Slider-Cranks as Compatibility Linkages for Parametrizing Center-Point Curves

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1 David H. Myszka Andrew P. Murray University of Dayton, Dayton, OH Slider-Cranks as Compatibility Linkages for Parametrizing Center-Point Curves 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 parametrized by a 4R compatibility linkage. In this paper, a slider-crank is presented as a suitable compatibility linkage to generate the center-point curve. Furthermore, 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. DOI: / 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 a fundamental element of more recent machine theory textbooks 5 7. Sandor and Erdman 8 presented an algebraic formulation for the center-point curve that is similar in form to a closure equation for a four-bar linkage. Furthermore, they introduced 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 showed that the curve can be parametrized based on the conceptual crank angle of the compatibility linkage. Chase et al. 9 observed that the shape of the centerpoint curve is dependent on the motion type of the compatibility linkage. McCarthy 10 used Sandor and Erdman s formulation and demonstrated that the opposite pole quadrilateral serves as a tangible compatibility linkage. In addition, he showed that crank angle of a compatibility linkage can be used to parametrize the center-point curve. Murray and McCarthy 11 stated 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 the existing results that will be used to develop a procedure to generate a slider-crank that serves as a compatibility linkage. The center-point curve is then parametrized 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. A benefit of a slider-crank compatibility linkage is that the closure equations are quadratic in the slider location and easily manipulated, whereas the equations for a four-bar may be manipulated into a quadratic, but are transcendental. Additionally a slidercrank compatibility linkage presents an alternative graphical method to generate a center-point curve by simply moving a slider along a straight line Contributed by the Mechanisms and Robotics Committee of ASME for publicationinthejournal OF MECHANISMS AND ROBOTICS. Manuscript received April 30, 2009; final manuscript received September 9, 2009; published online April 19, Assoc. Editor: Sundar Krishnamurty. 2 Compatibility Linkage In dealing with precision point synthesis, the location of the ith design position in the fixed frame is specified with a rotation angle i and a translation vector d i = d ix,d iy T. A rotation matrix is calculated as A i = cos i sin i sin i cos i 1 Any displacement of a rigid body from position j to position k, and vice versa, can be accomplished by a pure rotation about the 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 = A k A j A k d k d j + d k 2 Given four specified positions, six displacement poles exist P 12, P 13, P 14, P 23, P 24, and P 34. 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 P 12 P 23 P 34 P 14, P 12 P 24 P 34 P 13, and P 13 P 23 P 24 P 14. 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 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 intersections of lines along the opposite sides of an opposite pole quadrilateral are designated as Q ij -points, where the subscript designates the unshared subscripts of the two poles. For example, the intersection of P 12 P 14 and P 23 P 34 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 stated that the focus of the center-point curve is on the curve itself, which is not a general requirement of circular cubic curves. They presented a formulation for F using specialized coordinates. Beyer 3 described a geometric construction to locate the focus. Three circumscribed circles J, J, and J are constructed from P 23 P 24 Q 12, P 13 P 14 Q 12, and P 13 P 23 Q 34, respectively. The intersection of J, J, and J locates F. The construction is illustrated in Fig. 4. Journal of Mechanisms and Robotics Copyright 2010 by ASME MAY 2010, Vol. 2 /

2 Fig. 1 The displacement pole is the center of pure rotation from position j to position k 3 Algebraic Form of Center-Point Curves Burmester 1 showed that every point on a center-point curve must view the 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 P 1 P 2 and P 3 P 4 in the angle is tan = P 1 x, P 2 x P 1 x P 2 x = P 3 x, P 4 x 3 P 3 x P 4 x The vertical bars denote the 2 2 determinant of the matrix formed by the coordinates of the vectors. This is analogous to a two-dimensional cross product. Expanding Eq. 3 produces x + C 2 y x 2 + y 2 + C 3 x 2 + C 4 y 2 + C 5 xy + C 6 x + C 7 y + C 8 =0 4 The coefficients C i are in terms of the vertices P i : = q 1 q 2 q 3 + q 4 5 Fig. 3 Like the poles, Q points are on the center-point curve C 2 = 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 + p 1 + p 2 q 3 q 4 p 3 q 4 7 Fig. 2 Compatibility linkage can be used to generate a feasible center point Fig. 4 The principal focus can be located through a graphical construction / Vol. 2, MAY 2010 Transactions of the ASME

3 Fig. 5 A feasible center-point views opposite sides in equal, or supplementary, angles 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 C 5 =2 p 1 p 4 p 2 p 3 q 1 q 4 + q 2 q 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 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 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 Since the center-point equation is homogeneous, any set of coefficients KC i, where i=1,2,...,8, and K is constant, defines the same curve. Murray and McCarthy 11 stated that any four vertices that produce a set of coefficients KC i consistent with Eq. 4 serve as a compatibility linkage. Using the opposite pole quadrilateral as the compatibility linkage, P 1 =P 12, P 2 =P 14, P 3 =P 23, and P 4 =P 34. Equations 5 12 can be used to determine the 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 + C 2 y =0 13 which has a slope m = y/x = /C Alternative Compatibility Linkages Given a center-point curve, Murray and McCarthy 11 outlined a method to determine P 3 and P 4 for any P 1 and P 2 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 P 1, but leaving P 2 unchanged, corresponds to a change only in P 4. Likewise, selecting a new P 2, but leaving P 1 unchanged, corresponds to a change only in P 3. Equations 5 12 can be written as a linear combination of the components of a single vertex P j f i = R i p j + S i q j + T i KC 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 P 1A can be selected as any point on the center-point curve, and the corresponding vertex P 4A 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 equations 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, Fig. 6 Two compatibility linkages with different P 1 and P 4 vertices Ri1 S i1 C i1 R i2 S i2 C i2 R i3 S i3 C i3 p4a q 4A = Ti1 T i3 i2 K T 16 is used to readily solve for the corresponding vertex P 4A. In the same manner, this procedure applies to selecting an alternative P 2A on the center-point curve and solving for the corresponding P 3A. As an example, design positions are given as d 1 = 5, 10 T, 1 =200 deg; d 2 = 0,0 T, 2 = 100 deg; d 3 = 5,0 T, 3 =50 deg; and d 4 = 10, 5 T 4 =100 deg. Poles comprising a compatibility linkage are calculated as P 12 =P 1 = , T, P 23 =P 3 = , T, P 34 =P 4 = , T, and P 14 =P 2 = , T. Moving P 1 to an arbitrary point on the curve P 1A = , T, Eq. 16 is used with i 1 =3, i 2 =5, and i 3 =8 to determine the corresponding P 4A = , T. The two remaining vertices P 2 and P 3 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 P 1A is moved to infinity and P 2 is considered the fixed pivot, vertex P 3 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 give n = 1/m = p 1A /q 1A 17 As p 1A and q 1A become large, so do the coefficients in Eqs Therefore, the center-point equation will be normalized to deal with the numerical issues. If 0, Eq. 4 is divided by throughout, producing x + C 2 y x 2 + y 2 + C 3 x 2 + C 4 y 2 + C 5 xy + C 6 x + C 7 y + C 8 =0 observing that 18 Journal of Mechanisms and Robotics MAY 2010, Vol. 2 /

4 Fig. 7 A slider-crank that serves as a compatibility linkage Fig. 8 Slider-crank compatibility linkage from example 1 C 2 = p 1A + p 2 + p 3 p 4A q 1A q 2 q 3 + q 4A 1/q 1/q 1A 19 When q 1A, Eq. 19 states C 2 / = p 1A /q 1A =n. Repeating this process on Eqs 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 2 p 1A /q 1A p 4A 2q 4A = C 5 / 21 Rewriting Eqs. 20 and 21 gives 1 C 2 / 2 C 2 / p 2 q 4A = p 2 + p 3 + C 2 / q 2 + C 2 / q 3 + C 3 / C 5 / 22 which can be solved for the corresponding vertex P 4A associated with P 1A at infinity. It is observed that this vertex P 4A 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 P 2 =P 14. An alternative P 2A can be placed anywhere on the curve. However, Eq. 15 cannot be used with P 1A at infinity. Using a process similar to generating Eq. 22, Eqs. 8 and 12 can be divided by q 1A and rewritten factoring p 3A and q 3A, giving 1 p 3A C 2 / q 3A + C 4 / p 2 p 4 C 2 / q 2 + q 4 =0 23 d 4 = 1, 2 T, 4 =20 deg. Poles comprising the opposite pole quadrilateral are calculated as P 12 =P 1 = , T, P 23 =P 3 = , T, P 34 =P 4 = , T, and P 14 =P 2 = , T. The first two coefficients of the centerpoint curve, calculated from Eqs. 5 and 6, are = and C 2 = The slope of the asymptote, calculated from Eq. 14, ism= /C 2 = Moving P 1 to infinity, C 3,...,C 8 are determined from Eqs and substituted into Eq. 22 to determine the corresponding P 4 = , T. The two remaining vertices can be held as is, and the line of slide for P 3 will be perpendicular to the asymptote, having a slope n=c 2 / = The resulting compatibility linkage is given in Fig. 8. As with the four-bar compatibility linkage, the slider-crank can be used to graphically construct the center-point curve. As the slider displaced an arbitrary amount, a feasible center point can be located by determining the pole of the coupler displacement, as shown in Fig Parametrizing 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 = P 2 P 4 25 p 2A p 4A + C 2 / p 4A q 2A C 2 / p 2A q 4A + q 2A q 4A p 3A + C 2 / p 2A p 4A p 4A q 2A + p 2A q 4A + C 2 / q 2A q 4A q 3A + C 8 / =0 24 which can be used to determine the corresponding P 3A. Thus, there is a one-dimensional set of slider-crank compatibility linkages that define a center-point curve. Additionally, kinematic inversion allows P 3A to be considered the fixed pivot and P 2A to be the revolute attached to the prismatic. Both forms of the slidercrank will generate identical center-point curves. As an example, design positions are given as d 1 = 0,0 T, 1 =0 deg; d 2 = 1,0 T, 2 =5 deg; d 3 = 2, 1 T, 3 =10 deg; and Fig. 9 Slider-crank compatibility used to graphically generate a feasible center point / Vol. 2, MAY 2010 Transactions of the ASME

5 r 3 = P 3 P 4 26 The angle for the original configuration of link 3 and the angle of the slide direction of link 4 are 3o = tan 1 q 3 q 4 p 3 p 4 4o = tan 1 C In practice, a four-quadrant arctangent function should be used to ensure the proper angle. The offset distance is calculated as r 1 = P 3 P 2 sin tan 1 q 3 q 2 p 3 p 2 4o 29 The loop closure equations for a slider-crank linkage can be reduced to a well known equation for the coupler angle 12 as = sin r 1 r 2 sin 2 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 2 is C 2 = A A A o P 4 d + d 31 where d = P 3 + r 2 cos 2 + 4o sin 2 + 4o 32 and A o and A are determined by substituting 3o and 3 + 4o into Eq. 1, respectively. Once all substitutions are made, Eq. 31 defines a point on the center-point curve for every value of the crank angle 2. Thus, the center-point curve is parametrized based on the crank angle of a slider-crank linkage. The actual drivability and singularity conditions of the slider-crank do not affect this parametrization. Using complex vector notation, Eq. 31 can be rewritten into a form similar to McCarthy s 4R compatibility parametrization 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 Sec. 7. For the example presented in Fig. 8, Eqs were used to calculate the compatibility slider-crank dimensions: r 1 = , r 2 =7.1342, r 3 = , 3o =33.3 deg, and 4o = 53.1 deg. Using angular increments of 5 deg, the values of 2 were substituted into Eq. 30. Subsequently, the resulting values of 3 and 2 were substituted into Eq. 32, and then into Eq. 31 producing feasible center points. The resulting center-point curve parametrization by the crank angle 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 two circuits bicursal. Conversely, if the no link in the generating linkage is able to fully rotate, the center-point curve will have a single circuit unicursal. The condition 7 that a slider-crank will have a link that fully rotates is r 3 r 2 r 1 33 A slider-crank that produces an equality of Eq. 33 is considered a transition linkage by Murray Example 1: Unicursal, Center-Point Curve. For the example presented at the end of Sec. 5, r 1 = , r 2 =7.1342, and Fig. 10 A center-point curve is generated from a slider-crank compatibility linkage 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 Example 2: Bicursal, Center-Point Curve. Design positions are given as d 1 = 2,0 T, 1 =260 deg; d 2 = 2,2 T, 2 =120 deg; d 3 = 2,4 T, 3 =60 deg; and d 4 = 4,3 T, 4 =0 deg. Poles comprising the opposite pole quadrilateral are calculated as P 12 =P 1 = , T, P 23 =P 3 = , T, P 34 =P 4 = , T, and P 14 =P 2 = , T. The first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are = and C 2 = Moving P 1A to infinity, Eq. 22 is used to determine the corresponding P 4A = , T. The two remaining vertices can be held as is, and the line of slide for P 3 will be perpendicular to the asymptote, having a slope n=c 2 / = The resulting compatibility linkage has r 1 =1.2595, r 2 =8.6981, and r 3 = and is shown in Fig. 11 a. With these values, Eq. 33 is satisfied, indicating that a link in the generating slider-crank is able to fully rotate and resulting in a bicursal center-point curve Selecting P 2A = , T, Eqs. 23 and 24 can be used to determine the corresponding P 3A = , T. With these vertices, an in-line slider-crank generating linkage is created, as shown in Fig. 11 b. 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, 1 =120 deg; d 2 = 1, T, 2 = 75 deg; d 3 = 0,0 T, 3 = 45 deg; and d 4 = 1,1 T, 4 =45 deg. Poles comprising the opposite pole quadrilateral are calculated as P 12 =P 1 = , T, P 23 =P 3 = , T, P 34 =P 4 = , T, and P 14 =P 2 = , T. The first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are = and C 2 = Moving P 1A to infinity, Eq. 22 is used to determine the corresponding P 4A = , T. The two remaining vertices can be held as is, and the line of slide for P 3 will be perpendicular to the asymptote, having a slope n=c 2 / = The resulting compatibility linkage has r 1 =0.2534, r 2 =3.6919, and r 3 = and is shown in Fig. 12. With these values, Eq. 33 becomes an equality producing a transition compatibility slidercrank in which the links are not aligned and which generates a double-point, center-point curve. Journal of Mechanisms and Robotics MAY 2010, Vol. 2 /

6 Fig. 13 The slider-crank compatibility linkage from example 4 With these values, Eq. 33 becomes an equality producing a transition compatibility slider-crank in which the links are collinear and which generates a circle-degenerate, center-point curve. Notice in Fig. 13 that P 4A is located at the center of the circle. Beyer 3 identified that the focus of a circle-degenerate center-point curve is at the center of the circle. Fig. 11 The slider-crank compatibility linkage from example Example 4: Circle-Degenerate, Center-Point Curve. Design positions are given as d 1 = 0,0 T, 1 = 45 deg; d 2 = 1,1 T, 2 =45 deg; d 3 = 2, 1 T, 3 =120 deg; and d 4 = , T, 4 = 75 deg. Poles comprising the opposite pole quadrilateral are calculated as P 12 =P 1 = , T, P 23 =P 3 = , T, P 34 =P 4 = , T, and P 14 =P 2 = , T. The first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are C 2 = and C 2 = Moving P 1A to infinity, Eq. 22 is used to determine the corresponding P 4A = , T. The two remaining vertices can be held as is, and the line of slide for P 3 will be perpendicular to the asymptote, having a slope n =C 2 / = The resulting compatibility linkage has r 1 =8.6589, r 2 = , and r 3 = and is shown in Fig Example 5: Hyperbola-Degenerate, Center-Point Curve. Design positions are given as d 1 = 0,0 T, 1 = 100 deg; d 2 = 5,0 T, 2 =50 deg; d 3 = 10, 5 T, 3 =100 deg; and d 4 = , T, 4 =113 deg. Poles comprising the opposite pole quadrilateral are calculated as P 12 =P 1 = , T, P 23 =P 3 = , T, P 34 =P 4 = , , and P 14 =P 2 = , T. These poles form a parallelogram, and the first two coefficients of the center-point curve, calculated from Eqs. 5 and 6, are =C 2 =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. 14, and a circle at infinity. With =C 2 =0, Eqs. 5 and 6 dictate that as P 1 is moved to an alternate point P 1A, q 4A = q 2 + q 3 q 1A 34 p 4A = p 2 + p 3 p 1A 35 Therefore, moving P 1A to infinity, P 4A will also move to negative infinity. With two vertices at infinity, and the resulting compatibility linkage is an RPRP, shown as black in Fig. 15. The two remaining vertices are unaffected. For an equilateral hyperbola, C 4 = C 3, and the slopes of the two asymptotes 14 are Fig. 12 The slider-crank compatibility linkage from example 3 Fig. 14 Two compatibility linkage with different vertices from example / Vol. 2, MAY 2010 Transactions of the ASME

7 The significance of a slider-crank compatibility linkage is that the closure equations are quadratic and easily manipulated, whereas the equations for a four-bar are transcendental. Additionally a slider-crank compatibility linkage presents an alternative graphical method to generate a center-point curve by simply moving a slider along a straight line. Fig. 15 An RPRP compatibility linkage from example 5 n,m = C C5 +4C C 3 The asymptote slope n in Eq. 36 also represents the line of slide for the prismatic joints of the compatibility linkage of Fig. 15. 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 as black in Fig. 15 will generate the circle at infinity. The hyperbola is generated by the alternate assembly circuit, shown as gray in Fig Conclusions 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 serve as compatibility linkages. Furthermore, a parametrization of the center-point curve by the crank angle of the slider-crank 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, NJ, pp Beyer, R., 1963, Kinematic Synthesis of Mechanisms, translated by H. Kuenzel, McGraw-Hill, New York, pp Bottema, O., and Roth, B., 1979, Theoretical Kinematics, North-Holland, New York, pp Erdman, A., Sandor, G., and Kota, S., 2001, Mechanism Design: Analysis and Synthesis, 4th ed., Vol. 1, Prentice-Hall, Englewood Cliffs, NJ, pp Norton, R., 2008, Kinematics and Dynamics of Machinery, 4th ed., McGraw- Hill, New York, pp McCarthy, J. M., 2000, Geometric Design of Linkages, Springer-Verlag, New York, pp Sandor, G. N., and Erdman, A. G., 1984, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2, Prentice-Hall, Englewood Cliffs, NJ, pp Chase, T. R., Erdman, A. G., and Riley, D. R., 1985, Improved Center-Point Curve Generation Techniques for Four-Precision Position Synthesis Using the Complex Number Approach, ASME J. Mech. Des., 107 3, pp McCarthy, J. M., 1993, The Opposite Pole Quadrilateral as a Compatibility Linkage for Parameterizing the Center-Point Curve, ASME J. Mech. Des., 115 2, pp Murray, A., and McCarthy, J. M., 1997, Center-Point Curves Through Six Arbitrary Points, ASME J. Mech. Des., 119 1, pp Waldron, K., and Kinzel, G., 2004, Kinematics, Dynamics and Design of Machinery, Wiley, Hoboken, NJ, pp Murray, A. P., Turner, M. L., and Martin, D. T., 2008, Synthesizing Single DOF Linkages Via Transition Linkage Identification, ASME J. Mech. Des., 130 2, p Salmon, G., 1954, A Treatise on Conic Sections, 6th edition, Chelsea, New York, p Journal of Mechanisms and Robotics MAY 2010, Vol. 2 /

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