DETC SLIDER CRANKS AS COMPATIBILITY LINKAGES FOR PARAMETERIZING CENTER POINT CURVES

Transcription

1 Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information Proceedings in Engineering of IDETC/CIE Conference 2009 ASME 2009 International Design Engineering Technical Conferences & Computers IDETC/CIE and 2009 August 30 - September Information 2, 2009, insan Engineering Diego, California, Conference USA August 30 - September 2, 2009, San Diego, California, USA DETC SLIDER CRANKS AS COMPATIBILITY LINKAGES FOR PARAMETERIZING CENTER POINT CURVES David H. Myszka University of Dayton Dayton, Ohio dmyszka@udayton.edu Andrew P. Murray University of Dayton Dayton, Ohio murray@notes.udayton.edu ABSTRACT In synthesizing a planar 4R linkage that can achieve four positions, the fixed pivots are constrained to lie on a center-point curve. It is widely known that the curve can be parameterized by a 4R compatibility linkage. In this paper, a slider crank is presented as a suitable compatibility linkage to generate the centerpoint curve. Further, the center-point curve can be parametrized by the crank angle of a slider crank linkage. It is observed that the center-point curve is dependent on the classification of the slider crank. Lastly, a direct method to calculate the focus of the center-point curve is revealed. 1 INTRODUCTION A center-point curve is the locus of feasible fixed pivot locations for a planar RR dyad that will guide the coupler through four finitely separated positions. The theory, originally formulated by Burmester [1], is described in numerous classic sources [2 4] and continues to be an essential element of more recent machine theory textbooks [5 7]. Sandor and Erdman [8] present an algebraic formulation for the center-point curve that is similar in form to a closure equation for a four-bar linkage. Further, they introduce the concept of a compatibility linkage as a conceptual linkage whose solution for various crank orientations will generate points on the center-point curve. They show that the curve can be parameterized based on the conceptual crank angle of the compatibility linkage. Chase et al. [9] observe that the shape of the centerpoint curve is dependent on the motion type of the compatibility linkage. McCarthy [10, 11] demonstrates that the opposite pole quadrilateral serves as a tangible compatibility linkage. In addition, he shows that crank angle of a compatibility linkage can be used to parameterize the center-point curve. Murray and Mc- Carthy [12] state that there is a two dimensional set of quadrilaterals that can generate a given center-point curve. This paper explores the implications as a vertex of the compatibility linkage is selected at infinity. The resulting linkage becomes a slider-crank. Sections 2 and 3 review existing results which will be used to develop a procedure that generates a slider crank that serves as a compatibility linkage. The center-point curve is then parameterized by the crank angle of the slider-crank. The final section illustrates examples with different center-point curve types and the generating slider crank compatibility linkage. 2 COMPATIBILITY LINKAGE In dealing with precision point synthesis, the location of the i th design position in the fixed frame is specified with a rotation angle θ i and a translation vector d i = (d ix,d iy ) T. A rotation matrix is calculated as [ ] cos θi sin θ A i = i. (1) sin θ i cos θ i Any displacement of a rigid body from position j to position k, and vice versa, can be accomplished by a pure rotation about the 1 Copyright c 2009 by ASME

2 j 3 i 4 P ij Figure 1. Displacement pole. 4 x displacement pole P jk = P kj as shown in Fig. 1. The displacement pole is calculated as P jk = A j [A j A k ](d k d j ) + d j (2) = A k [A j A k ](d k d j ) + d k. Given four specified positions, six displacement poles exist (,3,4,3,4 and 4 ). The center-point curve passes through all displacement poles. An opposite pole quadrilateral is defined by four poles, such that the poles along the diagonal do not share an index. For the four position case, three different opposite pole quadrilaterals can be formed with vertices 3 4 4, 4 4 3, and An opposite pole quadrilateral can serve as a compatibility linkage [10]. The compatibility linkage can be used to graphically construct a center-point curve. As the crank of the compatibility linkage is displaced an arbitrary amount, a feasible center point can be located by determining the pole of the coupler displacement as shown in Fig. 2. The center-point curve is generated by tracking the coupler displacement pole between the original compatibility linkage and numerous assembly configurations. The intersection of lines along opposite sides of an opposite pole quadrilateral are designated as Q ij -points, where the subscript designates the uncommon positions defining the two poles. For example, the intersection of 4 and 3 4 is designated as Q 24. Along with the six poles, the center-point curve passes through the six Q ij points as shown in Fig. 3. Cubic curves have a principal focus, F, defined as the intersection of its tangents at the isotropic points. Bottema and Roth [4] state that the focus of the center-point curve is on the curve itself, which is not a general requirement of circular cubic curves. They present a formulation for F using specialized coordinates. Beyer [3] describes a geometric construction to locate the focus. Three circumscribed circles J, J, and J are constructed from 3 4 Q 12, 3 4 Q 12 and 3 3 Q 34, respectively. The intersection of J, J, and J locates F. The construction is illustrated in Fig. 4 Figure 2. Compatibility linkage used to generate a feasible center point. 3 ALGEBRAIC FORM OF CENTER-POINT CURVES Burmester showed that every point on a center-point curve must view opposite sides of a compatibility linkage in equal, or supplementary, angles as shown in Fig. 5. Vertices of the compatibility linkage are designated as points P i = (p i,q i ) T, i = 1,2,3,4. The condition that a point x on the center-point curve views sides and in the angle is = ( x),( x) ( x) ( x) = ( x),( x) ( x) ( x). (3) The vertical bars denote the 2 2 determinant of the matrix formed by the coordinates. Expanding Eq. 3 produces ( x + y)(x 2 + y 2 ) +C 3 x 2 +C 4 y 2 +C 5 xy (4) +C 6 x +C 7 y +C 8 = 0. The coefficients C i are in terms of the vertices P i : = q 1 q 2 q 3 + q 4, (5) = p 1 + p 2 + p 3 p 4, (6) C 3 = p 2 q 1 (p 3 + p 4 )(q 1 q 2 ) + p 1 q 2 + p 4 q 3 2 Copyright c 2009 by ASME

3 3 x Q 23 Q 13 3 Q 12 Q Q 24 Figure 5. Feasible center-point views opposite sides in equal, or supplementary, angles. 4 (p 1 + p 2 )p 4 q 3 p 1 p 2 (q 3 q 4 ) q 1 q 2 (q 3 q 4 ) + (p 1 + p 2 )p 3 q 4 + (q 1 q 2 )q 3 q 4, (10) C 7 = (p 2 p 1 )p 3 p 4 p 1 p 2 (p 4 p 3 ) (p 4 p 3 )q 1 q 2 Figure 3. Q 34 Q Points are on the center-point curve. p 4 (q 1 + q 2 )q 3 + p 3 (q 1 + q 2 )q 4 + (p 2 p 1 )q 3 q 4 +p 2 q 1 (q 3 + q 4 ) p 1 q 2 (q 3 + q 4 ), (11) C 8 = p 2 p 3 p 4 q 1 + p 1 p 3 p 4 q 2 + p 1 p 2 p 4 q 3 + p 4 q 1 q 2 q 3 p 1 p 2 p 3 q 4 p 3 q 1 q 2 q 4 p 2 q 1 q 3 q 4 + p 1 q 2 q 3 q 4. (12) Q 12 J 3 J F Since the center-point equation is homogeneous, any set of coefficients (K C i ), where i = 1,2,...,8, and K is constant, defines the same curve. Murray and McCarthy [12] state that any four vertices that produce a set of coefficients (K C i ) consistent with Eq. 4 serves as a compatibility linkage. Using the opposite pole quadrilateral as the compatibility linkage, =, = 4, = 3 and = 4. Equations 5 through 12 can be used to determine values for the coefficients of the center-point curve. As the center-point curve approaches infinity, the highest order terms in Eq. 4 dominate. The equation of the asymptote is x + y = 0, (13) Q 34 which has a slope J m = y/x = /. (14) Figure 4. Graphical construction of the principal focus. +(p 1 + p 2 )(q 3 q 4 ) p 3 q 4, (7) C 4 = p 2 q 1 + p 1 q 2 + (p 4 p 3 )(q 1 + q 2 ) + p 4 q 3 p 3 q 4 (p 2 p 1 )(q 3 + q 4 ), (8) C 5 = 2(p 1 p 4 p 2 p 3 q 1 q 4 + q 2 q 3 ), (9) C 6 = p 2 (p 3 + p 4 )q 1 + p 3 p 4 (q 1 q 2 ) p 1 (p 3 + p 4 )q 2 4 ALTERNATIVE COMPATIBILITY LINKAGES Given a center-point curve, Murray and McCarthy [12] outline a method to determine and for any and selected on the curve. Using that procedure it is observed that only the opposite vertices are dependent on each other. That is, selecting a new, but leaving unchanged, corresponds with a change to only. Likewise, selecting a new, but leaving unchanged, corresponds with a change to only. 3 Copyright c 2009 by ASME

4 Equations 5 through 12 can be written as a linear combination of a single vertex P j x f i = R i p j + S i q j + T i K C i = 0, (15) where i = 1,2,...,8 and R i,s i,t i = f (p k,q k ), k j. Starting with the opposite pole quadrilateral, an alternative compatibility vertex A can be selected as any point on the center-point curve and the corresponding vertex can be readily found. Values for p 1A,q 1A, p 2,q 2, p 3 and q 3 can be substituted into any three Eqs. 5 through 12. By factoring p 4A and q 4A the coefficients R i, S i, and T i of Eq. 15 are determined. Written in matrix form, R i 1 S i1 C i1 p 4A R i2 S i2 C i2 q 4A = T i 1 T i2 (16) R i3 S i3 C i3 K T i3 A is used to readily solve for the corresponding vertex. In the same manner, this procedure applies to selecting an alternative A, on the center-point curve, and solving for the corresponding A. As an example, design positions are given as d 1 = (5, 10) T θ 1 = 200 and d 2 = (0,0) T, θ 2 = 100, d 3 = (5,0) T, θ 3 = 50 and d 4 = (10, 5) T θ 4 = 100. Poles comprising a compatibility linkage are calculated as = = ( , ) T, 3 = = (2.5000,0.6699) T, 4 = = ( ,2.8613) T, and 4 = = (9.5977, ) T. Moving to an arbitrary point on the curve A = ( , ) T, Eq. 16 is used with i 1 = 3, i 2 = 5, and i 3 = 8 to determine the corresponding = ( , ) T. The two remaining vertices and remain unaffected. The resulting alternative compatibility linkage is shown in Fig SLIDER-CRANK COMPATIBILITY LINKAGES By selecting an alternative vertex of the compatibility linkage at infinity, along the asymptote of the center-point curve, the compatibility linkage becomes a slider crank mechanism as shown in Fig. 7. If vertex A is moved to infinity and is considered the fixed pivot, vertex becomes a revolute joint attached to a prismatic, where the line of slide is perpendicular to the asymptote of the center-point curve. Substituting the point (p 1A,q 1A ) into Eq. 14 and rearranging gives n = 1/m = p 1A /q 1A. (17) As p 1A and q 1A become large, so do the coefficients in Eqs. 5 through 12. Therefore the center-point equation will be normalized to deal with the numerical issues. If 0, Eq. 4 is divided Figure 6. by throughout, producing Observing that Compatibility linkage with alternate vertices. (x + y)(x 2 + y 2 ) + C 3 x 2 + C 4 y 2 + C 5 xy (18) = p 1A + p 2 + p 3 p 4A q 1A q 2 q 3 + q 4A + C 6 x + C 7 y + C 8 = 0. ( 1/q1A 1/q 1A ). (19) When q 1A, Eq. 19 states / = p 1A /q 1A = n. Repeating this process on Eqs. 7 through 12 produces a set of C i / s consistent with Eq. 18. Specifically, using Eqs. 7 and 9, p 2 p 3 p 4A + (p 1A /q 1A )q 2 +(p 1A /q 1A )q 3 (p 1A /q 1A )q 4A = C 3 /, (20) Rewriting Eqs. 20 and 21 gives 2(p 1A /q 1A )p 4A 2q 4A = C 5 /. (21) [ ]{ } 1 ( / ) p4a 2( / ) 2 q 4A [ ] p2 + p = 3 + ( / )q 2 + ( / )q 3 +C 3 / C 5 / (22) 4 Copyright c 2009 by ASME

5 asymptote - = 3 θ 4o r 1 = 4 θ 2 r 4 4 r 3 r 2 Figure 8. Slider-crank compatibility linkage from example 1. Figure 7. θ 3o θ 3 Slider-crank compatibility linkage. which can be solved for the corresponding vertex associated with A at infinity. It is observed that this vertex coincides with the focus of the center-point curve. Thus, Eq. 22 becomes a direct calculation for the focus of a center-point curve to complement the construction shown in Fig.4. Up to this point, the fixed pivot of the slider-crank compatibility linkage is = 4. An alternative A can be placed anywhere on the curve. However, Eq. 15 cannot be used with A at infinity. Using a process similar to generating Eq. 22, Eqs. 8 and 12 can divided by q 1A and rewritten factoring p 3A, and q 3A ), giving ( 1)p 3A ( / )q 3A +[ C 4 / p 2 p 4 ( / )(q 2 + q 4 )] = 0, (23) [p 2A p 4A + ( / )p 4A q 2A ( / )p 2A q 4A + q 2A q 4A ] p 3A +[( / )p 2A p 4A p 4A q 2A + p 2A q 4A + ( / )q 2A q 4A ]q 3A +(C 8 / ) = 0, (24) which can be used to determine the corresponding A. Thus, there is a one-dimensional set of slider-crank linkages that define a center-point curve. Additionally, kinematic inversion allows A to be considered the fixed pivot and A to be the revolute attached to the prismatic. Both forms of the slider-crank will generate identical center-point curves. As an example, design positions are given as d 1 = (0,0) T, θ 1 = 0, d 2 = (1,0) T, θ 2 = 5, d 3 = (2, 1) T θ 3 = 10 and d 4 = (1, 2) T, θ 4 = 20. Poles comprising the opposite pole quadrilateral are calculated as = = (0.5000, ) T, 3 = = ( , ) T, 4 = = (7.2150, ) T, and 4 = = (6.1713,1.8356) T. The first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are = and = The slope of the asymptote, calculated from Eq. 14, is m = / = Moving to infinity, Eq. 22 is used to determine the corresponding = ( ,0.8037) T. The two remaining vertices can be held as is, and the line of slide for will be perpendicular to the asymptote, having a slope n = / = The resulting compatibility linkage is given in Fig. 8 6 PARAMETERIZING THE CENTER-POINT CURVE WITH THE SLIDER-CRANK LINKAGE The link lengths of the compatibility slider crank linkage as shown in Fig. 7 are r 2 =, (25) r 3 =. (26) The angle for the original configuration of link 3 and the angle of the slide direction of link 4 are ( θ 3o = tan 1 q3 q 4 p 3 p 4 θ 4o = tan 1 ( C2 ), (27) ). (28) 5 Copyright c 2009 by ASME

6 In practice, a four-quadrant arctangent function should be used to insure the proper angle. The offset distance is calculated as [ ( ) ] r 1 = sin tan 1 q3 q 2 θ 4o. (29) p 3 p 2 The loop closure equations for a slider-crank linkage can be reduced to a well known equation for the coupler angle [5 7] as ( ) θ 3 = sin 1 r1 r 2 sinθ 2. (30) r 3 The two angles resulting from the inverse sine function pertain to the two assembly circuits of the slider crank linkage. Both configurations must be used to generate the center-point curve, and complex results due to the arcsine function are ignored. The displacement pole of the coupler from its initial position to an arbitrary position defined by θ 2 is Figure 9. linkage. Center-point curve generated from a slider-crank compatibility a single circuit (unicursal). The condition [11] that a slider-crank will have a link that fully rotates is where C(θ 2 ) = A θ [A θ A o ]( d θ ) + d θ, (31) [ ] cos(θ2 + θ d θ = + r 4o ) 2, (32) sin(θ 2 + θ 4o ) r 3 r 2 > r 1 (33) A slider crank that produces an equality of Eq. 33 is considered a transition linkage by Murray [13]. and A o and A θ are determined by is substituting θ 3o and (θ 3 + θ 4o ) into Eq. 2, respectively. Once all substitutions are made, Eq. 31 defines a point on the center-point curve for every value of the crank angle θ 2. Thus, the center-point curve is parameterized based on the crank angle of a slider crank linkage. The actual drivability and singularity conditions of the slider-crank does not affect this parameterization. Using complex vector notation, Eq. 31 can be rewritten into a form similar to McCarthy s 4R compatibility parameterization [10]. A benefit of this formulation is that a center-point curve can be classified by linkage type of the slider-crank that generates it, as will be discussed in the following section. Parameterization by crank angle was applied to the example presented in Fig. 8 with an anglular increment of 5. The resulting center-point curve is shown in Fig CURVE TYPES AND THE GENERATING SLIDER- CRANK Consistent with Chase, et al. [9], if a link in the generating linkage is able to fully rotate, the center-point curve will have a two circuits (bicursal). Conversely, if the no link in the generating linkage is able to fully rotate, the center-point curve will have 7.1 Example 1: Unicursal, Center-point Curve For the example presented a the end of Sec. 5, r 1 = , r 2 = , and r 3 = With these values, Eq. 33 is not satisfied, and no link in the generating slider-crank is able to fully rotate resulting in a unicursal center-point curve (Fig. 8). 7.2 Example 2: Bicursal, Center-point Curve Design positions are given as d 1 = ( 2,0) T, θ 1 = 260, d 2 = ( 2,2) T, θ 2 = 120, d 3 = (2,4) T θ 3 = 60 and d 4 = (4,3) T θ 4 = 0. Poles comprising the opposite pole quadrilateral are calculated as = = ( ,1.0000) T, 3 = = (1.7321, ) T, 4 = = (2.1340,1.7679) T, and 4 = = ( ,4.0173) T. The first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are = and = Moving A to infinity, Eq. 22 is used to determine the corresponding = (8.0768,1.5319) T. The two remaining vertices can be held as is, and the line of slide for will be perpendicular to the asymptote, having a slope n = / = The resulting compatibility linkage has r 1 = , r 2 = , and r 3 = and is shown in Fig. 10a. With these values, Eq. 33 is satisfied indicating that a link in the generating slider-crank is able to fully rotate resulting in a bicursal center-point curve 6 Copyright c 2009 by ASME

7 = 4 4 = 3 = 3 4 = 4 (a) Figure 11. Slider-crank compatibility linkage from example 3. A A slide for will be perpendicular to the asymptote, having a slope n = / = The resulting compatibility linkage has r 1 = , r 2 = , and r 3 = and is shown in Fig. 11. With these values, Eq. 33 becomes an equality producing a transition compatibility slider-crank where the links are not aligned, which generates a double-point, center-point curve. (b) Figure 10. Slider-crank compatibility linkage from example 2. Selecting A = ( ,2.5566) T, Eqs. 23 and 24 can be used to determine the corresponding A = (1.2984,0.6009) T. With these vertices, an in-line slider-crank generating linkage is created as shown in Fig. 10b. This is the most basic type of generating linkage. Note that the in-line slider-crank is not possible for unicursal center-point curve forms, as r 1 = 0 in Eq. 33 will always have a rotating link. 7.3 Example 3: Double-point, Center-point Curve Design positions are given as d 1 = (2, 1) T, θ 1 = 120, d 2 = ( 1, ) T, θ 2 = 75, d 3 = (0,0) T θ 3 = 45 and d 4 = (1,1) T θ 4 = 45. Poles comprising the opposite pole quadrilateral are calculated as = = (0.4895, ) T, 3 = = ( ,1.4457) T, 4 = = (0.0000,1.0000) T, and 4 = = (2.8032,0.6516) T. The first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are = and = Moving A to infinity, Eq. 22 is used to determine the corresponding = (2.0685,1.1399) T. The two remaining vertices can be held as is, and the line of 7.4 Example 4: Circle-degenerate, Center-point Curve Design positions are given as d 1 = (0,0) T, θ 1 = 45, d 2 = (1,1) T, θ 2 = 45, d 3 = (2, 1) T θ 3 = 120 and d 4 = (1.2412, ) T θ 4 = 75. Poles comprising the opposite pole quadrilateral are calculated as = = (0.0000,1.0000) T, 3 = = (2.8032,0.6516) T, 4 = = (1.7412, ) T, and 4 = = ( , ) T. The first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are = and = Moving A to infinity, Eq. 22 is used to determine the corresponding = (4.4507,1.6188) T. The two remaining vertices can be held as is, and the line of slide for will be perpendicular to the asymptote, having a slope n = / = The resulting compatibility linkage has r 1 = , r 2 = , and r 3 = and is shown in Fig. 12. With these values, Eq. 33 becomes an equality producing a transition compatibility slider-crank where the links are collinear, which generates a circle-degenerate, center-point curve. Notice in Fig. 12 that is located at the center of the circle. Beyer [3] identifies that the focus of a circle-degenerate center-point curve is at the center of the circle. 7 Copyright c 2009 by ASME

8 n 1 = 4 4 = 3 = 4 n 1 = 3 Figure 12. Slider-crank compatibility linkage from example 4. Figure 14. RPRP compatibility linkage from example Example 5: Hyperbola-degenerate, Center-point Curve Design positions are given as d 1 = (0,0) T, θ 1 = 100, d 2 = (5,0) T, θ 2 = 50, d 3 = (10, 5) T θ 3 = 100, d 4 = ( , ) T, and θ 4 = 113. Poles comprising the opposite pole quadrilateral are calculated as = P = (2.5000,0.6699) T, 3 = = ( ,2.8613) T, 4 = = ( , ), and 4 = = (4.4624, ) T. These poles form a parallelogram and the first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are = = 0. Therefore, the procedure using Eq. 22 is not applicable. Burmester [1] originally recognized that as the opposite pole quadrilateral is arranged as a parallelogram, the center-point curve will degenerate into an equilateral hyperbola, as shown in Fig. 13, and a circle at infinity. WithPC 3A 1 = = 0, Eqs. 5 and 6 dictate P 4A that as is moved to an alternate point A, towards F q 4A = q 2 + q 3 q 1A, (34) A 1 m χ A ζ n 1 A ζ Figure 13. Alternate compatibility linkage from example 5. p 4A = p 2 + p 3 p 1A. (35) χ The asymptote slope n in Eq. 36 also represents the line of slide for the prismatic joints of the compatibility linkage of Fig. 14. For this hyperbola example, n = The center-point curve is generated by tracking the coupler displacement pole as either prismatic joint is actuated. The assembly circuit shown bold red in Fig. 14 will generate the circle at infinity. The hyperbola is generated by the alternate assembly circuit, shown in a muted red in Fig. 14. A Therefore, moving A to infinity, will also move to negative infinity. With two vertices at infinity, and the resulting compatibility linkage is an RPRP, shown as bold red in Fig. 14. The two remaining vertices are unaffected. For an equilateral hyperbola C 4 = C 3 and the slopes of the two asymptotes (Salmon [14]) are n,m = C 5 ± C C2 3 (36) 2C 3 8 CONCLUSIONS towards F This paper explores the implications as a vertex of the compatibility linkage is selected at infinity. The resulting linkage is a slider-crank. A procedure is outlined to generate slider cranks that serves as a compatibility linkages. Further, a parameterization of the center-point curve by the crank angle of the slidercrank is given. REFERENCES [1] Burmester, L., 1886, Lehrbuch der Kinematic, Verlag Von Arthur Felix, Leipzig, Germany. [2] Hall, A. S., 1961, Kinematics and Linkage Design, Prentice-Hall, Englewood Cliffs, New Jersey, pp [3] Beyer, R., 1963, Kinematic Synthesis of Mechanisms, translated by H. Kuenzel, McGraw-Hill, New York, chapter IV. 8 Copyright c 2009 by ASME

9 [4] Bottema, O., Roth, B., 1979, Theoretical Kinematics, North-Holland Publishing Company, New York. [5] Erdman, A., Sandor, G., Kota, S., 2001, Mechanism Design: Analysis and Synthesis, Vol. 1, 4/e, Prentice-Hall, Englewood Cliffs, New Jersey. [6] Norton, R., 2008, Kinematics and Dynamics of Machinery, 4/e., McGraw Hill Book Company, New York. [7] Waldron, K., Kinzel, G., 2004, Kinematics, Dynamics and Design of Machinery, 2/e., John Wiley & Sons Inc., New Jersey. [8] Sandor, G. N., Erdman, A. G., 1984, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2, Prentice-Hall, Englewood Cliffs, New Jersey, pp [9] Chase, T. R., Erdman, A. G., Riley, D. R., 1985, Improved Center-point Curve Generation Techniques for Four-Precision Position synthesis Using the Complex Number Approach, Journal of Mechanical Design, vol. 107, No. 3, pp [10] McCarthy, J. M., 1993, The Opposite Pole Quadrilateral as a Compatibility Linkage for Parameterizing the Centerpoint Curve, Journal of Mechanical Design, vol. 115, No. 2, pp [11] McCarthy, J. M., 2000, Geometric Design of Linkages, Springer-Verlag, New York, p. 35. [12] Murray, A., McCarthy, J. M., 1997, Center-point Curves Through Six Arbitrary Points, Journal of Mechanical Design, vol. 119, No. 1, pp [13] Murray, A., Turner, M., Martin, D., 2008, Synthesizing Single DOF Linkages Via Transition Linkage Identification, Journal of Mechanical Design, vol. 130, No. 2, paper [14] Salmon, G., 1954, A Treatise on Conic Sections, 6th ed., Chelsea Pub. Co., New York, p Copyright c 2009 by ASME