- Email: [email protected]
Branch and circuit defect elimination in spherical four-bar linkages
PII:
Mech. Mach. Theory Vol. 33, No. 5, pp. 491±504, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00078-5
BRANCH AND CIRCUIT DEFECT ELIMINATION IN SPHERICAL FOUR-BAR LINKAGES K. C. GUPTA{ Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60607-7022, U.S.A.
A. S. BELOIU United Conveyor Corporation, Waukegan, IL 60085, U.S.A. (Received 4 November 1996; in revised form 15 May 1997) AbstractÐThis paper presents algebraic-geometrical methods to eliminate branch and circuit defects in the synthesis of spherical four-bar mechanisms. Regions containing potentially circuit defective mechanism (PCDM) designs have been identi®ed. Exclusion of the PCDM region then becomes a sucient condition for the elimination of the circuit defect. For branch defect elimination, the approach of Gupta and Tinubu (Journal of Mechanisms, Transmissions and Automation in Design, 1982, 105, 641± 647 [1]) has been extended for spherical four-bar mechanisms. Furthermore, regions which yield designs with fully rotatable input-links have been identi®ed. Examples have been given for three and four precision point function generators. # 1998 Elsevier Science Ltd. ZusammenfassungÐIn diesem Aufsatz wurden einige algebraisch-geometrische Methoden zur Beseitigung von Verzweigungs- und Umkreisdefekten, bei der Synthese von sphaÈrischen viergliedrigen Getrieben dargestellt. Dabei wurden Getriebenbereiche erzeugt bei denen hoÈchstwahrscheinlich Umkreisdefekte auftreten. Dann das Entfernen dieser Bereichen stellt eine genuÈgende Bedingung fuÈr die Beseitigung von Umkreisdefekten. FuÈr die Beseitigung von Verzweigungsdefekten wurde die Methode von Guppta und Tinubu (Journal of Mechanisms, Transmissions and Automation in Design, 1982, 105, 641±647 [1]) auch fuÈr sphaÈrische viergliedrige Getriebe erweitert. Weiterhin wurden die Bereiche festgestellt die voÈllig drehbare Antriebsglieder ermoÈglichen. Es wurden Beispiele fuÈr spaÈrische viergliedrige Getriebe angegeben, welche fuÈr drei und vier vorgeschriebene Stellungen die entsprechende UÈbertragungfunktionen erzeugen. # 1998 Elsevier Science Ltd.
1. INTRODUCTION
Spherical four-bar mechanism is a basic type of spatial mechanism. The existing literature considers its analysis, synthesis and applications [2±8]. Its type determination has also attracted considerable attention [3, 9±17]. This paper presents, for the ®rst time, a comprehensive algebraicgeometrical theory for synthesizing spherical four-bar mechanisms which are without the branch and circuit defects, and optionally, with fully rotatable input-links. De®nitions of branch and circuit defects which rely upon disassembly and reassembly of the linkage, or mode changes at toggle positions, often lead to controversial interpretations. In this paper, the following mathematically precise de®nitions have been adopted. If the closure equation of the linkage is F(f, c) = 0, where f and c are, respectively, the input and output angles, then it can be veri®ed that Fc0sin m, where m is the transmission angle. The sign of the partial derivative Fc, or equivalently, the sign of the interior-corner-angle from the follower to the coupler link, does not change for a ``branch'' of the linkage. The branch defect occurs in a design when the signs of all Fc at the design positions are not the same. The circuit defect appears in those designs which possess multiple disjointed ranges for input-link rotation and, in addition, have input-link positions which belong to dierent input rotation ranges at the speci®ed design positions. Viewed from this perspective, the branch and circuit defects are fundamentally dierent types of defects. In fact, it will be seen that the property of complete input-link rotatability and the existence of disjointed rotation ranges for the input-link are like the two {To whom all correspondence should be addressed. 491
492
K. C. Gupta and A. S. Beloiu
Fig. 1. Spherical four-bar linkage ABCG shown on sphere with center O. Also shown are the re¯ections of points A, B, C and G as A, B, C and G, respectively.
sides of a coin. Although this paper focuses speci®cally on the synthesis of spherical four-bar function generators, the formulation can be easily extended for the problems of rigid body guidance and path generation. 2. SPHERICAL MECHANISMS AND CLOSURE EQUATION
A spherical linkage has all the axes of the revolute joints cointersecting in one point. Figure 1 shows an example of the spherical four-bar linkage. The linkage is GCBA (drawn with thick lines), and the center of the sphere, where the extensions of the revolute joint axes intersect, is denoted by ``O''. The link lengths in this case are circular segments of great circles of the sphere (the circular segments which complete the great circles are drawn on Fig. 1 with dashed lines). Note that the axes of the joints, OG, OC, OB and OA (drawn with dash±dot lines), as well as the sphere itself, do not physically exist, but are used only for the mathematical formulation. Attached to each link is a coordinate system [5] and four such systems fully describe the geometry of the spherical four-bar linkage. This description will be called henceforth the ``DH description''. According to the description, the ith link in a chain is located between the joint axes (i ÿ 1) and i; the link coordinate system, Xi Yi Zi O, attached to the ith link has the Zi axis along the axis of the ith joint, and the Xi axis along the common perpendicular between the (i ÿ 1)th and the ith joint axes. The angle of twist for each link, ai ÿ 1, is de®ned as the angle between axes (Zi ÿ 1 4Zi), and the joint angle yi is the angle between axes (Xi 4Xi + 1). The positive direction of Xi, [Xi_(Zi ÿ 1, Zi)], is given by the direction of motion of a right-hand screw that rotates by ai about Xi. If we apply the general DH description to the spherical linkage in Fig. 2 we have the following link twists: a1 (input GOC), a2 (coupler COB), a3 (follower BOA) and a4 (frame AOG); note that a4 (from OA to OG) is not marked on Fig. 2. The joint angles are as follows: y1 (input angle), y2 (between input and coupler links), y3 (between coupler and follower links) and y4 (between follower and frame links). In Fig. 1 are shown the arcs ai with thick lines and the arcs (2p ÿ ai) with dashed lines, which completes the description for four great circles of the sphere. Two adjacent great circles intersect at two points, thus de®ning eight intersection points: G, C, B, A and G, C, B, A. By placing the revolute joints at four of the eight points, one can obtain 16 kinematically equivalent linkages [3, 4]. Any link of length ai is kinematically equivalent to the link of length (2p ÿ ai), therefore, when given four points on the surface of the sphere which de®ne a spherical linkage, the shorter path (on the surface of the sphere, from one point to another) will always be considered. For example, if we want to de®ne the link AB, then we will go from point A upward to point B directly, and not consider the arc AAB. This rule is to be observed for the above-mentioned 16 equivalent linkages, and therefore each of the links has a twist angle of less than p.
Branch and circuit defect elimination
493
Fig. 2. DH axes for a spherical four-bar linkage. Link twists are a1 (input GOC), a2 (coupler COB), a3 (follower BOA) and a4 (frame AOG). The input angle is y1 and the output angle is y4.
Eight linkages, out of these 16, are the re¯ections of the other eight, e.g. GCBA is the re¯ection of GCBA and GCBA is the re¯ection of GCBA; thus only eight kinematically equivalent linkages exist for a given set of link lengths [8, 11]. Table 1, included here for completeness, shows the seven supplementary linkages which can be obtained from the original linkage (GCBA), by replacing two or four of its link twist angles by their respective supplementary arcs [3]. In this paper, the analytical approach for circuit defect elimination and rotatability, which is to be presented in the next section, will yield linkages with the following property: a1$(0, p/2), a2$(0, p), a3$(0, p/2) and a4$(0, p). Under these assumptions, the possible designs can have even number, including zero, of obtuse links (Chiang's class I) as well as odd number of obtuse links (Chiang's class II) [3]. Since these two classes cover the totality of spherical linkages with twists in the range (0, p), the solution space is not restricted. It can also be seen directly from Table 1 that these conditions do not actually restrict the solution space. If it is necessary to design a linkage with a1>p/2 or a3>p/2, this will be possible by identifying an equivalent linkage with the help of Table 1. The loop closure equation for the spherical four-bar linkage in Fig. 2 can be written in the following form [5]: sin a1 sin y1 sin a3 sin y4 ÿ sin a3 cos a1 sin a4 sin a1 cos a4 cos y1 cos y4 cos a3 cos a1 cos a4 ÿ sin a1 sin a4 cos y1 ÿ cos a2 0:
1
Table 1. Equivalent spherical four-bar linkages Four-bar loop GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G)
a1
a2
GC p±GC p±GC GC GC GC p±GC p±GC
CB CB p±CB p±CB CB p±CB p±CB CB
Link length
a3
a4
BA BA BA p±BA p±BA BA p±BA p±BA
AG p±AG AG AG p±AG p±AG p±AG AG
494
K. C. Gupta and A. S. Beloiu
We will change the notation for the input and output angles as y1 fi ;
y4 ci ;
2
because we will identify by single subscripts the precision point number (or reference shaft angles when it is zero) and by double subscripts the angular increments of the input and output angles from the reference position (1st) to the ith position. Introducing the ®rst position as the reference position we have 3 fi f0 f1i ; ci c0 c1i : Note that f11=c11=0. Substituting Equations (2) and (3) into Equation (1), and dividing through by cos a1 cos a3 (a1$p/2 and a3$p/2) we obtain tan a1 tan a3 sin f0 f1i sin c0 c1i ÿ tan a3 sin a4 tan a1 cos a4 cos f0 f1i cos c0 c1i cos a4 ÿtan a1 sin a4 cos f0 f1i ÿ
cos a2 0: cos a1 cos a3
4
Equation (4) is the design equation for a spherical four-bar function generator, and it has to hold for each considered precision point, i.e. for i = 1, 2, . . . . We have ®ve design parameters, namely: a1, a2, a3, f0, c0; we will assume that a4 is prescribed by the application. To develop an algebraic closure equation, we switch to a new set of parameters as follows: A1 tan a3 cos c0 ;
A2 tan a3 sin c0 ;
A3 tan a1 cos f0 ;
A4 tan a1 sin f0 ;
A5 cos a4 ÿ
cos a2 : cos a1 cos a3
5
The transformations that express the old parameters in terms of the new ones are q A2 ÿ1 2 2 tan a3 A1 A2 ; c0 2 tan A1 tan a3 q A4 tan a1 A23 A24 ; f0 2 tanÿ1 A3 tan a1 cos a4 ÿ A5 cos a2 qq : 1 A21 A22 1 A23 A24
6
Note that a1 and a3 are in the range (0, p/2), but a2 and a4 can be in the range (0, p). As discussed earlier, in view of Table 1, this does not pose any limitation in the physical design space where all links can be considered within (0, p). Substituting the new parameters A1, . . . , A5 from Equation (5) into Equation (4) and expanding, we obtain the closure equation in a bilinear form: Ui A3 Vi A4 A5 Wi where
7
Ui sin f1i sin c1i ÿ cos a4 cos f1i cos c1i A1 sin f1i cos c1i cos a4 cos f1i sin c1i A2 ÿ sin a4 cos f1i ; Vi cos f1i sin c1i cos a4 sin f1i cos c1i A1 cos f1i cos c1i ÿ cos a4 sin f1i sin c1i A2 sin a4 sin f1i ; Wi A1 sin a4 cos c1i ÿ A2 sin a4 sin c1i :
8
Equation (7) is linear in terms of A3, A4, A5 and the functions Ui, Vi, Wi are polynomials of order 1 in terms of A1 and A2.
Branch and circuit defect elimination
495
3. SYNTHESIS
The speci®cations for the design of a spherical four-bar function generator for three precision points are given as the sequence of incremental angle pairs (0, 0), (f12, c12), (f13, c13); by writing the design Equation (7) three times, we produce the following system of equations: 2 32 3 2 3 W1 A3 U1 V1 1 4 U2 V2 1 54 A4 5 4 W 2 5 : 9 A5 W3 U3 V3 1 The system can be solved for the unknowns A3, A4, A5 in terms of A1, A2. Solving the system (9) by Cramer's rule, the solution is A3 where
U1 De U2 U3 U1 D4 U2 U3
D3 ; De V1 V2 V3 W1 W2 W3
A4
D4 ; De
A5
D5 ; De
W1 V 1 1 1 1 ; D 3 W2 V 2 1 ; W3 V 3 1 1 U 1 V 1 W1 1 1 ; D5 U2 V2 W2 : U 3 V 3 W3 1
10
11
It can be seen that, in terms of the variables A1, A2, the orders of De, D3, D4 are 2 and the order of D5 is 3. For the three precision point case, we can choose both parameters A1, A2, and then calculate the other three, A3, A4, A5, from Equations (10) and (11); the linkage parameters are then determined from Equation (6). For the four precision point design [given as (0, 0), (f12, c12), (f13, c13), (f14, c14)] we cannot choose both A1 and A2, because an additional condition must be met. Therefore we write the design Equation (7) four times, which gives a linear system of four equations in three unknowns (A3, A4, A5); for nontrivial solutions the determinant of the augmented matrix has to be zero, thus U1 V1 1 W1 U2 V2 1 W2 12 U3 V3 1 W3 0; U4 V4 1 W4 and if we substitute Ui, Vi, Wi, i = 1, . . . , 4 from Equation (8) into Equation (12), we obtain the expression of a cubic in the A1±A2 plane which will be used to determine one of the parameters, A1 or A2, after the other one has been chosen. 4. CIRCUIT DEFECT AND ROTATABILITY
The circuit defect and rotatability will be addressed in terms of the existence of the angle y3, which has the expression cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 cos f : 13 cos y3 sin a2 sin a3 For the equality (13) to hold, the right hand side term needs to be bounded within the interval [ ÿ1, 1]. If for a given set of link lengths the right hand side lies outside the [ÿ1, 1] interval at one position, f, then the considered linkage cannot physically reach that position. The extreme values for cos y3 occur when f = 0 and f = p, therefore the conditions for full rotatability are cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 1; ÿ1 sin a2 sin a3 ÿ1
cos a2 cos a3 ÿ cos a1 cos a4 ÿ sin a1 sin a4 1; sin a2 sin a3
14
496
K. C. Gupta and A. S. Beloiu
which are equivalent to cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 2 ÿ 1 0; sin a2 sin a3 2 cos a2 cos a3 ÿ cos a1 cos a4 ÿ sin a1 sin a4 2 ÿ 1 0: sin a2 sin a3 2
15
We can combine inequalities (15) into one single inequality. If we have two inequalities u E0 and v E0, then the combined inequality uv e0 would hold when either the original inequalities are both satis®ed or are both violated. We will multiply both inequalities (15) by (sin a2 sin a3)2 [knowing that sin a2 sin a3$0, since a2$(0, p) and a3$(0, p/2)]; thus, the condition for rotatability or circuit defect in spherical linkages is cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 2 ÿ sin a2 sin a3 2 cos a2 cos a3 ÿcos a1 cos a4 ÿ sin a1 sin a4 2 ÿ sin a2 sin a3 2 0:
16
Let us change the sign in inequality (16): cos a2 cos a3 ÿ cos a1 cos a4 sin a1 cos a4 2 ÿ sin a2 sin a3 2 ÿ cos a2 cos a3 ÿcos a1 cos a4 ÿ sin a1 sin a4 2 sin a2 sin a3 2 0:
17
The inequality (17) is satis®ed when both inequalities (15) are satis®ed, i.e. the input-link is fully rotatable. It is also satis®ed when both inequalities (15) are violated, i.e. f = 0 and f = p do not exist and the linkage has two disjointed ranges of input-link rotation; such linkages are potentially circuit defective mechanisms (PCDM). If inequality (17) is violated, then the linkage has partial rotatability with a single range for input-link rotation. The following substitutions are going to be made into inequality (17), taking into account Equations (6) and (10) q q 2 A2 A D23 D24 tan a1 3 4 sin a1 q q q ; 1 A23 A24 D2e D23 D24 1 tan a1 2 1 1 De cos a1 q q q ; 2 2 2 2 1 A3 A4 De D23 D24 1 tan a1 q A21 A22 tan a3 sin a3 q q ; 1 A21 A22 1 tan a3 2 1 1 cos a3 q q ; 2 1 A21 A22 1 tan a3 cos a4 ÿ A5 De cos a4 ÿ D5 cos a2 qq qq ; 2 2 2 2 2 1 A1 A2 1 A3 A4 1 A1 A22 D2e D23 D24 sin a2 2 1 ÿ cos a2 2 :
18
After carrying out the expansions, the following expression in A1, A2 results: 1 H16 0; D4e where H16 is a 16th order polynomial in A1, A2 and has the following form:
19
Branch and circuit defect elimination
497
H16 4 sin a4 2 De cos a4 ÿ D5 ÿ De cos a4 1 A21 A22 2 1 A21 A22 2 D23 D24 ÿ De cos a4 ÿ D5 ÿ De cos a4 1 A21 A22 2 sin a4 2 1 A21 A22 2 D23 D24 ÿ 1 A21 2 A22 D2e D23 D24 ÿ De cos a4 ÿ D5 2 A21 A22 :
20
It is obvious that the following inequality (21) is equivalent to Equation (19), and thus is equivalent to Equation (17). H16 0:
21
In order to eliminate the circuit defect and obtain a complete picture of the rotatability of a spherical four-bar linkage, one has to plot, in the plane A1±A2, the boundary curve H16=0, which de®nes three types of regions: . Type A (marked as ``*''): H16<0 and both inequalities (15) are satis®ed, which generates fully rotatable linkages. . Type B (marked as `` ÿ ''): H16<0 and both inequalities (15) are violated, which generates the PCDM region, i.e. the region in which the circuit defect may occur. . Type C (marked as `` + ''): H16>0, which means that one of the inequalities (15) is satis®ed and the other is violated, generating linkages with partial input link rotatability with a single range of input rotation. Although all designs in the PCDM region (type B) may not be circuit defective, if this region is excluded then the elimination of circuit defect is ensured. Thus, exclusion of the PCDM region is a sucient (but not necessary) condition for the elimination of circuit defect. 5. BRANCH DEFECT IDENTIFIER FOR SPHERICAL LINKAGES
The development for the branch defect identi®er (BDI) for spherical linkages will extend the approach that was presented in Ref. [1] to eliminate the branch defect in planar and spatial bimodal function generators. Let us consider the loop closure Equation (7) rewritten as follows: F fi ; ci Ui A3 Vi A4 A5 ÿ Wi 0;
22
with Ui, Vi, Wi obtained from Equation (8) and for i = 1, . . . , N. The function G(fi, ci) = (@F/ @ci) will be used to identify and separate the branches of the linkage. The designed linkage will be free of branch defect, if the function G will have the same sign for all precision points (i.e. all the values Gi(f1i, c1i), i = 1, . . ., N, will have the same sign). Whenever the signs of functions Gi are not the same, this will mean that not all precision points belong to the same branch; we will consider such a linkage branch defective, although the precision points might, in some cases, be eventually reachable without disassembly. Taking the partial derivative of Equation (22) with respect to c1i we obtain Gi
@Ui @Vi @Wi A3 A4 ÿ ; @c1i @c1i @c1i
which becomes, after substituting for A3, A4 from Equation (10), Gi
@Ui D3 @Vi D4 @Wi ÿ : @c1i De @c1i De @c1i
23
The functions Gi, i = 1, . . . , N, de®ne in the A1±A2 plane the necessary BDI for spherical linkages. Therefore, when making the choice for A1 (and for A2 as well in the three precision point design), the regions where not all the Gi have the same sign must be avoided, in order to avoid generating a branch defective mechanism. The expression for Gi in Equation (23) is not in a polynomial form, which is the preferred one, because the numerical method used for plotting obtains signi®cant computing time
498
K. C. Gupta and A. S. Beloiu
Fig. 3. Plot H16=0. Region A (marked as ``*'') contains designs with fully rotatable input links; region B (marked as `` ÿ '') contains PCDM designs and region C (marked as `` + '') contains partially rotatable designs with a single range for input rotation.
Fig. 4. Plot of the cubics DeGi for branch defect for three precision points.
Branch and circuit defect elimination
499
Table 2. Circuit defective design for A1=ÿ10.0, A2=2.00, three positions. a1=45.7048; a2=8.4928; a3=84.3998; a4=120.0008; f0=160.7528; c0=168.6908 f [deg] 160.752 170.752 180.752 190.752 200.752 210.752 220.752 230.752 240.752 250.752 260.752 270.752 280.752 290.752 300.752 310.752 320.752 330.752 340.752 350.752 360.752 370.752 380.752 390.752 400.752 410.752 420.752 430.752 440.752 450.752 460.752 470.752 480.752 490.752 500.752 510.752
Input
Output for branch 1 c [deg] c1i [deg]
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
163.205
191.524 194.902 199.636 205.784 216.699
137.930 138.944 144.493 152.540
354.515 Linkage Linkage Linkage 22.834 26.212 30.946 37.094 48.009 Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage 329.240 330.254 335.803 343.850
Output for branch 2 c [deg] c1i [deg] 168.690 does not exist does not exist does not exist 198.690 208.813 216.518 221.610 220.690 does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist 146.104 155.274 161.150 165.713
0.000
30.000 40.123 47.828 52.920 52.000
337.414 346.584 352.460 357.023
decreases by implementing a root search based on the Sturm sequence [18]. The following observation will solve this problem: if in a region all Gi have the same sign, then all (DeGi) will also have the same sign, and the region is marked as ``*''. The remainder is the branch defective region marked as `` ÿ ''. Therefore, we will plot in the A1±A2 plane the functions De Gi
@Ui @Vi @Wi D3 D4 ÿ De ; @c1i @c1i @c1i
24
which are cubics in terms of the variables A1, A2. Note that, in terms of A1, A2, the orders of D3 and D4 are two, and the orders of Ui, Vi, Wi are one (taking the partial derivatives of Ui, Vi, Wi with respect to c1i does not change their orders in terms of A1, A2).
6. DESIGN EXAMPLES
The design procedure according to the criteria developed so far will be as follows: . use the speci®ed twist angle of the ground link, a4; . develop, in the A1±A2 plane and using the precision points (0,0), (f12, c12), (f13, c13), the polynomial H16 from Equation (20) for circuit defect elimination; . for three precision points: *
develop, in the A1±A2 plane, the cubics (DeG1), (DeG2) and (DeG3) from Equation (24), and identify the regions where all of them have the same sign;
500
K. C. Gupta and A. S. Beloiu Table 3. Branch defective design for A1=8.0, A2= ÿ 4.00, three positions. a1=29.6848; a2=172.6958; a3=83.6208; a4=120.0008; f0=138.6748; c0=333.4358 f [deg] 138.674 148.674 158.674 168.674 178.674 188.674 198.674 208.674 218.674 228.674 238.674 248.674 258.674 268.674 278.674 288.674 298.674 308.674 318.674 328.674 338.674 348.674 358.674 368.674 378.674 388.674 398.674 408.674 418.674 428.674 438.674 448.674 458.674 468.674 478.674 488.674
*
Input
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
Output for branch 1 c [deg] c1i [deg] 348.136 351.944 355.480 359.137 3.435 8.757 14.674 20.433 25.435 29.089 30.115
336.110 343.548
14.701 18.509 22.045 25.702 30.000 35.322 41.239 46.998 52.000 55.654 56.680 Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage 2.675 10.113
Output for branch 2 c [deg] c1i [deg]
does does does does does does does does does does does does does does does does does does does does does does does
333.435 338.144 343.755 349.699 355.251 359.810 3.579 7.107 10.804 15.110 21.178 not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist 330.876 330.282
0.000 4.709 10.320 16.264 21.816 26.375 30.144 33.672 37.369 41.675 47.743
357.441 356.847
choose both A1 and A2 such that the branch defect is avoided, and then eliminate also the circuit defect, with the help of the plot of H16 (exclude the portion of the PCDM region in the branch-free-zone);
. for four precision points: *
* *
*
develop, in the A1±A2 plane, the cubics (DeG1), (DeG2), (DeG3) and (DeG4) from Equation (24), and identify the regions where all of them have the same sign; develop, in the A1±A2 plane, the four precision point cubic-Equation (12)]; choose, say, A1, and then determine A2 from the cubic Equation (12), such that the linkage is free of branch defect (i.e. select a segment of the cubic (12) in the branch-free zone); eliminate the circuit defect using the plot of H16 to exclude the portion of cubic Equation (12) in the branch-free zone which is also in the PCDM region.
The data for the precision points are (08, 08), (408, 308) (808, 528) and (1208, 608); for the three precision points case we will use the ®rst three of the four. Design results for three and four precision points for a4=1208 will be presented and analyzed. 6.1. Three precision points For this case, Fig. 3 shows the plot of H16=0 for circuit defect, and Fig. 4 shows the cubics (DeGi), for branch defect. On the plot in Fig. 4, some of the boundaries have been erased, because in this case we have actually overlapped three plots, therefore, when crossing one boundary, only one of the cubics changes sign and this does not necessarily represent a crossover from a branch free to a branch defective region (or vice versa).
Branch and circuit defect elimination
501
Fig. 5. Plot of the cubics DeGi for branch defect and cubic in Equation (12) for four precision points.
Both plots (and the following one) are at the same scale, so that, when trying to synthesize a linkage the plots can be overlapped, thus giving a complete picture of the characteristics of the regions. The following circuit defective design shows the existence of PCDM regions in branchfree zones. The analysis for the resulting linkage, if the choice is made at A1= ÿ 10.0, A2=2.0, is represented in tabular form in Table 2. The precision points are underlined and boldfaced; it can be seen that this linkage has two disjointed ranges of motion, and that precision points belong to dierent input ranges, thus being unreachable without disassembly. Note that the precision points belong to the same branch (2nd) of the linkage. This design is circuit defective, but it is free from branch defect. The next analyzed design is a partially rotatable linkage (H16>0, see Fig. 3), which suers from branch defect (DeGi do not all have the same signs, see Fig. 4), obtained for A1=8.0, A2=4.00. From the output column in Table 3 it can be seen that the precision points do not belong all to the same branch. This design has branch defect but not the circuit defect. Two designs presented in the next section can also be considered three precision point designs. If all regions marked `` ÿ '' (PCDM region B) in Fig. 3 and those marked `` ÿ '' (branch defective) in Fig. 4 are excluded, then in the remainder region the elimination of both branch and circuit defects is ensured. There is a large overlap between these two (PCDM and branch defective) regions. Yet what is important is that portions of PCDM region can exist outside of the branch defective region. For other design examples, see Ref. [18]. 6.2. Four precision points We will use for this case the plot for H16=0 from Fig. 3, because if we have resolved the rotatability and circuit defect for three precision points, the conclusions will also hold for four precision points. The cubics (DeGi) for four positions are plotted on Fig. 5; also on the same graph, shown is the cubic Equation (12), which is plotted using a larger diameter for the points and marked by the word ``cubic''. Table 4 presents the analysis for a four precision point design; the choice for A1 has been made in a branch defect free region, A1=3.0; from the three possible solutions for A2 [it can be seen that a vertical line intersects the cubic Equation (12) in three points], the choice was made
502
K. C. Gupta and A. S. Beloiu Table 4. Circuit defective design for A1=3.0; A21ÿ 3.19, four positions. a1=34.7978; a2=162.9338; a3=77.1488; a4=120.0008; f0=120.6098; c0=313.1908 f [deg] 120.609 130.609 140.609 150.609 160.609 170.609 180.609 190.609 200.609 210.609 220.609 230.609 240.609 250.609 260.609 270.609 280.609 290.609 300.609 310.609 320.609 330.609 340.609 350.609 360.609 370.609 380.609 390.609 400.609 410.609 420.609 430.609 440.609 450.609 460.609 470.609
Input
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 10.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
Output for branch 1 c [deg] c1i [deg] 347.142 349.623 351.751 353.570 354.932
18.018 27.155 35.148 41.950 47.394 51.241 53.087 51.995
324.133 335.102 340.397 344.166
33.952 36.433 38.561 40.379 41.742 Linkage Linkage Linkage 64.828 73.965 81.958 88.760 94.204 98.051 99.897 98.805 Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage 10.943 21.912 27.207 30.975
Output for branch 2 c [deg] c1i [deg]
does does does
does does does does does does does does does does does does does does does does
313.190 318.811 325.765 333.895 343.190 not exist not exist not exist 5.190 6.634 8.492 10.658 13.190 16.243 20.143 25.734 not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist 315.325 307.647 307.009 309.132
0.000 5.620 12.575 20.705 30.000
52.000 53.444 55.302 57.468 60.000 63.053 66.953 72.543
2.135 354.457 353.819 355.941
at A21ÿ 3.19. This is a circuit defective design, which originated from a branch defect-free region. A fully rotatable linkage is presented in Table 5, which was obtained for A1=1.0 and A212.56 and is free of any type of defects. Segments of the four precision point cubic Equation (12) which do not lie in the PCDM region (marked `` ÿ '' in Fig. 3) and in the branch defective region (marked `` ÿ '' in Fig. 5) lead to designs which are free from both branch and circuit defects. Other design examples can be found in Ref. [18]. 7. CONCLUSION
Algebraic-geometrical methods have been presented to design spherical four-bar mechanisms which are free from branch and circuit defects. The circuit defective designs may come from the PCDM regions (marked as `` ÿ '' in Fig. 3), while the branch defective designs come from the branch defect regions (marked as `` ÿ '' in Fig. 4). These two defective regions have considerable overlap, but their union is bigger than their intersection. The union of the PCDM and branch defective regions includes the cumulative defective region. If this cumulative defective region is avoided in three and four precision point synthesis, then the elimination of branch and circuit defects is ensured. The acceptable designs may have fully or partially rotatable input-links and the procedure to incorporate the rotatability requirement in synthesis has been developed. The formulation presented in this paper can be extended to the design of spherical four-bar mechanisms for rigid body guidance and path generation, and these extensions will be discussed in future work.
Branch and circuit defect elimination
503
Table 5. Fully rotatable defect free design for A1=1.0, A212.56, four positions. a1=43.1968; a2=113.3348; a3=70.0118; a4=120.0008; f0=193.4138; c0=68.6708 f [deg] 193.413 203.413 213.413 223.413 233.413 243.413 253.413 263.413 273.413 283.413 293.413 303.413 313.413 323.413 333.413 343.413 353.413 363.413 373.413 383.413 393.413 403.413 413.413 423.413 433.413 443.413 453.413 463.413 473.413 483.413 493.413 503.413 513.413 523.413 533.413 543.413
Input
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
Output for branch 1 c [deg] c1i [deg] 68.670 76.393 84.121 91.609 98.670 105.176 111.046 116.226 120.670 124.309 127.025 128.607 128.670 126.515 120.868 109.615 90.752 67.411 48.442 36.813 30.226 26.533 24.548 23.654 23.531 24.010 25.009 26.503 28.503 31.050 34.208 38.054 42.662 48.079 54.296 61.221
0.000 7.723 15.451 22.939 30.000 36.506 42.375 47.556 52.000 55.639 58.355 59.937 60.000 57.845 52.198 40.945 22.082 358.741 339.772 328.143 321.556 317.863 315.878 314.984 314.861 315.340 316.339 317.833 319.833 322.280 325.538 329.384 333.992 339.409 345.626 352.551
Output for branch 2 c [deg] c1i [deg] 310.031 315.707 320.568 324.649 328.017 330.751 332.918 334.570 335.727 336.377 336.458 335.835 334.245 331.188 325.701 315.966 299.417 276.577 255.527 241.975 234.817 231.728 231.184 232.336 234.721 238.083 242.280 247.233 252.889 259.200 266.096 273.466 281.141 288.895 296.460 303.575
241.361 247.037 251.898 255.979 259.347 262.080 264.248 265.900 267.057 267.707 267.788 267.165 265.575 262.518 257.031 247.296 230.747 207.906 186.857 173.305 166.147 163.058 162.514 163.666 166.051 169.413 173.610 178.563 184.219 190.530 197.426 204.796 212.471 220.225 227.790 234.905
REFERENCES 1. Gupta, K. C. and Tinubu, S. O., Synthesis of bimodal function generating mechanisms without branch defect. Journal of Mechanisms, Transmissions and Automation in Design, 1982, 105, 641±647. 2. Angeles, J., in Spatial Kinematic Chains. Springer, Berlin, 1982. 3. Chiang, C. H., in Kinematics of Spherical Linkages. Cambridge University Press, Cambridge, 1988. 4. Duy, J., in Analysis of Mechanisms and Robot Manipulators. Wiley, New York, 1980. 5. Hartenberg, R. S. and Denavit, J., in Kinematic Synthesis of Linkages. McGraw-Hill, New York, 1964. 6. Larochelle, P. and McCarthy, J. M., in Design of spatial mechanisms. ASME Design Technical Conference Tutorial/ Workshop, Irvine, CA, 1996. 7. Mohan, Rao A. V., Sandor, G. N., Kohli, D. and Soni, A. H., Closed form synthesis of spatial function generating mechanism for the maximum number of precision points. Journal of Engineering for Industry, 1973, 95, 725±736. 8. Soni, A. H. and Harrisberger, L., The design of spherical drag-link mechanism. ASME Journal of Engineering for Industry, 1967, 89, 171±181. 9. Barker, C. and Lo, J., Classi®cation of spherical four-bar mechanisms. ASME Paper 86-DET-144, 1986. 10. Duditza, F. L. and Dittrich, G., Die Bedingungen fuÈr die UmlauaÈhigkeit sphaÈrischer viergliedriger Kurbelgetriebe. Industrie Anzeiger, 1969, 91(71), 1687±1690. 11. Freudenstein, F., On the determination of the type of spherical mechanisms. In Contemporary Problems in the Theory of Machines and Mechanisms. USSR Academy of Sciences, 1965, pp. 193±196. 12. Gosselin, C. and Angeles, J., RepreÂsentation graphique de la reÂgion de mobilite des meÂcanismes plans et spheÂriques aÁ barres articuleÂs. Mechanism and Machine Theory, 1987, 22, 557±562. 13. Gupta, K. C., Rotatability considerations for spherical four-bar linkages with applications to robot wrist design. Journal of Mechanisms, Transmissions and Automation in Design, 1986, 108, 387±391. 14. Gupta, K. C. and Ma, R., A direct rotatability criterion for spherical four-bar linkages. Journal of Mechanical Design, 1995, 117, 597±600.
504
K. C. Gupta and A. S. Beloiu
15. Murray, A. P. and McCarthy, J. M., A linkage map for spherical four position synthesis. In Proceeding of the 1995 ASME Design Engineering Technical Conferences. Boston, MA, 1995. 16. Savage, M. and Hall, A. S., Unique description of all spherical four-bar linkages. Journal of Engineering for Industry, 1970, 6(3), 563±566. 17. Soni, A. H., Unique description of all spherical four-bar linkages (discussion). Journal of Engineering for Industry, 1970, 89, 177±181. 18. Beloiu, A. S., Elimination of circuit defects in planar and spherical linkages. Doctoral Dissertation, Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL, 1996.
Mech. Mach. Theory Vol. 33, No. 5, pp. 491±504, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0094-114X/98 $19.00 + 0.00 S0094-114X(97)00078-5
BRANCH AND CIRCUIT DEFECT ELIMINATION IN SPHERICAL FOUR-BAR LINKAGES K. C. GUPTA{ Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60607-7022, U.S.A.
A. S. BELOIU United Conveyor Corporation, Waukegan, IL 60085, U.S.A. (Received 4 November 1996; in revised form 15 May 1997) AbstractÐThis paper presents algebraic-geometrical methods to eliminate branch and circuit defects in the synthesis of spherical four-bar mechanisms. Regions containing potentially circuit defective mechanism (PCDM) designs have been identi®ed. Exclusion of the PCDM region then becomes a sucient condition for the elimination of the circuit defect. For branch defect elimination, the approach of Gupta and Tinubu (Journal of Mechanisms, Transmissions and Automation in Design, 1982, 105, 641± 647 [1]) has been extended for spherical four-bar mechanisms. Furthermore, regions which yield designs with fully rotatable input-links have been identi®ed. Examples have been given for three and four precision point function generators. # 1998 Elsevier Science Ltd. ZusammenfassungÐIn diesem Aufsatz wurden einige algebraisch-geometrische Methoden zur Beseitigung von Verzweigungs- und Umkreisdefekten, bei der Synthese von sphaÈrischen viergliedrigen Getrieben dargestellt. Dabei wurden Getriebenbereiche erzeugt bei denen hoÈchstwahrscheinlich Umkreisdefekte auftreten. Dann das Entfernen dieser Bereichen stellt eine genuÈgende Bedingung fuÈr die Beseitigung von Umkreisdefekten. FuÈr die Beseitigung von Verzweigungsdefekten wurde die Methode von Guppta und Tinubu (Journal of Mechanisms, Transmissions and Automation in Design, 1982, 105, 641±647 [1]) auch fuÈr sphaÈrische viergliedrige Getriebe erweitert. Weiterhin wurden die Bereiche festgestellt die voÈllig drehbare Antriebsglieder ermoÈglichen. Es wurden Beispiele fuÈr spaÈrische viergliedrige Getriebe angegeben, welche fuÈr drei und vier vorgeschriebene Stellungen die entsprechende UÈbertragungfunktionen erzeugen. # 1998 Elsevier Science Ltd.
1. INTRODUCTION
Spherical four-bar mechanism is a basic type of spatial mechanism. The existing literature considers its analysis, synthesis and applications [2±8]. Its type determination has also attracted considerable attention [3, 9±17]. This paper presents, for the ®rst time, a comprehensive algebraicgeometrical theory for synthesizing spherical four-bar mechanisms which are without the branch and circuit defects, and optionally, with fully rotatable input-links. De®nitions of branch and circuit defects which rely upon disassembly and reassembly of the linkage, or mode changes at toggle positions, often lead to controversial interpretations. In this paper, the following mathematically precise de®nitions have been adopted. If the closure equation of the linkage is F(f, c) = 0, where f and c are, respectively, the input and output angles, then it can be veri®ed that Fc0sin m, where m is the transmission angle. The sign of the partial derivative Fc, or equivalently, the sign of the interior-corner-angle from the follower to the coupler link, does not change for a ``branch'' of the linkage. The branch defect occurs in a design when the signs of all Fc at the design positions are not the same. The circuit defect appears in those designs which possess multiple disjointed ranges for input-link rotation and, in addition, have input-link positions which belong to dierent input rotation ranges at the speci®ed design positions. Viewed from this perspective, the branch and circuit defects are fundamentally dierent types of defects. In fact, it will be seen that the property of complete input-link rotatability and the existence of disjointed rotation ranges for the input-link are like the two {To whom all correspondence should be addressed. 491
492
K. C. Gupta and A. S. Beloiu
Fig. 1. Spherical four-bar linkage ABCG shown on sphere with center O. Also shown are the re¯ections of points A, B, C and G as A, B, C and G, respectively.
sides of a coin. Although this paper focuses speci®cally on the synthesis of spherical four-bar function generators, the formulation can be easily extended for the problems of rigid body guidance and path generation. 2. SPHERICAL MECHANISMS AND CLOSURE EQUATION
A spherical linkage has all the axes of the revolute joints cointersecting in one point. Figure 1 shows an example of the spherical four-bar linkage. The linkage is GCBA (drawn with thick lines), and the center of the sphere, where the extensions of the revolute joint axes intersect, is denoted by ``O''. The link lengths in this case are circular segments of great circles of the sphere (the circular segments which complete the great circles are drawn on Fig. 1 with dashed lines). Note that the axes of the joints, OG, OC, OB and OA (drawn with dash±dot lines), as well as the sphere itself, do not physically exist, but are used only for the mathematical formulation. Attached to each link is a coordinate system [5] and four such systems fully describe the geometry of the spherical four-bar linkage. This description will be called henceforth the ``DH description''. According to the description, the ith link in a chain is located between the joint axes (i ÿ 1) and i; the link coordinate system, Xi Yi Zi O, attached to the ith link has the Zi axis along the axis of the ith joint, and the Xi axis along the common perpendicular between the (i ÿ 1)th and the ith joint axes. The angle of twist for each link, ai ÿ 1, is de®ned as the angle between axes (Zi ÿ 1 4Zi), and the joint angle yi is the angle between axes (Xi 4Xi + 1). The positive direction of Xi, [Xi_(Zi ÿ 1, Zi)], is given by the direction of motion of a right-hand screw that rotates by ai about Xi. If we apply the general DH description to the spherical linkage in Fig. 2 we have the following link twists: a1 (input GOC), a2 (coupler COB), a3 (follower BOA) and a4 (frame AOG); note that a4 (from OA to OG) is not marked on Fig. 2. The joint angles are as follows: y1 (input angle), y2 (between input and coupler links), y3 (between coupler and follower links) and y4 (between follower and frame links). In Fig. 1 are shown the arcs ai with thick lines and the arcs (2p ÿ ai) with dashed lines, which completes the description for four great circles of the sphere. Two adjacent great circles intersect at two points, thus de®ning eight intersection points: G, C, B, A and G, C, B, A. By placing the revolute joints at four of the eight points, one can obtain 16 kinematically equivalent linkages [3, 4]. Any link of length ai is kinematically equivalent to the link of length (2p ÿ ai), therefore, when given four points on the surface of the sphere which de®ne a spherical linkage, the shorter path (on the surface of the sphere, from one point to another) will always be considered. For example, if we want to de®ne the link AB, then we will go from point A upward to point B directly, and not consider the arc AAB. This rule is to be observed for the above-mentioned 16 equivalent linkages, and therefore each of the links has a twist angle of less than p.
Branch and circuit defect elimination
493
Fig. 2. DH axes for a spherical four-bar linkage. Link twists are a1 (input GOC), a2 (coupler COB), a3 (follower BOA) and a4 (frame AOG). The input angle is y1 and the output angle is y4.
Eight linkages, out of these 16, are the re¯ections of the other eight, e.g. GCBA is the re¯ection of GCBA and GCBA is the re¯ection of GCBA; thus only eight kinematically equivalent linkages exist for a given set of link lengths [8, 11]. Table 1, included here for completeness, shows the seven supplementary linkages which can be obtained from the original linkage (GCBA), by replacing two or four of its link twist angles by their respective supplementary arcs [3]. In this paper, the analytical approach for circuit defect elimination and rotatability, which is to be presented in the next section, will yield linkages with the following property: a1$(0, p/2), a2$(0, p), a3$(0, p/2) and a4$(0, p). Under these assumptions, the possible designs can have even number, including zero, of obtuse links (Chiang's class I) as well as odd number of obtuse links (Chiang's class II) [3]. Since these two classes cover the totality of spherical linkages with twists in the range (0, p), the solution space is not restricted. It can also be seen directly from Table 1 that these conditions do not actually restrict the solution space. If it is necessary to design a linkage with a1>p/2 or a3>p/2, this will be possible by identifying an equivalent linkage with the help of Table 1. The loop closure equation for the spherical four-bar linkage in Fig. 2 can be written in the following form [5]: sin a1 sin y1 sin a3 sin y4 ÿ sin a3 cos a1 sin a4 sin a1 cos a4 cos y1 cos y4 cos a3 cos a1 cos a4 ÿ sin a1 sin a4 cos y1 ÿ cos a2 0:
1
Table 1. Equivalent spherical four-bar linkages Four-bar loop GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G) GCBA(G)
a1
a2
GC p±GC p±GC GC GC GC p±GC p±GC
CB CB p±CB p±CB CB p±CB p±CB CB
Link length
a3
a4
BA BA BA p±BA p±BA BA p±BA p±BA
AG p±AG AG AG p±AG p±AG p±AG AG
494
K. C. Gupta and A. S. Beloiu
We will change the notation for the input and output angles as y1 fi ;
y4 ci ;
2
because we will identify by single subscripts the precision point number (or reference shaft angles when it is zero) and by double subscripts the angular increments of the input and output angles from the reference position (1st) to the ith position. Introducing the ®rst position as the reference position we have 3 fi f0 f1i ; ci c0 c1i : Note that f11=c11=0. Substituting Equations (2) and (3) into Equation (1), and dividing through by cos a1 cos a3 (a1$p/2 and a3$p/2) we obtain tan a1 tan a3 sin f0 f1i sin c0 c1i ÿ tan a3 sin a4 tan a1 cos a4 cos f0 f1i cos c0 c1i cos a4 ÿtan a1 sin a4 cos f0 f1i ÿ
cos a2 0: cos a1 cos a3
4
Equation (4) is the design equation for a spherical four-bar function generator, and it has to hold for each considered precision point, i.e. for i = 1, 2, . . . . We have ®ve design parameters, namely: a1, a2, a3, f0, c0; we will assume that a4 is prescribed by the application. To develop an algebraic closure equation, we switch to a new set of parameters as follows: A1 tan a3 cos c0 ;
A2 tan a3 sin c0 ;
A3 tan a1 cos f0 ;
A4 tan a1 sin f0 ;
A5 cos a4 ÿ
cos a2 : cos a1 cos a3
5
The transformations that express the old parameters in terms of the new ones are q A2 ÿ1 2 2 tan a3 A1 A2 ; c0 2 tan A1 tan a3 q A4 tan a1 A23 A24 ; f0 2 tanÿ1 A3 tan a1 cos a4 ÿ A5 cos a2 qq : 1 A21 A22 1 A23 A24
6
Note that a1 and a3 are in the range (0, p/2), but a2 and a4 can be in the range (0, p). As discussed earlier, in view of Table 1, this does not pose any limitation in the physical design space where all links can be considered within (0, p). Substituting the new parameters A1, . . . , A5 from Equation (5) into Equation (4) and expanding, we obtain the closure equation in a bilinear form: Ui A3 Vi A4 A5 Wi where
7
Ui sin f1i sin c1i ÿ cos a4 cos f1i cos c1i A1 sin f1i cos c1i cos a4 cos f1i sin c1i A2 ÿ sin a4 cos f1i ; Vi cos f1i sin c1i cos a4 sin f1i cos c1i A1 cos f1i cos c1i ÿ cos a4 sin f1i sin c1i A2 sin a4 sin f1i ; Wi A1 sin a4 cos c1i ÿ A2 sin a4 sin c1i :
8
Equation (7) is linear in terms of A3, A4, A5 and the functions Ui, Vi, Wi are polynomials of order 1 in terms of A1 and A2.
Branch and circuit defect elimination
495
3. SYNTHESIS
The speci®cations for the design of a spherical four-bar function generator for three precision points are given as the sequence of incremental angle pairs (0, 0), (f12, c12), (f13, c13); by writing the design Equation (7) three times, we produce the following system of equations: 2 32 3 2 3 W1 A3 U1 V1 1 4 U2 V2 1 54 A4 5 4 W 2 5 : 9 A5 W3 U3 V3 1 The system can be solved for the unknowns A3, A4, A5 in terms of A1, A2. Solving the system (9) by Cramer's rule, the solution is A3 where
U1 De U2 U3 U1 D4 U2 U3
D3 ; De V1 V2 V3 W1 W2 W3
A4
D4 ; De
A5
D5 ; De
W1 V 1 1 1 1 ; D 3 W2 V 2 1 ; W3 V 3 1 1 U 1 V 1 W1 1 1 ; D5 U2 V2 W2 : U 3 V 3 W3 1
10
11
It can be seen that, in terms of the variables A1, A2, the orders of De, D3, D4 are 2 and the order of D5 is 3. For the three precision point case, we can choose both parameters A1, A2, and then calculate the other three, A3, A4, A5, from Equations (10) and (11); the linkage parameters are then determined from Equation (6). For the four precision point design [given as (0, 0), (f12, c12), (f13, c13), (f14, c14)] we cannot choose both A1 and A2, because an additional condition must be met. Therefore we write the design Equation (7) four times, which gives a linear system of four equations in three unknowns (A3, A4, A5); for nontrivial solutions the determinant of the augmented matrix has to be zero, thus U1 V1 1 W1 U2 V2 1 W2 12 U3 V3 1 W3 0; U4 V4 1 W4 and if we substitute Ui, Vi, Wi, i = 1, . . . , 4 from Equation (8) into Equation (12), we obtain the expression of a cubic in the A1±A2 plane which will be used to determine one of the parameters, A1 or A2, after the other one has been chosen. 4. CIRCUIT DEFECT AND ROTATABILITY
The circuit defect and rotatability will be addressed in terms of the existence of the angle y3, which has the expression cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 cos f : 13 cos y3 sin a2 sin a3 For the equality (13) to hold, the right hand side term needs to be bounded within the interval [ ÿ1, 1]. If for a given set of link lengths the right hand side lies outside the [ÿ1, 1] interval at one position, f, then the considered linkage cannot physically reach that position. The extreme values for cos y3 occur when f = 0 and f = p, therefore the conditions for full rotatability are cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 1; ÿ1 sin a2 sin a3 ÿ1
cos a2 cos a3 ÿ cos a1 cos a4 ÿ sin a1 sin a4 1; sin a2 sin a3
14
496
K. C. Gupta and A. S. Beloiu
which are equivalent to cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 2 ÿ 1 0; sin a2 sin a3 2 cos a2 cos a3 ÿ cos a1 cos a4 ÿ sin a1 sin a4 2 ÿ 1 0: sin a2 sin a3 2
15
We can combine inequalities (15) into one single inequality. If we have two inequalities u E0 and v E0, then the combined inequality uv e0 would hold when either the original inequalities are both satis®ed or are both violated. We will multiply both inequalities (15) by (sin a2 sin a3)2 [knowing that sin a2 sin a3$0, since a2$(0, p) and a3$(0, p/2)]; thus, the condition for rotatability or circuit defect in spherical linkages is cos a2 cos a3 ÿ cos a1 cos a4 sin a1 sin a4 2 ÿ sin a2 sin a3 2 cos a2 cos a3 ÿcos a1 cos a4 ÿ sin a1 sin a4 2 ÿ sin a2 sin a3 2 0:
16
Let us change the sign in inequality (16): cos a2 cos a3 ÿ cos a1 cos a4 sin a1 cos a4 2 ÿ sin a2 sin a3 2 ÿ cos a2 cos a3 ÿcos a1 cos a4 ÿ sin a1 sin a4 2 sin a2 sin a3 2 0:
17
The inequality (17) is satis®ed when both inequalities (15) are satis®ed, i.e. the input-link is fully rotatable. It is also satis®ed when both inequalities (15) are violated, i.e. f = 0 and f = p do not exist and the linkage has two disjointed ranges of input-link rotation; such linkages are potentially circuit defective mechanisms (PCDM). If inequality (17) is violated, then the linkage has partial rotatability with a single range for input-link rotation. The following substitutions are going to be made into inequality (17), taking into account Equations (6) and (10) q q 2 A2 A D23 D24 tan a1 3 4 sin a1 q q q ; 1 A23 A24 D2e D23 D24 1 tan a1 2 1 1 De cos a1 q q q ; 2 2 2 2 1 A3 A4 De D23 D24 1 tan a1 q A21 A22 tan a3 sin a3 q q ; 1 A21 A22 1 tan a3 2 1 1 cos a3 q q ; 2 1 A21 A22 1 tan a3 cos a4 ÿ A5 De cos a4 ÿ D5 cos a2 qq qq ; 2 2 2 2 2 1 A1 A2 1 A3 A4 1 A1 A22 D2e D23 D24 sin a2 2 1 ÿ cos a2 2 :
18
After carrying out the expansions, the following expression in A1, A2 results: 1 H16 0; D4e where H16 is a 16th order polynomial in A1, A2 and has the following form:
19
Branch and circuit defect elimination
497
H16 4 sin a4 2 De cos a4 ÿ D5 ÿ De cos a4 1 A21 A22 2 1 A21 A22 2 D23 D24 ÿ De cos a4 ÿ D5 ÿ De cos a4 1 A21 A22 2 sin a4 2 1 A21 A22 2 D23 D24 ÿ 1 A21 2 A22 D2e D23 D24 ÿ De cos a4 ÿ D5 2 A21 A22 :
20
It is obvious that the following inequality (21) is equivalent to Equation (19), and thus is equivalent to Equation (17). H16 0:
21
In order to eliminate the circuit defect and obtain a complete picture of the rotatability of a spherical four-bar linkage, one has to plot, in the plane A1±A2, the boundary curve H16=0, which de®nes three types of regions: . Type A (marked as ``*''): H16<0 and both inequalities (15) are satis®ed, which generates fully rotatable linkages. . Type B (marked as `` ÿ ''): H16<0 and both inequalities (15) are violated, which generates the PCDM region, i.e. the region in which the circuit defect may occur. . Type C (marked as `` + ''): H16>0, which means that one of the inequalities (15) is satis®ed and the other is violated, generating linkages with partial input link rotatability with a single range of input rotation. Although all designs in the PCDM region (type B) may not be circuit defective, if this region is excluded then the elimination of circuit defect is ensured. Thus, exclusion of the PCDM region is a sucient (but not necessary) condition for the elimination of circuit defect. 5. BRANCH DEFECT IDENTIFIER FOR SPHERICAL LINKAGES
The development for the branch defect identi®er (BDI) for spherical linkages will extend the approach that was presented in Ref. [1] to eliminate the branch defect in planar and spatial bimodal function generators. Let us consider the loop closure Equation (7) rewritten as follows: F fi ; ci Ui A3 Vi A4 A5 ÿ Wi 0;
22
with Ui, Vi, Wi obtained from Equation (8) and for i = 1, . . . , N. The function G(fi, ci) = (@F/ @ci) will be used to identify and separate the branches of the linkage. The designed linkage will be free of branch defect, if the function G will have the same sign for all precision points (i.e. all the values Gi(f1i, c1i), i = 1, . . ., N, will have the same sign). Whenever the signs of functions Gi are not the same, this will mean that not all precision points belong to the same branch; we will consider such a linkage branch defective, although the precision points might, in some cases, be eventually reachable without disassembly. Taking the partial derivative of Equation (22) with respect to c1i we obtain Gi
@Ui @Vi @Wi A3 A4 ÿ ; @c1i @c1i @c1i
which becomes, after substituting for A3, A4 from Equation (10), Gi
@Ui D3 @Vi D4 @Wi ÿ : @c1i De @c1i De @c1i
23
The functions Gi, i = 1, . . . , N, de®ne in the A1±A2 plane the necessary BDI for spherical linkages. Therefore, when making the choice for A1 (and for A2 as well in the three precision point design), the regions where not all the Gi have the same sign must be avoided, in order to avoid generating a branch defective mechanism. The expression for Gi in Equation (23) is not in a polynomial form, which is the preferred one, because the numerical method used for plotting obtains signi®cant computing time
498
K. C. Gupta and A. S. Beloiu
Fig. 3. Plot H16=0. Region A (marked as ``*'') contains designs with fully rotatable input links; region B (marked as `` ÿ '') contains PCDM designs and region C (marked as `` + '') contains partially rotatable designs with a single range for input rotation.
Fig. 4. Plot of the cubics DeGi for branch defect for three precision points.
Branch and circuit defect elimination
499
Table 2. Circuit defective design for A1=ÿ10.0, A2=2.00, three positions. a1=45.7048; a2=8.4928; a3=84.3998; a4=120.0008; f0=160.7528; c0=168.6908 f [deg] 160.752 170.752 180.752 190.752 200.752 210.752 220.752 230.752 240.752 250.752 260.752 270.752 280.752 290.752 300.752 310.752 320.752 330.752 340.752 350.752 360.752 370.752 380.752 390.752 400.752 410.752 420.752 430.752 440.752 450.752 460.752 470.752 480.752 490.752 500.752 510.752
Input
Output for branch 1 c [deg] c1i [deg]
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
163.205
191.524 194.902 199.636 205.784 216.699
137.930 138.944 144.493 152.540
354.515 Linkage Linkage Linkage 22.834 26.212 30.946 37.094 48.009 Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage 329.240 330.254 335.803 343.850
Output for branch 2 c [deg] c1i [deg] 168.690 does not exist does not exist does not exist 198.690 208.813 216.518 221.610 220.690 does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist does not exist 146.104 155.274 161.150 165.713
0.000
30.000 40.123 47.828 52.920 52.000
337.414 346.584 352.460 357.023
decreases by implementing a root search based on the Sturm sequence [18]. The following observation will solve this problem: if in a region all Gi have the same sign, then all (DeGi) will also have the same sign, and the region is marked as ``*''. The remainder is the branch defective region marked as `` ÿ ''. Therefore, we will plot in the A1±A2 plane the functions De Gi
@Ui @Vi @Wi D3 D4 ÿ De ; @c1i @c1i @c1i
24
which are cubics in terms of the variables A1, A2. Note that, in terms of A1, A2, the orders of D3 and D4 are two, and the orders of Ui, Vi, Wi are one (taking the partial derivatives of Ui, Vi, Wi with respect to c1i does not change their orders in terms of A1, A2).
6. DESIGN EXAMPLES
The design procedure according to the criteria developed so far will be as follows: . use the speci®ed twist angle of the ground link, a4; . develop, in the A1±A2 plane and using the precision points (0,0), (f12, c12), (f13, c13), the polynomial H16 from Equation (20) for circuit defect elimination; . for three precision points: *
develop, in the A1±A2 plane, the cubics (DeG1), (DeG2) and (DeG3) from Equation (24), and identify the regions where all of them have the same sign;
500
K. C. Gupta and A. S. Beloiu Table 3. Branch defective design for A1=8.0, A2= ÿ 4.00, three positions. a1=29.6848; a2=172.6958; a3=83.6208; a4=120.0008; f0=138.6748; c0=333.4358 f [deg] 138.674 148.674 158.674 168.674 178.674 188.674 198.674 208.674 218.674 228.674 238.674 248.674 258.674 268.674 278.674 288.674 298.674 308.674 318.674 328.674 338.674 348.674 358.674 368.674 378.674 388.674 398.674 408.674 418.674 428.674 438.674 448.674 458.674 468.674 478.674 488.674
*
Input
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
Output for branch 1 c [deg] c1i [deg] 348.136 351.944 355.480 359.137 3.435 8.757 14.674 20.433 25.435 29.089 30.115
336.110 343.548
14.701 18.509 22.045 25.702 30.000 35.322 41.239 46.998 52.000 55.654 56.680 Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage 2.675 10.113
Output for branch 2 c [deg] c1i [deg]
does does does does does does does does does does does does does does does does does does does does does does does
333.435 338.144 343.755 349.699 355.251 359.810 3.579 7.107 10.804 15.110 21.178 not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist 330.876 330.282
0.000 4.709 10.320 16.264 21.816 26.375 30.144 33.672 37.369 41.675 47.743
357.441 356.847
choose both A1 and A2 such that the branch defect is avoided, and then eliminate also the circuit defect, with the help of the plot of H16 (exclude the portion of the PCDM region in the branch-free-zone);
. for four precision points: *
* *
*
develop, in the A1±A2 plane, the cubics (DeG1), (DeG2), (DeG3) and (DeG4) from Equation (24), and identify the regions where all of them have the same sign; develop, in the A1±A2 plane, the four precision point cubic-Equation (12)]; choose, say, A1, and then determine A2 from the cubic Equation (12), such that the linkage is free of branch defect (i.e. select a segment of the cubic (12) in the branch-free zone); eliminate the circuit defect using the plot of H16 to exclude the portion of cubic Equation (12) in the branch-free zone which is also in the PCDM region.
The data for the precision points are (08, 08), (408, 308) (808, 528) and (1208, 608); for the three precision points case we will use the ®rst three of the four. Design results for three and four precision points for a4=1208 will be presented and analyzed. 6.1. Three precision points For this case, Fig. 3 shows the plot of H16=0 for circuit defect, and Fig. 4 shows the cubics (DeGi), for branch defect. On the plot in Fig. 4, some of the boundaries have been erased, because in this case we have actually overlapped three plots, therefore, when crossing one boundary, only one of the cubics changes sign and this does not necessarily represent a crossover from a branch free to a branch defective region (or vice versa).
Branch and circuit defect elimination
501
Fig. 5. Plot of the cubics DeGi for branch defect and cubic in Equation (12) for four precision points.
Both plots (and the following one) are at the same scale, so that, when trying to synthesize a linkage the plots can be overlapped, thus giving a complete picture of the characteristics of the regions. The following circuit defective design shows the existence of PCDM regions in branchfree zones. The analysis for the resulting linkage, if the choice is made at A1= ÿ 10.0, A2=2.0, is represented in tabular form in Table 2. The precision points are underlined and boldfaced; it can be seen that this linkage has two disjointed ranges of motion, and that precision points belong to dierent input ranges, thus being unreachable without disassembly. Note that the precision points belong to the same branch (2nd) of the linkage. This design is circuit defective, but it is free from branch defect. The next analyzed design is a partially rotatable linkage (H16>0, see Fig. 3), which suers from branch defect (DeGi do not all have the same signs, see Fig. 4), obtained for A1=8.0, A2=4.00. From the output column in Table 3 it can be seen that the precision points do not belong all to the same branch. This design has branch defect but not the circuit defect. Two designs presented in the next section can also be considered three precision point designs. If all regions marked `` ÿ '' (PCDM region B) in Fig. 3 and those marked `` ÿ '' (branch defective) in Fig. 4 are excluded, then in the remainder region the elimination of both branch and circuit defects is ensured. There is a large overlap between these two (PCDM and branch defective) regions. Yet what is important is that portions of PCDM region can exist outside of the branch defective region. For other design examples, see Ref. [18]. 6.2. Four precision points We will use for this case the plot for H16=0 from Fig. 3, because if we have resolved the rotatability and circuit defect for three precision points, the conclusions will also hold for four precision points. The cubics (DeGi) for four positions are plotted on Fig. 5; also on the same graph, shown is the cubic Equation (12), which is plotted using a larger diameter for the points and marked by the word ``cubic''. Table 4 presents the analysis for a four precision point design; the choice for A1 has been made in a branch defect free region, A1=3.0; from the three possible solutions for A2 [it can be seen that a vertical line intersects the cubic Equation (12) in three points], the choice was made
502
K. C. Gupta and A. S. Beloiu Table 4. Circuit defective design for A1=3.0; A21ÿ 3.19, four positions. a1=34.7978; a2=162.9338; a3=77.1488; a4=120.0008; f0=120.6098; c0=313.1908 f [deg] 120.609 130.609 140.609 150.609 160.609 170.609 180.609 190.609 200.609 210.609 220.609 230.609 240.609 250.609 260.609 270.609 280.609 290.609 300.609 310.609 320.609 330.609 340.609 350.609 360.609 370.609 380.609 390.609 400.609 410.609 420.609 430.609 440.609 450.609 460.609 470.609
Input
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 10.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
Output for branch 1 c [deg] c1i [deg] 347.142 349.623 351.751 353.570 354.932
18.018 27.155 35.148 41.950 47.394 51.241 53.087 51.995
324.133 335.102 340.397 344.166
33.952 36.433 38.561 40.379 41.742 Linkage Linkage Linkage 64.828 73.965 81.958 88.760 94.204 98.051 99.897 98.805 Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage Linkage 10.943 21.912 27.207 30.975
Output for branch 2 c [deg] c1i [deg]
does does does
does does does does does does does does does does does does does does does does
313.190 318.811 325.765 333.895 343.190 not exist not exist not exist 5.190 6.634 8.492 10.658 13.190 16.243 20.143 25.734 not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist not exist 315.325 307.647 307.009 309.132
0.000 5.620 12.575 20.705 30.000
52.000 53.444 55.302 57.468 60.000 63.053 66.953 72.543
2.135 354.457 353.819 355.941
at A21ÿ 3.19. This is a circuit defective design, which originated from a branch defect-free region. A fully rotatable linkage is presented in Table 5, which was obtained for A1=1.0 and A212.56 and is free of any type of defects. Segments of the four precision point cubic Equation (12) which do not lie in the PCDM region (marked `` ÿ '' in Fig. 3) and in the branch defective region (marked `` ÿ '' in Fig. 5) lead to designs which are free from both branch and circuit defects. Other design examples can be found in Ref. [18]. 7. CONCLUSION
Algebraic-geometrical methods have been presented to design spherical four-bar mechanisms which are free from branch and circuit defects. The circuit defective designs may come from the PCDM regions (marked as `` ÿ '' in Fig. 3), while the branch defective designs come from the branch defect regions (marked as `` ÿ '' in Fig. 4). These two defective regions have considerable overlap, but their union is bigger than their intersection. The union of the PCDM and branch defective regions includes the cumulative defective region. If this cumulative defective region is avoided in three and four precision point synthesis, then the elimination of branch and circuit defects is ensured. The acceptable designs may have fully or partially rotatable input-links and the procedure to incorporate the rotatability requirement in synthesis has been developed. The formulation presented in this paper can be extended to the design of spherical four-bar mechanisms for rigid body guidance and path generation, and these extensions will be discussed in future work.
Branch and circuit defect elimination
503
Table 5. Fully rotatable defect free design for A1=1.0, A212.56, four positions. a1=43.1968; a2=113.3348; a3=70.0118; a4=120.0008; f0=193.4138; c0=68.6708 f [deg] 193.413 203.413 213.413 223.413 233.413 243.413 253.413 263.413 273.413 283.413 293.413 303.413 313.413 323.413 333.413 343.413 353.413 363.413 373.413 383.413 393.413 403.413 413.413 423.413 433.413 443.413 453.413 463.413 473.413 483.413 493.413 503.413 513.413 523.413 533.413 543.413
Input
f1i [deg] 0.000 10.000 20.000 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000 110.000 120.000 130.000 140.000 150.000 160.000 170.000 180.000 190.000 200.000 210.000 220.000 230.000 240.000 250.000 260.000 270.000 280.000 290.000 300.000 310.000 320.000 330.000 340.000 350.000
Output for branch 1 c [deg] c1i [deg] 68.670 76.393 84.121 91.609 98.670 105.176 111.046 116.226 120.670 124.309 127.025 128.607 128.670 126.515 120.868 109.615 90.752 67.411 48.442 36.813 30.226 26.533 24.548 23.654 23.531 24.010 25.009 26.503 28.503 31.050 34.208 38.054 42.662 48.079 54.296 61.221
0.000 7.723 15.451 22.939 30.000 36.506 42.375 47.556 52.000 55.639 58.355 59.937 60.000 57.845 52.198 40.945 22.082 358.741 339.772 328.143 321.556 317.863 315.878 314.984 314.861 315.340 316.339 317.833 319.833 322.280 325.538 329.384 333.992 339.409 345.626 352.551
Output for branch 2 c [deg] c1i [deg] 310.031 315.707 320.568 324.649 328.017 330.751 332.918 334.570 335.727 336.377 336.458 335.835 334.245 331.188 325.701 315.966 299.417 276.577 255.527 241.975 234.817 231.728 231.184 232.336 234.721 238.083 242.280 247.233 252.889 259.200 266.096 273.466 281.141 288.895 296.460 303.575
241.361 247.037 251.898 255.979 259.347 262.080 264.248 265.900 267.057 267.707 267.788 267.165 265.575 262.518 257.031 247.296 230.747 207.906 186.857 173.305 166.147 163.058 162.514 163.666 166.051 169.413 173.610 178.563 184.219 190.530 197.426 204.796 212.471 220.225 227.790 234.905
REFERENCES 1. Gupta, K. C. and Tinubu, S. O., Synthesis of bimodal function generating mechanisms without branch defect. Journal of Mechanisms, Transmissions and Automation in Design, 1982, 105, 641±647. 2. Angeles, J., in Spatial Kinematic Chains. Springer, Berlin, 1982. 3. Chiang, C. H., in Kinematics of Spherical Linkages. Cambridge University Press, Cambridge, 1988. 4. Duy, J., in Analysis of Mechanisms and Robot Manipulators. Wiley, New York, 1980. 5. Hartenberg, R. S. and Denavit, J., in Kinematic Synthesis of Linkages. McGraw-Hill, New York, 1964. 6. Larochelle, P. and McCarthy, J. M., in Design of spatial mechanisms. ASME Design Technical Conference Tutorial/ Workshop, Irvine, CA, 1996. 7. Mohan, Rao A. V., Sandor, G. N., Kohli, D. and Soni, A. H., Closed form synthesis of spatial function generating mechanism for the maximum number of precision points. Journal of Engineering for Industry, 1973, 95, 725±736. 8. Soni, A. H. and Harrisberger, L., The design of spherical drag-link mechanism. ASME Journal of Engineering for Industry, 1967, 89, 171±181. 9. Barker, C. and Lo, J., Classi®cation of spherical four-bar mechanisms. ASME Paper 86-DET-144, 1986. 10. Duditza, F. L. and Dittrich, G., Die Bedingungen fuÈr die UmlauaÈhigkeit sphaÈrischer viergliedriger Kurbelgetriebe. Industrie Anzeiger, 1969, 91(71), 1687±1690. 11. Freudenstein, F., On the determination of the type of spherical mechanisms. In Contemporary Problems in the Theory of Machines and Mechanisms. USSR Academy of Sciences, 1965, pp. 193±196. 12. Gosselin, C. and Angeles, J., RepreÂsentation graphique de la reÂgion de mobilite des meÂcanismes plans et spheÂriques aÁ barres articuleÂs. Mechanism and Machine Theory, 1987, 22, 557±562. 13. Gupta, K. C., Rotatability considerations for spherical four-bar linkages with applications to robot wrist design. Journal of Mechanisms, Transmissions and Automation in Design, 1986, 108, 387±391. 14. Gupta, K. C. and Ma, R., A direct rotatability criterion for spherical four-bar linkages. Journal of Mechanical Design, 1995, 117, 597±600.
504
K. C. Gupta and A. S. Beloiu
15. Murray, A. P. and McCarthy, J. M., A linkage map for spherical four position synthesis. In Proceeding of the 1995 ASME Design Engineering Technical Conferences. Boston, MA, 1995. 16. Savage, M. and Hall, A. S., Unique description of all spherical four-bar linkages. Journal of Engineering for Industry, 1970, 6(3), 563±566. 17. Soni, A. H., Unique description of all spherical four-bar linkages (discussion). Journal of Engineering for Industry, 1970, 89, 177±181. 18. Beloiu, A. S., Elimination of circuit defects in planar and spherical linkages. Doctoral Dissertation, Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL, 1996.