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Avoiding singular configurations in finite position synthesis of spherical 4R linkages
Mechanism and Machine Theory 35 (2000) 451±462
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Avoiding singular con®gurations in ®nite position synthesis of spherical 4R linkages J. Michael McCarthy a,*, R. Mohan Bodduluri b a
Department of Mechanical Engineering, University of California, Irvine, Irvine, CA 92697, USA b Accuracy Inc., Santa Clara, CA 95054, USA Received 7 April 1993; received in revised form 8 October 1998
Abstract In this paper we consider the generalization of planar recti®cation theory to spherical 4R linkages. The goal is to ensure that the result of a ®nite position synthesis is a linkage that does not have a `branching problem.' Branching defects limit the usefulness of a linkage and the ability to remove them in the design process is of fundamental importance. The primary results of planar recti®cation theory, Filemon's construction and Waldron's `three circle diagram,' are found to have direct analogies in spherical 4R synthesis theory. In this case, however, we obtain three quadric cones, not circles, as our `spherical version' of Waldron's three circle diagram. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The process of `®nite position synthesis' of a planar 4R linkage constructs RR open chains compatible with speci®ed locations of a moving bodyÐR denotes a revolute, or hinged, joint. Two RR chains connected to the moving body forms a one degree-of-freedom closed chain that can be assembled in each of the speci®ed locations [9]. To be useful the linkage should be able to move between these positions. The goal of `recti®cation theory' is to ensure, as part of the design process, that this movement can occur. Filemon [5] introduced a construction guiding the selection of the moving pivot for the driving RR crank, for a previously select driven crank, that ensures the coupler of a crank-
* Corresponding author. Tel.: +1-714-856-6893; fax: +1-714-856-7966. E-mail address: [email protected] (J.M. McCarthy) 0094-114X/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 9 9 ) 0 0 0 0 5 - 1
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rocker passes through the speci®ed goal positions. Filemon's construction actually solves a broader problem ensuring that a double-rocker linkage does not hit a singular, or jamming, con®guration between the goal positions. For these linkages, a singular con®guration is acceptable at the beginning and end of the range of movement of coupler, after it has passed through the goal positions. However, a singular con®guration occurring between two goal positions forces an interchange of the role of the driving and driven cranks, which is unacceptable. Unfortunately, Filemon's construction may yield no acceptable driving moving pivots, depending on the initial choice of the driven moving pivot. To address this problem, Waldron [11] introduced the `three circle diagram' that guides the selection of the moving pivot of the driven crank. This expanded the reach of recti®cation theory to other planar linkages types. Spherical crank-rocker linkages, like their planar counterparts, have two modes of assembly. In the process of ®nite position synthesis of these mechanisms [3,4,10], the designer may inadvertently choose pivots such that certain positions are reached in one assembly mode while the remaining occur in the other assembly. A linkage design with this property is said to have a `branch defect' [11]. Avoiding linkage designs with a branch defect is often called `the branching problem' [7,8]. In this paper, we provide a direct extension of this recti®cation theory to the ®nite position synthesis of spherical 4R linkages. A spherical version of Filemon's construction identi®es the driving crank moving axes that avoid singular con®gurations, in the same way as it does for planar linkages; and a spherical version of Waldron's `three circle diagram' identi®es the set of driven pivots that guarantee a solution to Filemon's construction [5,11].
2. Synthesis for three speci®ed orientations A spherical 4R linkage is constructed by rigidly connecting the ¯oating links of two spherical RR chains to form a coupler (Fig. 1). The four axes of these two RR chains must intersect at a single ®xed point, c. With this con®guration, the linkage guides the coupler in a rotational movement about c. Finite position synthesis of a spherical 4R linkage begins with the speci®cation of a set of goal orientations for a workpiece, M. Let, M1, M2 and M3, be referenced frames locating these orientations relative to a ®xed frame F. Associated with each is a 3 3 rotation matrix, [Aj], that de®nes the transformation of a coordinate vector, w, in M to coordinates Wj in F [6]: W j Aj w:
1
If the point, w, is the axis of the moving pivot of an RR chain, then its coordinates in F must maintain a constant angle, r, relative to the ®xed axis, G of the chain. This is represented by the three constraint equations: G W j j GkW j j cos r,
j 1,2,3:
2
Note that vWjv=v[Aj]wv=vwv, so the right side of Eq. (2) is the same for each equation. This
J.M. McCarthy, R.M. Bodduluri / Mechanism and Machine Theory 35 (2000) 451±462
453
Fig. 1. A spherical 4R linkage.
means we can subtract the ®rst equation from the remaining two and obtain: G W2 ÿ W1 0, G W3 ÿ W1 0:
3
Geometrically, these equations de®ne the planes that are the perpendicular bisectors of the segments W2ÿW1 and W3ÿW1, respectively. The ®xed axis G is the line of intersection of these two planes, given by the cross-product: G k W2 ÿ W1 W3 ÿ W1 :
4
The coecient k is an arbitrary constant that is used to normalize the length of G. Thus, the ®xed pivot, G of a spherical RR chain (obtained for three goal orientations) is a vector function of the coordinates of the coordinates of moving pivot, w, measured in M: G w k A2 ÿ A1 w A3 ÿ A1 w:
5
The design of a spherical 4R linkage can be viewed as the two step process: 1. First choose the moving axis, wo, of the `driven' crank, and compute the associated ®xed axis, Go (the subscript `o' represents `output'); and then 2. Choose the moving axis, wi, of the `driving' crank and compute its ®xed axis, Gi (`i' represents `input').
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3. Singular con®gurations We now determine the movement characteristics of a particular 4R linkage design Chiang [2]. For convenience, let O=Gi and C=Go be the coordinates of the two ®xed pivots; and let A=W1i and B=W1o be the coordinates in F of the moving pivots. The angular dimensions of the 4R linkage are given by: ground: cos g O C, driving crank: cos a O A, driven crank: cos b C B, coupler: cos Z A B:
6
Note that we assume the length of the vectors O, A, B, and C have been normalized to one. We now introduce the frame F' adapted for the purpose of deriving the input/output equation for this linkage. Orient F' so its z-axis is along O and its x-axis is in the direction O C. Let y be the dihedral angle from OC to OA. Similarly, let c be the angle from OC to OB. The result is that, in F', A and B have the coordinates: 8 9 8 9 sin b sin c < sin a sin y = < = A ÿcos y sin a , B ÿcos g cos c sin b ÿ cos b sin g : 7 : ; : ; cos a cos b cos g ÿ cos c sin b sin g The fact that AB=cos Z yields an equation that we solve to determine the output angle c in terms of the input angle y: A y cos c B y sin c C y,
8
where A y sin a sin b cos g cos y ÿ cos a sin b sin g, B y sin a sin b sin y, C y cos Z ÿ sin a cos b sin g cos y ÿ cos a cos b cos g: An explicit formula c(y ) is easily obtained by dividing both sides of Eq. (8) by introducing the angle d, such that: A cos d p , 2 A B2
B sin d p : 2 A B2
9
p A2 B 2 , and
10
This angle locates the diagonal plane, CA, of the linkage and is measured about C from OC. The solution to Eq. (8) is given by
J.M. McCarthy, R.M. Bodduluri / Mechanism and Machine Theory 35 (2000) 451±462
C c y arctan d2arccos p : 2 A B2
455
11
Clearly, for any given value of y there are two possible values for c. These are the two modes of solution for the linkage (Fig. 2). A singular con®guration is de®ned by the condition A 2+B 2ÿC 2=0, in which case, Eq. (11) has a single solution. Geometrically, this occurs when the moving pivot B falls on the linkage diagonal plane CA. The interaction of singular con®gurations with the modes of assembly yields three basic situations that can be distinguished by the range of movement of the input crank: y. There are three cases: 1. Fully rotatable input movement: There is no limit on the input angle y which means the mechanism is a crank-rocker or drag-link mechanism. In this case, there are no singular con®gurations and the two modes of assembly are completely distinct. It is possible for the design positions to be on dierent assemblies, which means the linkage simply cannot reach the other set of design positions unless the linkage is physically reassembled in the other mode. 2. One angular range of input movement: The input angle rocks between two bounding values 2y. Here there are two singular con®gurations located symmetrically above and below the OC plane. The linkage is a non-Grasho double rocker, and the singular con®gurations separate the two solution modes. In this case, if the design positions are on separate solution modes, then the driving and driven cranks must exchange roles in order to guide the linkage through the remaining positions. 3. Two angular ranges of input movement: In this case, the input angle rocks between a set of bounding values y 1 R y R y 2 above the OC plane which is one assembly, and between
Fig. 2. Two assemblies of spherical 4R linkage, (a) mode 1; (b) mode 2.
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another set of bounding values, ÿy 1 ry r ÿy 2, symmetrically located below this plane which forms the second assembly. This is the case of the rocker-crank and Grasho doublerocker linkages. Even if the design positions are on the same solution mode, they may be separated by a singular con®guration.
4. Filemon's construction Filemon [5] introduced a construction that identi®es moving pivots for the driving crank that ensure the design positions for a crank-rocker are on the same assembly. Given a choice for the moving pivot of the driven crank, wo, the associated ®xed pivot Go is known, and it is possible to determine the movement required of this crank to reach the design positions. Viewed from the coupler a line along this crank sweeps out two wedge shaped regions centered on wo. If the designer selects the driving moving pivot wi from outside of these regions, then the linkage cannot take on a singular con®guration between the design positions. Filemon showed this strategy ensures that the design positions are on the same assembly mode for a crank-rocker. This procedure can be applied to the design of spherical linkages by simply considering the joint axes of the linkage as de®ning planes through the ®xed point c, rather than lines. Assume that we have selected a driven moving pivot, wo. It takes the positions Wjo=[Aj]wo in the ®xed frame F in the three design positions. We view the driven crank from the moving body by inverting the positions of the ®xed pivot Go, given by: goj ATj G:
12
Let z12 and z23 be the angles measured around wo from g1o to g2o and from g2o to g3o, respectively. They are dihedral angles between the planes, L1, L2 and between L2, L3, respectively, that pass through gjo and wo (and the origin c). We now determine the angular range of movement of the driven crank, t, relative to the coupler. Assuming the angles zjk are between 2p, we have: 1. If sgnz12=sgnz23, then t=vz12v+vz23v; 2. If sgnz12 $sgnz23, then t=vzmaxv, where zmax is the angle with the greater magnitude. The angle, t, is bounded by two of the planes Lj and de®nes a pair wedge-shaped regions in the moving frame M. If the driving moving pivot, wi, is selected outside of this wedge-shaped region, then the resulting spherical 4R linkage must pass through the design positions before it hits a singular con®guration. This is the spherical version of Filemon's [5] construction (Fig. 3). Waldron [11,12] notes that, in the plane, it is possible for the range t=vz12v+vz23v to be greater than p. In this case, the region excluded by Filemon's construction is the entire moving body. This same situation occurs for spherical linkages. Our generalization of Waldron's three circle diagram solves this problem.
J.M. McCarthy, R.M. Bodduluri / Mechanism and Machine Theory 35 (2000) 451±462
457
Fig. 3. The spherical version of Filemon's construction: The planes L1 and L3 bound the region excluded from selection for driving moving pivots wi.
5. The spherical pole triangle Waldron [11,12] introduced a construction that identi®es the driven pivots wo for which the range of angular movement t is guaranteed to be less than p, thus ensuring a solution to Filemon's construction for the driving moving pivots wi. Waldron's construction depends on a fundamental kinematic relationship between the moving pivot of a planar RR chain and the relative position poles of the speci®ed planar design positions. An identical relationship exists for the moving axis of a spherical RR chain, and the relative rotation axes of the speci®ed design orientations. Given a pair of rotations [Aj] and [Ak] de®ning the frames Mj and Mk, the relative rotation [Ajk] from Mj to Mk is de®ned by: Ajk Ak ATj :
13
In our case, we obtain the three relative rotations [A12], [A23], and [A13]. Associated with each of these is a rotation axis Sjk and rotation angle fjk. The three axes, S12, S23, and S13 form what is known as the spherical pole triangle [1].
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In Filemon's construction, we have inverted the relationship between the ®xed and moving frames by considering the movement of the ®xed frame relative to the coupler. This is applied to the spherical pole triangle as follows. Consider the three inverted rotation matrices, [A Tj], j = 1, 2, 3. The relative rotations computed using these inverted rotations, yield the relative inverted rotation matrices: Rjk ATk Aj :
14
Notice that a relative inverted rotation is not simply the inverse of the relative rotation in Eq. (13). For our case, we have the relative inverted rotations [R12], [R23] and [R13]. Denote the rotation axes of these matrices as s12, s23, and s13. They form the spherical image pole triangle in M. If the moving frame is placed in the ®rst design position, M1, then the relative inverted rotations can be transformed to F to de®ne: R1jk A1 ATk Aj AT1 :
15
In this case, it is easy to determine that S12=[A1]s12 and S13=[A1]s13; that is, in the ®rst position M1, the relative poles S1k coincides with s1k. On the other hand, the relative pole s23 appears as the re¯ection of S23 through the plane containing S12 and S13. Waldron's construction is based on the planar version of the fact that each moving axis w views the two image poles sjn and snk in the rotation angle zjk/2 of the crank relative to the coupler. The proof for this statement is found in Bottema and Roth [1]. We use this to ®nd the equivalent to Waldron's three circle diagram for spherical 4R linkages. 6. The spherical version of Waldron's three circle diagram Waldron [11] notes that Filemon's construction in the plane has no solution for driving moving pivots, wi, when any one of the angles z12, z23, or z13 is greater than or equal to p. Waldron de®ned the set of points that view the sides of the pole triangle in p/2 and obtained three circles. Points outside of these circles are driven pivots for which Filemon's construction will ®nd satisfactory driving moving pivots. An even number of crossing of these circles also identi®es a region of acceptable driving moving pivots. In what follows, we de®ne and analyze the generalization of these circles. The relative rotation angle zjk of an spherical RR crank about its moving pivot, w, as the ¯oating link moves from orientation Mj to Mk can be computed directly from the rotation axes sjn and snk. The general result is that the dihedral angle between the plane Pj containing wsj and plane Pk containing wsk is zjk/2 (Fig. 4). zjk wo sjn wo snk cos : 16 j wo sjn kwo snk j 2 The points x for which zjk=p can now be seen to lie on three quadric cones: Cjk : x sjn x snk 0:
17
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459
Fig. 4. The moving pivot w views the axes s12 and s23 in the angle z13/2.
It is easy to see that each of the three cones Cjk passes through the pair of poles sjn and snk in a con®guration complete analogous to Waldron's three circle diagram. In our case, however, the cones, C12, C23 and C13, are general quadrics, and not circular (Fig. 5).
7. The cones Cjk We now study the shape of the cones Cjk. Let [Sjn] and [Snk] be the matrices obtain from the poles sjn and snk such that [Sjn]y=sjn y and [Snk]y=snk y, then Eq. (17) can be written as the quadratic form: Cjk :xT S Tjn Snk x 0:
18
The coecient matrix [S TjnSnk] is not symmetric but we can transpose this equation and add it to the original to obtain: Cjk :xT S Tjn Snk S Tnk Sjn x xT C x 0:
19
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Fig. 5. The spherical version of Waldron's three circle diagram: The three quadrics de®ne regions excluded from selection for the driven moving pivots wo.
The following analysis applies to each of these three quadrics C12, C23, and C13, therefore we will not index the matrix [C ] of the quadratic form in Eq. (19). The matric [C ] is symmetric and has the eigenvectors: e1 snk ÿ sjn ,
e2 sjn snk ,
and e3 sjn snk :
20
To see this we compute directly: Cjk e1 cos a 1e1 ,
Cjk e2 2 cos ae2 ,
and Cjk e3 cos a ÿ 1e3 ,
21
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461
where a is the angle between the vectors sjn and snk, that is cos a=sjnsnk. Please note that here a is used dierently than in Eq. (6). We now introduce a coordinate frame, M', with its x-axis directed along e1, its y-axis along e2, and its z-axis directed along e3. The result is the axes sjn and snk have the local coordinates: 9 8 9 8 a> a> > > > > sin > > > > > > 2> > > > ÿsin 2 > > = = < < 0 sjn , and s : 22 0 nk > > > > > > > > > > a> > > > cos a > > ; > : > ; : cos 2 2 Substituting this into Eq. (19), we obtain the canonical form of the quadric cone Cjk: a a a a ÿ z2 sin 2 0: Cjk :x 2 cos 2 y2 cos 2 ÿ sin 2 2 2 2 2
23
The intersection of this cone with the z = 1 plane yields an ellipse with semi-diameters a and b along the x and y directions, respectively, given by: a tan 2 a 2 : a2 tan 2 , and b2 a 2 1 ÿ tan 2 2
24
Notice that a is the angular width of the cone in the x±z plane. Introduce the angle, b, de®ned so tan b/2=b is the angular height of this cone in the y±z plane. When a is small the cone is circular in shape. As a increases, the angle b also increases, but at a faster rate, to the point that when a=p/2, we have b=p. In this case, the sheet on in the front half of the sphere contacts its antipodal sheet on the back of the sphere at the y-axis, 2e2. In fact, the cone becomes two planes intersecting along e2, separated by the angle a. As a increases beyond p/2 the coecient of y 2 changes sign, and the ellipse in the z = 1 plane becomes a hyperbola. In this case, we view the cone by determining its intersection with the x = 1 plane which is seen to be an ellipse surrounding the x-axis. Thus, the original cone has separated into left and right sheets. Finally, as a approaches p, the left and right sheets become circular in cross-section. We conclude that if a R p/2, then axes outside the cone Cjk are to be used for Filemon's construction: and when a > p/2, it is the axes inside of the cones that are to be used.
8. Conclusion In this paper, we show that central results of planar recti®cation theory, which guides the synthesis of singularity free planar 4R linkages, can be extended to spherical 4R linkage synthesis. Generalization of Filemon's construction and Waldron's three circle diagram provide a means to identify moving axes that ensure the coupler of a spherical 4R linkage passes through the speci®ed orientations while avoiding singular con®gurations. The focus here is on
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synthesis for three speci®ed orientations, however, the results apply to any number of goal orientations. References [1] O. Bottema, B. Roth, Theoretical Kinematics, North Holland Publ. Co. (reprinted 1990 by Dover Publications, New York), 1979. [2] C.H. Chiang, Kinematics of Spherical Mechanisms, Cambridge University Press, 1988. [3] H.J. Dowler, J. Duy, D. Tesar, A generalized study of three multiply separated positions in spherical kinematics, Mechanism and Machine Theory 11 (1976) 395±410. [4] H.J. Dowler, J. Duy, D. Tesar, A generalized study of four and ®ve multiply separated positions in spherical kinematics, Mechanism and Machine Theory 13 (4) (1978) 409±436. [5] E. Filemon, Useful ranges of centerpoint curves for design of crank-and-rocker linkages, Mechanism and Machine Theory 7 (1972) 47±53. [6] J.M. McCarthy, Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA, 1990. [7] J.M. Prentis, The pole triangle, Burmester theory, and order and branching problemsÐII: the branching problem, Mechanism and Machine Theory 26 (1) (1991) 31±39. [8] C.F. Reinholtz, G.N. Sandor, J. Duy, Branching analysis of spherical RRRR and spatial RCCC mechanisms, ASME J. Mechanisms, Transmissions and Automation in Design 108 (1986) 481±486. [9] G.N. Sandor, A.G. Erdman, Advanced Mechanism Design: Analysis and Synthesis, vol. 2, Prentice-Hall, New Jersey, 1984. [10] C.H. Suh, C.W. Radclie, Kinematics and Mechanisms Design, Wiley, New York, 1978. [11] K.J. Waldron, Elimination of the branch problem in graphical Burmester mechanism synthesis for four ®nitely separated positions, ASME Journal of Engineering for Industry 98 (1) (1976) 176±182. [12] K.J. Waldron, Graphical solution of the branch and order problems of linkage synthesis for multiply separated positions, ASME Journal of Engineering for Industry 99 (3) (1977) 591±597.
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Avoiding singular con®gurations in ®nite position synthesis of spherical 4R linkages J. Michael McCarthy a,*, R. Mohan Bodduluri b a
Department of Mechanical Engineering, University of California, Irvine, Irvine, CA 92697, USA b Accuracy Inc., Santa Clara, CA 95054, USA Received 7 April 1993; received in revised form 8 October 1998
Abstract In this paper we consider the generalization of planar recti®cation theory to spherical 4R linkages. The goal is to ensure that the result of a ®nite position synthesis is a linkage that does not have a `branching problem.' Branching defects limit the usefulness of a linkage and the ability to remove them in the design process is of fundamental importance. The primary results of planar recti®cation theory, Filemon's construction and Waldron's `three circle diagram,' are found to have direct analogies in spherical 4R synthesis theory. In this case, however, we obtain three quadric cones, not circles, as our `spherical version' of Waldron's three circle diagram. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The process of `®nite position synthesis' of a planar 4R linkage constructs RR open chains compatible with speci®ed locations of a moving bodyÐR denotes a revolute, or hinged, joint. Two RR chains connected to the moving body forms a one degree-of-freedom closed chain that can be assembled in each of the speci®ed locations [9]. To be useful the linkage should be able to move between these positions. The goal of `recti®cation theory' is to ensure, as part of the design process, that this movement can occur. Filemon [5] introduced a construction guiding the selection of the moving pivot for the driving RR crank, for a previously select driven crank, that ensures the coupler of a crank-
* Corresponding author. Tel.: +1-714-856-6893; fax: +1-714-856-7966. E-mail address: [email protected] (J.M. McCarthy) 0094-114X/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 9 9 ) 0 0 0 0 5 - 1
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rocker passes through the speci®ed goal positions. Filemon's construction actually solves a broader problem ensuring that a double-rocker linkage does not hit a singular, or jamming, con®guration between the goal positions. For these linkages, a singular con®guration is acceptable at the beginning and end of the range of movement of coupler, after it has passed through the goal positions. However, a singular con®guration occurring between two goal positions forces an interchange of the role of the driving and driven cranks, which is unacceptable. Unfortunately, Filemon's construction may yield no acceptable driving moving pivots, depending on the initial choice of the driven moving pivot. To address this problem, Waldron [11] introduced the `three circle diagram' that guides the selection of the moving pivot of the driven crank. This expanded the reach of recti®cation theory to other planar linkages types. Spherical crank-rocker linkages, like their planar counterparts, have two modes of assembly. In the process of ®nite position synthesis of these mechanisms [3,4,10], the designer may inadvertently choose pivots such that certain positions are reached in one assembly mode while the remaining occur in the other assembly. A linkage design with this property is said to have a `branch defect' [11]. Avoiding linkage designs with a branch defect is often called `the branching problem' [7,8]. In this paper, we provide a direct extension of this recti®cation theory to the ®nite position synthesis of spherical 4R linkages. A spherical version of Filemon's construction identi®es the driving crank moving axes that avoid singular con®gurations, in the same way as it does for planar linkages; and a spherical version of Waldron's `three circle diagram' identi®es the set of driven pivots that guarantee a solution to Filemon's construction [5,11].
2. Synthesis for three speci®ed orientations A spherical 4R linkage is constructed by rigidly connecting the ¯oating links of two spherical RR chains to form a coupler (Fig. 1). The four axes of these two RR chains must intersect at a single ®xed point, c. With this con®guration, the linkage guides the coupler in a rotational movement about c. Finite position synthesis of a spherical 4R linkage begins with the speci®cation of a set of goal orientations for a workpiece, M. Let, M1, M2 and M3, be referenced frames locating these orientations relative to a ®xed frame F. Associated with each is a 3 3 rotation matrix, [Aj], that de®nes the transformation of a coordinate vector, w, in M to coordinates Wj in F [6]: W j Aj w:
1
If the point, w, is the axis of the moving pivot of an RR chain, then its coordinates in F must maintain a constant angle, r, relative to the ®xed axis, G of the chain. This is represented by the three constraint equations: G W j j GkW j j cos r,
j 1,2,3:
2
Note that vWjv=v[Aj]wv=vwv, so the right side of Eq. (2) is the same for each equation. This
J.M. McCarthy, R.M. Bodduluri / Mechanism and Machine Theory 35 (2000) 451±462
453
Fig. 1. A spherical 4R linkage.
means we can subtract the ®rst equation from the remaining two and obtain: G W2 ÿ W1 0, G W3 ÿ W1 0:
3
Geometrically, these equations de®ne the planes that are the perpendicular bisectors of the segments W2ÿW1 and W3ÿW1, respectively. The ®xed axis G is the line of intersection of these two planes, given by the cross-product: G k W2 ÿ W1 W3 ÿ W1 :
4
The coecient k is an arbitrary constant that is used to normalize the length of G. Thus, the ®xed pivot, G of a spherical RR chain (obtained for three goal orientations) is a vector function of the coordinates of the coordinates of moving pivot, w, measured in M: G w k A2 ÿ A1 w A3 ÿ A1 w:
5
The design of a spherical 4R linkage can be viewed as the two step process: 1. First choose the moving axis, wo, of the `driven' crank, and compute the associated ®xed axis, Go (the subscript `o' represents `output'); and then 2. Choose the moving axis, wi, of the `driving' crank and compute its ®xed axis, Gi (`i' represents `input').
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3. Singular con®gurations We now determine the movement characteristics of a particular 4R linkage design Chiang [2]. For convenience, let O=Gi and C=Go be the coordinates of the two ®xed pivots; and let A=W1i and B=W1o be the coordinates in F of the moving pivots. The angular dimensions of the 4R linkage are given by: ground: cos g O C, driving crank: cos a O A, driven crank: cos b C B, coupler: cos Z A B:
6
Note that we assume the length of the vectors O, A, B, and C have been normalized to one. We now introduce the frame F' adapted for the purpose of deriving the input/output equation for this linkage. Orient F' so its z-axis is along O and its x-axis is in the direction O C. Let y be the dihedral angle from OC to OA. Similarly, let c be the angle from OC to OB. The result is that, in F', A and B have the coordinates: 8 9 8 9 sin b sin c < sin a sin y = < = A ÿcos y sin a , B ÿcos g cos c sin b ÿ cos b sin g : 7 : ; : ; cos a cos b cos g ÿ cos c sin b sin g The fact that AB=cos Z yields an equation that we solve to determine the output angle c in terms of the input angle y: A y cos c B y sin c C y,
8
where A y sin a sin b cos g cos y ÿ cos a sin b sin g, B y sin a sin b sin y, C y cos Z ÿ sin a cos b sin g cos y ÿ cos a cos b cos g: An explicit formula c(y ) is easily obtained by dividing both sides of Eq. (8) by introducing the angle d, such that: A cos d p , 2 A B2
B sin d p : 2 A B2
9
p A2 B 2 , and
10
This angle locates the diagonal plane, CA, of the linkage and is measured about C from OC. The solution to Eq. (8) is given by
J.M. McCarthy, R.M. Bodduluri / Mechanism and Machine Theory 35 (2000) 451±462
C c y arctan d2arccos p : 2 A B2
455
11
Clearly, for any given value of y there are two possible values for c. These are the two modes of solution for the linkage (Fig. 2). A singular con®guration is de®ned by the condition A 2+B 2ÿC 2=0, in which case, Eq. (11) has a single solution. Geometrically, this occurs when the moving pivot B falls on the linkage diagonal plane CA. The interaction of singular con®gurations with the modes of assembly yields three basic situations that can be distinguished by the range of movement of the input crank: y. There are three cases: 1. Fully rotatable input movement: There is no limit on the input angle y which means the mechanism is a crank-rocker or drag-link mechanism. In this case, there are no singular con®gurations and the two modes of assembly are completely distinct. It is possible for the design positions to be on dierent assemblies, which means the linkage simply cannot reach the other set of design positions unless the linkage is physically reassembled in the other mode. 2. One angular range of input movement: The input angle rocks between two bounding values 2y. Here there are two singular con®gurations located symmetrically above and below the OC plane. The linkage is a non-Grasho double rocker, and the singular con®gurations separate the two solution modes. In this case, if the design positions are on separate solution modes, then the driving and driven cranks must exchange roles in order to guide the linkage through the remaining positions. 3. Two angular ranges of input movement: In this case, the input angle rocks between a set of bounding values y 1 R y R y 2 above the OC plane which is one assembly, and between
Fig. 2. Two assemblies of spherical 4R linkage, (a) mode 1; (b) mode 2.
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another set of bounding values, ÿy 1 ry r ÿy 2, symmetrically located below this plane which forms the second assembly. This is the case of the rocker-crank and Grasho doublerocker linkages. Even if the design positions are on the same solution mode, they may be separated by a singular con®guration.
4. Filemon's construction Filemon [5] introduced a construction that identi®es moving pivots for the driving crank that ensure the design positions for a crank-rocker are on the same assembly. Given a choice for the moving pivot of the driven crank, wo, the associated ®xed pivot Go is known, and it is possible to determine the movement required of this crank to reach the design positions. Viewed from the coupler a line along this crank sweeps out two wedge shaped regions centered on wo. If the designer selects the driving moving pivot wi from outside of these regions, then the linkage cannot take on a singular con®guration between the design positions. Filemon showed this strategy ensures that the design positions are on the same assembly mode for a crank-rocker. This procedure can be applied to the design of spherical linkages by simply considering the joint axes of the linkage as de®ning planes through the ®xed point c, rather than lines. Assume that we have selected a driven moving pivot, wo. It takes the positions Wjo=[Aj]wo in the ®xed frame F in the three design positions. We view the driven crank from the moving body by inverting the positions of the ®xed pivot Go, given by: goj ATj G:
12
Let z12 and z23 be the angles measured around wo from g1o to g2o and from g2o to g3o, respectively. They are dihedral angles between the planes, L1, L2 and between L2, L3, respectively, that pass through gjo and wo (and the origin c). We now determine the angular range of movement of the driven crank, t, relative to the coupler. Assuming the angles zjk are between 2p, we have: 1. If sgnz12=sgnz23, then t=vz12v+vz23v; 2. If sgnz12 $sgnz23, then t=vzmaxv, where zmax is the angle with the greater magnitude. The angle, t, is bounded by two of the planes Lj and de®nes a pair wedge-shaped regions in the moving frame M. If the driving moving pivot, wi, is selected outside of this wedge-shaped region, then the resulting spherical 4R linkage must pass through the design positions before it hits a singular con®guration. This is the spherical version of Filemon's [5] construction (Fig. 3). Waldron [11,12] notes that, in the plane, it is possible for the range t=vz12v+vz23v to be greater than p. In this case, the region excluded by Filemon's construction is the entire moving body. This same situation occurs for spherical linkages. Our generalization of Waldron's three circle diagram solves this problem.
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457
Fig. 3. The spherical version of Filemon's construction: The planes L1 and L3 bound the region excluded from selection for driving moving pivots wi.
5. The spherical pole triangle Waldron [11,12] introduced a construction that identi®es the driven pivots wo for which the range of angular movement t is guaranteed to be less than p, thus ensuring a solution to Filemon's construction for the driving moving pivots wi. Waldron's construction depends on a fundamental kinematic relationship between the moving pivot of a planar RR chain and the relative position poles of the speci®ed planar design positions. An identical relationship exists for the moving axis of a spherical RR chain, and the relative rotation axes of the speci®ed design orientations. Given a pair of rotations [Aj] and [Ak] de®ning the frames Mj and Mk, the relative rotation [Ajk] from Mj to Mk is de®ned by: Ajk Ak ATj :
13
In our case, we obtain the three relative rotations [A12], [A23], and [A13]. Associated with each of these is a rotation axis Sjk and rotation angle fjk. The three axes, S12, S23, and S13 form what is known as the spherical pole triangle [1].
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In Filemon's construction, we have inverted the relationship between the ®xed and moving frames by considering the movement of the ®xed frame relative to the coupler. This is applied to the spherical pole triangle as follows. Consider the three inverted rotation matrices, [A Tj], j = 1, 2, 3. The relative rotations computed using these inverted rotations, yield the relative inverted rotation matrices: Rjk ATk Aj :
14
Notice that a relative inverted rotation is not simply the inverse of the relative rotation in Eq. (13). For our case, we have the relative inverted rotations [R12], [R23] and [R13]. Denote the rotation axes of these matrices as s12, s23, and s13. They form the spherical image pole triangle in M. If the moving frame is placed in the ®rst design position, M1, then the relative inverted rotations can be transformed to F to de®ne: R1jk A1 ATk Aj AT1 :
15
In this case, it is easy to determine that S12=[A1]s12 and S13=[A1]s13; that is, in the ®rst position M1, the relative poles S1k coincides with s1k. On the other hand, the relative pole s23 appears as the re¯ection of S23 through the plane containing S12 and S13. Waldron's construction is based on the planar version of the fact that each moving axis w views the two image poles sjn and snk in the rotation angle zjk/2 of the crank relative to the coupler. The proof for this statement is found in Bottema and Roth [1]. We use this to ®nd the equivalent to Waldron's three circle diagram for spherical 4R linkages. 6. The spherical version of Waldron's three circle diagram Waldron [11] notes that Filemon's construction in the plane has no solution for driving moving pivots, wi, when any one of the angles z12, z23, or z13 is greater than or equal to p. Waldron de®ned the set of points that view the sides of the pole triangle in p/2 and obtained three circles. Points outside of these circles are driven pivots for which Filemon's construction will ®nd satisfactory driving moving pivots. An even number of crossing of these circles also identi®es a region of acceptable driving moving pivots. In what follows, we de®ne and analyze the generalization of these circles. The relative rotation angle zjk of an spherical RR crank about its moving pivot, w, as the ¯oating link moves from orientation Mj to Mk can be computed directly from the rotation axes sjn and snk. The general result is that the dihedral angle between the plane Pj containing wsj and plane Pk containing wsk is zjk/2 (Fig. 4). zjk wo sjn wo snk cos : 16 j wo sjn kwo snk j 2 The points x for which zjk=p can now be seen to lie on three quadric cones: Cjk : x sjn x snk 0:
17
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459
Fig. 4. The moving pivot w views the axes s12 and s23 in the angle z13/2.
It is easy to see that each of the three cones Cjk passes through the pair of poles sjn and snk in a con®guration complete analogous to Waldron's three circle diagram. In our case, however, the cones, C12, C23 and C13, are general quadrics, and not circular (Fig. 5).
7. The cones Cjk We now study the shape of the cones Cjk. Let [Sjn] and [Snk] be the matrices obtain from the poles sjn and snk such that [Sjn]y=sjn y and [Snk]y=snk y, then Eq. (17) can be written as the quadratic form: Cjk :xT S Tjn Snk x 0:
18
The coecient matrix [S TjnSnk] is not symmetric but we can transpose this equation and add it to the original to obtain: Cjk :xT S Tjn Snk S Tnk Sjn x xT C x 0:
19
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Fig. 5. The spherical version of Waldron's three circle diagram: The three quadrics de®ne regions excluded from selection for the driven moving pivots wo.
The following analysis applies to each of these three quadrics C12, C23, and C13, therefore we will not index the matrix [C ] of the quadratic form in Eq. (19). The matric [C ] is symmetric and has the eigenvectors: e1 snk ÿ sjn ,
e2 sjn snk ,
and e3 sjn snk :
20
To see this we compute directly: Cjk e1 cos a 1e1 ,
Cjk e2 2 cos ae2 ,
and Cjk e3 cos a ÿ 1e3 ,
21
J.M. McCarthy, R.M. Bodduluri / Mechanism and Machine Theory 35 (2000) 451±462
461
where a is the angle between the vectors sjn and snk, that is cos a=sjnsnk. Please note that here a is used dierently than in Eq. (6). We now introduce a coordinate frame, M', with its x-axis directed along e1, its y-axis along e2, and its z-axis directed along e3. The result is the axes sjn and snk have the local coordinates: 9 8 9 8 a> a> > > > > sin > > > > > > 2> > > > ÿsin 2 > > = = < < 0 sjn , and s : 22 0 nk > > > > > > > > > > a> > > > cos a > > ; > : > ; : cos 2 2 Substituting this into Eq. (19), we obtain the canonical form of the quadric cone Cjk: a a a a ÿ z2 sin 2 0: Cjk :x 2 cos 2 y2 cos 2 ÿ sin 2 2 2 2 2
23
The intersection of this cone with the z = 1 plane yields an ellipse with semi-diameters a and b along the x and y directions, respectively, given by: a tan 2 a 2 : a2 tan 2 , and b2 a 2 1 ÿ tan 2 2
24
Notice that a is the angular width of the cone in the x±z plane. Introduce the angle, b, de®ned so tan b/2=b is the angular height of this cone in the y±z plane. When a is small the cone is circular in shape. As a increases, the angle b also increases, but at a faster rate, to the point that when a=p/2, we have b=p. In this case, the sheet on in the front half of the sphere contacts its antipodal sheet on the back of the sphere at the y-axis, 2e2. In fact, the cone becomes two planes intersecting along e2, separated by the angle a. As a increases beyond p/2 the coecient of y 2 changes sign, and the ellipse in the z = 1 plane becomes a hyperbola. In this case, we view the cone by determining its intersection with the x = 1 plane which is seen to be an ellipse surrounding the x-axis. Thus, the original cone has separated into left and right sheets. Finally, as a approaches p, the left and right sheets become circular in cross-section. We conclude that if a R p/2, then axes outside the cone Cjk are to be used for Filemon's construction: and when a > p/2, it is the axes inside of the cones that are to be used.
8. Conclusion In this paper, we show that central results of planar recti®cation theory, which guides the synthesis of singularity free planar 4R linkages, can be extended to spherical 4R linkage synthesis. Generalization of Filemon's construction and Waldron's three circle diagram provide a means to identify moving axes that ensure the coupler of a spherical 4R linkage passes through the speci®ed orientations while avoiding singular con®gurations. The focus here is on
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synthesis for three speci®ed orientations, however, the results apply to any number of goal orientations. References [1] O. Bottema, B. Roth, Theoretical Kinematics, North Holland Publ. Co. (reprinted 1990 by Dover Publications, New York), 1979. [2] C.H. Chiang, Kinematics of Spherical Mechanisms, Cambridge University Press, 1988. [3] H.J. Dowler, J. Duy, D. Tesar, A generalized study of three multiply separated positions in spherical kinematics, Mechanism and Machine Theory 11 (1976) 395±410. [4] H.J. Dowler, J. Duy, D. Tesar, A generalized study of four and ®ve multiply separated positions in spherical kinematics, Mechanism and Machine Theory 13 (4) (1978) 409±436. [5] E. Filemon, Useful ranges of centerpoint curves for design of crank-and-rocker linkages, Mechanism and Machine Theory 7 (1972) 47±53. [6] J.M. McCarthy, Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA, 1990. [7] J.M. Prentis, The pole triangle, Burmester theory, and order and branching problemsÐII: the branching problem, Mechanism and Machine Theory 26 (1) (1991) 31±39. [8] C.F. Reinholtz, G.N. Sandor, J. Duy, Branching analysis of spherical RRRR and spatial RCCC mechanisms, ASME J. Mechanisms, Transmissions and Automation in Design 108 (1986) 481±486. [9] G.N. Sandor, A.G. Erdman, Advanced Mechanism Design: Analysis and Synthesis, vol. 2, Prentice-Hall, New Jersey, 1984. [10] C.H. Suh, C.W. Radclie, Kinematics and Mechanisms Design, Wiley, New York, 1978. [11] K.J. Waldron, Elimination of the branch problem in graphical Burmester mechanism synthesis for four ®nitely separated positions, ASME Journal of Engineering for Industry 98 (1) (1976) 176±182. [12] K.J. Waldron, Graphical solution of the branch and order problems of linkage synthesis for multiply separated positions, ASME Journal of Engineering for Industry 99 (3) (1977) 591±597.