Proceedings of A Symposium Commemorating the Legacy, Works, and Life of Sir Robert Stawell Ball Upon the 100th Anniversary of A Treatise on the Theory of Screws July 9-11, 2000 University of Cambridge, Trinity College © 2000

ON RECIPROCAL LINKAGES J Eddie Baker School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney 2052, Australia

Abstract The concept of reciprocity for infinitesimal screws arose from that of the rate of working, and so involved simultaneously a twist screw and an action screw. Its definition has since been widened to apply to screws of a single type, a particular benefit being the treatment of special configurations in linkage analysis. A further development has been the tentative suggestion that kinematic loops of higher connectivity-sum might be categorised by identifying the screws reciprocal to those defined by their articulations. An extension of this proposal is the thought that certain chains could be regarded as reciprocal to each other. We exemplify the idea by adhibition to the thirteen four-bar linkages of Delassus. INTRODUCTION Among the considerations of screw system theory is the general expression for the rate of working and the consequent notion of reciprocity [Ball, 1900; Hunt, 1978; Phillips, 1984, 1990] when this rate is zero. A twist screw of pitch ht and an action screw of pitch ha are thereby said to be reciprocal when

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(ht + ha ) cos λ = d sin λ , d being the length of the common normal between the two instantaneous screw axes and λ the angle included between them. An equivalent, but more versatile, equation is that which arises from the scalar product of two motors put forward by [v. Mises, 1924a, 1924b], and which I choose to call the "Mises product", although it appears in various guises and under other names in the literature. The condition may be written as

) ) St o Sa = ω • M * + f • V * = 0 ,

(1)

where the twist and action motors are, respectively,

) St = ω ,V * = (ω , htω + P t × ω ), ) Sa = f , M * = (f , haf + P a × f ),

( (

) )

V = P t ×ω

M = Pa ×f ,

to employ fairly familiar notation. Now, eqn. (1) may be taken to define reciprocity between two screws without reference to the rate of working, and the entities concerned might both be motion screws, for example. Even if there is no ready physical referent, one can speak by extension of reciprocal screw systems; if the screws are all of the same character, there is no dimensional inconsistency among the scalar products. This purely formal definition has found application in determining stationary, uncertainty and other special configurations of spatial linkages, as well as in the establishment of gross mobility [Baker, 1978a, 1980, 1985, 1986, 1993, 1995a, 1995b, 2000].

Latterly it has been deployed in

suggesting that linkages of higher connectivity-sum might be categorised by identifying the screws reciprocal to the screw systems defined by the loops’ joint screws [Baker, Wohlhart and Yu, 1994; Baker, 1997].

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A fecund area for the investigation of screw systems and their reciprocals is provided by the group of thirteen four-bar linkages isolated by [Delassus, 1900, 1902, 1922], which have been fully delineated only in recent times [Baker, 1975].

For each loop, we set out its defined 3-system and,

consequently, the reciprocal 3-system. On this basis, relationships among the loops are established, implying the notion of "reciprocal linkages". Some cases are found to be "self-reciprocal" and connections are revealed too with linkages outside the group. It should be mentioned at the outset that [Hunt, 1978] has described several of the Delassus linkages in terms of the screw systems presented below from a geometrical perspective. The solutions are designated d.1 - d.13 by [Waldron, 1969] and we adopt this notation. We make frequent references to the comprehensive catalogue of screw systems of [Hunt, 1978] and, for a more concise treatment, to the restricted one of [Waldron, 1969]. There is a caution to be observed in the categorisation of screw systems, perhaps trite to the initiated, but worthy of a pause for those unfamiliar with the practice. It is the basis of distinctions made below among various loops. If there are two different sets of screws each belonging to the same nominated "special" screw system, the members of one set might not be linearly dependent on those of the other.

A chosen screw system is

sometimes better regarded as a family of screw systems of the same definable type, each member of the family distinguishable by particular characteristics of basis screws, usually the pitches. However attractive or convenient, special screw systems are somewhat arbitrary subsets of the major divisions based on the number of independent screws. This fact is highlighted in the differences between Hunt’s and Waldron’s sub-systems. It

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is well, too, to keep in mind generally the point made by [Hunt, 1978] that the existence of a certain screw system does not guarantee the possibility of a corresponding linkage with full-cycle mobility; an appropriate chain is subject to the feasibility of physical construction which permits the continuous embodiment of the screw system. THE ANALYSIS Solutions d.1 - d.4 Solution d.1, given as H-H-H-H-, contains four parallel helical joints of equal pitch. Solutions d.2 - d.4, given respectively as P-H-H-H-, P-P-H-H- and P-HP-H-, all include parallel screws of equal pitch and prismatic joints directed at rightangles to the screws. Beginning with d.1, we may set up the four screws of pitch h, the first three of them regarded as defining screws, as shown in Fig. 1. Their unit motors can be written as follows.

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) $1 =  0    0  1    0  0    h  

) $2 =  0    0 1    b2  0   h  

) $3 =  0     0   1     b3  - a   3  h   

) $4 =  0     0   1     b4  - a   4  h   

In order for the linkage to be mobile, the fourth screw must belong to the screw system defined by the other three. That is, the following equations must have a non-trivial solution; clearly, they do.

a3 ω 3 + a4 ω 4 = 0 b2 ω 2 + b3 ω 3 + b4 ω 4 = 0

ω1 + ω 2 + ω 3 + ω 4 = 0 These conditions, based upon instantaneous relationships, are only necessary ones. Establishment of gross mobility requires determination of displacement-closure equations [Baker, 1975] or some equivalent technique. For the prismatic joint-screw (0, i) or (0, j) to be included in the same screw system, we require, respectively, that

ω1 + ω 2 + ω 3 = 0 b2 ω 2 + b3 ω 3 + V5 = 0 a3 ω 3 = 0 or

ω1 + ω 2 + ω 3 = 0 b2 ω 2 + b3 ω 3 = 0 a3 ω 3 + V6 = 0 .

The sets of equations are obviously satisfiable, and so solutions d.1 - d.4

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give rise to the same screw system, Hunt’s fifth special 3-system and one akin to the particular case of Waldron’s system 3(c) for which φ1 = π/2 = φ2, but generalised in that the finite-pitch screw need not have pitch zero. [In Waldron’s scheme, the φi are inclinations of the infinite-pitch defining screws to the zero-pitch one.] Let us now consider a screw denoted by ) S r = (α , β , γ , δ , ε , ζ )

which is to be reciprocal to each of the screws given above. Then we must have

ζ + hγ = 0 ζ + b2α + hγ = 0 ζ + b3α − a3 β + hγ = 0 , from which we find

α =0 = β whence

ζ = − hγ ,

) $ r = (0 , 0 , 1, η , θ , − h ) .

That is, the reciprocal system can be defined by three screws directed in the sense of k with pitch -h, a system of the same recognised type as its progenitor. Hence, for any linkage from the d.1 - d.4 grouping, we can name a reciprocal linkage and reciprocal grouping. Solution d.5 The next case for consideration is the spherical linkage, a particular instance of Hunt’s second special 3-system and of Waldron’s system 3(a). It is readily

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identified by three mutually orthogonal screws of zero pitch passing through the origin, represented by ) $ 1 = (i , 0 )

) $2 = ( j, 0)

) $ 3 = (k , 0 ) .

The system is self-reciprocal. Solution d.6 This case is the spatial 4-slider, its screw system belonging to Hunt’s sixth special 3-system and Waldron’s system 3(e). It is easily represented by the three translational screws ) $ 1 = (0 , i )

) $ 2 = (0 , j )

) $ 3 = (0 , k )

and is self-reciprocal. Solution d.7 The next case, consisting of two pairs of coaxial screws, is an unusual one. Illustrated in Fig. 2, the four screws are represented by the unit motors

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) $1 =  0    0 1    0 0   h   1

) $2 =  0    0 1   0 0   h   2

) $3 =  0    0 1   b 0   h   3

) $4 =  0    0 1   b 0   h  .  4

Because the following set of equations has a non-trivial solution, mobility is immediately established as possible.

ω1 + ω 2 + ω 3 + ω 4 = 0 ω3 + ω4 = 0 h1ω1 + h2 ω 2 + h3 ω 3 + h4 ω 4 = 0 Taking, as before, a reciprocal screw of the form ) S r = (α , β , γ , δ , ε , ζ ) ,

it is required that

ζ + h1γ = 0 ζ + h2 γ = 0 ζ + bα + h3 γ = 0 . We cannot allow h2 = h1 or b = 0 without introducing part-chain mobility, and so

α =γ =ζ =0, whence

) $ r = (0 , 1, 0 , η , θ , 0 ) .

Such a screw is directed in the sense of j, has arbitrary pitch θ and arbitrary

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location -ηk. Pictured in Fig. 3, the system cannot be applied to a full-cycle mobile loop, because the screws could only transitorily be located in the same plane.

We observe, however [Hunt, 1978; Baker, 1978b], that d.7 is kinematically equivalent, external to its coaxial combinations, to the P-P- linkage, governed by Hunt’s special 1-system and Waldron’s system 1(b).

The system is

definable by the single translational screw

(0, k ) . The reciprocal system, Hunt’s special 5-system and Waldron’s system 5(a), has the typical member

(α , β , 0 , δ , ε , ζ ) and includes the various 6-H parallel-screw linkages. Solution d.8 A similar interpretation to that of the previous case is appropriate here. The

10

linkage comprises a pair of coaxial screws and two prismatic joints lying in planes parallel to each other and to the helical joints. It functions, outside the coaxial combination, as a 3-slider loop, governed by Hunt’s third special 2system and Waldron’s system 2(d). Its defining motors in this context might be stated as ) $ 1 = (0 , i )

) $ 2 = (0 , j ) ,

whence a typical member of the reciprocal 4-system would be represented by

(0 ,0 , γ , δ , ε , ζ ) . Governed by this system, Hunt’s third special 4-system and Waldron’s system 4(a), are the various 5-H parallel-screw loops. Solution d.9 Here the linkage takes the projected shape of a kite, with four screws parallel and the pitches of those in the plane of symmetry equal to each other and to the arithmetic mean of the remaining two. Depicted in Fig. 4, the screws may be represented by the motors

) $1 =          (h  2

)  $2 =  0     0   1    0  b2    a  0  3  h   + h3 ) / 2   2 0 0 1

) $3 =  0     0   1     b2  - a   3 h   3 

) 0 $4 =   0   1  b4   0   (h + h ) / 3  2

        2  .

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Possible mobility of the loop is established by finding that there is a non-trivial solution of the equations

ω1 + ω 2 + ω 3 + ω 4 = 0 b2ω 2 + b2 ω 3 + b4 ω 4 = 0 ω2 − ω3 = 0 (h2 + h3 )ω1 / 2 + h2ω 2 + h3ω 3 + (h2 + h3 )ω 4 / 2 = 0 . Now investigating the reciprocal system, we have, in terms of the components of our generic reciprocal-screw motor, the requirements h2 + h3 γ =0 2 ζ + b2α + a3 β + h2 γ = 0 ζ + b2α − a3 β + h3 γ = 0 .

ζ

Hence,

+

12

)  h − h2 h +h  $ r =  0 , 3 γ , γ , δ , ε , − 2 3 γ  . 2a 3 2   So two of its defining screws can be represented as

(0, i )

(0, j )

with a third as

 h3 − h2 h + h3  0 , , 1, 0 , 0 , − 2 2a 3 2 

  , 

this last denoting a screw of pitch − (h2 + h3 ) / 2 located at

  h − h 2   3 2 +1     2 a3   

−½

h22 − h32 i. 4 a3

For a reason to be made clear we postpone further comment on the screw systems and proceed with the next three linkages. Solution d.10 As portrayed in Fig. 5, this chain is parallelogrammatic in projection, the four screw joints are parallel and the sums of pitches of diagonally opposite screws are equal. The screw motors can be written as

) $1 =  0    0 1    0 0   h   1

) $2 =  0    0 1    b2  0   h   2

) $3 =  0     0   1     b3  - a   3 h   3 

) 0  $4 =    0     1    b2 + b3    - a3   h + h −h  , 3 1  2

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whence the possibility of mobility is readily seen. The generic reciprocal screw must accommodate the equations

ζ + h1 γ = 0 ζ + b2α + h2 γ = 0 ζ + b3α − a3 β + h3 γ = 0 . Two defining screws of the reciprocal system can again be represented by

(0, i )

(0, j ) ,

a third by

  h1 − h2 h3 − h1 b3 [h1 − h2 ]  , ,1,0 ,0 , − h1  . + a3 a3 b2   b2 The latter has pitch -h1 and is situated at

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−½

 h − h  2  h − h b (h − h )  2   h − h b (h − h )  h − h 1 2 3 1 + 3 1 2  + 1  1 3 − 3 1 2  i + 1 2   + a b a3 b2  a3 b2  b2  2   3   a3

 j h1 . 

Solution d.11 Displayed in Fig. 6, this loop consists of four parallel screw joints, antiparallelogrammatic in projection, with pitches of alternate screws equal. The unit screw motors are given by

) $1 =  0     0   1     0  - a   1 h   1 

) $2 =  0    0 1    0 a   1 h   1

) $3 =  0     0   1     b3  - a   3 h   3 

) $4 =  0    0 1    b3  a   3 h  ,  3

and we see that mobility is easily established as possible.

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Investigating the reciprocal screw system, it is necessary to satisfy the equations

ζ − a1 β + h1γ = 0 ζ + a1 β + h1γ = 0 ζ + b3α − a3 β + h3 γ = 0 . Two independent defining screws of the system can again be denoted by

(0, i )

(0, j ) ,

a third by

 h1 − h3   , 0 , 1, 0 , 0 , − h1  .  b3  The last has pitch -h1 and is positioned at

 h − h  2  1 3   + 1 b  3  

−½

h1 (h1 − h3 ) j . b3

Solution d.12 This case, limned in Fig. 7, comprises two parallel helical joints of equal pitch and two sliders bilaterally symmetric with respect to the plane containing the screws. The representative motors are

) $1 =  0    0  1    0  0    h  

) $2 =  0    0 0   l  m   n  

) $3 =  0    0 0   − l  m   n  

) $4 =  0    0 1    b 0   h  .  

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The necessary conditions for mobility, obviously possessing a non-trivial solution, are given by

ω1 + ω 4 lV2 − lV3 + bω 4 mV2 + mV3 hω1 + nV2 + nV3 + hω 4

=0 =0 =0 = 0.

Determination of the reciprocal screw system lies in solving the equations

ζ

+ hγ = 0 lα + mβ + nγ = 0 − lα + mβ + nγ = 0 .

Once more we find that

(0, i )

(0, j )

represent two independent defining screws. A third is denoted by

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n    0 , − , 1, 0 , 0 , − h  , m   so that its pitch is -h and it is situated at

 n2   2 +1 m 

−½

h

n i. m

The group d.9 - d.12 We have shown that the four chains d.9 - d.12 give rise to the same type of reciprocal screw system, Hunt’s ninth special 3-system or Waldron’s system 3(c). Scrutiny of solution d.12 reveals that its joint screws also belong to this reciprocal system, and so then do the joint screws of the other three loops. That is, any of the four chains defines a screw system and its reciprocal which both belong to the categories named. The directly defined systems can be distinguished, nevertheless, and this is easiest achieved by reference to Waldron’s classification.

His category 3(c) depends on one zero-pitch

screw in a certain position and two infinite-pitch screws prescribed by their orientations with respect to the former. Taking loop d.9, for example, the zero-pitch screw

(0 , 0 ,1, b0 , − a0 , 0 ) is located by solving the equations

ω1 + ω 2 + ω 3 = ω 0 b2ω 2 + b3ω 3 = b0ω0 a3ω 2 − a3ω 3 = − a0ω0 (h2 + h3 )ω1 / 2 + h2ω2 + h3ω3 = 0. We find that the line of zero-pitch screws is given by

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a0 =

h2 + h3 a3 . h2 − h3

One of the infinite-pitch screws, represented by

(0 , 0 , 0 , sφ x , 0 , cφ x ) , is determined by solving the equations ω1 + ω 2 + ω 3 = 0 b2ω 2 + b3ω 3 = sφ x a3ω 2 − a3ω 3 = 0

(h2 + h3 )ω1 / 2 + h2ω2

+ h3ω 3 = cφ x ,

from which

φx =

π . 2

The other, denoted by

(0 , 0 , 0 , 0 , sφ

y

, cφ y ) ,

is obtained from the equations

ω1 + ω 2 + ω 3 = 0 b2ω 2 + b3ω 3 = 0 a3ω 2 − a3ω 3 = sφ y

(h2 + h3 )ω1 / 2 + h2ω2

+ h3ω 3 = cφ y ,

so that

φ y = tan −1

2 a3 . h2 − h3

We can examine the other loops in the same way to find the following results.

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d.10: The line of zero-pitch screws is represented by

 b b a0   h3 − h1 + 3 [h1 − h2 ] + 0 (h2 − h1 ) + h1 = 0 b2 a3   b2 and the infinite-pitch screws defined by

cot φ x =

h2 − h1 b2

cot φ y =

 1  b2 − b3 b  h1 + 3 h2 − h3  . a3  b2 b2 

d.11: The line of zero-pitch screws is prescribed by

b0 =

h1 b3 h1 − h3

and the infinite-pitch screws by

cot φ x =

h1 − h3 b3

φy =

π . 2

d.12: The line of zero-pitch screws is determined by a0 =

m h n

and the infinite-pitch screws specified by

φx =

π 2

cot φ y =

n . m

We may relate these direct systems to their respective reciprocals in a somewhat similar fashion.

If we denote the three defining screws of the

reciprocal system by a zero-pitch motor

(α , β , γ , δ , ε , ζ )

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together with

(0, i )

(0, j ) ,

then we require that the following equations be satisfied.

ζ + b0α − a0 β = 0 sφ xα + cφ x γ = 0 sφ y β + cφ y γ = 0 αδ + βε + γζ = 0 So the motor of the zero-pitch screw can be written as

 1  − , − 1 , 1, δ , ε , ζ  tφ tφ y γ γ γ x 

 ,  

b δ ε a + =γ 0 − 0  tφ x tφ y  tφ x tφ y

 .  

where

This screw is located by

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 Xr Y  r Z  r

i tφ x tφ y   = Pr = γ t 2 φ x + t 2 φ y + t 2 φ x t 2 φ y - tφ y   δ 

(

=

=

)

tφ x tφ y

γ (t φ x + t 2 φ y + t 2 φ x t 2 φ y ) 2

tφ x tφ y

γ (t φ x + t 2 φ y + t 2 φ x t 2 φ y ) 2

j - tφ x ε

k tφ x tφ y γ

     − γtφ x  b0 − a0  − εtφ x tφ y    tφ    x tφ y     δtφ tφ + γtφ  b0 − a0   x y y      tφ x tφ y    − εtφ y + δtφ x       εtφ x   1 + t 2φ y  −δ − tφ y     δtφ y 1 + t 2φ x   ε+ tφ x    − εtφ y + δtφ x .    

(

(

)

)

That is, Pr is defined by the planes Xtφ x − Ytφ y = a0 tφ x − b0 tφ y

and Xtφ y + Ytφ x = Ztφ x tφ y .

Solution d.13 This final four-bar is the Bennett linkage, of which much has been written. Its joint screws belong to the particular instance of the general 3-system for which the pitch is zero. At any configuration of the loop they are generators of one regulus of an hyperboloid, and the reciprocal system consists of zeropitch screws which are generators of the complementary regulus, thereby defining a reciprocal Bennett chain. A discussion of the locations of the links and joint axes of the Bennett loop on its associated quadric surfaces is

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presented in [Baker, 1988]. Other loops [Waldron, 1973a, 1973b] has compiled a near-complete catalogue of mobile, single-loop linkages with two, three or four members, featuring all lower pairs but helical ones.

Several of the more simply prescribed cases can be

analysed in the same way as above, although reciprocal linkages cannot be named in every instance, for lack of knowledge of relevant loops. Some equivalences with foregoing solutions can be established and other interconnections made. Since the publication of Waldron’s work many other linkages have been isolated, generally of greater complexity, however, and so there is ample material for further investigation of the topic raised in this paper. REFERENCES Baker, J.E., 1975, Mobility Analyses of Spatial Linkages, Doctoral Dissertation, The University of New South Wales, Sydney. Baker, J.E., 1978a, "On the Investigation of Extrema in Linkage Analysis, using Screw System Algebra", Mechanism and Machine Theory, Vol. 13, No. 3, pp. 333-343. Baker, J.E., 1978b, "On Coaxial Screws in Spatial Linkages", Mechanism and Machine Theory, Vol. 13, No. 3, pp. 345-349. Baker, J.E., 1980, "Screw System Algebra Applied to Special Linkage Configurations", Mechanism and Machine Theory, Vol. 15, No. 4, pp. 255265.

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Baker, J.E., 1985, "Linkage Kinematics and Conformational Analysis", Tetrahedron, Vol. 41, No. 1, pp. 121-128. Baker, J.E., 1986, "Limiting Positions of a Bricard Linkage and their Possible Relevance to the Cyclohexane Molecule", Mechanism and Machine Theory, Vol. 21, No. 3, pp. 253-260. Baker, J.E., 1988, "The Bennett Linkage and its Associated Quadric Surfaces", Mechanism and Machine Theory, Vol. 23, No. 2, pp. 147-156. Baker, J.E., 1993, "A Geometrico-Algebraic Exploration of Altmann’s Linkage", Mechanism and Machine Theory, Vol. 28, No. 2, pp. 249-260. Baker, J.E., 1995a, "On Testing for Gross Mobility of New Kinematic Loops with Screw System Algebra", Mechanism and Machine Theory, Vol. 30, No. 5, pp. 679-693. Baker, J.E., 1995b, "On the 6-Hinge Loops in Bricard’s Line-Symmetric and Plane-Symmetric Octahedra", Proceedings of the Ninth World Congress on the Theory of Machines and Mechanisms, Milano, Italy, August 29September 2, Vol. 2, pp. 1494-1498. Baker, J.E., 1997, "The Single Screw Reciprocal to the General PlaneSymmetric Six-Screw Linkage", Journal for Geometry and Graphics, Vol. 1, No. 1, pp. 5-12. Baker, J.E., 2000, "On Equivalent Motions in Spatial Linkages", Mechanism and Machine Theory, Vol. 35, No. 7, in press. Baker, J.E., Wohlhart, K. and Yu H.-c., 1994, "On the Single Screw

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Reciprocal to the General Line-Symmetric Six-Screw Linkage", Mechanism and Machine Theory, Vol. 29, No. 1, pp. 169-175. Ball, R.S., 1900, A Treatise on the Theory of Screws, C.U.P., Cambridge. Delassus, E., 1900, "Sur les Systèmes Articulés Gauches, Première Partie", Annales Scientifiques de l'Ecole Normale Supérieure de Paris, Series 3, Vol. XVII, pp. 445-499. Delassus, E., 1902, "Sur les Systèmes Articulés Gauches, Deuxième Partie", Annales Scientifiques de l'Ecole Normale Supérieure de Paris, Series 3, Vol. XIX, pp. 119-152. Delassus, E., 1922, "Les Chaînes Articulées Fermées et Déformables à Quatre Membres", Bulletin des Sciences Mathématiques, Series 2, Vol. 46, No. 1, pp. 283-304. Hunt, K.H., 1978, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford. Phillips, J., 1984, Freedom in Machinery Volume 1 Introducing screw theory, C.U.P., Cambridge. Phillips, J., 1990, Freedom in Machinery Volume 2 Screw theory exemplified, C.U.P., Cambridge. von Mises, R., 1924a, "Motorrechnung, ein neues Hilfsmittel in der Mechanik", Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 4, No. 2, pp. 155-181.

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von Mises, R., 1924b, "Anwendungen der Motorrechnung", Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 4, No. 3, pp. 193-213. Waldron, K.J., 1969, The Mobility of Linkages, Doctoral Dissertation, Stanford University, Palo Alto. Waldron, K.J., 1973a, "A Study of Overconstrained Linkage Geometry by Solution of Closure Equations - Part 1. Method of Study", Mechanism and Machine Theory, Vol. 8, pp. 95-104. Waldron, K.J., 1973b, "A Study of Overconstrained Linkage Geometry by Solution of Closure Equations - Part II. Four-bar Linkages with Lower Pair Joints other than Screw Joints", Mechanism and Machine Theory, Vol. 8, pp. 233-247.