Oblique circular torus, Villarceau circles, and four types of Bennett linkages

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Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

Chung-Ching Lee and Jacques M. Hervé

Oblique circular torus, Villarceau circles and four types of Bennett linkages

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Oblique circular torus, Villarceau circles,

and four types of Bennett linkages

Chung-Ching Lee

1

and Jacques M. Herve´

2

Abstract

The oblique circular torus (OCT) and its main geometric properties are introduced. Intrinsic vector calculation is utilized to mathematically describe the OCT. The coordinate-free approach leads to the algebraic equation of an OCT in a privileged Cartesian reference frame. The OCT equation is used to confirm a theorem of Euclidean geometry. In a broad category of OCT, through any point five circles can be drawn on the surface, namely the parallel of latitude and four circular generatrices whose planes pass through the OCT center of symmetry. In the special case of a right circular torus, the Villarceau theorem is verified. Next, consider the four RRS open chains whose S spherical-joint centers move on the same OCT and their possible in-parallel assemblies in single-loop RRRS chains. From a category of the foregoing RRRS chains, a new derivation of the amazing Bennett 4R linkage is proposed. Two kinds of Bennett linkages are further verified and each kind contains two enantiomorphic or symmetric linkages. Four types of Bennett linkages associated with one OCT are established by uniquely specifying the link twist as an acute value. Two cases of special type, rectangular and equilateral configurations, are also confirmed.

Keywords

Oblique circular torus, Villarceau circles, circular generatrix, Bennett linkage, enantiomorphic

Date received: 30 November 2012; accepted: 16 April 2013

Introduction

The surface generated by the rotation of a circle around a fixed axis that belongs to the circle plane is called a torus or toric surface. This is the standard acceptation of the word torus. In fact, the word torus is also used to designate the obtained revolute surfaces when the axis of revolution does not lie on the circle plane. In this way, the shape of the surface may look like a flattened torus. However, the locution flattened torusis not mathematically accurate and one may be misled into thinking that it is the toroidal surface obtained by revolving an ellipse that is a flattened circle. A toroidal surface or toroid is a revolute sur-face swept out by a closed curve, which is not neces-sarily a circle. When a generatrix is a circle, the surface is called circular toroid.

In what follows, our emphasis is on the oblique circular torus (abbreviated to OCT), which is the revolute surface generated by a circle whose plane intersects the axis of revolution at one point. Moreover, the center of the rotating circle is assumed to belong to the common perpendicular to the rotat-ing circle axis and the axis of revolution. The foot of the perpendicular drawn from the rotating circle

center onto the axis of revolution is a fixed point of the OCT axis and the plane that is perpendicular to the axis at this point is the fixed equatorial plane of the OCT. The intersections of the OCT with other fixed planes perpendicular to the OCT axis are called parallels of latitude, which are obviously coaxial circles. The intersections of the OCT with half-planes containing the revolution axis are curves called merid-ians. Clearly, the meridians are congruent by rotation around the revolution axis. In the standard torus, the meridians are circles, but in an OCT, they are neither circles nor ellipses. The standard torus is a special case of OCT and can be termed a right circular torus. In 1848, Yvon Villarceau discovered an amazing

1_{Department of Mold and Die Engineering, National Kaohsiung}

University of Applied Sciences, Kaohsiung, Taiwan, Republic of China

2_{Ecole Centrale des Arts et Manufactures, Grande Voie des Vignes,}

Chatenay-Malabry, France Corresponding author:

Chung-Ching Lee, Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences, 415 Chien Kung Road, Kaohsiung, 80782, Taiwan, Republic of China.

Email: cclee@cc.kuas.edu.tw

Proc IMechE Part C:

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geometric property of the torus: a plane that is bitan-gent to a torus intersects the surface along two con-gruent circles. The circles are symmetric with respect to a plane containing the axis of revolution. These two circles are called the Villarceau circles.1 Our paper verifies that Villarceau’s theorem corresponds to a special case of a general theorem on the OCT. Through any point of a broad category of OCT, one can draw on the OCT surface five circles, which are the parallel of latitude, and two couples of congruent circular generatrices whose planes pass through the OCT center of symmetry. A plane that is bitangent to an OCT intersects the surface along a couple of congruent circles, which generalize the Villarceau circles. The intersection of an OCT by a plane belonging to another family of planes, which is not bitangent to the OCT, is also a couple of con-gruent circles. With the help of this theorem of Euclidean geometry on OCT, this article presents a new way to derive the well-known Bennett linkage, which is movable with one-degree-of-freedom (1-DoF) of finite mobility.

The Bennett linkage is the only movable four-revo-lute chain with non-parallel and non-intersecting revolute R joint axes. The link twist angles and the link lengths, which are mathematically related by one constraint equality, characterize it. It can be viewed as a warped hinged parallelogram and its opposite link lengths and link twist angles are equal. G. T. Bennett2 first discovered this four-revolute mechanism in 1903. E. Borel3found it independently only one year later after Bennett. R. Bricard4also noticed this outstand-ing findoutstand-ing. Specifically, this linkage disobeys the Chebyshev–Gruebler–Kuzbach formula5 and it was qualified as paradoxical in the terminology of J. M. Herve´ scientific work.6Paradoxical mobility is subject to geometric conditions that cannot be expressed without using Euclidean (or Pythagorean) metric.7 The geometric figure of four points and four segments of straight lines, whose alternate sides are equal in length, was called isogram by Bennett.8The locution of skew isogram mechanism was used to describe this linkage. The connection between the Bennett linkage and the Villarceau theorem1 was first detected by Bricard9 and was confirmed by F. E. Myard.10 A noteworthy work of Krames11 provided further insights into the relation between Bennett’s linkage and Villarceau’s circles. Ref. 11 presented that the motion generated by the coupler of Bennett mechan-ism is line symmetric and the line of symmetry lies on a regulus of a hyperboloid of one sheet. However, the signs of Bennett equality are not discussed in Krames’ paper. Besides, the Cartesian equation of an OCT appears nowhere and the Bennett relation is not derived by logical deduction from the OCT equation. The existence of two categories of OCT is also ignored. Two kinds of Bennett mechanisms were elu-cidated in a published work.12Hereinafter, a new geo-metric derivation of Bennett 4R linkage will be done

by considering RRS open chains whose spherical S joint center moves on the same OCT.

This article proceeds as follows. First, an OCT is the surface trajectory of a point in a 2-dof motion generated by a serial concatenation of two revolute R joints. From a parametric representation obtained through the product of two rotations, a Cartesian equation of an OCT is derived. That equation further provides three other ways to generate the same sur-face by rotating circles. The four circular generatrices of an OCT are associated with four RRS open chains producing the same Cartesian equation of an OCT. Some assemblies of two RRS open chains yield a geo-metrical derivation of four types of Bennett linkages having the same link lengths and the same absolute value of the link twist angles. An appropriate conven-tion for orientating vectors and angles is adopted to discuss special cases comprehensively as well.

The OCT

Definition of an OCT

Let us consider the serial concatenation of two revo-lute R pairs shown in Figure 1, which is sometimes called also a two-hinge dyad in the literature on mech-anism theory. The open RR chain includes three rigid bodies connected in series by two revolute R joints or hinges. Each of the two R joints (or R pairs) is char-acterized by its axis and each axis is determined by the datum of any one of its points and a unit vector par-allel to the axis in an initial home configuration of the RR chain. Assuming that the two axes are not paral-lel, there is a unique common perpendicular to the two axes. It is convenient to choose two points on the common perpendicular to specify the two R axes. When the first fixed R pair is locked, a point P belonging to the end body of a two-hinge dyad traces a circle on a plane, which is perpendicular to the second R axis and is located at a distance d of the common perpendicular. The distance d is called offset. When the fixed R is unlocked, the point P traces a surface of revolution, which is called

Figure 1. The circular toroid obtained as a surface trajectory in a RR dyad.

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a general circular toroid, as shown in Figure 1. In the noteworthy paper, Fichter and Hunt13proved that a plane that is bitangent to a general circular toroid intersects the surface along a couple of congruent cir-cles. The pertinent proof is established by using the complex projective completion of the Euclidean space. In what follows, one will not consider imaginary geo-metric entities but will focus on the special case with a zero oﬀset.

When the oﬀset (d) is zero and two R axes are not twisted by a right angle (jj 6¼ 90_{), the circular toroid}

becomes an OCT depicted in Figure 2. The case (jj ¼ 90_{) corresponds to the right circular torus}

that is the standard torus. An OCT is produced by revolving an inclined circle (circular generatrix) whose plane contains the common perpendicular of the circle axis and the revolution axis. It is proved that, gener-ally, the revolution of other circular generatrices lying on planes containing the OCT center produces the same OCT. Because of the obvious symmetry of the revolute surface with respect to any plane containing the revolution axis, a second generatrix is simply the plane-symmetric of the ﬁrst one. In the case (jj ¼ 90_{), the foregoing two circular generatrices}

are meridians and the other two circles are those of Villarceau’s theorem.

Refer to Figure 2; (O, i, j, k) is a Cartesian ortho-normal frame of reference whose origin O is the inter-section of the ﬁrst R axis and the common perpendicular of the two R axes. The unit vector k is parallel to the ﬁrst R axis, which is characterized by (O, k). The unit vector i is parallel to the common perpendicular of the two R axes. The point B is the intersection of the common perpendicular and the second R axis. The vector (OB) ¼ B O is equal to bi. The number b, b > 0, is the length (or Euclidean norm) jjOBjj of the bar (OB). The unit vector s is parallel to the axis of the second R pair. In the con-catenation of two R pairs, the axes are twisted by an angle around the common normal OB. It is worth noticing that through the usual convention of hand-edness, the vector i orients the angle . An observer

(Ampe`re’s manikin in electromagnetism theory) lying along the axis (O, i) with O at his feet and i going from his feet towards his head sees the counterclockwise rotations with positive angles whereas the clockwise rotations have negative angles. It is important to note that the same given rotation has two representations. If one is a rotation of angle around an axis parallel to i, then the other one is the rotation of angle around (i). That way, without loss of generality, the angle of twist in the oriented bar (OB) can be, and therefore is assumed to be acute, 0 < jj 4 /2. Specifying the link twist as an acute angle will uniquely determine the proper twist of link.

In this RR chain, the first R pair has the fixed axis (O, k) and the second R pair has the axis (B, s). A point M is attached at the end body in such a way that the segment (BM) is collinear with (OB) at the initial home posture of the RR chain: vector(BM) ¼ M B ¼ ai with a ¼ jjBMjj. One can notice that jj is the angle between the equatorial plane of the OCT and the planes of the circular tra-jectories of M when the fixed R is locked. Assuming that the fixed axis (O, k) is vertical, the angle jj ¼ jff(k, s)j is the common slope of a family of planes, which contain a circular generatrix of the OCT.

Cartesian equation of an OCT

Following two steps, one can obtain the variable pos-ition of the point M, which moves with respect to the fixed frame (O, i, j, k), and its 2-dof motion mathem-atically. In the first step, the second R rotates around the axis (B, s) with an angle ’ while the first R keeps its home pose ( ¼ 0); the initial position Mo of M

becomes Mv, which is expressed by

Mo!Mv¼B þ expð’ sÞðBMoÞ ð1Þ

Using

s ¼ expð iÞk ¼ Sin j þ Cos k ð2aÞ v ¼ expð iÞj ¼ Cos j þ Sin k ¼ s i ð2bÞ ðOBÞ ¼ bi, ðBMoÞ ¼ai, ðOMoÞ ¼ða þ bÞi ð2cÞ

then, OMv

ð Þ ¼ðOBÞ þ ðBMvÞ ¼ ðOBÞ þ expð’ sÞðBMoÞ

¼bi þ aðSin ’ s i þ Cos ’ iÞ ¼bi þ aðSin ’ v þ Cos ’ iÞ

¼bi þ a½Sin ’ðCos j þ Sin kÞ þ Cos ’ i ¼ ðb þ aCos’Þi þ aSin ’ Cos j þ aSin ’ Sin k

ð3Þ This is a parametric representation of the circular trajectory of Mvwith only the rotation in the second

R pair.

Figure 2. An oblique circular torus obtained as a surface trajectory in a RR dyad.

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In the second step, once the second R pair is locked (angle ’ keeps its value), the ﬁrst R pair rotates around the axis (O, k) by an angle and the point position Mvis transformed into MT.

Mv!MT¼O þ expð kÞðOMvÞ ð4Þ

Note that the rotation around (O, k) with an angle yields

i ! Cos i þ Sin j ð5aÞ j ! Sin i þ Cos j ð5bÞ

k ! k ð5cÞ

Hence, OMT

ð Þ ¼ ðb þ a Cos ’ÞðCos i þ Sin jÞ þa Sin ’ Cos ðSin i þ Cos jÞ þa Sin ’ Sin k ð6Þ The coordinates (X, Y, Z) of the point MT

belong-ing to the OCT are functions of the two parameters and ’, which are angles of rotation in the open RR chain:

X ¼ ðb þ a Cos ’ÞCos a Sin ’ Cos Sin Y ¼ ðb þ a Cos ’ÞSin þ a Sin ’ Cos Cos Z ¼ a Sin ’ Sin

ð7Þ

Eliminating and ’ from equation (7) yields the Cartesian equation of the surface as follows,

ðX2þY2þZ2b2a2Þ2þ 4b

2

Sin2Z

2_¼_4a2_b2 _ð8Þ

This is an algebraic equation for the surface trajec-tory of M located at the end of the last link.

Other circular generatrices of an OCT

The surface modeled by equation (8) is an OCT. From this quartic equation, one can notice that replacing b by a and by a second angle does not change the surface equation provided that the equality a2 Sin2 ¼ b2 Sin2 is satisﬁed. Consequently, another RR bar with a common perpendicular of length a (instead of b) and a twist angle verifying Sin2 ¼(a/b)2Sin2produces the same surface trajec-tory for a point located at the distance b of the second R axis (instead of a in the original RR system). For a given (a/b)2 Sin2, when assuming (a/b)2Sin24 1 , (a/b)Sinjj 4 1 , Sinjj 4 b/a, the equation of unknown , Sin2 ¼(a/b)2Sin2, has two solutions that are of opposite signs under the assumption

2(/2, þ /2). Noteworthy is the fact that all the systems of arbitrary data (a, b, jj) are not valid for the replacement of b by a and by . For instance, if b ¼1 and a ¼ 2, then the inequality Sinjj 4 1/ 2 , jj 4 /3 limiting the possible choices of jj and jj5 /3 gives OCTs, which do not allow the forego-ing replacement. When b/a 5 1 , b 5 a, the surface is a ring OCT or a horn OCT; the inequality Sinjj 4 b/ a is always veriﬁed and jj is equal to Sin1 [(a/ b)Sinjj]. Here, Sin1denotes the inverse function of the Sine function rather than 1/Sin. This inverse func-tion is multivalued and becomes single valued when its value is restricted to belonging to the domain (/ 2, /2). Hence, under the assumption 2 (/2, /2), jj ¼ Sin1[(a/b)Sinjj] yields one value for jj. If b/ a <1 , b < a, then the surface belongs to a category of OCT divided into two subcategories. Either the angle jj is small enough to satisfy Sinjj 4 b/a or jjis such as Sinjj > b/a; there is no angle jj satisfy-ing a2Sin2 ¼ b2Sin2. Consequently, when the given system (a, b, jj) of two lengths and one angle is appropriate, there are four RR open chains producing the same OCT for a point M of the outermost body in the triplet of rigid bodies. The metric properties of the four chains are symbolized by (b, ; a), (a, ; b), (b, ; a), and (a, ; b) where the combination (b, ; a) characterizes the original mechanism used to obtain the OCT equation.

Theorems on OCT circular generatrices

In Figure 3, through an arbitrary point Q on the sur-face of the OCT, ﬁve circles drawn to the OCT sursur-face intersect. They are the parallel of latitude and four circular generatrices. One can state a theorem of Euclidean geometry: for a broad family of OCTs, an OCT has four circular generatrices, which lie on planes containing the OCT center. From the datum of one circular generatrix of an OCT, the other three circular generatrices can be derived.

That theorem is complemented by the property: any plane of slope, either jj or jj and passing through the OCT center that is the origin O, intersects the surface along a couple of congruent circles. As the

Figure 3. Five circles on the surface of OCT.

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Conclusions

The OCT is a revolute surface, which is less special than the standard torus but is more special than the general circular toroid. From the OCT equation, two categories of OCTs are detected. In a broad category of OCTs, the revolute surface has four circular gen-eratrices whose planes contain the OCT center of sym-metry. Because of the symmetry about any plane containing the axis of revolution, only two genera-trices are not congruent circles amid the four circles. The Villarceau theorem corresponds to a special case of OCT. The mechanical generation of the OCTs belonging to the appropriate category leads to special RRRS chains in a ﬁrst step and to the Bennett 4R linkage in a second step. From the OCT equation, a new formulation of the so-called Bennett equality (or index) tying link lengths and twist angles in the four bars of Bennett linkage is derived and proposed to be an improvement of what is available in already pub-lished literature. Based on the speciﬁcation of a coher-ent convcoher-ention for oricoher-enting the bars and the twist angles together with the choice of an appropriate range for the twist angles, the discussion of the improved Bennett equality results in the discrimin-ation of four related types for the non-equilateral Bennett linkages. The four types are sorted into two kinds; each kind contains a pair of enantiomorphic

linkages. Apparently, for the ﬁrst time, the enantio-merism in mechanisms is taken into account, expli-citly. There are only two types of rectangular Bennett linkages and one type of non-degenerated equilateral Bennett linkages. Probably, the presenta-tion of hybrids of Bennett linkages needs to be revisited.

Funding

The authors are very thankful to the National Science Council for supporting the research under grants NSC 100-2221-E-151-022 and 101-2221-E-151-017.

References

1. Villarceau Y. The´ore`me sur le tore. Nouvelles Annales de Mathe´matiques1848; 7: 345–347.

2. Bennett GT. A new mechanism. Engineering 1903; 76: 777–778.

3. Borel E. Me´moires des savants e´trangers: Me´moire sur les de´placements a` trajectories sphe´riques. Comptes Rendus1904; 23: 56.

4. Bricard R. Sur une proprie´te´ de l’hyperboloı¨de orthog-onal et sur un syste`me articule´. Nouvelles Annales de Mathe´matiques1906; 4: 69–80.

5. Selig JM. Geometrical foundations of robotics. Singapore: World Scientific, 2000, pp.39–56.

6. Herve´ JM. Analyse structurelle des me´canismes par groupes de de´placements. Mech Mach Theory 1978; 13: 437–450.

7. Lee C-C and Herve´ JM. Synthesize new 5-bar paradox-ical chains via the oblique circular cylinder. Mech Mach Theory2011; 46: 784–793.

8. Bennett GT. The skew isogram mechanism. Proc London Math Soc1913; 13: 151–173.

9. Bricard R. De´monstrations e´le´mentaires des proprie´te´s fondamentales du tore. Nouvelles Annales de Mathe´matiques1924; 5: 308–313.

10. Myard FE. Sur les chaıˆnes ferme´es a` quatre couples rotoı¨des non concourants, de´formables au premier degre´ de liberte´. Isogramme torique. Comptes-Rendus de l’Acade´mie des Sciences de Paris, Cine´matique Applique´e1931; 192: 1194–1196.

11. Krames J. Zur geometrie des Bennett’schen mechanis-mus. Sitzungsber., Abt. II, o¨sterr. Akad. Wiss., Math.-Naturw. Kl1937; 146: 145–158.

12. Herve´ JM and Dahan M. The two kinds of Bennett’s mechanisms. In: Proceedings of the sixth IFToMM world congress on theory of machines and mechanisms, New Delhi, India, 15–20 December 1983, Vol. 1. New Delhi: Wiley Eastern Ltd, pp.116–119.

13. Fichter EF and Hunt KH. The fecund torus, its bitan-gent-circles and derived linkages. Mech Mach Theory 1975; 10: 167–176.

14. Lee C-C and Herve´ JM. Discontinuously movable seven-link mechanisms via group-algebraic approach. Proc IMechE, Part C: J Mechanical Engineering Science2005; 219: 577–587.

Partitioned mobility: Type A(+) ≅ Type A(-)

Two postures in : Type B(+) ≅ Type B(-) (a)

(b)

Figure 13. Equilateral Bennett linkages. (a) Partitioned mobility: Type A(þ) ﬃ Type A(). (b) Two postures in Type B(þ) ﬃ Type B().