be easily determined from +al and b ,

4bl.

(7)

-

Ll

mlm& -co

+ Lb1

2

+

2

Equation (7) is a non-linear equation and has two unknown variables $~~i and +bi which determine the values of parameters L,, , Lbl, x0,, and yol as shown in Equations (2)-(5). Using the Newton-Ralphon method and choosing suitable initial values of rpal and 4bi, Equation (7) can be solved.

Determination of the following circular arcs In this section, we show how to determine the parameters of the (i+l)tb circular arc after those of the ith circular arc

-Yo,i12=lPi+l

(8)

-PiI

where (x0.i. Y,,i) and (Xo,i+i, Yo,i+i) are the coordinates of 0; and Oi+i. The coordinates (Xi,Yi) of the point Ci can be calculated with the following Equation.

1 xi=xo,i+l

Yi=Yo,i+l

At the starting point (x0, yO) of the spiral, the absolute value of the interpolation error is equal to E/2 as explained above. Thus Equation (6) can be obtained as follows:

-xo,i)2+(Yo,i+l

-

-

-xo,i)

Pi+lCxo,i+l

Pi+1

-Pi

(9)

Pi+l(Yo,i+l Pi+1

-Yo,i) -Pi

I

Therefore, we can solve Fl and F2 of Equation values of ‘#Ja,i+iand 4b,i+l. Fl(~a,i+l,~b,i+l)ALa,i+l

-Lb,i+l

=

7

(10) for the

E=O

F2(4a,i+lT4b,i+l)‘J cYo,i+l-Xo,i)2+b’o,i+l-Yo,i12

-

lpi+] -_Pil=O

(10)

As Equation (7), Equation (10) is a non-linear equation and has only two unknown variables +a,i+i and @b,i+i which determine the values of parameters L,,i+ 1, Lb,i+ 1, X,i+lr method and Yo,i+l and Pi+,. Using the Newton-Ralphson starting at the initial values of $a,i+i and +b,i+i as shown in Appendix B, Equation (10) can easily be solved. The position of center point and the radius of the (i+l)th circular arc cm be easily determined from 4n,i+i and +b,i+i. Using Equation (lo), each circular arc is sequentially determined one after another until ei+ i(dhr) I E/2. ci+i(4N) is the interpolation error of the (i+l)th circular arc at the end point of the spiral. Total number of the circular arcs is i + 1. The calculation of the interpolation error is shown in Appendix A. In addition, 4i is defined as the curve parameter of the correspondent point on the spiral with the tangent point Ci between the ith and the (i+l)th circular arcs. We can solve non-linear Equation (11) for the value of 4i after +a,i+i and +b,i+i has been determined. yi _ yt4ij = _ {xi -x(4i)}x’(4i) Y’(4i)

(11)

753

Optimal circular arc interpolation:

Hua Qiu et al.

(b)

(a)

Circular

arc

Cd)

Cc)

Figure 3 Relation between contour curve and circular arc\ in the optimal circular arc interpolation; (a) the case of a curve with monotonic increasing and positive radius of curvature. (b) the case of a curve with monotonic decreasing and positive radius of curvature. (c) the case of a curve with monotonic decreasing and negative radius of curvature, (d) the case of a curve with monotonic increasing and negative radius of curvature.

Using the Newton-Ralphson method and the initial value of 4; as shown in Appendix B, we can easily solve Equation (11). Because the spiral is continuous and the connection between the ith and the (i + 1)th circular arcs is tangentcontinuous, there is no extremum point of the interpolation error within the range from the second error extremum point of the ith circular arc to the first error extremum point of the (i + 1)th circular arc. So it is guaranteed that the absolute value of the interpolation error at the point C, is not larger than desired interpolation accuracy E/2. From the definition of the minimax interpolation error and Figure 3, it is found out that, at either extremum points of the interpolation error, a circular arc lies on the outside of spiral curve and its bending lever is larger than that of the spiral. At the another one, the relation is just opposite. Therefore, the following relation about the radius pi of the ith circular arc is obtained: P(&,,) < t- Pi < P(&I.,)

754

These aspects limit the application of the above algorithm in industry since they bring about serious damage to the robustness of the algorithm.

(12)

where p(4) is the radius of curvature of a point with curve parameter r$ on the spiral. The circular arcs are compatible with the spiral in concavity and convexity, and each of their radii is close to the middle value of the radius of curvature in the corresponding part of the spiral respectively. Therefore, the interpolation looks very smooth and natural. The following aspects of the above algorithm must be explained. l

For a short spiral curve, Equation (7) does not hold probably since we can not find out two extremum points of interpolation error on the spiral where the interpolation error is equal to -t E/2 respectively. Based on the same reason, Equation (10) does not hold probably for the last circular arc when the corresponding part of the spiral is too short. Although some techniques can be used to solve these problems, we have omitted the details because it hardly fits our purpose which aims at a robust and universal algorithm for engineering application. The circular arcs obtained by the above algorithm are not continuous at the joining of two spiral segments of a continuous contour curve in general.

Experience is needed on how to determine the appropriate initial values of $,,, and 4,,, for solving Equation (7). An inappropriate initial value may result in no solution when the Newton-Ralphson method is used.

Circular arc interpolation several spiral segments

for a curve consisting of

In this section, an improved algorithm is proposed for perfectly resolving the above-mentioned problems. The proposed algorithm has a high computation efficiency. Firstly, an additional condition is defined so as to make the connection of circular arcs between two spiral segments of a continuous contour curve tangent-continuous. This condition is that, for a spiral segment, the first and the last circular arcs are tangential to the spiral at the start and the end points respectively. Therefore, the first and the last (Nth) circular arcs in every spiral segment satisfy the

Optimal circular arc interpolation:

following

equation:

-

(%I -

under the condition as follows:

xb1)2

+ (Yol

-Ybl)*

T

\

-

c/~~,sJ

(&Xv-

A =

(x,,v -Gv)*

;x()

+ 6’0~ - YLZN)*

xbA’)2-6kN-YbN)2?

$0

WIner e, for a spiral with monotonic curvature, there is

increasing

radius

>

of

(14)

and for a spiral with monotonic ture, there is

decreasing

of e 5 E, we develop a new algorithm

Step A. Using two circular arcs to interpolate a spiral segment. That is, setting N equal to 2, using e instead of E and solving simultaneous equations, Equation (13) and the second formulation of Equation (lo), for 4,, , and4b.2 Or +b, 1 and +a,29 respectively, and e. If e 5 E, the computation is stopped, otherwise, going to Step B. Step B. Using three circular arcs to interpolate the spiral. That is, setting N equal to 3, using e instead of E and solving simultaneous equations consisted of Equations (10) and (13) for r/~~,,and 4b,N, or +b,, and 4 respectively, and 4,~ and 4b 2 and e. If e 5 E, the c:gputation is stopped, otherwise, going to Step C. Step C. Sequentially solving Equation (13) and Equation (10) so as to determine each circular arc one after another from two directions, the direction A from 4,, to 4N and the direction B from 4,., to do, respectively. In the first case, the circular arcs are numbered as 1, 2, . . ..j. ... and in the latter as 1, 2, ..., k, .... Step D. When the Jth circular arc in the direction A intersects with or is tangential to the Kth circular arc in the direction B, the required number of circular arcs is decided as N = J+K. For a tangential case, the computation is halted and the resulting parameters are rewritten as follows:

(13) F2(&,,,

Hua Qiu et a/.

radius of curva-

Equations (14) and (15) indicate that the interpolation error at the start point of the spiral is one of two error extrema of the first circular arc, and the interpolation error at the end point is one of two error extrema of the last circular arc. Thus, in order to satisfy Equation (13), the number of interpolation circular arcs is at least 2 for a spiral segment. Introduction of Equation (13) brings about a new problem that we have to make 2N - 2 unknown variables satisfy 2N - 1 constraint equations and thus determine N circular arcs. Here, the unknown variables are 4a,i and +b,i(i = 1,2, ..., N). Both of them have been known from Equation (14) or (15). The constraints are, N equations on the extremum of interpolation error being imposed on every circular arc (Equation (13) and the first formula of Equation (lo)), and N - 1 equations on the continuity between every two adjacent circular arcs (the second ones of Equation (10)). It cannot be implemented generally. For resolving this problem, we take interpolation error value in whole spiral

‘N=J+K e=E 4d

4a,i= 1

(i,j=

4 a,k (i=J+

1, 2, ..., J) K -k+

1, k= 1,2;,

4&j (i,j= 1, 2, . . . . J) +b,i

= 4b,k

(i=J+

K -k+

1, k=1,2;.

(16) For an intersecting case, going to Step E. Step E. using e instead of E and solving the simultaneous equations of Equations (10) and (13) by the Newton-Ralphson method with the following initial values:

N=J+K 4N-40

4’0? = a,1 4’? b,r=

40 + d4a, j - 40)

(17)

(i,j=1,2;..,J)

4N- a(4N - 4a, k ) (i=J+K-k+

1, k= 1,2;..,K)

40+d4b,jm40)

(i,j=l,&...,J)

4N-a(4N-4b,k)

(i=J+K-k+l,

segment, denoted as e, as a new unknown variable instead of the desired interpolation accuracy E in the 2N - 1 constraints, and solve simultaneous equations consisted of these constraints for 4a,ir 4b.i and e. In order to reduce the difficulty in solving the simultaneous equations, which are non-linear, and make the number of circular arcs fewest

k=l,2;..,K)

where the superscript value.

symbol

“ (O)’‘indicates

an initial

In this algorithm, each circular arc satisfies the minimax approximation for a spiral segment, except the first and the last one whose shapes guarantee the tangent-continuous

755

Optimal circular arc interpolation:

Hua Qiu et a/.

connection of the circular arcs at the joint of two adjacent spiral segments. Therefore. in the sense of the fewest number of circular arcs being relative to a desired interpolation accuracy and the ever present tangent-continuous circular arc segments for a continuous contour curve, the circular arc interpolation is optimal. Moreover, introducing Step A and Step B before Step C guarantees that a correct solution can always be obtained for various situations of the desired interpolation accuracy and the length of spiral segment. At the same time, the difficulty to determine initial values of unknown variables for solving Equation (13) in Step C has been overcome, as also shown in Appendix B. In this algorithm, the Newton-Ralphson method is used for solving all non-linear equations. All initial values of unknown variables have been formulated in a simple and universal form as shown in Appendix B and Equation ( 17). From the interpolation results for many types of planar contours, such as various cam profiles and NURBS curves, it has been proved that all these initial values are very robust. For a complex curve contour. we must decide at first the construction of spiral segments and the interval of every spiral on curve parameter $J, and then sequentially carry out the interpolation for every spiral segment. We adopt a technique”. calculating the curvature radius of sample points on the contour curve and comparing the values of the curvature radius at two adjacent points, to break up a contour curve into the spiral segments. In the calculation of the radius of curvature, second-order differentials ~“(4) and ~“(4) can be obtained by numerical differentiation or other numerical techniques with respect to ~‘(4) and v’(4) on 4. On the other hand, if the contour curve itself is continuous, the interpolation curve consisted of circular arcs obtained by this algorithm is also continuous. One of the both is tangent to the another at the start and at the end points. The maximum distance between the both curves along normal direction of the contour curve. that is. the extremum of the interpolation error, is just the same as that between each offset curve of them with the same offset distance. Therefore, the parameters of interpolation circular arcs for an offset profile can be directly obtained from those for the original contour curve through the compensation function of the cutter radius in NC apparatus. Moreover, the interpolation errors for the offset profiles are all just the same as those for the original contour curve.

EXAMPLES

The interpolation results in relation to the desired interpolation accuracy are shown in Tuble I. Figure 5a shows an example of the interpolation error calculated for cam profile and Figure .5b shows an actual test result of the profile using a three-dimensional coordinate measuring machine for a duralumin work piece finished by a machining center. It should be noticed that the conventional sign representation is used for the profile error in these figures. That is, if a circular arc lies outside the cam profile, the error is defined as a positive, otherwise as a negative. The interval of measuring point is about 0.5 mm and the number of measuring points is 241 on the cam profile. From the comparison of Figure 5a with Figure _Yb,it is found that the local undulation caused by the circular arc interpolation coincides well with each other on the both. The interpolation error in Figure 5b is about ? 1.8 pm, approximately same as the calculated one, but a global wave caused by the motion error of the machining center’” is observed too.

OF INTERPOLATION

Profile interpolation

of a planar cam

In this section. the contour manufacturing of a translating cam with a reciprocating roller follower is selected as an example for further proving the proposed algorithm. Pitch curve of the cam is a modified sine curve3, the translating stroke and the maximum displacement of the follower are respectively 120 mm and 40 mm as shown in Figure 4a. The pitch curve is defined by parameter d. which is translating displacement of the cam itself, and it is the offset curve of the cam profile. Thus. it is necessary only to interpolate the pitch curve. The pitch curve consists of four spiral segments, I, II, III, and IV that are defined in four intervals ondasOmmsd5 ISmm. lSmm~d~60mm,60mm 5 d 5 105 mm and 105 mm 5 d 5 120 mm, as illustrated in Figure 4b. Although the pitch curve is symmetrical about its center point, the interpolation was carried out for every spiral segment in turn. as an evaluation for the algorithm. 756

(b)

Figure 4 Cam used in the example: (a) cam profile, (b) radius ot’curvature p(d) of pitch curve.

Interpolation BCzier form

error of a quadratic NURBS curve in

In this section. both of the interpolation results, respectively obtained with the proposed algorithm and a biarc algorithm presented by Meek and Walton7, are compared for a quadratic NURBS curve in the Bkzier form. Table

1

interpolation results of the cam profile

Dewed interpolation accuracy (pm) IO0 40 10 IO 3 2 T

No of circular arc\

Obtained interpolation error ’ (pm)

IO I2 I2 I6 I8 24 28

.I Obtained interpolation error: Intervals

18.14 18.14 18.14 3.44 3.44 0.54 I .20

I &IV/Intervals

142.36 / 14.76 / 14.76 16.78 13.66 /IO.96 I .42

II&Ill.

Optimal circular arc interpolation:

I

(/rm)

Table 2

L__.--l____.J.

-4

I

__..I___..

0

~~~~~~~~~_.L..........~

15

60

;

;

&mm)

results of the quadratic

Obtained interpolation

arcs

By Meek and Walton ” 2 3 4 6 8 12 16

120

(a)

e(d) (w)

Ellipse spiral

L....~..._-J.....;...-...-.--.....__--____--’-

-4

0

60

15

d(mm)

105

r

Hyperbora spiral

Figure 5 Interpolation error of cam profile corresponding to a desired accuracy of 4 pm: (a) computed error, (b) measured error of a finished work piece.

Q(t) =

NURBS curve is presented

as follows7:

(1 - t)2w$o + 2t( 1 - t)w1P1 + t2w2P2 (1 - tywo + 2t( 1 - t)w1 + t2w2

where PO, PI, P2 are control vertices, wo, wI,w2 are positive weights, under the condition of 0 5 t I 1. For a set of the control vertices and weights, (PO. wo) = (20,20,1), (PI, wl) = (150,250,1), (P2, w2) = (320,300,1), the curve is a parabola spiral. For a set of the control vertices and weights, (PO, wo) = (20,20,1), (PI, wl) = (150,250,0.9), (P2, wz) = (320,300,1), the curve is an ellipse spiral. For the third set, (PO, wo) = (20,20,1), (PI wl) = (150, 250, l.l), (P2, w2) = (280, 300, l), the curve is a hyperbola spiral. These three spiral curves are illustrated in Figure 6. The interpolation results are shown in Table 2. For the same desired interpolation accuracy or the same number of circular arcs, the number of circular arcs or the interpolation error obtained by the proposed algorithm is about 314 or about l/4- l/2 of that obtained by Meek and Walton’s algoritbm7, respectively, In other words, the fewer number of the circular arcs is required in the proposed algorithm in order to satisfy a desired interpolation accuracy. Figure 7 shows an example of the interpolation error obtained by the

Pz for hyperbc ala,

Parabola

I ar parabola ellipse spirals

0.002857 1.841

1.1127055 0.2171204 0.0759157 0.0188478 0.0073046 0.0019925 0.000807 1

0.2130 0.02846 0.003577 3.042

1.3056089 0.2528825 0.0881863 0.0218569 0.00854676 0.0023099 0.0009359

0.3297 0.03 168 0.003534

CONCLUSIONS The optimal circular arc interpolation has the following features. l

l

and

0.02439

proposed algorithm, where eight circular arcs are used and “ X" mark indicates the end of a circular arc. In the calculation, a personal computer with a 486 sx CPU and a i487 sx over drive processor was used. The algorithm is programmed with PTO-FORTRAN-77. The differential values of non-linear functions with the Newton-Ralphson method were obtained using a numerical difference technique. The value of each non-linear function could be conver ed at less than 10m9 mm in the first example or less than lo- $ in the second example, after only several times of interactive computation, using initial values presented in Appendix B and Equation (17). The calculation time is less than 1.5 s in all examples.

YS

300 .

1.0770945 0.2085 114 0.0727478 0.0180464 0.0069949 0.0019089 0.0007735

0.2357

2 3 4 6 8 12 16

error(e /2)

By the authors

2.186

2 3 4 6 8 12 16

(b)

The quadratic

NURBS curves

No of circular

Parabora spiral

I 105

Interpolation

Hua Qiu et a/.

proposed in this paper

For every spiral segment of a continuous contour curve, the obtained interpolation error is minimax which satisfies the desired accuracy and the required number of the circular arcs is fewest simultaneously. At the joining point of two spiral segments, the connection of the circular arcs is tangent-continuous. The circular arcs are compatible with the contour curve in concavity and convexity. The radius of every circular arc is close to the middle value of the radius of curvature in the corresponding part of the contour curve. Therefore, the interpolation looks very smooth and natural.

spiral 0.008

Ellipse

spiral

0.004 0 -0.004 -0.008 0

Figure

6

Spiral curves in the example.

0.2

0.4

0.6

0.8

Figure 7 Interpolation error obtained by the proposed eight circular arcs for the parabola spiral curve.

t

1

algorithm

with

757

Optimal circular arc interpolation:

l

l

Hua Qiu et al.

The algorithm is relatively simple and with a high conputation efficiency. Its implementation programming is easy. The initial values used to solve the non-lineal equations are not only simple but also robust for engineering applications. The obtained interpolation data can be directly applied on a NC machine tools for machining a planar curve contour or its offset protiles without any other additional operations.

Therefore. it can be concluded that the propoacd algorithm is very practical and useful for the tool path generation in accurate manufacturing of curve contours. The algorithm has been evaluated and validated by the actual examples ot NC contour manufacturing. In addition. if the algorithm is implemented into CAD system. the profile data can be defined as circular arcs on the design stage. So the data will be more advantageous than the point data for recording profiles.

(a)

APPENDIX A: CALCULATION OF CIRCULAR ARC INTERPOLATION ERROR The center is 0, and the radius is p, l’or the ith circular arc. the length of 00, is R, and the angle of 00, relative to the positive direction of.~ axis is 19,as shown in Figure 8. The normal line through a point A whose curve parameter is 4 on the contour curve intersects with the circular arc at point B. The coordinates of A are {a($),~(~)). The interpolation error .zi($) at the point A is presented as follow\:

09 Figure 8 Calculation of interpolation error in relation to a circular arc; (a) the ca\eof a positive radius of curvature. (b) the case of a negative radius of curvature.

segment. That is, setting N to 2 and solving the simultaneous equations combining Equation (13) and the second formula of Equation (10) for 4h,, and 4(,.?. or $,,, I and 4,,,?, respectively, and e. The initial values are given by the following equations:

>

where the upper sign is for a positive value and the lowe sign is for a negative value of radius curvature, in compound signs.

(Bib)

6’

APPENDIX B: INITIAL VALUES NON-LINEAR EQUATIONS

TO RESOLVE

Using the Newton-Kalphaon method to solve non-lineal equations, it is normally difficult to select a suitable initial value for an unknown variable. A poor initial value may result in no resolution. We suggest the following procedure to determine the initial values of the unknown variables in the proposed algorithm. In Step A. two circular arcs are used to interpolate a spiral

758

(Bla)

101

=E

WI

where the superscript symbol ““)“’ indicates a initial value and the added number ” - a” and “ - b” of the equations are relative to a spiral with a monotonic increasing or a monotonic decreasing radius of curvature, respectively. This is same for other equations in this appendix. The interpolation error obtained on this step is written as r”‘. In Step B, three circular arcs are used to interpolate the spiral. That is. setting N equal to 3 and solving the simultaneous equations combining Equations (I 3) and ( 10) for @Jo,, and $,,.:, or $,., and #J,,,~respectively, and +o,2r $),.? and e.

Optimal circular arc interpolation:

The initial values are given by the following

equations:

The robustness of the above initial values has been confirmed with many interpolation examples for various types of contour curves.

(0)

4bl = (4%- 4om

L I

Hua Qiu et a/.

I$(‘) a2 = 3(&, - do)/8

c$(‘) b2 =5(&r,,

-

I$(‘) 03 =7(&j

- $~~)/8 4ow

2. 3.

3(& - $o)/8

Wb)

f#I(O) a2 = 5( r&v f&(O) b3 =

REFERENCES 1.

4(O) al = (@JN t$‘) b2 =

9o)/8

4.

+0)/s

5.

Bedi, S., Ah, I. and Quan, N., Advanced interpolation technques for NC. machines. Trans. ASME, J. Eng. Znd., 1993, 115, 329-336. Bolton, K.M., Biarc curves. Cornput:Aided Des., 1975, 7(2), 89-92. Chen, F.Y., Mechanics and Design of Cam Mechanisms. Pergamon Press, New York, 1982, pp. 85-91. Kosugi, M. and Teranishi, T., Construction of a curve segment with two circular arcs. Trans. ZEZCE, 1977, 60-D(1 I), 944-95 1. Kosugi, M., Local curve fitting procedures using circular arcs. Trans. ZEZCE, 1977, 60-D(12),

7( $N - &)/8

,(a) zz 0.3eQ’

Marciniak, K. and Putz, B., Approximation of spirals by piecewise curves of fewest circular arc segments. Cornput:Aided Des., 1994.

I.

Meek, D. S. and Walton, D. .I., Approximating quadratic NURBS curves by arc splines. Cornput.-Aided Des., 1993, 25(6), 371-376. Norton, R.L., Effect of manufacturing method on dynamic performance of cams - an experimental study (part 1, eccentric cams). Mech. Mach. Theory, 1988,23(3), 191-199. Norton, R.L., Effect of method on dynamic performance of cams an experimental study (part II, double dwell cams). Mech. Much.

(B4)

In Step C, each circular arc is sequentially determined in two directions, from do to 4N and from 4~ to 40, as same as that explained in the context. The initial values for solving Equation (13) is presented by the following equations:

16(2),

8.

9.

4;’ = 4;’ (from40 to4~) 1

87-90.

Theory,

#$’ = (p$’ (from $N to Cpo)

1023-1030.

6.

1988,23(3),

201-208.

IO. Nishioka, M., Contouring of planar cam (2nd report, contour constituted by circular arcs). Trans. Jap. Sot. Mech. Eng., 1994, 60X(580), 427 I-4275.

4(O) al (from r#~~to 4N) crl = $(3) > 4(O) to do) bl = ~5~~) h3 (from9,

(B5b)

where the superscript symbol “(3)” indicates a result obtained on Step B. The initial values +a,i+I and 4b,i+i for solving Equation (10) can be decided by the following equations:

Parkinson,

Il.

D.B. and Moreton,

Cornput-Aided

D.N., Optimal

Des., 1991, 23.41

biarc - curve fitting.

I-419.

12.

Qiu, H., Yomamoto, J. and Hirakawa, I., A study on the algorithm and the accuracy of the circular interpolation for NC contour manufacturing of involute tooth profiles. Bull. of the Faculty of Engineering

13.

Qiu, H., Ozaki, H., Sato, E., Suzuki, T.. Oho, A. and Ariura, Y., An analysis using offset curves for profiles, manufacturing and errors of plane cams. JSME Inter. J., 1993, 36-C(1), 110-I 18. Qiu, H., Ohoka, M., Yamamoto, J. and Hirakawa, I., Manufacturing accuracy of contours with a machining center (on accuracy of circular interpolation). Bull. of the Faculty of Engineering Kyushu Sangyo

Kyushu Sangyo

14.

University,

15.

16.

University,

1993,30,45-5

I.

1994, 31, 27-34.

Ralston, A. and Rabinowitz, P., A First Course in Numerical Analysis. McGraw-Hill Book Company (Japanese translation by Toda, H. and Ono, H., Brain Book Pub. Co. Ltd., Vol. 2, 1986, pp. 295-298). Schonherr, J., Smooth biarc curves. Comput.-Aided Des., 1993,25(6). 365-370.

17.

,$‘O! a,r+l =r#,&I c#&‘! b,r+l

=C#I

+2A 18.

(from40 to4~)

a,~

+A (B6b)

c#J”! a,r+l

(from t$‘! b,r+

h

to 40)

I =4jbi-24,

20.

where +a,i and ~b,i are the parameters for the ith circular arc and their values have been determined earlier, and A is: (B7) The initial follows: +!O)=

I

$