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Synthesis of fully rotatable R-S-S-R linkages
0094-114X/83 $3.00+.00 PergamonPress Ltd.
Mechanism and Machine Theory Vol. 18, No. 3, pp. 199--205,1983
Printed in Great Britain.
SYNTHESIS OF FULLY ROTATABLE R-S-S-R LINKAGES K. C. GUPTA and S. M. K. KAZEROUNIAN Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60680, U.S.A. (Received/or publication 15 October 1982) Abstract--Algebraic-geometrical techniques to design fully rotatable R-S-S-R linkages with transmission angle control are presented. The concept of upper bounding Cos/~ where/~ is the transmission angle, is used to obtain feasible regions for linkages with full inputqink rotatability. The proposed feasible regions have algebraic boundaries. This concept was developed earlier for the precision synthesis of planar four-bar linkage and it has now been extended to the precision synthesis of R-S-S-R linkage. INTRODUCTION
$4 is follower axial position, 0 is crank angle, 00 is initial crank angle, ~b is follower angle, ~bo is initial follower angle, A0~ is incremental rotation of crank at position i and A~b~is incremental rotation of follower at position i. We set a~ = 1. The displacement equation can be written as [11]:
In the precision point synthesis of a spatial R-S-S-R linkage, the full rotatability of its input link is sometimes desired in addition to the usual function generation requirements. A controlled transmission angle and a prior knowledge of the class of the linkage (crank rocker or drag link) may also be desired. Unfortunately, no simple criteria, analogous to Grashof's criteria for planar four-bar linkage, exists for R-S-S-R linkage [2]. In some recent works [3-8] analytical or geometrical methods for analysis of the mobility of R-S-S-R mechanism are presented. These methods lead to mobility criteria which are rather complex to apply even when the linkage parameters arc known numerically. What is required for algebraic synthesis is one or more explicit algebraic conditions in linkage parameters only which are not too complex. This would enable the designer to have a full insight into the set of linkages that satisfy his requirements. A method for the evaluation of force transmission characteristics (transmission index) for various spatial linkages is presented in Ref. [9]. This index can be used in the analysis of a known mechanism or in the iterative synthesis of these mechanisms. In this work we present an algebraic-geometrical method for designing function generating R-S-S-P, mechanisms with precision type specifications, fully rotatable input link and controlled transmission angle. The notion of establishing an upper bound for cos /z and deriving a sufficient condition for rotatability from it [10] is extended to the case of spatial R-S-S-R. As a result, a feasible region with algebraic boundary is obtained which guarantees that any point in the region corresponds to a unique crank-type R-S-S-R function generator which satisfies the transmission angle criteria max Icos ~,l-< k in addition to its precision requirements. The problems of branching and type determination (crank rocker vs drag link) are also considered.
P~A4+ U~A5+ V~A6+ A 7 - We = 0
(1)
where Pi = 2{(sin A~bicos A0i - cos a sin A0i cos A~b~)A2 - (cos A0~ cos A~, + cos a sin A0, sin A~bi)A, + sin a sin AOiA3+ cos A0i } U~ = 2{(sin A0~ cos A~b~- cos a cos A0~ sin A~b~)A, (sin A~blsin A0~ -
+ cos a cos A0~ cos A~b,)A2+ sin a cos AO~A3- sin A0~} V~= 2 sin a{A, sin A$i + A2 cos A$~} W~= - 2{A2 sin A~b,- At cos h4,~}
DISPLACEMENT EQUATION
Figure 1 shows a R-S-S--R mechanism. The design parameters and variables shown in the figure are defined as follows: a, is frame link length, a2 is crank length, a3 is coupler length, a4 is follower length, a is angle between input and output axes, $2 is crank axial position,
¥4
Fig. I. Parameters of'the R-S-S-R linkage. 199
(2)
200
K. C. GUPTAand S. M. K. KAZEROUNIAN
and A:s are defined as:
where
A, = a 4 c o s 4,0 A2 = a4 sin 4'0
y = 2a3a4
A3 = 54
or
A4 = a2 cos 0o A~ = a2 sin Oo A s : $2 A7 = (12 + a2,- a~ + a~ + S~ + S42 + 2S2S4 cos a).
COS2/,t = 1 - cos28 = y2 _ 4p,2 _ 4Q,2 + 4 U .2 y2
(3)
Design parameters (AI-A7) have a one-to-one correspondence with the linkage parameters (at - a4, S~, $2,
0,1, 4,0, a): a,= l
Equation (9) defines the transmission angle # in terms of linkage design parameters (al - a4, S1, $2, a, 0o, 4,0) and also the input angle 0. It is in agreement with the results given in Ref. [9]. This equation can be rearranged as:
a2=(A~ + A42)t/2
a4 = (A~ + A~) '/2 a3 = {i + a~ + a~ + S~ + S~ + 2S2S4 cos a _ A7}1/2 00 = tan ' (A~/A4) $2 = As 0o = t a n - '
(9)
(A2/A,).
S4 = A3
(4)
Equation (1) is linear in parameters A4, As, As and As. For specified a and A3, Pi, Ui, Vi and W~ are linear functions of A, and A2.
c°s 2 # = 1 (A* cos 20 + B* sin 20 + C* cos 0 + D* sin 0 Y
+E*+y2)
(10)
Expressions for A * - E * and y in terms of both sets of design parameters (i.e. a , - a4, S,, $2, a, 0o, 4,0 and A, - AT) are given in Appendix A. The condition that controls the transmission angle over a complete cycle is Icosl~l
(11)
Substituting for cos2# from eqn (10) into eqn (11): TRANSMISSION ANGLE
The closure equation for R-S-S-R mechanism can he written as [I]:
2 1 cos # = ~ (A'cos 20 + B* sin 20 + C* cos 0 + D* sin 0 + E* + ,,/2)< k 2
P* cos 4' + Q* sin d' = U* where P* = a4(a, + a2cos 0)
Q* = a4(a2 cos a sin 0 - $2 sin a) Ug~
l (a~ - 2 + a,2 + S~ + $24- a32 + 2S2S4 cos a
+ 2aza, cos 0 + 2a2S4 sin a sin 0)
(6)
We define deviation angle 8 as the angle between the velocity of the follower S-joint (C in Fig. 1) and the static force exerted by the coupler link on this joint. Because the coupler is a two-force member, the transmission angle # is defined by /~ = 90° - 8. After some manipulation we obtain
Reduction of condition (12) to a set of conditions which do not contain the input angle 0 and which involve A*, B* . . . . in a simple way is not an easy matter. The latter requirement is essential for developing an algebraicgeometrical procedure which is computationaily manageable. On the other hand, it is easier to establish an upper bound for cos2/z which does not involve the input angle 0. This upper bound can then be used to arrive at a sufficient (but not necessary) condition for rotatability and transmission angel control. It should be noted that a sufficient condition will guarantee rotatability and transmission angle control; therefore it is very useful in design in spite of the fact that it leaves out some admissible designs. We know that for all values of 0 A* cos 20 + B* cos 20 < (A .2 + B*2) 1/2 and
a3a4 cos 8 = - P* sin 0 + Q* cos 0
(7)
where P* and Q* are defined by eqn (6). Equations (5) and (7) together yield: cos z 8 = 4P'2 + 4 Q~* 2 - 4 U .2 y
(12)
(5)
(8)
(13) C* cos 0 + D* sin 0 <- (C .2 + D*2) 112
From the fact that (E* + y2)/y2 is the mean value of cos 2 # in eqn (12), and in view of (11), we conclude that E*+ y 2 - 0 for all 0.
(14)
201
Synthesis of fully rotatable R-S-S-R linkages From inequalities (13) and (14) it follows that inequality 1 ((A,2 + B*2)v2 + (C.2 + D.2)v2 + E* + y 2) <- k 2 3,
(15)
is a sufficient condition for inequality (12) to hold. Now by Cauchy's inequality
Both of the above inequalities are algebraic functions of design parameters (A1 - A7) and are also free from the input and output angles. These provide an alternate set of sufficient conditions for (I1). The feasible region is defined by the intersection of two inequalities (a) and (b) in (22). This alternate condition was found to be sharper than the one given by inequality (17). FOUR PRECISIONPOINT SYNTHESIS
~ ((A*2 + B'2)~/2 + (C'2+ D*2)v2 + E* + 32) < - ~ [A .2 + B .2 + C .2 + D .2 + (E* + 32)2]~/2
(16)
From inequalities (15) and (16), a sufficient condition for the transmission control condition (11) is found to be:
A*2 + B*2 + C*2 + D*2 + E*2 + 2E*y2 + 3,4-'j ,_43,4<-0 (17) This inequality is algebraic in design parameters A , - A7 and it does not contain the input angle 0. An alternate sufficient condition can be found after arranging equation (12) as follows. A* cos 20 + B* sin 20 + C* cos 0 + D* sin 0
-< 3,Zk2- E* - 32
Let the parameters a and A3 be selected according to the physical requirements or preferences of the application of the R-S-S-R linkage. For example, in generating a symmetric function we require A3 = 0, and when we want to replace two bevel gears in a small range of angular motion by a R-S-S-R mechanism, we require that a be equal to the angle between the axis of two bevel gears. The precision specifications in general are (AO~,Ad'~), i = 1, 4. From eqn (1) it follows that:
E" v' vl ilrA41rwll / A q /W2/ /A~/=IW~/
P4
LA7J LW4A
U4 V4
(23)
The unknown parameters A4, As, A6, A7 can be determined in terms of A, and A2, which are implicitly contained in Pi, U, V, W, by means of eqn (2), as follows
(18)
A4
for all 0 Clearly, for this inequality to be meaningful, the righthand side must be positive, i.e. 3,2k2 - E* - 32 _>0
(19)
A~
D4(At, A2) Os(A1, A2) A~ = - D(AI, A2) D(A1, A,) D6(A,,A2) D(A,, A2)
A7
D7(AI,A2) D(A,, A2)
o = IP,V,v, 11 o, = l w, u, v, 11
LHS < (A .2 + B*2) '/2 + (C .2 + D*2) '/2. < ~/2(A .2 + B .2 + C .2 + D'22) v2 (Cauchy's inequality) (20) Then, in addition to inequality (19), a sufficient condition for (18) is: X/2(A*2 + B*2+ C *2"+D'2)1/2-< 3,2k2- E* - 72 (21) Squaring both sides of (21), which is permissible in view of (19), we can write conditions (19) and (21) as
It may be verified that [11] D = 0 and /)6=0 are general cubic curves in A1 - A2 plane,/)4 = 0 and Ds = 0 are cubic curves which contain a linear factor and a quadratic factor (A~ + A~), and D7 = 0 is a quartic curve which contains a real quadratic factor and the factor (A~ + A~). Substituting the expressions for A4 - A7 from eqn (24) into the eqns (A.2) of Appendix A, A * . . . E* and y2 are found as follows I
A* = ~ [ - 2(D] + Ds2) 2(A .2 + B .2 + C .2 + D .2) - (3,2k2 - E* - 3,2)2_<0. (22b)
(25)
D7=IP, U,V,W,I
(22a)
and
(24)
The 4 x 4 determinants D,/)4, Ds,/)6, D7 are defined as
The left-hand side (LHS) has an upper bound (see (13)), i.e.
E* + 3,2 - k23,2 ~ 0
P2 v2 v2 P3 v~ v3
x (A~ sin 2 a + A~ sin2a + A~ sin 2 a - 1)]
202
K. C. GUPTAand S. M. K. KAZEP.0UNIAN This quartic degenerates to two quadratics. One of these has real roots and the other one is A~ + A~ = 0. Any point (A,, A2) on the segment of T4 = 0 which lies in the region defined by F,~-<0 or the one defined by Gs-<0 and H,6 -- 0 guarantees a completely rotatable input link with a specified maximum deviation of transmission angle from 90°.
B* = ~ [4(D] + D~)(A3) sin a]
C* : ~ [-4(D~ + D~)~/2(2A~ D + 2A~D - D7)]
D* = ~ [4(D] + D~) ''2
ELIMINATIONOF BRANCHDEFECT
x (2A~ D, sin a cos a + 2A~D6 sin a cos a + A3D7 sin a)] !
E* = ~ [- 2(D] + D~) x (A~ + A~2 cos 2 a + A22 + A22 COS2 a - A~ sin 2 a - 1) - 4(A~ + A~)D 2 - 4(A~ + A~)(D~, sin 2 a) + D721 !
y2 = ~_~ [4(A~ + A~)(D z + D2A~ + D2A~ + D:A~ + D] + D~2+D62 - DD7 + 2A3DD6 cos or].
(26)
If we substitute these values for A * - E * and 32 into the inequality (17), and drop a factor ((A 2 + A2)/D2) 2, we get a feasible region bounded by a 16th order polynomial in A~ - A2 plane.
The problem of elimination of branch defect in the precision synthesis of R-S-S-R mechanisms has been considered separately in Ref. [11]. A mechanism will have branch defect if the sign of sin/~, i = l, N, changes at the design positions. This implies that at the boundary of the branching region sin /~k = 0, where k is a design position. However, the feasible region guarantees that max Icos ~I -< k-< I. Therefore, the branching region boundary may not pass through the feasible region for four precision point designs. In other words, all mechanisms corresponding to points in one loop of the feasible region either have branch defect or are free from it. Therefore, it is enough to check one mechanism in each loop of the feasible region. If we use conditions (22) this invariance property is true for the intersection of the two regions and not for any of them separately. For five precision points, the preceding statements are valid if the term "feasible region" is replaced by "feasible segment".
(27)
F,6(A,, A2) <-O.
TYPE DETERMINATION
This feasible region can be improved if we use the set of conditions (22). Substituting for A * - E * and ?2, and dropping a factor (A~ + A~)ID2 from (22a) and (22b), we obtain a mapping of these conditions in A , - A2 plane. The feasible region is the intersection of the two regions, one bounded by an 8th order curve (from (22a)) and another bounded by a 16th order curve (from (22b)), i.e. G d A . A2) <-0
(28a)
HI6(A,, A2) <-O.
(28b)
and
Each point (A,, A2) in the feasible re#on given by either (27) or (28) guarantees a fully rotatable input link and max Icos #l -< k.
FIVE PRECISIONPOINT SYNTHESIS
The synthesis procedure for four precission points described in the previous section can be extended to five precision points. The five precision point specifications are (A0, A&), i = 1, 5. The regions are obtained as before by using the first four precision points, that is either by the inequality F,6-< 0, or by the inequalities G8 <-0 and H,6_< 0. In addition we must consider the quartic curve defined by the following 5 x 5 determinant [11]: T4=IP~U~V~I W,I--0
i=1,5
(29)
Two types of R-S-S-R mechanisms can exist in the feasible regions or segments obtained in the previous sections. These are double crank mechanisms (drag link) and crank rocker mechanisms. Points that lie on the boundary that separates crank rocker from drag link correspond to mechanisms that can be considered both as drag link and crank rocker mechanisms. The limit of the range of the output link in a crackrocker mechansim is reached when the input link, coupler link and input axis of rotation lie in the same plane. For this position we can say that if the input link moves in any direction, the output link will move in one direction. For the aforementioned mechanisms, which can he characterized both as crank-rocker and drag link mechanisms, whenever the crank rocker mode limiting position is reached (i.e. input crank, input axis and coupler are coplanar), we can say that the output link must be able to rotate in both directions when the input link rotates in one direction (otherwise it could not be a drag link). Analogous to what we said about the input crank before, at this position the output crank, output axis and coupler also lie in a plane (the same thing occurs at input side at this position). This means that Icos #1 = 1. Such a mechanism cannot exist in the interior of the feasible region which guarantees that [cos[.,~x-< k <-1. Therefore, all mechanisms corresponding to points in a particular loop of a feasible region (or a branch of feasible segment) are all either crank-rocker mechanisms or drag-link mechanisms. It is necessary and sufficient to check only one mechanism in each loop (or segment) by using one of the known criterion [3-5].
203
Synthesis of fully rotatable R-S-S-R linkages
of these mechanisms which correspond to points marked in Fig. 2. Figure 3 shows the influence of parameter k upon the feasible region obtained from conditions (28). When k = l, the approximate feasible regions given by conditions (27) or (28) are significantly smaller than the region given by an exhaustive mapping of the condition (11). A possible explanation is that the number of linkages for which max Icos #[-', 1 is very large and many of these linkages are not included in the approximate feasible regions. On the other hand, for k < l the approximate feasible region represents a substantial portion of the exact feasible region obtained by exhaustive mapping, as shown in Fig. 2. If the mechanism is to meet a fifth precision point given by 0,5 = 125.57° and ~b,~= -94.735 °, the curve T4 (29) must be developed and points (A~, A,) should be chosen on the segments of T4 which lie in an appropriate region of Fig. 2. As an example, point P5 in Table l is on the curve T4 [11] and its corresponding R-S-S-R mechanism meets the transmission angle requirement with k = 0.866 in addition to meeting the five aforementioned precision points.
If we use inequalities (22), this invariance property is true for the intersection of two regions, and not for any of them alone. NUMERICAL EXAMPLE
We want to design a function generator with fully rotatable input link such that max Icos # [ < k where k = 0.866. Precision specifications are OIl=O.O ° $,,=-0.0 ° O,z = 19.410 dh2 = -5.1250 0,3 = 53.030
~)13 =
-30.165 °
0,4 = 91.850 ~b,4= -69.6950 Let us assume a = 900 and A3 = 1. Following the method presented in previous sections, the feasible region is the intersection of G~< and H,6-<0 in the A,-A2 plane (conditions (28)). Figure 2 shows this feasible region. All mechanisms corresponding to points in this region are free from branch defect and are crank rockers. An alternative method is to develop the feasible region F,6<-0 (condition (27)), but generally this region is smaller (see Fig. 2) than the feasible region given by condition (28). For comparison, the exact region corresponding to the condition (11) was also obtained by an exhaustive and time consuming search; it is shown in Fig. 2. For any point in one of the feasible regions shown in Fig. 2, there is a unique mechanism. Table I shows some
CONCLUSION
A method has been presented to design R-S-S-R function generating mechanisms for precision type specifications with fully rotatable input links and controlled transmission angle. The problem of branching and type determination are also discussed and resolved for crank mechanisms. The feasible region for four precision points is bounded by a 16th order c u r v e Fi6-<0, or through the improvement outlined the feasible region can
'k 2 6 • EXACT REGION
3
--F16(0 G8
-3
_6
-9 Fig. 2. Feasible regions for crank-rocker R-S-S-R linkages without branch defect.
204
K. C. GUPTAand S. M. K. KAZEROUNIAN
Table 1. Rotatable four precision point R-S-S-R linkages Parameters
P1
P2
P3
P4
P5
A1
-0.2
-1.0
-1.8
-3.0
-1.3215
A2
-2.2
-1.0
-1.8
-1.8
-1.0933
Shaft angle e
90 °
90 °
90 °
90 °
90 ~
Frame link length, a I
1.0
1.0
1.0
1.0
1.0
Input crank length, a 2
1.799
5.84
3.987
4.98
4.666
Coupler length, a 3
3.377
11.57
7.159
7.26
8.466
Follower l e n g t h , a4
2.209
1.41
2.546
3.55
1.715
Crank axial position, S 2
2.706
-9.99
-5.273
-8.69
1.0
Follower axial position, S 4
1.0
1.0
-7.09
i 1.0
1.0
i
Input reference angle 8° {deg.)
53.26
212.09
206.79 200.38
209.08
Output reference angle ¢o (deg.)
264.80
225.0
225.0
219,60
~x
Icos~1
0.829
0.724
1212.29
0.720
0.834
0.827
/
/
2
/ / / /
/ \
k --
/
/
/
/
/
I /
- -
--
:
A1
'866
k=l.
/
/
/ _,2
Fig. 3. Influence of k upon the feasible region given by Gs < 0 and H~6 < 0.
be found which is the intersection of two regions bounded by a 16th order and an 8th order curve (H,6 < 0 and Gs<0). For five precision points, feasible quadratic segments can be obtained from eqn (29) along with the condition (27) or (28). This algebraic-geometrical method represents a significant extension of the technique which was first developed in the context of planar four-bar linkage.
Acknowledgement--The financial support of the National Science Foundation under the grant CME 79-23193 is gratefully acknowledged. REFERENCES 1. R. S. Hartenberg and J. Denavit Kinematics Synthesis of Linkages. McGraw-Hill, New York (1964). 2. K. H. Hunt Kinematic Geometry o[ Mechanisms. Clarendon Press, Oxford (1978).
Synthesis of fully rotatable R-S-S-R linkages 3. M. Skreiner, J. Mechanisms, 2, 415-427 (1967). 4. F. Freudenstein and I. S. Kiss, ./. Engng Industry 91,220-224 (1969). 5. H. Noll J. Mech. 4, 145-157 (1969). 6. F. C. O. Sticher ./. Mechanisms $ 393-415 (1970). 7. F. C. O. Sticher J. Mechanisms 6, 303-339 (1971). 8. F. Freudenstein and E. J. F. Primrose, .L Engng [or Industry, Trans. ASME, Series B. 98, 1285-1288 (1976). 9. G. Sutherland and B. Roth ./. Engng/or Industry, Trans. ASME, Series B 95, 589-597 (1973). 10. K. C. Gupta Trans. ASME, I. Appl. Mech., Series E 45, 415-421 (1978). 11. K. C. Gupta and S. Tinubu Synthesis of bimodal function generating mechanism without branch defect. ASME Paper No. 82-DET-85.
205
D* = 4a~(2a~S2 sin a cos a + A7S4 sin a) E* = - 2a~(a~ + a~ cos 2 a - ~ sin 2 a - a 2) - 4a~a~ - 4~a~ sin2 a + A~,
y2 =
4a~a~ (A1)
A * . . . E* and y 2 in terms of (A, - A7): (set a~ = 1) A* = 2(A~+ A~)(A~ sin a + A~ sin a + A3 sin2 a - 1) B* = 4(A~+ A~)(A3) sin a
c * = - 4(/d + A~) '~2 (2A~ + 2 A ~ - AT) D* = 4(A~ + A~)'/2 (2A~A6 sin a cos a + 2A~A6 sin a cos a
+ A3A 7 sin a) E* = - 2(A2~+ A~)(A~ + A~ cos 2 a + A~ + A22 cos 2 a - A~ sin2 a
APPENDIX A A * . . . E* and 7 in terms of (a's, $2, S4, 0o, cko and a):
- 1) - 4(A~2 + A~) - 4(A~ + A~)(A 2 sin 2 a) + A~
A* = - 2a~(a~ sin 2 a + S~ sin 2 a - a~) y2 = 4(I + A~ + A~ + A~ + A 2 + A~ + Ag + A3A6 cos a - A7)(A~ + A~) (A2)
B* = 4a~(atS4 sin a)
C* = - 4a2al(2a 2 - A7)
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Mechanism and Machine Theory Vol. 18, No. 3, pp. 199--205,1983
Printed in Great Britain.
SYNTHESIS OF FULLY ROTATABLE R-S-S-R LINKAGES K. C. GUPTA and S. M. K. KAZEROUNIAN Department of Mechanical Engineering, University of Illinois at Chicago, Chicago, IL 60680, U.S.A. (Received/or publication 15 October 1982) Abstract--Algebraic-geometrical techniques to design fully rotatable R-S-S-R linkages with transmission angle control are presented. The concept of upper bounding Cos/~ where/~ is the transmission angle, is used to obtain feasible regions for linkages with full inputqink rotatability. The proposed feasible regions have algebraic boundaries. This concept was developed earlier for the precision synthesis of planar four-bar linkage and it has now been extended to the precision synthesis of R-S-S-R linkage. INTRODUCTION
$4 is follower axial position, 0 is crank angle, 00 is initial crank angle, ~b is follower angle, ~bo is initial follower angle, A0~ is incremental rotation of crank at position i and A~b~is incremental rotation of follower at position i. We set a~ = 1. The displacement equation can be written as [11]:
In the precision point synthesis of a spatial R-S-S-R linkage, the full rotatability of its input link is sometimes desired in addition to the usual function generation requirements. A controlled transmission angle and a prior knowledge of the class of the linkage (crank rocker or drag link) may also be desired. Unfortunately, no simple criteria, analogous to Grashof's criteria for planar four-bar linkage, exists for R-S-S-R linkage [2]. In some recent works [3-8] analytical or geometrical methods for analysis of the mobility of R-S-S-R mechanism are presented. These methods lead to mobility criteria which are rather complex to apply even when the linkage parameters arc known numerically. What is required for algebraic synthesis is one or more explicit algebraic conditions in linkage parameters only which are not too complex. This would enable the designer to have a full insight into the set of linkages that satisfy his requirements. A method for the evaluation of force transmission characteristics (transmission index) for various spatial linkages is presented in Ref. [9]. This index can be used in the analysis of a known mechanism or in the iterative synthesis of these mechanisms. In this work we present an algebraic-geometrical method for designing function generating R-S-S-P, mechanisms with precision type specifications, fully rotatable input link and controlled transmission angle. The notion of establishing an upper bound for cos /z and deriving a sufficient condition for rotatability from it [10] is extended to the case of spatial R-S-S-R. As a result, a feasible region with algebraic boundary is obtained which guarantees that any point in the region corresponds to a unique crank-type R-S-S-R function generator which satisfies the transmission angle criteria max Icos ~,l-< k in addition to its precision requirements. The problems of branching and type determination (crank rocker vs drag link) are also considered.
P~A4+ U~A5+ V~A6+ A 7 - We = 0
(1)
where Pi = 2{(sin A~bicos A0i - cos a sin A0i cos A~b~)A2 - (cos A0~ cos A~, + cos a sin A0, sin A~bi)A, + sin a sin AOiA3+ cos A0i } U~ = 2{(sin A0~ cos A~b~- cos a cos A0~ sin A~b~)A, (sin A~blsin A0~ -
+ cos a cos A0~ cos A~b,)A2+ sin a cos AO~A3- sin A0~} V~= 2 sin a{A, sin A$i + A2 cos A$~} W~= - 2{A2 sin A~b,- At cos h4,~}
DISPLACEMENT EQUATION
Figure 1 shows a R-S-S--R mechanism. The design parameters and variables shown in the figure are defined as follows: a, is frame link length, a2 is crank length, a3 is coupler length, a4 is follower length, a is angle between input and output axes, $2 is crank axial position,
¥4
Fig. I. Parameters of'the R-S-S-R linkage. 199
(2)
200
K. C. GUPTAand S. M. K. KAZEROUNIAN
and A:s are defined as:
where
A, = a 4 c o s 4,0 A2 = a4 sin 4'0
y = 2a3a4
A3 = 54
or
A4 = a2 cos 0o A~ = a2 sin Oo A s : $2 A7 = (12 + a2,- a~ + a~ + S~ + S42 + 2S2S4 cos a).
COS2/,t = 1 - cos28 = y2 _ 4p,2 _ 4Q,2 + 4 U .2 y2
(3)
Design parameters (AI-A7) have a one-to-one correspondence with the linkage parameters (at - a4, S~, $2,
0,1, 4,0, a): a,= l
Equation (9) defines the transmission angle # in terms of linkage design parameters (al - a4, S1, $2, a, 0o, 4,0) and also the input angle 0. It is in agreement with the results given in Ref. [9]. This equation can be rearranged as:
a2=(A~ + A42)t/2
a4 = (A~ + A~) '/2 a3 = {i + a~ + a~ + S~ + S~ + 2S2S4 cos a _ A7}1/2 00 = tan ' (A~/A4) $2 = As 0o = t a n - '
(9)
(A2/A,).
S4 = A3
(4)
Equation (1) is linear in parameters A4, As, As and As. For specified a and A3, Pi, Ui, Vi and W~ are linear functions of A, and A2.
c°s 2 # = 1 (A* cos 20 + B* sin 20 + C* cos 0 + D* sin 0 Y
+E*+y2)
(10)
Expressions for A * - E * and y in terms of both sets of design parameters (i.e. a , - a4, S,, $2, a, 0o, 4,0 and A, - AT) are given in Appendix A. The condition that controls the transmission angle over a complete cycle is Icosl~l
(11)
Substituting for cos2# from eqn (10) into eqn (11): TRANSMISSION ANGLE
The closure equation for R-S-S-R mechanism can he written as [I]:
2 1 cos # = ~ (A'cos 20 + B* sin 20 + C* cos 0 + D* sin 0 + E* + ,,/2)< k 2
P* cos 4' + Q* sin d' = U* where P* = a4(a, + a2cos 0)
Q* = a4(a2 cos a sin 0 - $2 sin a) Ug~
l (a~ - 2 + a,2 + S~ + $24- a32 + 2S2S4 cos a
+ 2aza, cos 0 + 2a2S4 sin a sin 0)
(6)
We define deviation angle 8 as the angle between the velocity of the follower S-joint (C in Fig. 1) and the static force exerted by the coupler link on this joint. Because the coupler is a two-force member, the transmission angle # is defined by /~ = 90° - 8. After some manipulation we obtain
Reduction of condition (12) to a set of conditions which do not contain the input angle 0 and which involve A*, B* . . . . in a simple way is not an easy matter. The latter requirement is essential for developing an algebraicgeometrical procedure which is computationaily manageable. On the other hand, it is easier to establish an upper bound for cos2/z which does not involve the input angle 0. This upper bound can then be used to arrive at a sufficient (but not necessary) condition for rotatability and transmission angel control. It should be noted that a sufficient condition will guarantee rotatability and transmission angle control; therefore it is very useful in design in spite of the fact that it leaves out some admissible designs. We know that for all values of 0 A* cos 20 + B* cos 20 < (A .2 + B*2) 1/2 and
a3a4 cos 8 = - P* sin 0 + Q* cos 0
(7)
where P* and Q* are defined by eqn (6). Equations (5) and (7) together yield: cos z 8 = 4P'2 + 4 Q~* 2 - 4 U .2 y
(12)
(5)
(8)
(13) C* cos 0 + D* sin 0 <- (C .2 + D*2) 112
From the fact that (E* + y2)/y2 is the mean value of cos 2 # in eqn (12), and in view of (11), we conclude that E*+ y 2 - 0 for all 0.
(14)
201
Synthesis of fully rotatable R-S-S-R linkages From inequalities (13) and (14) it follows that inequality 1 ((A,2 + B*2)v2 + (C.2 + D.2)v2 + E* + y 2) <- k 2 3,
(15)
is a sufficient condition for inequality (12) to hold. Now by Cauchy's inequality
Both of the above inequalities are algebraic functions of design parameters (A1 - A7) and are also free from the input and output angles. These provide an alternate set of sufficient conditions for (I1). The feasible region is defined by the intersection of two inequalities (a) and (b) in (22). This alternate condition was found to be sharper than the one given by inequality (17). FOUR PRECISIONPOINT SYNTHESIS
~ ((A*2 + B'2)~/2 + (C'2+ D*2)v2 + E* + 32) < - ~ [A .2 + B .2 + C .2 + D .2 + (E* + 32)2]~/2
(16)
From inequalities (15) and (16), a sufficient condition for the transmission control condition (11) is found to be:
A*2 + B*2 + C*2 + D*2 + E*2 + 2E*y2 + 3,4-'j ,_43,4<-0 (17) This inequality is algebraic in design parameters A , - A7 and it does not contain the input angle 0. An alternate sufficient condition can be found after arranging equation (12) as follows. A* cos 20 + B* sin 20 + C* cos 0 + D* sin 0
-< 3,Zk2- E* - 32
Let the parameters a and A3 be selected according to the physical requirements or preferences of the application of the R-S-S-R linkage. For example, in generating a symmetric function we require A3 = 0, and when we want to replace two bevel gears in a small range of angular motion by a R-S-S-R mechanism, we require that a be equal to the angle between the axis of two bevel gears. The precision specifications in general are (AO~,Ad'~), i = 1, 4. From eqn (1) it follows that:
E" v' vl ilrA41rwll / A q /W2/ /A~/=IW~/
P4
LA7J LW4A
U4 V4
(23)
The unknown parameters A4, As, A6, A7 can be determined in terms of A, and A2, which are implicitly contained in Pi, U, V, W, by means of eqn (2), as follows
(18)
A4
for all 0 Clearly, for this inequality to be meaningful, the righthand side must be positive, i.e. 3,2k2 - E* - 32 _>0
(19)
A~
D4(At, A2) Os(A1, A2) A~ = - D(AI, A2) D(A1, A,) D6(A,,A2) D(A,, A2)
A7
D7(AI,A2) D(A,, A2)
o = IP,V,v, 11 o, = l w, u, v, 11
LHS < (A .2 + B*2) '/2 + (C .2 + D*2) '/2. < ~/2(A .2 + B .2 + C .2 + D'22) v2 (Cauchy's inequality) (20) Then, in addition to inequality (19), a sufficient condition for (18) is: X/2(A*2 + B*2+ C *2"+D'2)1/2-< 3,2k2- E* - 72 (21) Squaring both sides of (21), which is permissible in view of (19), we can write conditions (19) and (21) as
It may be verified that [11] D = 0 and /)6=0 are general cubic curves in A1 - A2 plane,/)4 = 0 and Ds = 0 are cubic curves which contain a linear factor and a quadratic factor (A~ + A~), and D7 = 0 is a quartic curve which contains a real quadratic factor and the factor (A~ + A~). Substituting the expressions for A4 - A7 from eqn (24) into the eqns (A.2) of Appendix A, A * . . . E* and y2 are found as follows I
A* = ~ [ - 2(D] + Ds2) 2(A .2 + B .2 + C .2 + D .2) - (3,2k2 - E* - 3,2)2_<0. (22b)
(25)
D7=IP, U,V,W,I
(22a)
and
(24)
The 4 x 4 determinants D,/)4, Ds,/)6, D7 are defined as
The left-hand side (LHS) has an upper bound (see (13)), i.e.
E* + 3,2 - k23,2 ~ 0
P2 v2 v2 P3 v~ v3
x (A~ sin 2 a + A~ sin2a + A~ sin 2 a - 1)]
202
K. C. GUPTAand S. M. K. KAZEP.0UNIAN This quartic degenerates to two quadratics. One of these has real roots and the other one is A~ + A~ = 0. Any point (A,, A2) on the segment of T4 = 0 which lies in the region defined by F,~-<0 or the one defined by Gs-<0 and H,6 -- 0 guarantees a completely rotatable input link with a specified maximum deviation of transmission angle from 90°.
B* = ~ [4(D] + D~)(A3) sin a]
C* : ~ [-4(D~ + D~)~/2(2A~ D + 2A~D - D7)]
D* = ~ [4(D] + D~) ''2
ELIMINATIONOF BRANCHDEFECT
x (2A~ D, sin a cos a + 2A~D6 sin a cos a + A3D7 sin a)] !
E* = ~ [- 2(D] + D~) x (A~ + A~2 cos 2 a + A22 + A22 COS2 a - A~ sin 2 a - 1) - 4(A~ + A~)D 2 - 4(A~ + A~)(D~, sin 2 a) + D721 !
y2 = ~_~ [4(A~ + A~)(D z + D2A~ + D2A~ + D:A~ + D] + D~2+D62 - DD7 + 2A3DD6 cos or].
(26)
If we substitute these values for A * - E * and 32 into the inequality (17), and drop a factor ((A 2 + A2)/D2) 2, we get a feasible region bounded by a 16th order polynomial in A~ - A2 plane.
The problem of elimination of branch defect in the precision synthesis of R-S-S-R mechanisms has been considered separately in Ref. [11]. A mechanism will have branch defect if the sign of sin/~, i = l, N, changes at the design positions. This implies that at the boundary of the branching region sin /~k = 0, where k is a design position. However, the feasible region guarantees that max Icos ~I -< k-< I. Therefore, the branching region boundary may not pass through the feasible region for four precision point designs. In other words, all mechanisms corresponding to points in one loop of the feasible region either have branch defect or are free from it. Therefore, it is enough to check one mechanism in each loop of the feasible region. If we use conditions (22) this invariance property is true for the intersection of the two regions and not for any of them separately. For five precision points, the preceding statements are valid if the term "feasible region" is replaced by "feasible segment".
(27)
F,6(A,, A2) <-O.
TYPE DETERMINATION
This feasible region can be improved if we use the set of conditions (22). Substituting for A * - E * and ?2, and dropping a factor (A~ + A~)ID2 from (22a) and (22b), we obtain a mapping of these conditions in A , - A2 plane. The feasible region is the intersection of the two regions, one bounded by an 8th order curve (from (22a)) and another bounded by a 16th order curve (from (22b)), i.e. G d A . A2) <-0
(28a)
HI6(A,, A2) <-O.
(28b)
and
Each point (A,, A2) in the feasible re#on given by either (27) or (28) guarantees a fully rotatable input link and max Icos #l -< k.
FIVE PRECISIONPOINT SYNTHESIS
The synthesis procedure for four precission points described in the previous section can be extended to five precision points. The five precision point specifications are (A0, A&), i = 1, 5. The regions are obtained as before by using the first four precision points, that is either by the inequality F,6-< 0, or by the inequalities G8 <-0 and H,6_< 0. In addition we must consider the quartic curve defined by the following 5 x 5 determinant [11]: T4=IP~U~V~I W,I--0
i=1,5
(29)
Two types of R-S-S-R mechanisms can exist in the feasible regions or segments obtained in the previous sections. These are double crank mechanisms (drag link) and crank rocker mechanisms. Points that lie on the boundary that separates crank rocker from drag link correspond to mechanisms that can be considered both as drag link and crank rocker mechanisms. The limit of the range of the output link in a crackrocker mechansim is reached when the input link, coupler link and input axis of rotation lie in the same plane. For this position we can say that if the input link moves in any direction, the output link will move in one direction. For the aforementioned mechanisms, which can he characterized both as crank-rocker and drag link mechanisms, whenever the crank rocker mode limiting position is reached (i.e. input crank, input axis and coupler are coplanar), we can say that the output link must be able to rotate in both directions when the input link rotates in one direction (otherwise it could not be a drag link). Analogous to what we said about the input crank before, at this position the output crank, output axis and coupler also lie in a plane (the same thing occurs at input side at this position). This means that Icos #1 = 1. Such a mechanism cannot exist in the interior of the feasible region which guarantees that [cos[.,~x-< k <-1. Therefore, all mechanisms corresponding to points in a particular loop of a feasible region (or a branch of feasible segment) are all either crank-rocker mechanisms or drag-link mechanisms. It is necessary and sufficient to check only one mechanism in each loop (or segment) by using one of the known criterion [3-5].
203
Synthesis of fully rotatable R-S-S-R linkages
of these mechanisms which correspond to points marked in Fig. 2. Figure 3 shows the influence of parameter k upon the feasible region obtained from conditions (28). When k = l, the approximate feasible regions given by conditions (27) or (28) are significantly smaller than the region given by an exhaustive mapping of the condition (11). A possible explanation is that the number of linkages for which max Icos #[-', 1 is very large and many of these linkages are not included in the approximate feasible regions. On the other hand, for k < l the approximate feasible region represents a substantial portion of the exact feasible region obtained by exhaustive mapping, as shown in Fig. 2. If the mechanism is to meet a fifth precision point given by 0,5 = 125.57° and ~b,~= -94.735 °, the curve T4 (29) must be developed and points (A~, A,) should be chosen on the segments of T4 which lie in an appropriate region of Fig. 2. As an example, point P5 in Table l is on the curve T4 [11] and its corresponding R-S-S-R mechanism meets the transmission angle requirement with k = 0.866 in addition to meeting the five aforementioned precision points.
If we use inequalities (22), this invariance property is true for the intersection of two regions, and not for any of them alone. NUMERICAL EXAMPLE
We want to design a function generator with fully rotatable input link such that max Icos # [ < k where k = 0.866. Precision specifications are OIl=O.O ° $,,=-0.0 ° O,z = 19.410 dh2 = -5.1250 0,3 = 53.030
~)13 =
-30.165 °
0,4 = 91.850 ~b,4= -69.6950 Let us assume a = 900 and A3 = 1. Following the method presented in previous sections, the feasible region is the intersection of G~< and H,6-<0 in the A,-A2 plane (conditions (28)). Figure 2 shows this feasible region. All mechanisms corresponding to points in this region are free from branch defect and are crank rockers. An alternative method is to develop the feasible region F,6<-0 (condition (27)), but generally this region is smaller (see Fig. 2) than the feasible region given by condition (28). For comparison, the exact region corresponding to the condition (11) was also obtained by an exhaustive and time consuming search; it is shown in Fig. 2. For any point in one of the feasible regions shown in Fig. 2, there is a unique mechanism. Table I shows some
CONCLUSION
A method has been presented to design R-S-S-R function generating mechanisms for precision type specifications with fully rotatable input links and controlled transmission angle. The problem of branching and type determination are also discussed and resolved for crank mechanisms. The feasible region for four precision points is bounded by a 16th order c u r v e Fi6-<0, or through the improvement outlined the feasible region can
'k 2 6 • EXACT REGION
3
--F16(0 G8
-3
_6
-9 Fig. 2. Feasible regions for crank-rocker R-S-S-R linkages without branch defect.
204
K. C. GUPTAand S. M. K. KAZEROUNIAN
Table 1. Rotatable four precision point R-S-S-R linkages Parameters
P1
P2
P3
P4
P5
A1
-0.2
-1.0
-1.8
-3.0
-1.3215
A2
-2.2
-1.0
-1.8
-1.8
-1.0933
Shaft angle e
90 °
90 °
90 °
90 °
90 ~
Frame link length, a I
1.0
1.0
1.0
1.0
1.0
Input crank length, a 2
1.799
5.84
3.987
4.98
4.666
Coupler length, a 3
3.377
11.57
7.159
7.26
8.466
Follower l e n g t h , a4
2.209
1.41
2.546
3.55
1.715
Crank axial position, S 2
2.706
-9.99
-5.273
-8.69
1.0
Follower axial position, S 4
1.0
1.0
-7.09
i 1.0
1.0
i
Input reference angle 8° {deg.)
53.26
212.09
206.79 200.38
209.08
Output reference angle ¢o (deg.)
264.80
225.0
225.0
219,60
~x
Icos~1
0.829
0.724
1212.29
0.720
0.834
0.827
/
/
2
/ / / /
/ \
k --
/
/
/
/
/
I /
- -
--
:
A1
'866
k=l.
/
/
/ _,2
Fig. 3. Influence of k upon the feasible region given by Gs < 0 and H~6 < 0.
be found which is the intersection of two regions bounded by a 16th order and an 8th order curve (H,6 < 0 and Gs<0). For five precision points, feasible quadratic segments can be obtained from eqn (29) along with the condition (27) or (28). This algebraic-geometrical method represents a significant extension of the technique which was first developed in the context of planar four-bar linkage.
Acknowledgement--The financial support of the National Science Foundation under the grant CME 79-23193 is gratefully acknowledged. REFERENCES 1. R. S. Hartenberg and J. Denavit Kinematics Synthesis of Linkages. McGraw-Hill, New York (1964). 2. K. H. Hunt Kinematic Geometry o[ Mechanisms. Clarendon Press, Oxford (1978).
Synthesis of fully rotatable R-S-S-R linkages 3. M. Skreiner, J. Mechanisms, 2, 415-427 (1967). 4. F. Freudenstein and I. S. Kiss, ./. Engng Industry 91,220-224 (1969). 5. H. Noll J. Mech. 4, 145-157 (1969). 6. F. C. O. Sticher ./. Mechanisms $ 393-415 (1970). 7. F. C. O. Sticher J. Mechanisms 6, 303-339 (1971). 8. F. Freudenstein and E. J. F. Primrose, .L Engng [or Industry, Trans. ASME, Series B. 98, 1285-1288 (1976). 9. G. Sutherland and B. Roth ./. Engng/or Industry, Trans. ASME, Series B 95, 589-597 (1973). 10. K. C. Gupta Trans. ASME, I. Appl. Mech., Series E 45, 415-421 (1978). 11. K. C. Gupta and S. Tinubu Synthesis of bimodal function generating mechanism without branch defect. ASME Paper No. 82-DET-85.
205
D* = 4a~(2a~S2 sin a cos a + A7S4 sin a) E* = - 2a~(a~ + a~ cos 2 a - ~ sin 2 a - a 2) - 4a~a~ - 4~a~ sin2 a + A~,
y2 =
4a~a~ (A1)
A * . . . E* and y 2 in terms of (A, - A7): (set a~ = 1) A* = 2(A~+ A~)(A~ sin a + A~ sin a + A3 sin2 a - 1) B* = 4(A~+ A~)(A3) sin a
c * = - 4(/d + A~) '~2 (2A~ + 2 A ~ - AT) D* = 4(A~ + A~)'/2 (2A~A6 sin a cos a + 2A~A6 sin a cos a
+ A3A 7 sin a) E* = - 2(A2~+ A~)(A~ + A~ cos 2 a + A~ + A22 cos 2 a - A~ sin2 a
APPENDIX A A * . . . E* and 7 in terms of (a's, $2, S4, 0o, cko and a):
- 1) - 4(A~2 + A~) - 4(A~ + A~)(A 2 sin 2 a) + A~
A* = - 2a~(a~ sin 2 a + S~ sin 2 a - a~) y2 = 4(I + A~ + A~ + A~ + A 2 + A~ + Ag + A3A6 cos a - A7)(A~ + A~) (A2)
B* = 4a~(atS4 sin a)
C* = - 4a2al(2a 2 - A7)
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