Mechanism and Machine Theory

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1 Mechanism and Machine Theory 45 (21) Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: Pole arrangements that introduce prismatic joints into the design space of four- and five-position rigid-body synthesis David H. Myszka, Andrew P. Murray University of Dayton, Dayton, OH 45469, United States article info abstract Article history: Received 5 June 29 Received in revised form 7 April 21 Accepted 12 April 21 Available online 26 May 21 Keywords: Rigid-body guidance Planar synthesis Center-point curves Compatibility linkage Although a general five-position, rigid-body guidance problem admits a discrete number of revolute revolute (RR) dyads, this paper identifies arrangements of five task positions that result in a center-point curve. For these special arrangements, a one-dimensional set of revolute-prismatic (RP) dyads exist to achieve the task positions. Other five-position arrangements are identified where a one-dimensional set of prismatic-revolute (PR) dyads exist to achieve the task positions. For a general case of five task positions, neither PR nor RP dyads are possible. In a general case of four-position rigid-body guidance problems, a unique PR dyad and RP dyad exist. Four-position arrangements are identified where the associated center-point curve includes the line at infinity and admits a PR dyad with a line of slide in any direction. Likewise, arrangements of the four positions are identified where the circle-point curve includes the line at infinity, permitting a one-dimensional set of RP dyads. These special four and five positions lead to dyads that can be coupled to solve a rigid-body guidance synthesis problem with a PRRP or RPPR device which is generally not possible. These solutions are particularly useful in design situations where actuation through a prismatic joint is desired. 21 Elsevier Ltd. All rights reserved. 1. Introduction This paper presents arrangements of four- and five-position rigid-body guidance problems where sets of linkages can be synthesized that contain prismatic joints. Design situations frequently arise where actuating a mechanism with a prismatic joint offers significant advantages. For example, in cases that involve large driving forces, hydraulic actuators may be ideal. In cases that require precise and repeatable positional accuracy, electromechanical linear actuators are well-suited. Lead screw actuators are ideal for applications where back driving must be eliminated. For four general positions, unique PR and RP dyads exist to achieve the task positions. For five general positions, no prismatic joint chains are available. By adjusting general task positions into the arrangements discussed in this paper, linkages with prismatic joints are introduced into the design space. In rigid-body guidance, the location of the ith 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 A i = cosθ i sinθ i : ð1þ sinθ i cosθ i Corresponding author. Tel.: ; fax: addresses: dmyszka@udayton.edu (D.H. Myszka), murray@udayton.edu (A.P. Murray) X/$ see front matter 21 Elsevier Ltd. All rights reserved. doi:1.116/j.mechmachtheory

2 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) 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, where P jk = A j ½A j A k Šðd k d j Þ + d j = A k ½A j A k Šðd k d j Þ + d k : ð2þ Given four specified positions, six displacement poles exist (P 12, P 13, P 14, P 23, P 24 and P 34 ). An opposite pole quadrilateral is defined by four poles such that opposite sides have the form P ij P ik and P mj P mk [1]. 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 34 P 14. Note that as the special arrangements to be discussed in the remainder of the paper are formed with one opposite pole quadrilateral, the others will also form that same arrangement. Therefore, it is sufficient to focus on a single opposite pole quadrilateral. To that end, the development presented in the remaining sections of the paper will focus on the quadrilateral P 12 P 23 P 34 P 14. A center-point curve is the locus of feasible fixed pivot locations for a planar revolute revolute (RR) dyad that will guide the end-link through four finitely separated positions. The theory, originally formulated by Burmester [2], is described in numerous classic sources [3 5] and continues to be an essential element of more recent machine theory textbooks [6 8]. The equation of a center-point curve has the form ðc 1 x + C 2 yþðx 2 + y 2 Þ + C 3 x 2 + C 4 y 2 + C 5 xy + C 6 x + C 7 y + C 8 =; ð3þ and is a special case of a circular cubic. Keller [9] shows that the constants C i are not independent and functions of the coordinates of an opposite pole quadrilateral. Thus, the opposite pole quadrilateral dictates the shape of the center-point curve. Sandor and Erdman [1] 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. McCarthy [11] shows that the opposite pole quadrilateral serves as a compatibility linkage and uses its crank angle to parameterize the center-point curve. Chase et al. [12] observe that the form of the center-point curve is dependent on the motion type of the compatibility linkage. Murray and McCarthy [13] state that there is a two-dimensional set of quadrilaterals that can generate a given center-point curve. Myszka and Murray [14] show that slidercranks can also be used as compatibility linkages and that their motion dictates the form of the center-point curve. For each point on the center-point curve a corresponding moving pivot exists. These points form the circle-point curve which is defined on the moving plane. That is, similar to a center-point curve, the locus of moving pivots that will guide the end-link of an RR dyad through four task positions yields a circle-point curve. A circle-point curve is the kinematic inverse of the center-point curve and also takes the form of Eq. (3). The established method to synthesize a 4R linkage that reaches a specified set of positions is to individually generate two RR dyads [1,6 8]. As dictated by the center-point or circle-point curve, a one-dimensional set of RR dyads are available. Connecting the end-links of the two dyads yields a single-degree-of-freedom linkage. The coupling of two open chains that reach a given set of task positions ensures that the 4R chain can be assembled in each of these positions. However, the interaction of the two chains Fig. 1. Compatibility structure for five-position synthesis.

3 1316 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) forms a linkage that can suffer from circuit, branch or order defects. Solution rectification refers to the elimination of defects from a mechanism synthesis result. Balli and Chand provide a comprehensive overview of linkage defects and solution rectification [15]. For a general case of four-position rigid-body guidance problems, Eq. (3) has a single asymptote. This implies that a unique PR dyad exists to achieve the task positions. The line of slide of the P joint is perpendicular to the asymptote [1]. The PR dyad can be joined with any other dyad to form a single-degree-of-freedom linkage. Similarly, the circle-point curve has a single asymptote. Thus, one unique RP dyad exists to achieve a general set of four task positions. For a general set of five task positions, a discrete number of locations (, 2 or 4) are possible for a fixed pivot of an RR dyad. The fixed pivot locations correspond to the intersection of all five center-point curves generated by taking every four-position combination. With five positions, 1 poles exist (P 12, P 13, P 14, P 15, P 23, P 24, P 25, P 34, P 35 and P 45 ). Three center-point curves will intersect at a pole as the two positions that define the pole are used in the generation of the center-point curves. The triangle formed by displacement poles P 23, P 34 and P 35 is considered to be a platform connected to ground by the RR dyads P 12 P 13, P 14 P 34 and P 15 P 35. Murray and McCarthy [16] call this a compatibility platform. In general, a compatibility platform is a structure with zero degrees of freedom, yet it can exhibit alternate assembly configurations. A compatibility platform is shown with the black lines in Fig. 1. An alternative assembly configuration of this structure is shown as gray lines. As Murray and McCarthy describe, the fixed pivot of an RR dyad compatible with five task positions is a pole of an assembly of the compatibility platform relative to its initial configuration. One such fixed pivot for an RR dyad to achieve all five positions is shown as S in Fig. 1. Note that S is coincident with the displacement pole of the gray configuration relative to the black configuration. For a general case of five-position guidance problems, no PR or RP dyads are able to achieve the task positions [1]. The study presented in this paper identifies pole arrangements that introduce a one-dimensional family of solutions to rigidbody guidance that includes a prismatic joint. Specifically, pole arrangements for four- and five-position cases are identified that have a center-point at infinity, allowing a PR dyad in any direction to achieve the design positions. Additionally, pole arrangements for four- and five-position cases that have a circle-point at infinity allow an RP dyad to be selected from anywhere along a centerpoint curve to achieve the design positions. Myszka and Murray [17] outline a process to determine the task positions associated with the special pole arrangements. With the aforementioned restrictions on four- and five-position guidance, adjusting the positions such that the poles are configured into the special arrangements will introduce a one-dimensional set of dyads that include prismatic joints. Adjusting a set of arbitrary positions into these special arrangements is an optimization problem and not the focus of this paper, although one possible strategy is identified. The remainder of the paper is organized as follows. Section 2 identifies arrangements of four task position poles that generate center-point circle-point curves that include a line at infinity. Section 3 presents a method to synthesize an RP dyad and illustrates that as the center-point curve includes the line at infinity, a one-dimensional set of RP dyads exist. Section 4 presents a method to synthesize a PR dyad and illustrates that as the circle-point curve includes the line at infinity, a one-dimensional set of PR dyads exist. Section 5 presents five-position pole arrangements that have center-point and circle-point curves that include a line at infinity. Section 6 presents a strategy to shift positions such that the poles align into a desired arrangement. 2. Special arrangements of four-position poles Beyer [4] states that when the opposite pole quadrilateral consists of parallel opposite sides having equal lengths, the centerpoint curve includes the line at infinity. Myszka and Murray [17] extend this to show that when the opposite pole quadrilateral Fig. 2. Opposite pole quadrilateral forming a parallelogram.

4 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) consists of non-parallel opposite sides having equal lengths, the circle-point curve includes the line at infinity. The constraint for an opposite pole quadrilateral having equal sides is jp 12 P 14 j = jp 23 P 34 j; jp 12 P 23 j = jp 14 P 34 j: ð4þ The pole arrangement that satisfies Eq. (4) introduces prismatic joints into the design space of four-position rigid-body guidance problems. Recalling that coincident poles are not under consideration here, the only opposite pole quadrilateral shapes that satisfy Eq. (4) are the parallelogram, rhombus, crossed parallelogram, and folded rhombus. Those four possibilities are described in the following sections Opposite pole quadrilateral forming a parallelogram One solution to Eq. (4) generates an opposite pole quadrilateral exhibiting two pairs of sides that are parallel and the same length, forming a parallelogram. Burmester [2] originally recognized that as the opposite pole quadrilateral forms a parallelogram, the center-point curve degenerates into an equilateral hyperbola as shown in Fig. 2. As the center-point curve tends to an equilateral hyperbola, the curve also contains the line at infinity [4]. Also shown as the dotted curve is the circle-point curve drawn relative to the first position, which has a circle-degenerate form. When both pairs of parallel sides have the same length, the opposite pole quadrilateral forms a rhombus, which is a special case of the parallelogram. In this case, the center-point curve becomes a limiting case of a hyperbola and appears as intersecting lines that also includes the line at infinity [17] Opposite pole quadrilateral forming a crossed parallelogram A second solution to Eq. (4) generates an opposite pole quadrilateral exhibiting two pairs of equal length sides that are not parallel. This shape is sometimes referred to as a crossed parallelogram. A compatibility linkage taking the form of a crossed parallelogram will result in a center-point curve that takes a circle-degenerate form as shown in Fig. 3 [17]. Also shown as the dotted curve is the circle-point curve drawn relative to the first position, which tends to an equilateral hyperbola. As with the center-point curve, a hyperbolic-degenerate circle-point curve includes the line at infinity. When both pairs of sides in a crossed parallelogram have the same length, one set of opposite vertices become coincident and the opposite pole quadrilateral forms a folded rhombus. In this case, the circle-point curve becomes a limiting case of a hyperbola and appears as intersecting lines that also includes the line at infinity [17]. Fig. 3. Opposite pole quadrilateral forming a crossed parallelogram.

5 1318 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) Synthesizing PR dyads Fig. 4. General PR dyad. For a general case of four finitely separated positions, one unique PR dyad can be synthesized. A general PR dyad is shown in Fig. 4. The location of the moving pivot relative to the fixed frame is Z i = A i z + d i : ð5þ Alternatively, the location of the moving pivot relative to the fixed frame as constrained by the slide is cosϕ Z i = G + L i ; ð6þ sinϕ where L i = L i. Combining Eqs. (5) and (6), for two positions i and j, gives cosϕ ða j A i Þz + ðd j d i Þ = L ij ; ð7þ sinϕ where L ij =L i L j. For four finitely separated positions, three independent versions of Eq. (7) can be expressed as 2 ½A 4 A 3 Š ½A 4 A 2 Š 6 4 ½A 4 A 1 Š cosϕ sinϕ cosϕ sinϕ 38 d 4 d 3 >< d 4 d 2 7 cosϕ 5 sinϕ d 4 d 1 >: z x z y L 34 L 24 L >= =: ð8þ >; An equation of the form in Eq. (8) has a non-trivial solution if the matrix loses rank. Thus, a PR solution exists only if the determinant of the matrix in Eq. (8) is zero. For a general set of positions, the determinant of the matrix takes the form, D 1 sinϕ + D 2 cosϕ =; ð9þ and a single value of ϕ ( ϕbπ) is possible. The line of slide defined by ϕ will be perpendicular to the asymptote of the centerpoint curve. See McCarthy [1] or Sandor and Erdman [1] for details on PR synthesis of four general task positions. As the centerpoint curve approaches infinity, the highest order terms in Eq. (3) dominate. The equation of the asymptote is C 1 x + C 2 y =; ð1þ which has a slope m = y = x = C 1 = C 2 : ð11þ

6 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) Fig. 5. A PR dyad can be oriented in any direction when the center-point curve is a hyperbola. When the center-point curve tends to a hyperbola, as the opposite pole quadrilateral forms a parallelogram, the center-point curve exhibits two asymptotes and includes the line at infinity. For this special arrangement where the center-point curve includes the line at infinity, D 1 =D 2 =, and the determinant of the matrix in Eq. (8) is zero for any ϕ. This confirms that any five of the six equations can be solved for z, L 14, L 24, L 34 with an arbitrary ϕ. The solution will be consistent with the unused equation. Therefore, for these situations a PR dyad to achieve the four positions can be synthesized with the line of slide in any direction. As an example, PR dyads are synthesized at arbitrary directions of 47 and 152 to achieve positions where the opposite pole quadrilateral forms a rhombus are shown in Fig. 5a and b, respectively. The two dyads are joined to form a double-slider (PRRP) linkage to achieve the four target positions that are shown in Fig. 5c. Although the center-point curve degeneracies are thoroughly discussed in the literature, the authors are not aware of previous examples detailing the arbitrary line of slide for the P joint in a PR chain. 4. Synthesizing RP dyads For a general case of four finitely separated positions, one unique RP dyad can be synthesized. A general RP dyad is shown in Fig. 6. Similar to the PR dyad, the location of the prismatic joint can be represented through alternative vector paths, written relative to the moving reference frame M i as A T i sinψ ðg d i Þ = z + l i : ð12þ cosψ

7 132 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) Fig. 6. General RP dyad. Writing Eq. (12) for two positions i and j and subtracting gives A T i AT j G + l ij sinψ cosψ = A T i d i AT j d j : ð13þ where l ij = l i l j. For the general case of four finitely separated positions, three independent versions of Eq. (13) can be expressed as 2 ½A 4 A 3 Š T sinψ cosψ ½A 4 A 2 Š T 6 4 ½A 4 A 1 Š T sinψ cosψ sinψ cosψ A T 3 d 3 AT 4 d 4 A T 2d 2 A T 4d 4 A T 1d 1 A T 4d >< 7 5 >: G x G y l 34 l 24 l >= =: ð14þ >; An RP solution exists only if the determinant of the matrix in Eq. (14) is zero. For a general set of positions, as in the case of the PR dyad, a single value of ψ ( ψbπ) is possible. When the opposite pole quadrilateral forms a crossed parallelogram or folded rhombus, the circle-point curve is a hyperbola that exhibits two asymptotes and the line at infinity. For this special pole arrangement where the circle-point curve includes the line at infinity, the determinant of the matrix in Eq. (14) is zero for any ψ. This confirms that any five of the six equations can be solved for G, l 14, l 24, l 34 with an arbitrary ψ. The solution will be consistent with the unused equation. The corresponding fixed pivot G will always lie along the circular portion of the center-point curve. Therefore, an RP dyad to achieve the special arrangement of four positions can be synthesized for any point along the circular portion of the center-point curve. As an example, two RP dyads are synthesized from arbitrary points on the center-point curve illustrated in Fig. 3, as shown in Fig Special arrangements for the poles of five positions This section shows that pole arrangements associated with a compatibility platform that is able to move present a onedimensional set of dyads that are able to reach the five task positions. The only known configuration where the platform gains a degree of freedom is the E-quintet paradox [7] where the cranks are parallel and have equal lengths. Conditions that are related to this movable platform are described in the following sections.

8 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) Fig. 7. Two RP dyads are synthesized from arbitrary points on the center-point curve Case 1: movable compatibility platform The first arrangement of a five-position guidance problem that admits a one-dimensional set of dyads has a compatibility platform that contains three equal length and parallel cranks. As in the general case, this compatibility platform can have different assembly configurations revealing fixed pivots for an RR dyad. In addition, the symmetry allows the structure to move as a triple crank. The motion of the platform is an orbital translation where each point on the platform traces a circle, and the displacement poles form a line at infinity. Like the four-position cases where the center-point curve includes a line at infinity, a PR dyad can be synthesized to achieve the five positions with the line of slide in any arbitrary direction. As an example of a compatibility platform with motion, five positions that generate a movable compatibility platform are shown in Fig. 8a. Note that an alternate assembly configuration and the pole of the compatibility platform are also shown. An RR dyad, synthesized from the pole of the platform is shown in Fig. 8b. Two PR dyads are synthesized at arbitrary directions of 57 and 162, and shown in Fig. 8c and d, respectively Case 2: alternate configuration of the movable compatibility platform A second arrangement of five-position poles that admits a one-dimensional set of dyads is revealed if a movable compatibility platform is placed in an alternate configuration. An alternate configuration is shown in gray in Fig. 8a. As described earlier, fixed pivot locations for the five positions correspond to the pole of the compatibility platform with another configuration. In this case, the other configuration is able to move, and the displacement pole becomes a curve. Thus, a center-

9 1322 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) Fig. 8. Five positions with a compatibility structure that has motion. point curve for five positions is obtained and takes the form of a circle. However, the corresponding circle point for all feasible center points is at infinity. As with the four-position arrangements where the circle-point curve included the line at infinity, an RP dyad can be constructed from any point on the center-point curve to achieve the five task positions. The center-point curve for these five positions is shown in Fig. 9a. Two RP dyads are synthesized from arbitrary points on the center-point curve and are shown in Fig. 9b and c, respectively. The prescription for pole locations that generate an alternative configuration of a movable compatibility platform is shown in Fig. 1. The three poles P 23, P 34, and P 35 define a circle. An arbitrary distance l can be used from P 23 to place P 12 on the circle. Then, using the same length l, P 14 and P 15 are placed on the circle in the same rotational direction as P 12. With this arrangement of poles, the design space includes an infinite number of RP dyads to solve the five-position rigid-body guidance. 6. A strategy for adjusting the positions In situations where actuation through a prismatic joint is preferred, the poles of the original positions should be evaluated to determine their proximity to the special arrangements that admit prismatic joints. Thus, a set of shifted positions defined by d i and θ i are desired that are minimally moved from the original positions defined by d i and θ i, yet place the poles in one of the special arrangements discussed.

10 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) Fig. 9. Center-point curve for five task positions.

11 1324 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) Fig. 1. Pole locations that generate a center-point curve for five task positions. The process of moving positions can be viewed as a minimization problem where the cost function is any valid metric on SE(2) that quantifies the distance between the shifted and original task positions. An example is, MINIMIZE : f ðd i ; θ i n Þ = i =1 d * i d T i d * i d i + wjθ * i θ i j; ð15þ where w is a weighting factor on the rotations. Functional constraints are established from the conditions that describe the desired special arrangement of the poles. For four positions, constraints are derived directly from the parallelogram conditions of Eq. (4), P * 12 P* T 14 P * 12 P * 14 P * 23 P* T 34 P * 23 P * 34 =; P * 12 P* T 23 P * 12 P * 23 P * 14 P* T 34 P * 14 P * ð16þ 34 =; where P jk are the poles of the shifted positions. Introducing only PR dyads into the design space, an additional constraint is specified to ensure that opposite sides remain parallel, tan 1 P * 23y P* 12y P * A tan 1 P * 34y P* 14y 23x P* 12x P * A =: 34x P* 14x ð17þ Eqs. (1) and (2) are used to relate the shifted position coordinates d ix, d iy, and θ i to the poles P jk used in the constraints of Eqs. (16) and (17). The optimization generates a set of shifted positions. The designer must evaluate the cost function f in Eq. (15) to assess whether the solution is acceptable and produces a workable design. 7. Conclusions The study presented in this paper identifies four-position pole arrangements where the associated center-point curve includes the line at infinity. Five-position pole arrangements were identified where the possible center points also include the line at infinity. When the center-point curve includes the line at infinity, a PR dyad with line of slide in any direction can be synthesized to achieve the design positions. Further, four-position pole arrangements were identified where the associated circle-point curve includes the line at infinity. Five-position pole arrangements were identified that generates a center-point curve and the possible circle points also include the line at infinity. With a circle-point curve that includes a line at infinity, an RP dyad originating anywhere on a center-point curve can be synthesized to achieve the design positions. A one-dimensional set of dyads that include a prismatic joint are introduced into the design space by adjusting the positions in four- and five-position guidance problems such that the poles are configured into the special arrangements. References [1] J.M. McCarthy, Geometric Design of Linkages, Springer-Verlag, New York, 2. [2] L. Burmester, Lehrbuch der Kinematic, Verlag Von Arthur Felix, Leipzig, Germany, [3] A.S. Hall, Kinematics and Linkage Design, Prentice-Hall, Englewood Cliffs, New Jersey, [4] R. Beyer, Kinematic Synthesis of Mechanisms, McGraw-Hill, New York, 1963 translated by H. Kuenzel. [5] O. Bottema, B. Roth, Theoretical Kinematics, North-Holland Publishing Company, New York, [6] A. Erdman, G. Sandor, S. Kota, Mechanism Design: Analysis and Synthesis, Vol. 1, 4/e, Prentice-Hall, Englewood Cliffs, New Jersey, 21. [7] R. Norton, Kinematics and Dynamics of Machinery, 4/e, McGraw Hill Book Company, New York, 28.

12 D.H. Myszka, A.P. Murray / Mechanism and Machine Theory 45 (21) [8] K. Waldron, G. Kinzel, Kinematics, Dynamics and Design of Machinery, 2/e, John Wiley & Sons Inc, New Jersey, 24. [9] R.E. Keller, Sketching rules for the curves of Burmester mechanism synthesis, Journal of Engineering for Industry 87 (5) (1965) [1] G.N. Sandor, A.G. Erdman, Advanced Mechanism Design: Analysis and Synthesis, Vol. 2, Prentice-Hall, Englewood Cliffs, New Jersey, [11] J.M. McCarthy, The opposite pole quadrilateral as a compatibility linkage for parameterizing the center-point curve, Journal of Mechanical Design 115 (2) (1993) [12] T.R. Chase, A.G. Erdman, D.R. Riley, Improved center-point curve generation techniques for four-precision position synthesis using the complex number approach, Journal of Mechanical Design 17 (3) (1985) [13] A.P. Murray, J.M. McCarthy, Center-point curves through six arbitrary points, Journal of Mechanical Design 119 (1) (1997) [14] D.H. Myszka, A.P. Murray, Slider-Cranks as compatibility linkages for paramterizing center point curves, Journal of Mechanisms and Robotics 2 (2) (21). [15] S.S. Balli, S. Chand, Defects in link mechanisms and solution rectification, Mechanism and Machine Theory 37 (22) [16] A.P. Murray, J.M. McCarthy, Determining Burmester points from the analysis of a planar platform, Journal of Mechanical Design 117 (2A) (1995) [17] D.H. Myszka, A.P. Murray, Identifying sets of four and five positions that generate distinctive center-point curves, Proc. of the ASME Design Engineering Technical Conference, 29.