Convex cones in screw spaces

Mechanism

www.elsevier.com/locate/mechmt

Mechanism and Machine Theory 40 (2005) 710–727

andMachine Theory

Convex cones in screw spaces

Dimiter Zlatanov a,*, Sunil Agrawal b, Clement M. Gosselin c

a Musashi Institute of Technology, Department of Mechanical Systems Engineering,

Tamazutzsumi 1-28-1, Tokyo 158-8557, Japanb Department of Mechanical Engineering, University of Delaware, Newark, DE, USA

c Departement de Genie Mecanique, Universite Laval Quebec, Quebec, Canada

Received 17 July 2003; received in revised form 5 November 2004; accepted 23 November 2004

Available online 5 March 2005

Abstract

This work examines different screw systems and analyzes the possible subsets spanned by the various

choices of a screw basis when the intensities of the wrenches, applied along the screws of the basis, are

not allowed to change sign. Such sets arise in cable robotics, grasping and assembly. Convex screw spaces

exhibit a surprising geometric variety, often even inside the same screw system.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Screw systems; Convex cones; Cable robots

1. Introduction

In some applications, such as cable-driven robots, grasping or pushing, the force, that can beapplied by an actuator on the end-effector, cannot reverse its direction. Similar situations occur inassembly planning [1,2]. In such cases, the possible resultant wrenches do not span a vector space,but rather a convex cone in the complete screw system. Linear combinations with nonnegative

0094-114X/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mechmachtheory.2004.11.004

* Corresponding author.

E-mail addresses: [email protected] (D. Zlatanov), [email protected] (S. Agrawal), gosselin@

gmc.ulaval.ca (C.M. Gosselin).

mailto:[email protected]

mailto:[email protected]

mailto:gosselin@ gmc.ulaval.ca

mailto:gosselin@ gmc.ulaval.ca

https://www.researchgate.net/publication/239665585_An_Extension_of_Screw_Theory?el=1_x_8&enrichId=rgreq-285c58af-1b2e-4504-aa88-4e0d8aca513f&enrichSource=Y292ZXJQYWdlOzIyOTMxOTg0MztBUzoxMDM4OTM2NjkyNTMxMjVAMTQwMTc4MTU4OTE4NQ==

D. Zlatanov et al. / Mechanism and Machine Theory 40 (2005) 710–727 711

coefficients are referred to as conic combinations, and their union forms the conic hull of the gen-erating vectors (as opposed to their linear hull (or span), obtained by all linear combinations).

This paper studies the geometric properties of the conic hulls of 2 or 3 linearly independentscrews. The arising convex cones are classified up to a change of reference frame and describedvia (arguably) clear and intuitive geometric representations. The results provide a designer or ana-lyst of such mechanical systems with the valuable ability to visualize a set of possible resultantwrenches and estimate how it would change when the generators are modified. As a byproduct,the work provides new ways of looking at the linear spans of screws and thus improves under-standing and simplifies the visualization of some of the more complex screw systems.

It suffices to consider the normalized screws in a cone. A wrench, f, is normalized (or has unitintensity) when it either is a unit couple, f = l = (0,m), jmj = 1, or has a unit force component,f = (f,m), jfj = 1. (Usually, u, l and f denote a pure force/moment and general wrench, respec-tively.) For our purposes, it is important to distinguish two normalized screws which coincidebut have opposite directions. The set of directed normalized elements of a k-system of screws dou-ble-covers RPk�1, i.e., it can be identified with two copies of the (k � 1)-dimensional projectivespace, one for each direction.

Consider a convex cone inRn. It consists of rays from the origin, which can be identifiedwith theirintersections with the unit sphere. Thus, the cone can be described by a convex shape on the (n � 1)-sphere. Alternatively, a ray can be described by its intersection point with a hyperplane away fromthe origin, Fig. 1.When a ray is parallel to the hyperplane, we identify it with a ‘‘point at infinity.’’ Ifthe ray points away from the hyperplane, we use the intersection point of the opposite ray. The con-vex hull of independent generators includes no more than one unit vector on a given line. Hence,such a cone can be described as a subset of RPn�1. The hull of a basis corresponds to a generalizedsimplex in the hyperplane, Fig. 1. This set consists of all points that are either in the intersection of nhalf-hyperplanes, or in the intersection of their complements. The simplex is called externalwhen ithas two connected components, Fig. 1(a). The internal (n � 1)-simplex is the usual convex hull of npoints, Fig. 1(b). Fig. 5 shows examples of generalized triangles (2-simplices) with or without ver-tices at infinity.We can use the orientation (positive or negative) of the (n � 1)-simplex in n-space todistinguish between the oppositely directed conic hulls.

For a screw n-space, we use similar ideas to construct a faithful (n � 1)-dimensional image on aplane or sphere and describe a conic hull via its characteristic simplex. This is easy to do in general

(a) (b)

Fig. 1. Conic hulls in R3 as generalized simplices in RP2.

712 D. Zlatanov et al. / Mechanism and Machine Theory 40 (2005) 710–727

(as described above), the challenge is to obtain a map with an immediate visual correspon-dence between the image and the actual location and pitch of the screws in the usual Cartesianspace.

The oriented generalized simplices in the image space represent the conic hulls in the screw sys-tem. To classify the conic hulls one needs to describe the orbits, in the space of simplices, of thetransformations induced by the Euclidean group. Obviously, it is enough to consider the Euclid-ian symmetries of the screw system, i.e. the rigid-body displacements that leave the system fixedwhile causing some of its elements to change places.

Following (with variations) Gibson and Hunt [3] (see also [4] as well as [5]) we label a screwsystem by its dimension (2 or 3); I or II, indicating whether or not there are screws of more thanone finite pitch; a letter (from A to D) denoting the number (from 0 to 3) of independent infinite-pitch screws; and, where needed, some geometric parameters (angle and pitch values), e.g. 2–IB(c).To obtain a conic-hull classification we append a list of additional parameters, e.g.,2–IB(c){h1,h2,+}. For brevity, we refer to screws of infinite pitch, zero pitch, and pitch h as1-screws, 0-screws, and h-screws, respectively.

2. Two-systems

Hunt [6] distinguishes one general and five special 2-systems. In each, we consider the pairs ofnormalized generators and analyze their conic hulls, Cone(f1,f1). For three 2-systems, the convexcones are best visualized as an arc on a circle; for the rest, an interval of a line is used.

2.1. Two-systems with circular representations

In two special 2-systems, 2–IIC (planar translations) and 2–IIA(h) (a planar pencil of h-screws),a convex cone is immediately seen as a ‘‘pie-slice’’ (a sector of a circle) in the plane of the screws. Aconic hull is described, up to an isometry, by the angle between the two generators. Hence, weclassify these cones as 2–IIC{/} and 2–IIA(h){/}, / 2 [0,p).

The screws in the general two-system 2–IA(hx,hy), hx < hy, (a cylindroid) are not coplanar, how-ever, they are parallel to a (horizontal) plane and project bijectively on a planar pencil. It is con-venient, therefore, to identify a cone as part of a circle. Unlike in 2–IIC or 2–IIA(h), the pitch andelevation vary around the circle, hence, the angle, /, between the generators is not sufficient toidentify the cone. Also needed is the angle, h1 2 ½0; p�/

2� [ ½p � /;p � /

2�, from the first principal

screw of the cylindroid to the first generator. Thus, a cone class is given by 2–IA(hx,hy){h1,/}.It contains four conic hulls in general (two if the generators are at equal angles from a principalscrew). These are carried into one another by the half-turn rotations about the coordinate axeswhich form the isotropy group of 2–IA(hx,hy).

It should be noted that a different (more sophisticated) circular representation of 2–IA(hx,hy)and its conic hulls can be obtained using Ball�s circular diagram of the cylindroid (see [6]). (In fact,in our novel image of 3–IB0(h�), Section 3.4.2, the subspaces of type 2–IIA are mapped into similarcircular diagrams.)

In the other 2-systems, almost all screws are parallel and circular images are not helpful.

https://www.researchgate.net/publication/243695030_Kinematic_Geometry_of_Mechanism?el=1_x_8&enrichId=rgreq-285c58af-1b2e-4504-aa88-4e0d8aca513f&enrichSource=Y292ZXJQYWdlOzIyOTMxOTg0MztBUzoxMDM4OTM2NjkyNTMxMjVAMTQwMTc4MTU4OTE4NQ==


https://www.researchgate.net/publication/276954475_Geometry_of_screw_systems-1?el=1_x_8&enrichId=rgreq-285c58af-1b2e-4504-aa88-4e0d8aca513f&enrichSource=Y292ZXJQYWdlOzIyOTMxOTg0MztBUzoxMDM4OTM2NjkyNTMxMjVAMTQwMTc4MTU4OTE4NQ==


2.2. Two-systems with linear representations

2.2.1. 2–IB(c)In Hunt�s original classification [6], 2–IB(c) is the fourth special 2-system. It is generated by a

0-screw, (f, r · f), and an 1-screw, (0,m), forming an angle c 2 ð0; p2Þ. It represents an ‘‘infinitely-

long cylindroid’’ formed by what will be referred to as a linear array of screws [6,7]. These arecoplanar parallel screws whose finite pitches vary linearly, h = xcot c, with their distance, x, (mea-sured along f · m) from the 0-screw generator. The possible convex cones in 2–IB(c) can be clas-sified as follows:

(i) 2–IB(c){h1,h2,+}. Generated by two screws, f1, f2, with the same direction and differentpitches, h1 < h2. This cone includes all screws lying between, and with the same directionas, the generators, Figs. 2 and 3(a). The pitch varies linearly from h1 to h2. The set includesno 1-screws.

(ii) 2–IB(c){h1,h2,�}. Generators with pitches h1 < h2 have opposite directions, Figs. 2 and 3(b).The generated h-screws are on all axes outside the strip between the generators. Each feasibleh-screw is directed as the closest generator. The cone includes one normalized 1-screw. Its

(a) (b)

(c) (d)

Fig. 2. Convex cones in 2–IB(c).



(a) (b)

(c) (d)

Fig. 3. Characteristic simplices in 2–IB(c).


direction forms angle c < p2with the direction of the larger-pitch screw, i.e when projected on

the plane, it points as the h2-screw. The 1-screw component perpendicular to the plane is acouple that would be obtained by equal-intensity forces along the generators.

(iii) 2–IB(c){h1,+1}. Generated by an h1-screw and an 1-screw at an angle c < p2. The axes of

the cone fill the half-plane bounded by the h1-generator and lying in the direction of the posi-tive pitch gradient, Figs. 2 and 3(c). In this part of the linear array the pitch is h1 or higher.

(iv) 2–IB(c){h2,�1}. An 1-screw and an h2-screw, at angle p � c > p2. This is the lower-pitch

half of the linear array, Figs. 2 and 3(d).

Figs. 2 and 3 show the system from different view-points. In the latter, the characteristic simplexof each cone can be seen very clearly: it is either an internal interval (i); an external interval (ii); aninterval with an end-point at positive (iii) or negative (iv) infinity. (Note that we illustrate a direc-ted 1-screw by an arrow with a pyramid-shaped head.) A half turn about a common normal is asymmetry of the system which allows us to identify conic hulls with opposite directions.

Presenting exhaustive proofs of the multiple assertions contained in (i)–(iv), as well as in sub-sequent cone-type descriptions, would make the text too long and cumbersome. The key ideas areprovided in the text, but most details are left out. The main algebraic technique, at least for 2-sys-tems, is to represent the cone generators in a convenient reference frame, and express an elementof the cone as f = k1f1 + k2f2, ki P 0. The geometric location of f for different values of ki is thenanalyzed. Different geometric considerations facilitate the analysis and often make explicit equa-tions superfluous.

For (ii), with Oy along f1 and Ox along the common normal towards higher pitches, we have:

f1 ¼ ðj; h1jÞ; f2 ¼ ð�j;�h2j� dkÞ
where d ¼ ðh2 � h1Þ tan c is the distance between the screws. This yields:
f ¼ ððk1 � k2Þj; ðk1h1 � k2h2Þj� k2dkÞ ¼aðj; hj� k2

a dkÞ : k1 > k2

k1dsin c ð0;�j cos c � k sin cÞ : k1 ¼ k2

að�j; hj� k2a dkÞ : k1 < k2

8>><>>:

ð1Þ

where a ¼j k1 � k2 j; h ¼ 1a ðk1h1 � k2h2Þ. The higher- and lower-pitch parts of 2–IB(c){h1,h2,�}

are obtained when k1 > k2 and k1 < k2, respectively. When k1 = k2 the1-screw element of the coneis generated.

(a)

(b)

(c)

(d)

Fig. 4. Conic hulls in 2–IB0.


2.2.2. 2–IIB(h) and 2–IB0

These systems can be viewed as special cases of 2–IB(c), obtained when c becomes p2and 0. The

description of their cones is similar, only simpler. In 2–IIB(h), the 1-screw is normal to the plane,the pitch gradient is zero and all screws have the same pitch h. There are only three types of cones:2–IIB(h){d,+} and 2–IIB(h){d,�}, where d is the distance between the generators, correspond tocases (i) and (ii) above; while 2–IIB(h){h,1} corresponds to (iii) and (iv) (which are identical whenc ¼ p

2). Eq. 1 is valid for 2–IIB(h){d,�} as well.

In 2–IB0, screws of all pitches share the same axis. This makes it rather difficult to visually dis-tinguish between different cones. Yet, the one-dimensional representation is straightforward andanalogous to 2–IB(c). Screws are mapped as points on a pitch axis, with associated directionsalong or against a chosen vector on their common physical axis. The four cone types (analogousto 2–IB(c)) are shown in Fig. 4. Each cone class contains two oppositely-directed conic hulls,which can be identified by a p-rotation about a normal to the common screw axis. Note thatthe induced symmetry on the 1-dimensional image space reverses directions but keeps all pointsfixed.

3. Three-systems

The ten special and one general types of 3-systems are divided into four groups, depending onthe chosen geometric representation.

3.1. Spherical representations

In two special 3-systems all screws are of the same pitch: 3–IID (all 1-screws) and 3–IIA (all h-screws through the origin). They share an obvious spherical symmetry and a cone is described by atriangle on a sphere: 3–IID{/1,/2,/3} or 3–IIA{/1,/2,/3}, where the cyclic triple of parameters,angles in [0,p), are the arcs of the triangle sides. In general, the opposite hull belongs to a differentclass, {/1,/3,/2}.

The general system, 3–IA(hx,hy,hz), (or the special system where two of the principal pitches areequal) has screws with varying location and (finite) pitch. However, just as 3–IID or 3–IIA(h), 3–IA(hx,hy,hz) contains exactly one (up to sign reversal) normalized screw with every direction in

(a) (b)

(c) (d)

(e)

Fig. 5. Different conic hulls in 3–IIC(h).


space. (This is obvious since {(i,hxi),(j,hyj),(k,hzk)} is a basis.) Therefore, the directed normalizedscrews of the system can be mapped bijectively as points on the unit sphere and a conic hull ofthree generators is represented as a spherical triangle. In this case, triangles of the same shapeand different locations cannot be identified, so a conic hull can be described by 3–IA(hx,hy,hz){nA1A2A3}, where Ai are points on the unit sphere. Each class has exactly four such hulls, whichcan be carried into one another by a half turn about a coordinate axis.

3.2. Points-on-a-plane representations

3.2.1. 3–IIC(h)

Systems of type 3–IIC(h) are spaces analogous to the system of planar motion (or constraint),obtained when h = 0. The screws are: all 1-screws parallel to a (horizontal) plane, Oxy, and allh-screws perpendicular to it. Every point in the Oxy plane (including points at infinity) corre-sponds to two normalized screws (one for each direction). Thus, three elements, f1, f2 and f3,


in 3–IIC(h) define a triangle in the plane, possibly with some of its vertices at infinity, Fig. 5. Thefollowing types of conic hulls are possible:

(i) 3–IIC{a1,a2,a3,+}. The generators are h-screws, with the same direction. The cone is defined,up to isometry, by the cyclic triple of the sides of the triangle, i.e., the distances between thescrews, (a1,a2,a3). (A cyclic permutation leaves the cone-class unchanged.) The cone consistsof h-screws passing through all points of the internal triangle with vertices the projections ofthe generators, Fig. 5(a).

(ii) 3–IIC{a1,a2,a3,�} One h-screws, f3, which faces side a3, points opposite to the other two.The cone includes h-screws passing through the points of the external triangle with finite-length side a3, as well as a pie-slice of 1-screws, Fig. 5(b). The generated h-screws aredirected as the adjacent generator(s). The boundaries of the infinite-pitch pie slice areperpendicular to the infinite-length sides of the external triangle.

(iii) 3–IIC{a,a,+}. An1-screw is at an angle, a 2 [0,p), from the normal to the plane of the othertwo, finite-pitch, generators with identical directions, Fig. 5(c). The cone is defined by aninternal triangle with a vertex at infinity. The h-screws pass through every point of the trian-gle. Apart from the generator, there are no 1-screws.

(iv) 3–IIC{a,a,�}. Same as above, however, the generating h-screws have opposite directions.Now, the characteristic simplex is an external triangle with a vertex at infinity, Fig. 5(d).A range of 1-screws is also generated and shown as a sector with angle a.

(v) 3–IIC{a}. Two 1-screw generators and one h-screw. The characteristic triangle now has twoinfinite vertices and becomes simply a pie-slice, extended indefinitely, Fig. 5(e).

The remarks at the end of Section 2.2 apply here as well—explicit proofs of the above have beenomitted. To verify, one could analyze the expression for f = k1f1 + k2f2 + k3f2, ki P 0. An easierand more enlightening approach is to consider pairs of generators and construct their conic hullsusing the results of Section 2, and thus obtain the boundaries of the 3-cone. For example, in case(ii) (Fig. 5(b)) we know from the study of 2–IIB(h) that Cone(f1,f2) is given by the vertical screwsthrough side a3 (see Fig. 3(a)). Combining each of those with f1 to obtain a 2-hull as in Fig. 3(b)obviously will result in Fig. 5(b).

Apart from the planar displacements, a p-rotation about any horizontal axis induces a symme-try of the space of hulls. It identifies the opposite of a hull defined by {a1,a2,a3} with one definedby {a2,a1,a3}. For types (iii) and (iv) the opposite hull class is obtained when the angle is p � a.

3.2.2. 3–IC(c)The system 3–IC(c) is spanned by a plane of1-screws and an h-screw at an angle c to the plane,

c 2 ð0; p2Þ. Suppose that the vertical z axis is parallel to the h-screw, while the x axis is along the

unique horizontal 1-screw. Then, the finite-pitch screws in the system are all vertical. Those ofthem that share a common plane normal to Ox will have the same pitch, and, in fact, will forma 2–IIB(h) system, whose pitch h will vary linearly when the plane moves along Ox. On the otherhand, in all vertical planes parallel to Ox there will be identical two-subsystems of the 2–IB(c)type, i.e., linear arrays of screws, with the same pitch gradient of cot c.

Seen from above, the picture is similar to that of 3–IIC(h). Every point in Oxy, includingthose at infinity, corresponds to a unique pair of oppositely-directed normalized screws. Unlike


in 3–IIC(h), here the h-screws have different pitches and the directions in Oxy are only projections,rather than the actual directions, of the 1-screws. However, three generators define a generalizedtriangle in Oxy just as in 3–IIC(h), and the possible cone-types are determined by the internal andexternal triangles exactly as in 3–IIC(h), Fig. 5(a)–(e).

Additional parameters are required to specify a cone-class up to isometry, since in 3–IC(c) onemust know not only the shape and size of the characteristic triangle, but also its location. Trans-lation parallel to Oy leaves the cone-class invariant, so only two more parameters are needed, forexample the finite pitches of two generators. We agree that (for the y coordinates of f1, f2) y1 6 y2and the triangle sides are listed counter-clockwise when looking against Oz. Then the conic hulltypes are: (i) 3–IC(c){a1,a2,a3,h1,h2,+} (ii) 3–IC(c){a1,a2,a3,h1,h2,�i} (i = 1,2,3); (iii) 3–IC(c){a,a,h1,h2,+}, a 2[0,p); (iv) 3–IC(c){a,a,h1,h2,�}, p � a 2 (�p,p); and (v) 3–IC(c){h,a1,a},a 2 ð0; p

2Þ. A half turn about the y axis is also a symmetry sending the opposite of a type

(i)–(ii), (iii)–(iv) or (v) hull into that of {a2,a1,a3,h2,h1}, {a,�a,h2,h1}, or {h,�a1,a}.

3.3. Lines-on-a-plane representations

3.3.1. 3–IIB(h)This system consists of coplanar horizontal h-screws and vertical 1-screws. 3–IIB(0) is the

system of wrenches applied by planar grippers and cable robots.There is an obvious planar representation: the elements of the system are mapped into directed

lines in Oxy (rather than the ‘‘directed points’’ that we used in systems up to now). The two nor-malized 1-screws are identified with the two ‘‘directions’’ of ‘‘the line at infinity.’’ Any threescrews, f1, f2 and f3, define a triangle in the plane: the sides of the triangle are along the screwaxes. (When two of the axes are parallel, we have a triangle with a vertex at infinity.) The screwsdefine the internal (or a specific external) triangle when the vertical components, miz, of their mo-ments with respect to a point inside that triangle have the same sign. In other words, the uniquegeneralized triangle defined by the generators consists of the points which are on the same side(left or right) of all three directed lines, Figs. 6–8. This triangle is the characteristic simplex for3–IIB(h), however, it defines the generated Cone(f1,f2,f3) in a manner different from the direc-ted-point representations that we used for previous screw systems. In the present case, the triangle

Fig. 6. A convex cone of type 3–IIB(h){a1,a2,a3,+}.

(a) (b)

Fig. 7. Convex cones of types 3–IIB(h){a1,a2,a3,�} and 3–IIB(h){a,a,+}.

(a) (b)

Fig. 8. Convex cones of type 3–IIB(h){a,a,�} and 3–IIB(h){a,+}.


defines a ‘‘forbidden area,’’ and the screws in the cone are only those whose directed lines in Oxy

avoid the area.This may not be intuitively clear, so we sketch the proof. First we show that an h-screw, f, with

a point, O, in the generalized-triangle interior cannot be a conic combination of the generators.Indeed, let f = k1f1 + k2f2 + k3f2. If the origin is at O, the vertical component, mz, of the momentof f = (f,m) is zero, hence

Pikimiz ¼ 0. If O is in the triangle all miz have the same sign and there-

fore there are two ki with opposite signs. Conversely, suppose the f axis misses the triangle com-pletely. Then, the axis intersects at least two of the h-screw generators at different points(otherwise f passes through a triangle vertex, possibly at infinity). There is a point, Oi, where f

will intersect only the h-screw axis of fi, say i = 1,2. Choosing O1 as the origin we havem1z = 0, hence m2z m3z < 0 and therefore k2k3 > 0. Similarly k1k3 > 0, so the ki have the same sign.Therefore, either f or �f is in Cone(f1,f2,f3). It is not difficult to check separately the special caseswhen f is an 1-screw or passes through a vertex and complete the picture of Cone(f1,f2,f3),namely all system wrenches missing the characteristic triangle and ‘‘trying to rotate the plane’’in the same sense as the generators, if any triangle point were fixed.


Looking at the possible triangles we classify the conic hulls:

(i) 3–IIB(h){a1,a2,a3,+}. The cyclic triple of directed lines delimits, clockwise, an internal trian-gle with sides a1, a2 and a3, Fig. 6. An h-screw in every direction is possible, and can be arbi-trarily far from (but only on one side of) the triangle. The cone includes an 1-screw.

(ii) 3–IIB(h){a1,a2,a3,�}. The three directed lines delimit an external triangle, f3 pointing ‘‘thewrong way,’’ Fig. 7(a). All feasible screws must pass between the two area components ofthe external triangle and in the same direction as the generators. There are no 1-screws.

(iii) 3–IIB(h){a,a,+}. Two h-screws have opposite directions. They intersect the third generatorat an angle a and cut a length a from it. The characteristic simplex bounded by the generatorsis an internal triangle with a vertex at infinity, Fig. 7(b). The screws lie outside the boundedarea. There is an 1-screw.

(iv) 3–IIB(h){a,a,�}. As in (iii), however, two generators have the same direction. The boundedshape is an external triangle with an infinite vertex, Fig. 8(a). The elements of the conic hullare all h-screws and pass between the two corners formed by the generators.

(v) 3–IIB(h){a}. One of the generators is an 1-screw, while the other two form an angle a. All h-screws that miss the pie-slice delimited by the generators are in the generated cone, Fig. 8(b).

Since p-rotations about axes in Oxy are system symmetries, the above classes also contain hullswhere the screws ‘‘go around’’ the characteristic triangle counterclockwise. Hulls with oppositedirections are related as in 3–IIC(h).

3.3.2. 3–IB(h,c) and 3–IB(hx,hy)

The seventh-special three-system, 3–IB(h,c), is spanned when an 1-screw, (0,m), is added, atan angle c 2 ð0; p

2Þ, to a (horizontal) planar pencil of h-screws in Oxy. The system�s finite-pitch

screws are all parallel to Oxy. The screws with any given direction, at an angle h 2 [0,p) from

Fig. 9. The screws with a given direction in 3–IB(h,c).


Ox, form a linear array (a 2–IB(ch) system with cos ch ¼ cos c cos h) in a plane, rh, through O, in-clined at an angle b from Oxy, Fig. 9. The angle b is zero for screws parallel to Ox, and r0 is Oxy.The inclination increases with h, cos a ¼ sin cð1� cos2ccos2hÞ�

12, and reaches its maximum for

lines parallel to Oy ðh ¼ p2Þ when b ¼ p

2� c.

From this, it is clear that every line in Oxy is the projection of a unique screw axis in 3–IB(h,c).Therefore, we can use the lines in Oxy as a faithful image of this system, just as we did with 3–IIB(h). Unlike 3–IIB(h), here the directed lines in the representation plane correspond to screwsof different pitches and elevations, but the one-to-one correspondence is maintained.

The third special three-system, 3–IB(hx,hy), is spanned by the two principal screws of a 2–IA(hx,hy) cylindroid and an 1-screw perpendicular to them. Oxy is chosen as the central planeof the cylindroid. The1-screw has the effect of reproducing the 2–IA(hx,hy) system at every pointof Oxy. The result is that all finite-pitch screws in 3–IB(hx,hy) occupy horizontal planes offset at adistance at most

hy�hx2

from Oxy. All screws with a given direction, defined by its angle h with Ox,have the same pitch, hh, and form a (2–IIB(hh) system in a) horizontal plane rh. Similarly to 3–IB(h,c) there is a one-to-one correspondence between the finite-pitch screws of the system andtheir projections on the Oxy plane.

In both 3–IIB(h,c) and 3–IB(hx,hy), the projections of three generators form a triangle (internal orexternal, with infinite vertices or not), which defines their conic hull. The projections in Oxy of thefeasible screwsmust not intersect the bounded area and, at any of its points, the projections onOz oftheir moments must have the same sign as those of the generators. Therefore, the ‘‘top view’’ of theconvex cones in both 3–IIB(h,c) and 3–IB(hx,hy) is as described in the discussion of 3–IB(h,c) andshown in Figs. 6–8. The three-dimensional picture will be different, but can be reconstructed fromthe planar image, Fig. 10. The characteristic triangle extrudes vertically as an infinitely high prism,which trims each of the planes rh, removing the screw axes that intersect the prism interior.

Each of these 3-systems allows different internal symmetries and requires different sets ofparameters to classify a convex cone up to a change of frame. In 3–IB(hx,hy), the characteristictriangle in Oxy needs to be specified in terms of its size, shape and orientation, /, in the plane,

Fig. 10. The screws with a given direction in a cone of type 3–IBðh; p4ÞfMA1A2A3;þg.


while its position is not important. In 3–IB(h,c), there is only one internal symmetry—p-rotationabout Oy. The exact location (position and orientation, or the coordinates of the vertices,A1A2A3) of the triangle must be given. Every 3–IB(h,c) class contains exactly two hulls with oppo-site directions and simplexes reflected by Oyz.

Fig. 10 shows the prism extruded from the characteristic triangle of Cone(f1,f2,f3) of class 3–IBðh; c ¼ p

4ÞfMA1A2A3;þg and the screws in the cone with axes in the direction defined by h ¼ p

4.

The sides of the triangle are at angles h1 ¼ p4, h2 ¼ 5p

6, h3 ¼ 2p

3and pass through points

0;ffiffi2

p

2

� �; ð1; 0Þ; 0;� 1

2

, respectively.

3.4. Pitch-to-elevation representations

In this subsection we consider 3–IC0 (the tenth special three-system) and 3–IB0(h�) (the eighth).In each of them, there is at least one axis (Ox) with screws of all pitches. Both the spherical andplanar representations, considered thus far, are inadequate.

3.4.1. 3–IC0

This can be viewed as a limit case of 3–IC(c) for c = 0. The finite pitch generator is in the planeof 1-screws. All the different planes of same-pitch screws, observed in 3–IC(c), merge into oneand 3–IC0�s finite-pitch screws are all parallel and in a single plane. We choose this to be Oxz,and all the screw axes to be horizontal, i.e., parallel to Ox. On each axis there are screws of allpitches. The 1-screws are all horizontal.

The 2-dimensional representation is given by the h–z plane, which we superimpose with bothOxz and Oxy. Each point, (h,z), corresponds to a unique pair of normalized screws with oppositedirections. The two are distinguished by attaching at each point a direction, right or left, depend-ing on whether the actual screw is along or against Ox. The points at infinity of the representationplane are associated with the1-screws of the system. For this purpose, the h direction is identifiedwith Ox while the vertical coordinate corresponds to Oy. This is physically meaningful since,when a force (or wrench) parallel to Ox is moved to infinity in the z direction, it becomes a puremoment alongOy. The conic hull of three screws, f1, f2 and f3, is an internal or external triangle inthe representation plane.

The system symmetries are translations and p-rotations about vertical and horizontal axes inOyz. While translations parallel to Ox have no effect on the cones and the image space, the othersymmetries identify hulls with triangles obtained from one another by vertical translation; mirrorsymmetry about a horizontal axis in Oyz; or by inverting the directions of all screws.

The cones in 3–IC0 are classified like those in 3–IC(c), listed at the end of Section 3.2: (i)3–IC0{a1,a2,a3,h1,h2,+} an internal triangle; (ii) 3–IC0{a1,a2,a3,h1,h2,�i} an external triangle,generator i has an opposite direction, Fig. 11; (iii) 3–IC0{Dz,h,h1,h2,+} an internal trianglewith a vertex at infinity, an 1-screw generator and two finite-pitch screws at a distance Dz; (iv)3–IC0{Dz,h,h1,h2,�} an infinite external triangle; (v) 3–IC0{h,a} two vertices at infinity.

3.4.2. 3–IB 0(h�)The 3–IB0(h�) system is obtained as a limit when c ! p

2in 3–IB(h�,c). It is spanned by a hori-

zontal planar pencil of h�-screws in Oxy and an 1-screw along Ox. A convenient system basis is

Fig. 11. A convex cone of type 3–IC0{a1,a2,a3,h1,h2,�3}.

Fig. 12. The 3–IB0(ho) system.


(i, 0),(j,h�j),(0, i). A normalized finite-pitch screw has coordinates ðcos h; sin h;mxÞ in this basis andis given by

f ¼ ðf;mÞ ¼ ði cos h þ j sin h;mxiþ h�j sin hÞ: ð2Þ
We can think of this system as composed of all vertical planes through the origin, each containinga linear array of parallel, horizontal screws, Fig. 12. The pitch gradient in such a plane depends onthe angle, h, between the plane and Ox. In the Oyz plane h ¼ p
2

the gradient is zero and all screws

have pitch h�. When h nears 0 or p, dhdz approaches �1, and in Oxz the linear array degenerates to

a line, Ox, on which lie screws of all pitches.We note that in each horizontal plane, for every value of z 5 0, there is a unique screw of every

finite pitch, h. (In fact, we have h � h� = �zcot h.) We can therefore use the h—z plane toconstruct a two dimensional image of the system.

We change variables from (h,z) to Z = �z, H = h � h� and denote the (H,Z) coordinate planeby PHZ . In its tangent bundle, TPHZ , let Q ¼ fðw; fÞ 2 TPHZ j w ¼ � j w j f; j f j¼ 1g. The ele-ments of Q are ‘‘directed points’’ or unit vectors attached at points of PHZ . At each pointw = (H,Z)5 (0,0) there are two elements of Q, directed to and from the origin, a feature that will


be referred to as negative or positive polarity. At w = 0 there are elements of Q in all directions.Finally, we supplement Q with two ‘‘points at infinity,’’ Q1 ¼ Q [ fi1; o1g. When jwj ! 1 thelimit of a sequence of directed points will be i1 or o1 depending on whether the vectors f aredirected in or out.

Now, each normalized finite-pitch screw f = (f,m) in 3–IB0(h�) is mapped into qðfÞ ¼ ðw; fÞ 2 Q,where w = (H,Z). It can be shown that for f as in Eq. (2) this gives

w ¼ ððmx � h� cos hÞ cos h; ðmx � h� cos hÞ sin hÞf ¼ ðcos h; sin hÞ

In fact, every q 2 Q can be described without ambiguity by (h,H), h 2 (�p,p]. The image of the1-screw along (or against) Ox is o1 (or i1). The map provides a very simple and intuitivedescription of 3–IB0(h�). Given a point q 2 Q1 one can immediately obtain the pitch,h = h� + H, the elevation, z = �Z, and also the orientation of the real screw, since the f componentof q is exactly the unit force component of the corresponding wrench.

Moreover, if q(fi) = (wi, fi), i = 1,2, then Span (f1,f2) maps onto the line or circle through 0 inPHZ passing through w1, w2 and, if a wi = 0, tangent to fi. (Such lines and circles can be seen inFigs. 13–18.) The 2-subsystem is mapped into a line if it is a linear array of screws, and into acircle if it is a cylindroid. When f sweeps once through all directed normalized screws in a cylin-droid, the image point, w, goes around the circle twice, each time with opposite polarity.

The conic hull of two screws, Cone (f1,f2), will map into the unique arc or interval (possiblyexternal) between w1 and w2, which allows f1 to merge continuously into f2. For three generators,Cone (f1,f2,f3) is given by the ‘‘triangle’’ bounded by the sides Cone (fi,fj) and containing noother image points of Span (fi,fj). (The triangle perimeter is the unique path, connecting the threepoints along the lines or circles, on which f changes continuously. The triangle interior contains nopoints on the lines and circles.) The triangle can be either internal, if its interior is connected, orexternal, if it has two components touching at 0.

The described correspondence between conic hulls and curvilinear triangles can be proven with-out major difficulty, but hardly in the limits of the present paper. The key is to establish the map-ping of 2-hulls onto arcs, which is best understood as a close analogue of Ball�s circular diagram of

(a) (b)

Fig. 13. Convex cones of type 3–IB0(ho){nw1w2w3,+}.

(a) (b)

Fig. 14. Convex cones of type 3–IB0(ho){nw1w2w3,�}.

(a) (b)

Fig. 15. Convex cones of type 3–IB0(ho){nw1w2w3f3,±}.

(a) (b)

Fig. 16. Convex cones of type 3–IB0(ho){nw1f2f3}.


a cylindroid. Then, one has to show that these arcs (and/or intervals) bound the hull�s image andthat the image interior can have no points on the circles/lines of the sides.

Just as in the general 3-system, the 3–IB0(h�) symmetries form the 4-element group of p-rota-tions about the coordinate axes. A half-turn about Ox or Oz induces a mirror symmetry about the

Fig. 17. Convex cones of type 3–IB0(ho){nw1w21, 0}.

(a) (b)

Fig. 18. Convex cones of type 3–IB0(ho){nw1w21,±1}.


H axis or an inversion of polarity in PHZ . A class has generally four hulls (two if the characteristictriangle has an H-axis symmetry).

When classes are grouped together according to the appearance of the characteristic simplex,one discovers remarkable variety, even in each of the groups listed below. (Indeed, in cases(i)–(v), a cone may or may not contain 1-screws.) We denote the h� pencil in Oxy by A�.

(i) 3–IB0(h�){nw1w2w3,+}. Three generators not in A�, with finite pitch and the same polarity.The image is an internal triangle not containing the origin, Fig. 13. The cone may or may notcontain an 1-screw, as the two examples show.

(ii) 3–IB0(h�){nw1w2w3,�}. One of the points has an opposite polarity to the other two. Theimage is an external triangle. The cone includes screws in A�, whose feasible directionsare bounded by the tangents to the two sides of the triangle that include 0, Fig. 14.

(iii) 3–IB0(h�){nw1w2f3,+}. One generator maps to (0, f3). The other two have the same polarity.There are no screws in A� apart from the generator, Fig. 15(a).

(iv) 3–IB0(h�){nw1w2f3,�}. The two generators not in A� have opposite polarity. The boundedtriangle is external, Fig. 15(b). The convex cone will contain a pie-slice of A�.

(v) 3–IB0(h�){nw1f2f3}. Two of the generators are in A�. The whole Cone(f2,f3) maps at theorigin. The bounded shape is an internal ‘‘triangle’’ with only two vertices, Fig. 16.


(vi) 3–IB0(h�){nw1w21, 0}. Two screws of opposite polarity and an 1-screw. The image is anexternal triangle with a vertex at infinity, Fig. 17.

(vii) 3–IB0(h�){nw1w21,+1}. Two finite generators have the same polarity as an 1-screw. Aninternal infinite triangle is formed, Fig. 18(a).

(viii) 3–IB0(h�){nw1w21,�1}. The 1-screw has opposite polarity to the other generators. This isanother external triangle, Fig. 18(b).

(ix) 3–IB0(h�){nw1f21,+1}. The 1-screw and the generator not in A� have the same polarity.An internal triangle similar to (vii).

(x) 3–IB0(h�){nw1f21,�1}. The 1-screw and the generator not in A� have opposite polarity.An external triangle similar to (viii).

(xi) 3–IB0(h�){nf1f21}. A pie-slice in PHZ .

4. Conclusions

The complex geometry of screw systems requires a classification much more sophisticated thana simple dimension count. The underlying reason is the composite character of screw quantitiesand the non-Euclidian nature of the vector spaces they span. This also causes the great varietyof conic hulls. The classification of these sets, presented herein, can be the theoretical foundationfor the study of many problems in robotics and mechanisms where sign reversal of twists orwrenches is not allowed. The lower-dimension images of screw systems and their convex setscan be a valuable practical visualization tool in the design and use of such systems.

It should be noted that the classification of 2- and 3-cones cannot be applied directly to 5- and4-cones, as this is done with screw systems via reciprocity. Nevertheless, when cones of four ormore independent screws are studied, the presented results will be essential, since they allowthe visualization of various sections of a multidimensional cone.

References

[1] M.S. Ohwovoriole, B. Roth, An extension of screw theory, Trans. ASME J. Mech. Des. 103 (1981) 725–735.

[2] E. Zussman, M. Shoham, E. Lenz, Automatic assembly planning—A kinematic approach by means of screw

theory, ASME Manuf. Rev. 5 (4) (1992) 293–304.

[3] C.G. Gibson, K.H. Hunt, Geometry of screw systems, Mech. Mach. Theory 25 (1) (1990) 1–27.

[4] J.M. Selig, Geometrical Methods in Robotics, Springer, 1996 (Chapter 8).

[5] P.S. Donovan, C.G. Gibson, On the hierarchy of screw systems, Acta Appl. Math. 32 (3) (1993) 267–296.

[6] K.H. Hunt, Kinematic Geometry of Mechanisms, Oxford University Press, 1978.

[7] J. Phillips, Freedom in Machinery 1–2, Cambridge University Press, 1984–1990.

Date post:	14-May-2023
Category:	Documents
Upload:	independent
View:	0 times
Download:	0 times

Convex cones in screw spaces

Documents