A Study of Curvature for Single Point Contact Griffis

8/2/2019 A Study of Curvature for Single Point Contact Griffis

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A study of curvature for single point contact

Michael Griffis

The Eigenpoint Company, P.O. Box 1708, High Springs, FL 32655, USA

Received 26 July 2002; received in revised form 9 May 2003; accepted 10 June 2003

Abstract

The study of curvature focuses on the case of two bodies (e.g. gears) that touch in a single point, where

additionally at least one virtual body is introduced that contains as fixed elements, the point of contact, its

tangent, and the contact normal. The equations of relative motion as well as the new idea of higher-order

reciprocity are used to investigate what happens between the gears and the virtual bodies in order to de-

termine various curvature properties (to third-order).

2003 Elsevier Ltd. All rights reserved.

1. Introduction

This is a study of tracking direct contact in each of two bodies, where the bodies always touchin a single point and where this point of contact moves in each body as relative motion proceeds.

Other types of contact are not considered, and other necessary links are, for the most part, notemphasized, since the relative motion (instantaneous twist, etc.) between the touching bodies is

considered given. A simple example, prevalent here, is a pair of gears, an input and an output,where a tooth from one touches a tooth from the other in a single point throughout a meshingcycle. A more involved example considers a robotic finger touching an object in a single point

during a manipulation sequence.

Focusing on the gears, the point of contact would trace a (spatial) curve on each tooth surface ifmotion were to proceed, e.g. a point path on the input gear tooth surface, or more simply, an input

gear point path. Now, let us say the smooth input gear tooth surface is known very well (given).

Further, consider that, at the first instant, the point of contact on that surface is given. Then, acontact normal (line of action) and a tangent plane can be determined. Finally, consider that the

tangent for the input gear point path is given, lying in the tangent plane. As it will be seen, this

E-mail address: [email protected] (M. Griffis).

0094-114X/$ - see front matter

2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0094-114X(03)00094-6

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means the geodesic torsion and normal curvature of the input gear point path are also known

(easily determined).The objective here then is to take these givens and determine the other curvature properties we

can, such as the geodesic torsion and normal curvature for the output gear point path as well as

the geodesic curvature for both point paths. The end result, a collection of EulerSavary-typeequations for point path envelopes, follows, which highlight second- and third-order properties ofthe surfaces in contact. The tangents for the input and output gear point paths may be aligned

(special) or they may not be (general). In the end, effectively, it seems the results are an implicationof Dooners Third Law of Gearing [1].

The approach adopted here to achieve the objective is founded upon classical differential geo-metry and screw theory. Dijksman [2] demonstrates in the plane a technique of adding a virtual

Nomenclature

$,_

$$,

$$ instantaneous screw axis for instantaneous twist of output gear relative to input geartogether with two derivatives (time-rates-of-change)$1, $2 lines of rotation for input and output gears relative to ground$n line normal to tooth surfaces at the point of contact (line of action)

$T, $p reference lines at the point of contact lying in the tangent plane dual vector operator

A A Ao screw coordinates where A is the direction part and Ao is the moment partS, _SS, SS coordinates of instantaneous twist of output gear relative to input with two derivativesS1, S2 coordinates of lines of rotation (normalized)T, n, p coordinates of$T, $n, and $pTNB three mutually perpendicular intersecting lines (tangent, normal, and bi-normal) de-

fining a body that tracks a point which traces a spatial curveVd coordinates of instantaneous twist of TNB-based body relative to the body of the

curves, j torsion and curvature of spatial curveTnp three mutually perpendicular intersecting lines (path tangent, surface normal, and

second line in tangent plane) defining a second body that tracks a point whose path

lies on a surfaceV x T coordinates of the instantaneous twist of Tnp-based body relative to the body of

the surface (note T T 1)t, c, k geodesic torsion, geodesic curvature, normal curvature of curve lying on surface

/ cut angle describing orientation between Tnp and TNB bodies about commonT

a spin angle describing orientation of Tnp bodies about common normal n_ss1, _ss2 propagation velocity of point of contact in the input and output gears0 prime denotes differentiation with respect to arc length (either s1 or s2)t, c, k skewed versions of t, c, k (properties modified by a spin angle a) time rate of change or dot product of screw coordinates: A B A B reciprocal product of screw coordinates: A B A Bo Ao B

1392 M. Griffis / Mechanism and Machine Theory 38 (2003) 13911411


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(fictitious) body to the gear pair in order to facilitate analysis. This technique is applied to the

spatial case here in order to provide a number of necessary equations, where a particular virtualbody is attached to the point of contact, the contact normal and a point path tangent. Remaining

necessary equations are obtained by utilizing the new concept of higher-order reciprocity (see also[3]), which ensures to higher-order that the contact normal (line of action) is reciprocal to theinstantaneous screw axis (ISA) describing how the output gear is moving with respect to the input.In other words, higher-order reciprocity ensures to higher-order that the force along the contact

normal is a constraint force acting between the two gears.The use of higher-order reciprocity is believed to be novel and, in the end, in stark contrast with

the typical approach, which uses derivatives of an equation of meshing (see Chapter 8 of [4] and

Footnote 1 of this article). The equation of meshing is a vectoral equation which says that thecontact force is normal to the relative velocity at the point of contact. On the face of it, this

appears to be equivalent to reciprocity, when the point of contact is used as the origin. However,higher-order versions of these are not equivalent, and subtle but important differences exist be-

tween the two approaches.Certainly, there have been other curvature studies in this arena, but none has sufficiently ad-

dressed the single point contact problem to third order, and none has demonstrated curvature

relationships in the forms shown here. There has been work done on the line contact problem,where relative curvature between the gear tooth surfaces was obtained. (see [5]). Dooner [1] uti-lizes a curvature cylindroid but does not study the single point contact problem in any detail.

Dyson [6] analytically investigates with minimal geometrical significance the relative curvatureproperties found in hypoid gearing. A simple relation is found between principal radii of relativecurvature and the motion of the point of contact. (However, Dysons work is restricted to a

constant speed ratio, while here we are not.) Dyson also discusses the apparent free design choice

for the travel direction of the point of contact, suggesting that an optimum should consider Hertzstresses, minimum oil film thickness, and flash temperature.

In Rimon and Burdick [7], the emphasis is placed on studying point-path derivatives in order to

describe second-order mobility. There, the second condition of the second-order roll-slide mo-tions appears to be equivalent to a derivative of the equation of meshing. Cai and Roth [8] by wayof contact mode constraints and Montana [9] by way of contact equations employ approaches

that are fundamentally different from the current study. Each of [8,9] does assume that relativemotion is sensed, which is reasonable considering the robotic gripping application. (Here it isgiven.)

1.1. The gearing scenario

Before delving into curvature straight away, let us look at the well known gearing scenario firstand then extract what is actually needed. First, imagine a typical spatial scenario where we have

an input gear, an output gear, and a fixed frame. Furthermore, let us say that the input gearrotates at an input speed x1 about a first line $1 fixed in the frame and that the output gear rotatesat an output speed x2 about a second fixed line $2 which is generally disposed. The ratio x2=x1 isknown as the instantaneous gear ratio of the gear pair.

Let us further imagine that we know at an instant the single point of contact between thetwo gears. (Other kinds of contact are neglected here.) This point contact gives rise to a

M. Griffis / Mechanism and Machine Theory 38 (2003) 13911411 1393


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five-degree-of-freedom joint (kinematic pair) between the two gears, and there will be a contact

normal $n through the point of contact that is perpendicular to the two gear teeth in mesh there,one tooth being on the input gear and the other tooth being on the output gear. Essentially, this

defines a spatial three-link mechanism, where the instantaneous twist between the output gear andthe input gear about the instantaneous screw axis $ is located by

S x2S2 x1S1; 1where S2 are the normalized coordinates of$2, S1 are the normalized coordinates of$1, and S arethe coordinates of the instantaneous twist that acts on $. The geometry of this relationship is bestdescribed by Balls cylindroid [10], denoted in this case by Cg, which is defined by the lines $1 and

$2. Note that here and throughout, it is assumed that the point of contact does not lie on $.Balls reciprocity has been elegantly applied to this scenario to say that $ and $n are reciprocal

because a force along the line of action (contact normal) between the two gears is a constraint

force which does no work. Algebraically, this means that S n 0, where n are the coordinates of$n and where is the reciprocal product operator. Furthermore, for the three-link mechanismunder study, this together with (1) means that the instantaneous gear ratio is given by the geo-metrical relationship x2=x1 S1 n=S2 n.

Note that (1) can also be written as x2S2 x1S1 S, which puts x2S2 as the coordinates of acomposite twist, that of the output gear relative to the fixed frame. Now, while either form isvalid, I want to re-emphasize here that it is the instantaneous twist of the output gear relative to

the input gear which is given, namely $ located by S. Moreover, it is the relative motion of theoutput gear with respect to the input gear that is given, which identifies that at least two deriv-atives _$$, $$, located by _SS, SS, are also given. The derivative _$$ denotes the time-rate-of-change of$ as

seen by an observer standing on the input gear, and $$ similarly denotes the time-rate-of-change of

_$$. See Bokelberg et al. [11] for what this means geometrically, where their differential screw agreeswith _$$ when S S 1. Time-based derivatives are used here, but it will be shown that time willnaturally be associated with scalar speeds and derivatives in a way that clearly identifies the geo-

metry contained herein (namely, curvatures).

1.2. Differential geometry on a surface

Of primary interest here is what happens to the point of contact and the contact normal asmotion proceeds. Let us confine our attention to one of the gears first, say the input gear. Here,

the point of contact traces some spatial curve in the input gear, namely the input gear point path,which will always be on the actual tooth surface. The tangent to the curve traces a ruled surface

(developable) that intersects the tooth surface. Also, the contact normal will always be perpen-dicular to this tooth surface at the point of contact, and as motion proceeds, the contact normaltraces its own surface in the input gear. All of this together creates an ideal scenario to apply the

well known differential geometry of a spatial curve traced on a surface.Brand [12] provides much insight in how to relate what happens to a spatial curve with what

happens at the surface where it is traced. His Fig. 130 is reproduced here with permission as Fig. 1

to illustrate this. Let us first examine this in a general way without application to the gears understudy.



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Consider that for a spatial curve, there exists the well known TNB triad of lines which rideswith the point, where T is the tangent, N the normal in the plane of osculation, and B the so-calledbi-normal. Brand provides further insight about specifically this by identifying a body with this

triad and identifying an instantaneous twist between the TNB-based body and the body in whichthe curve is traced. When the origin of the reference system is the point, then the coordinates forthis twist are Vd d T, where it is assumed that the point travels at unit speed in the tangentdirection, i.e. T T 1. The direction part for these coordinates is the well-known Darbouxvector, d sT jB, where s and j are torsion and curvature and where T and B are the nor-malized tangent and bi-normal directions, respectively.

Returning to the figure, one can see that the point path normal N is not, in general, normal tothe surface where the curve is traced. The surface normal n is separated from N by /, what I call acut angle. Here, let us associate a second body with a Tnp triad and see how this Tnp-based bodymoves relative to the TNB-based body. The two bodies so disposed are always connected to one

another by the point as it traces its curve, and they always share the tangent line T. They simplyrotate relative to one another about the tangent line T by a changing angle / as the point traces itspath. Again using the point as the origin of the reference, the coordinates of the composite twist

are V x T, where again the point travels at unit speed, i.e. T T 1 or _ss 1. Here, thedirection part for these coordinates is x tT cn kp, where t, c, and k are geodesic torsion,geodesic curvature, and normal curvature respectively, where n is the normalized direction of thesurface normal, and where p T n. Finally, Brand relates the geodesic torsion, geodesic cur-vature, and normal curvature with torsion, curvature, and cut angle / by the following:

t : s

d/ds

; geodesic torsion

c : j sin/ js/; geodesic curvaturek : j cos/ jc/; normal curvature

2

where the scalar s is arc-length but the scalar s/ is the sine of the angle /.

A final comment about this that proves useful regards a well-known theorem concerninggeodesic torsion and normal curvature. Consider that we know everything about a smooth sur-face, that we have chosen a point on it, and that we have chosen a tangent direction for T.

Theorem 1 states that the geodesic torsion and normal curvature are the same for all curves on agiven surface which pass through the same point and have the same tangent. In this situation, /,

Fig. 1. Brands Fig. 130 reprinted with permission of Sarah Brand Minor.



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j, and s vary across such a group of curves, while all curves in such a group maintain the same t

and k pair.

1.3. Back to gearing

Let us apply Brands development first specifically to the input gear and assume that the input

gear tooth surface is known very well. This means that we know geodesic torsion and normalcurvature (t and k) when given the point of contact and the tangent direction for the propagation

of the point of contact in the input gear.In summary, the givens here are the instantaneous screw axis and derivatives ($, _$$, and $$) that

describe the relative motion between the two gears, the point of contact between the two gears, the

(reciprocal) contact normal $n that identifies the line of action, and the geodesic torsion, normalcurvature, and tangent direction for the input gear point path. From this, we shall find the cor-responding curvature properties for the output gear point path (t, k, and tangent direction) and

the geodesic curvatures (c) for the point paths of both gears. This will be used to determinecurvature and torsion for the two related spatial curves.

2. General concepts

2.1. Classification of point path trajectories

Now, the Tnp-based body described above proves to be an important element in the analysis ofwhat happens between the two gears at the point of contact. The foregoing described what this

body is in relation to the input gear point path. The Tnp-based body is a virtual body that existshere solely for analysis.

In fact, it can also be said that, in general, the output gear has its own Tnp-based body thatfollows the point of contact as it traces a different curve on the toothed surface of the output gear

(output gear point path). We shall refer to the case where the two Tnp-based bodies are differentas the general case. However, it is conceivable that the two Tnp-based virtual bodies are thesame body throughout the motion, and we shall refer to this as the special case.

Dijksman [2] has effectively applied this technique to some extent in order to relate curvaturesbetween planar spur gears. Fig. 2 is Dijksmans Fig. 2.36 reproduced here with permission in orderto demonstrate this. In this figure, the input gear is Body 1, the output gear is Body 2, and the

fixed frame is Body 0. It can be argued that only the special case exists in planar motion, and forthe figure, this leaves the sole virtual body to be shown as Body 3. The contact point R and the

moving common normal are fixed in Body 3 throughout the relative motion.In order to help differentiate between the two cases, a reference has been established. Because

the point of contact and the instantaneous screw axis (between the two gears) are known, then the

corresponding null plane can be found (see [13]). This null plane contains a pencil of lines throughthe point of contact, a member of which must be the contact normal, $n. The line perpendicular tothe null plane through the point of contact is the reference line $T that will be used here.

Now, the point in the output gear that is instantaneously the point of contact has a velocity(relative to the corresponding point in the input gear) in the direction of $T due to the relative



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motion between the two gears. Therefore, when we use the point of contact as the origin of thereference, then the normalized coordinates T T 0 locate $T and the coordinatesS S So S vpT 3

locate $ and thereby specify the instantaneous twist. Here vp is the relative speed between points in

the two gears instantaneously at the point of contact. Additionally, the line of reference, $T, liesin the tangent plane that is tangent to the two gear tooth surfaces at the point of contact. The

intersection of the tangent plane and the null plane defines the line $p. Fig. 3 illustrates theseelements. Finally, let us use the same origin of reference and identify the normalized coordinatesn n 0 to locate $n and the normalized coordinates p p 0 to locate $p.

Note that we will observe from the input body, but we have chosen the reference frame to be atthe point of contact in order to simplify equations. This type of thing is typically done (e.g. see [11]

and [14, pp. 2734]). Therefore, our resulting equations will only be valid near this beginninglocation, since the actual point and axes will move away. Generalizing can be done, but that is leftfor the reader.

Now, as it turns out for the special case, the T line for the single Tnp-based body aligns itselfwith $T throughout the motion (Fig. 4). On the other hand, for the general case, the first T line forthe Tnp-based body of the input gear, the second T line for the Tnp-based body of the output

gear, and the $T reference line do not align but do share the point of contact and lie in the tangentplane. In both cases, the n lines for the Tnp-based bodies align themselves with the contact normal

Fig. 2. Dijksmans Fig. 2.36 reprinted with permission of Cambridge University Press and E.A. Dijksman.



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$n. For the general case, spin angles (a1 and a2) are introduced in order to locate the Ts of the input

and output Tnp-based bodies, each measured in the tangent plane towards $T (see Fig. 5, wherethe p lines are not shown).

2.2. Use of products

Some well known relations that are used throughout are provided here without reference.The reciprocal product between two screws A and B having coordinates A A Ao and

B

B

Bo

is denoted by

and is defined by A

B

A

Bo

Ao

B. If both sets of coordinates

are normalized, it can be shown that the product yields A B hA hB cosbAB dAB sinbAB,where hA and hB are the pitches of the screws, and where bAB and dAB are the included angle and

perpendicular distance between the two screws. The reciprocal product is also referred to as theKlein form.

The dot product between the two screws is denoted by and is defined by A B A B. Ifboth sets of coordinates are normalized, it can be shown that the product yields A B cosbAB.The dot product, which is also referred to as the Killing form, can be used to normalize coor-

dinates by dividing the coordinates by the norm kAk ffiffiffiffiffiffiffiffiffiffiA Ap ffiffiffiffiffiffiffiffiffiffiA Ap . Note that the dotproduct used here is not the dualized version that has been used in the past (e.g. Brand). Also,

Fig. 3. Reference system for single point contact.



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note that an alternate norm kAko ffiffiffiffiffiffiffiffiffiffiA o A

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiAo Aop has also been used herein so that themoment part is unitized thereby permitting scaling by translational velocity

_ss

to obtain in-

stantaneous twist coordinates.The cross product between the two screws is denoted by and is defined by

A B A B A Bo Ao B. The result is another screw whose axis is in general thecommon perpendicular between A and B. The pitch of this screw is hA hB dAB cotbAB.

The cross product can be useful when the derivative of a screw is desired. Consider that the

time-rate-of-change of screw A is denoted by the screw _AA which has coordinates _AA, where _AA isinherently associated with a particular body. Here, this is emphasized by identifying an observerstationed on that body. If the observer sees the reference (axes, screws, lines) fixed, then he stands

on (say) body X and the reference axes are fixed to X. In that case, the derivative is located bysimply the derivative of the coordinates, an apparent time-rate-of-change. On the other hand, if theobserver sees the reference axes moving too, like in the case where they are attached to a moving

body Y, then the cross product comes into play.First suppose that (i) the reference for the coordinates A of screw A is attached to body Y and

(ii) body Y twists relative to body X on screw $ whose coordinates are S measured from the samereference on Y. Now, first of all, whenever screw A is fixed in body Y, then an observer standing

on body X will see an _AA whose coordinates are _AA S A. Second, whenever a screw (say) B is notfixed in body Y, then an observer standing on body X will see a _BB whose coordinates contain anextra part, _BB _BB@ S B, where the extra part _BB@ are the coordinates of a screw _BB@. The extrapart is the time-rate-of-change of screw B as seen by a second observer standing on body Y (theapparent time-rate-of-change seen by the second observer).

Fig. 4. Special case of single point contact.



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2.3. Higher-order reciprocity

In anticipation of its use, higher-order reciprocity is introduced here. First, consider that two

screws A and B move continuously in a body specified by functions of a single variable (e.g.time). Also, consider that the two screws are not only reciprocal at a given instant but that theyare reciprocal continuously. Therefore, at the next instant, A moves infinitesimally to A , Bmoves infinitesimally to B, and the two screws A and B are also reciprocal. For this to

happen, there must be additional restrictions placed on the time-rates-of-change for the twoscrews.For example, given A B 0, where A and B are the coordinates of A and B, respectively, then

this means, first of all, that

A _BB _AA B 0; 4

where _AA are the coordinates of the first time-rate-of-change _AA of A as seen by an observer standing

on the body and where _BB are the coordinates of the first time-rate-of-change _BB of B as seen by thesame observer. This means that the two reciprocal products of the four screws A, B, _AA, and _BB,

Fig. 5. General case of single point contact.



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taken in the pairs A, _BB and B, _AA, must cancel each other. This is referred to as first-order reci-procity.

This can have geometrical meaning whenever a variable can be expressed as a function of time,

e.g. y ft, where here tdenotes time. Then (4) has geometrical meaning when it is written in theform

hA h _BB cosbA _BB dA _BB sinbA _BB xh _AA hB cosb _AAB d_AAB sinb _AAB 0;where x is a geometrical function resulting from the elimination of the time-dependent variable y.

One may continue this process by differentiating again, which for second order yields

A BB 2 _AA _BB AA B 0; 5where AA are the coordinates of the second time-rate-of-change AA of A as seen by the observer and

where BB are the coordinates of the second time-rate-of-change BB of B as seen by the observer. Thisis referred to as second order reciprocity, and similar statements can be made regarding cancel-

lation of pair-wise reciprocal products as well as geometrical meaning associated with the removalof time as the independent variable.

3. Details of single point contact

3.1. Special case

For the special case, the Tnp-based virtual bodies for both the input gear and output gear pointpaths are the same throughout the motion. Accordingly, the T and $T lines are the same line, andn corresponds with $n. The special case is characterized by the fact that the point of contact tracestwo curves, one in each gear, that are continuously tangent in the direction of $T.

This effectively establishes a virtual three-link mechanism, consisting of the input gear (Body 1),the output gear (Body 2), and the Tnp-based virtual body (Body 3) (see Fig. 4). Looking at themobility between the three bodies pair-wise, there are three instantaneous twists belonging to a

two-system (defining another cylindroid denoted by C1, where C1 6 Cg, see above). The threetwists must belong to a two-system because the virtual mechanism needs mobility in order for thespecial case to be repeated at the next instant when the point of contact moves to its next location

in the two bodies.

The linearly dependent coordinates of these twists are related by

_ss1V1 _ss2V2 S; 6where V1 t1T c1n k1p T, where V2 t2T c2n k2p T, where t1, c1, and k1 are thegeodesic torsion, geodesic curvature, and normal curvature of the input gear point path, where t2,c2, and k2 are the same for the output gear point path, and where _ss1 and _ss2 are the speeds at whichthe point of contact is traveling in the input and output gears along their respective paths. From

(6), four scalar equations may be obtained by using dot and reciprocal products of screw coor-dinates:



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_ss1V1 _ss2V2 S T;_ss1V1 _ss2V2 S n;

_ss

1V

1 _ss

2V

2 S

p;

_ss1V1 _ss2V2 S T:

7

Another condition for the special case to be repeated at the next instant is that the pencil oflines in the null plane must move to the neighboring pencil of lines in the next null plane that is

determined by the next point of contact and the next instantaneous screw axis. This ensures thatreciprocity between the contact normal and instantaneous screw axis is preserved at the nextinstant. For this to happen, the reciprocal pair $n and $ must exhibit higher-order reciprocity, and

also the reciprocal pair $p and $ must do the same. Recall that the lines $n and $p establish thepencil of lines in the null plane (see Fig. 3).

First-order reciprocity for the reciprocal $n and $ pair and also for the reciprocal $p and $ pair

yields the following scalar equations:

S _nn _SS n _ss1S n0 _SS n 0;S _pp _SS p _ss1S p0 _SS p 0;

8

where _nn and _pp are the coordinates for the time-rates-of-change _$$n and _$$p of the lines $n and $p asseen by an observer standing on the input gear, and where _nn _ss1n0 and _pp _ss1p0. 1

Now, $n and $p are fixed lines in the Tnp-based body that each instantaneously twists relative tothe input gear on a screw whose twist coordinates are _ss1V1. This means the cross product can beused in order to establish Frenet-like screw coordinate relations

T0 dTds1

V1 T k1n c1p 0;

n0 dnds1

V1 n k1T t1p p;

p0 dpds1

V1 p c1T t1n n;

9

where, for example, the coordinates for the time-rate-of-change _TT of T (and $T) as seen by anobserver standing on the input gear are given by _TT _ss1T0, and where the prime is used to denotedifferentiation with respect to arc length of the associated surface curve (for the above, it is s1).

1 _SS _SS _SSo are the coordinates of the time-rate-of-change of twist. (I call it the accelerator, Brand [12] calls it theacceleration motor, and Bokelberg et al. [11] call it the differential screw.) Indeed, the moment part of the original

twist is equivalent at each instant to the velocity of a point (say So vp, see (3)). However, the moment part _SSo of thederivative is not equivalent to acceleration or the time-rate-of-change of the point s velocity d=dtvp ap. In otherwords, in general, _SSo 6 ap, and in fact, typically, _SSo ap vpT S (see (3) and [12, p. 127]). This means higher-orderreciprocity (e.g. (8)) is usually not equivalent to what is commonly seen in literature, which is a time-rate-of-change of a

simple vector dot product, d=dtvp n 0, where ap is used in the expansion. This latter thing appears to be pointdependent (not coordinate invariant), unless a special frame of reference is used or unless the two approaches otherwise

gain equivalence. More on this is forthcoming in a later article.



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Using (3) and (9), the six scalar equations given in (7) and (8) can be re-written in the following

way:

0 _ss2t2 _ss1t1 S T;0 _ss2c2 _ss1c1 S n;0 _ss2k2 _ss1k1 S p;0 _ss2 _ss1 vp;0 _SSo n _ss1S p _ss1k1vp;0 _SSo p _ss1S n _ss1c1vp:

10

The six equations are linear in the unknowns c1, c2, k2, t2, _ss1, and _ss2, so the solution is unique.

A reduction can be made of the above set of equations for the planar scenario to obtain the

EulerSavary equation for envelopes. First, the fifth equation can be used to find _ss1 as a functionofq1, qc, and v cosw, where q1 1=k1, qc vp=x, and v cosw _SSo n=x, and where x S p.Next, the third equation can be used to find _ss2 as a function of vp, q2, q1;qc, and v cosw, whereq2 1=k2. Finally, the following can be obtained, which is the equivalent planar EulerSavaryequation for envelopes:

1

qc q1

1qc q2

cosw x

v; 11

where w is essentially the complement of the angle typically used, and where v is the propagation

speed of the instant center.

Since t, c, and k are now known for both bodies, what remains now is to determine the cur-vature j, torsion s, and the cut angle / for the two related spatial curves that are being traced on

the tooth surfaces. From (2), one may obtain j and / straightaway. However, because s is ahigher-order property, we are required to differentiate again to obtain it, which introduces a host

of other unknowns.First of all, differentiating the bottom two equations of (2) with respect to the generic arc length

s yields

k0 j0c/ js/s js/t;c0 j0s/ jc/s jc/t;

12

where the relationship /0 t s has been used (see (2)), and where again the prime is used todenote differentiation with respect to arc length of the associated surface curve. (Note that (12)may be applied to either the input gear or the output gear.) From the above equations, one candetermine torsion s and the arc-length-rate-of-change j0 of curvature j whenever given the arc-length-rates-of-change k0 and c0 of normal and geodesic curvature (as well as the lower orderproperties). Therefore it remains to determine k0 and c0 (and also t0) for both the input and outputgear point paths, which leads to at least six higher-order unknowns.

Actually, the unknowns ss1 and ss2 are also added as more equations are developed, whichtentatively brings the total to eight higher-order unknowns. However, we can reduce this back to



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six by using a Theorem 2 that is essentially a higher-order version of Theorem 1 (see Section 1).

Theorem 2 states that the arc-length-rates-of-change (t0 and k0) of geodesic torsion and normalcurvature are the same for all curves on a given surface which pass through the same point, have

the same tangent, and have the same arc-length-rate-of-change in tangent (see Appendix A, whereT and T0 set k and k0). Therefore, let us say there are six unknowns, c01, c

02, k

02, t

02, ss1, and ss2, where

the prime in c01 denotes differentiation with respect to s1 and where the prime in c02, k

02, and t

02

denotes differentiation with respect to s2.

Eq. (6) can be differentiated to yield an equation relating the coordinates for time-rates-of-change of instantaneous twists as seen by an observer standing on the input gear

ss1V1 _ss21V01 ss2V2 _ss22V02 _ss2S V2 _SS; 13

where V01 t01T c01n k01p 0 and V02 t02T c02n k02p 0. In the above, the _ss2S V2 _ss1SV1

was necessary to add because the first two terms on the right-hand side of (13) correspond to

the time-rate-of-change of an instantaneous twist as seen by an observer standing on the outputgear, and the addition adjusted this to account for our observer, who stands on the input gear. Eq.(13) can be projected in the same way as (7) to create four new scalar equations.

The two additional scalar equations are obtained by using second order reciprocity, whichensures to a higher-order that the special case is repeated at the next instant by requiring to ahigher-order that the current pencil of lines in the null plane moves to the neighboring pencil of

lines of the next null plane. This can be obtained by applying (5) or differentiating (8) again wherein either case the same observer (input gear) is used for derivatives, which yields

S

nn

2 _SS

_nn

SS

n

S

ss1n

0

_ss21n

00

2_ss1 _SS

n0

SS

n

0;

S pp 2 _SS _pp SS p S ss1p0 _ss21p00 2_ss1 _SS p0 SS p 0; 14

where nn ss1n0 _ss21n00 and pp ss1p0 _ss21p00. In total, this yields six new scalar equations that aregiven by the following:

_ss21t01 ss1t1 _ss22t02 ss2t2 _ss1k1S n c1S p _SS T;

_ss21c01 ss1c1 _ss22c02 ss2c2 _ss1t1S p k1S T _SS n;

_ss21k01 ss1k1 _ss22k02 ss2k2 _ss1c1S T t1S n _SS p;

ss1

ss2

_

SSo T;0 ss1S p k1vp _ss21c1t1 k01vp 2_ss21t1S n _ss21c1S T

2_ss1 _SS p _ss1t1 _SSo p _ss1k1 _SSo T SSo n;0 ss1c1vp S n _ss21k1t1 c01vp 2_ss21t1S p _ss21k1S T

2_ss1c1 _SSo T _ss1 _SS n _ss1t1 _SSo n SSo p;

15

where the following second-order Frenet-like screw coordinate relations were used in the last twoequations:



15/21

T00 d2T

ds21 c21 k21T t1c1 k01n k1t1 c01p k1p c1n;

n00 d2

nds21

t1c1 k01T k21 t21n c1k1 t01p c1T 2t1n;

p00 d2p

ds21 k1t1 c01T c1k1 t01n t21 c21p k1T 2t1p:

16

Here (15) is linear in the unknowns, c01, c02, k

02, t

02, ss1, and ss2, which again yields a unique solution

set. This solution set together with the solution of (12) for both gears yields the other requiredcurvature properties.

3.2. General case

For the general case, each gear has its own Tnp-based body throughout the relative motion,which provides a second virtual body in this development. These two virtual bodies are lockedtogether at the point of contact, and they share the same surface normal throughout the relative

motion. However, they will spin relative to one another about the common surface normal.Considering that we have the triad of lines T1n1p1 corresponding to the input gear point path and

triad of lines T2n2p2 corresponding to the output gear point path, this means that n1 and n2correspond with $n and that T1, p1, T2, and p2 all lie in the tangent plane defined by $T and $p.

Additionally, in order to keep track of this, a third virtual body is introduced giving us a total offive bodies. Let us denote this last body using the triad of lines Tnp, where T tracks with $T and n

tracks with $n. The general case is characterized by the fact that the point of contact traces twocurves, one in each gear, whose tangents lie in the tangent plane intersecting continuously at thepoint of contact (see Fig. 5).

This effectively establishes a virtual five-link mechanism, consisting of the input gear (Body 1),

the output gear (Body 2), the Tnp-based virtual body (Body 3), and the T1n1p1-based and T2n2p2-

based virtual bodies (Bodies 4 and 5) that correspond with the paths that the point of contacttraces in the input and output gears, respectively. Looking at the mobility between the five bodies

pair-wise, one can see that there are tentatively five instantaneous twists. With an assumption, thiscan be simplified.

First, let us introduce two spin angles a1 and a2, which locate T1 and T2. We measure a1 from T1to T and a

2from T

2to T, each about the common normal. (Note, Fig. 5 shows these angles as

negative.) Let us assume that these angles are implicit functions of arc length so that _aa1 a01 _ss1 and_aa2 a02 _ss2. This permits us to easily create a pair of composite twists. The first composite twistbetween Bodies 3 and 1 is the sum of the twist between Bodies 3 and 4 and the twist betweenBodies 4 and 1. The second composite twist between Bodies 3 and 2 is the sum of the twist be-

tween Bodies 3 and 5 and the twist between Bodies 5 and 2.Since we know have the two composite twists together with the instantaneous twist between

Bodies 2 and 1, the problem essentially reverts back to what we had in the special case. That is, we

have three links and three twists that belong to a two-system, which gives the virtual mechanismmobility so that it can repeat the general case at the next instant.



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The linearly dependent coordinates of these twists are also related by (6), but now where V1 are

the coordinates of the composite twist between Bodies 3 and 1 and V2 are the coordinates of thecomposite twist between Bodies 3 and 2, said twist coordinates given by the following:

V1 t1T c1n k1p ca1T sa1p;V2 t2T c2n k2p ca2T sa2p;

17

where ca1 cosa1, sa1 sina1, ca2 cosa2, and sa2 sina2. In the above, the star-basedproperties are given by the following:

t1 t1ca1 k1sa1;c1

c1

a01;

k1 t1sa1 k1ca1;t2 t2ca2 k2sa2;c2 c2 a02;k2 t2sa2 k2ca2;

18

which essentially introduces a skewed set of curvature properties for the two surfaces.It is helpful to review what are the givens in the general case. Recall that the instantaneous twist

and its derivatives are given to describe the relative motion between Bodies 2 and 1. Also, the

point of contact together with the contact normal ($n) are givens, which together with the in-stantaneous screw axis ($) establishes the reference line ($T) and the tangent plane. Finally, it isnecessary to establish the desired direction of T1 in the tangent plane, which means that a1 must begiven. The angle a1 is required in order to establish t1 and k1, which using (18) thereby establishest1 and k

1 . In the end, this will leave us with seven unknowns, five (t

2, c

2, k

2 , a2, and _ss2) for the

output body and two (c1 and _ss1) for the input body. Now, using a method similar to (7) applyingthose dot products and reciprocal product to (6), five scalar equations in these seven unknowns

can be obtained where the fifth is given by _ss1V1 _ss2V2 S p 0.Based on the same reasoning used in the special case, first order reciprocity must be satisfied,

which means that the two equations in (8) must hold for the general case as well. For the general

case implementation, however, the following first-order Frenet-like screw coordinate relations for

the Tnp-based body (Body 3) should now be used:

T0 V1 T k1 n c1p sa1n;n0 V1 n k1 T t1p sa1T ca1p;

p0 V1 p c1T t1n ca1n:19

Using (3), (17), and (19), the seven scalar equations given by (7), (8) and ( _ss1V1 _ss2V2 S p 0can be re-written in the following way:



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0 _ss2t2 _ss1t1 S T;0 _ss2c2 _ss1c1 S n;0

_ss

2k

2 _ss

1k

1 S

p;

0 _ss2ca2 _ss1ca1 vp;0 _ss2sa2 _ss1sa1;0 _SSo n _ss1ca1S p _ss1sa1S T _ss1k1 vp;0 _SSo p _ss1ca1S n _ss1c1vp:

20

In the above, the seven equations are non-linear in the unknowns t2, c2, k

2 , a2, _ss2, c

1, and _ss1.

However, there is a unique solution set.

Since a, t, c, and k are now known for point paths in both gears, it remains to determine t, c,and k and subsequently /, j, and s for both gears in order to completely solve the general case.

Eq. (18) yields t1,k1, t2, and k2 but demonstrates that a01 and a02 are needed in order to determine c1and c2. Also, it was shown in the special case that additional differentiation was required anywayto determine torsion for both curves.

By examination of (20), one can see that differentiation yields new variables a0, t0, c0, k0, and ssfor both the output and input gears. Now as it has been our pattern regarding knowledge of theinput body surface, we would hope that t01 and k

01 are known. However, after application of

Theorem 2 and upon analysis of the derivative of (18)

t01 t01ca1 k01sa1 t1sa1a01 k1ca1a01;c01 c01 a001;k01 t01sa1 k01ca1 t1ca1a01 k1sa1a01;t02 t02ca2 k02sa2 t2sa2a02 k2ca2a02;c02 c02 a002;k02 t02sa2 k02ca2 t2ca2a02 k2sa2a02

21

one can see that a01 is required together with k01 and t

01 in order to obtain t

01 and k

01 . Accordingly, let

us require that arc-length-rate-of-change of the spin angle a01 for the input gear be specified, whichis consistent with (20) and establishes k01 and t

01. Therefore, there are seven unknowns for the

higher-order equations in the general case, a02, t02 , c

02 , k

02 , and ss2 for the output gear and c

01 and ss1

for the input gear, where the primes in the first group corresponds to differentiation with respect

to s2 and where the prime in the second group corresponds to differentiation with respect to s1.The technique used to generate the higher-order equations in the general case will follow closely

that used for the special case. Specifically (13) remains valid, except for where now V01 and V0

2 are

given for the general case by

V01 t01 T c01 n k01 p sa1a01T ca1a01p;V02 t02 T c02 n k02 p sa2a02T ca2a02p:

22

Applying the same five projections (three dot products and two reciprocal products) yields fivenew scalar equations in the higher-order unknowns.



18/21

Based on the same reasoning used in the special case, second order reciprocity must be satisfied,

which means that the two equations in (14) must hold for the general case as well. For the generalcase implementation, however, the following second-order Frenet-like screw coordinate relations

for the Tnp-based body (Body 3) should now be used:

T00 d2T

ds21 c21 k21 T t1c1 k01 n k1 t1 c01 p

2k1sa1T c1 a01ca1n k1 ca1 t1sa1p;

n00 d2n

ds21 t1c1 k01 T k21 t21 n c1k1 t01 p

c1 a01ca1T 2k1sa1 t1ca1n c1 a01sa1p;

p00 d2p

ds21 k1 t1 c01 T c1k1 t01 n t2

1 c2

1 p k1 ca1 t1sa1T c1 a01sa1n 2t1ca1p:

23

Assembling the five kinematic higher-order equations together with the two higher-order reci-

procity equations yields the desired seven scalar equations

_ss21t01 ss1t1 _ss22t02 ss2t2 _ss1k1 S n c1S p _SS T;

_ss21c01 ss1c1 _ss22c02 ss2c2 _ss1t1S p k1 S T _SS n;

_ss21k01 ss1k1 _ss22k02 ss2k2 _ss1c1S T t1S n _SS p

ss1ca1 _ss21sa1a01 ss2ca2 _ss22sa2a02 _SSo T _ss1sa1S n;ss1sa1 _ss21ca1a01 ss2sa2 _ss22ca2a02 _SSo p _ss1c1vp ca1S n;0 ss1ca1S p sa1S T k1 vp _ss21c1t1 k01 vp

2_ss21k1sa1 t1ca1S n _ss21sa1c1 a01S p _ss21ca1c1 a01S T 2_ss1ca1 _SS p _ss1sa1 _SS T _ss1t1 _SSo p _ss1k1 _SSo T SSo n;

0 ss1c1vp ca1S n _ss21k1 t1 c01 vp _ss21sa1c1 a01S n 2_ss21t1ca1S p _ss21k1 ca1 t1sa1S T

2_ss

1c

1

_SSo

T

_ss1ca1

_SS

n

_ss1

t1

_SSo

n

SSo

p:

24

The above equations are linear in the unknowns, a02, t02 , c

02 , k

02 , and ss2 for the output gear and

c01 and ss1 for the input gear, and they can be easily solved.The results of solving (20) and (24) can be used together with (18) and (21) (and (2) and (12) for

both gears) to solve for all curvature properties for both gears, thereby solving the general case.However, the second arc-length rates of change a001 and a

002 for the input and output gears are

required in the process. We could say that a001 should be a given, which is consistent with thederivation here. For a002, however, it appears we must either differentiate yet again or simply de-clare it as a free choice (at least to second order motion and second order reciprocity). After all,



19/21

the rate-of-change (c02) of geodesic curvature can be modified to accommodate any freely chosena002 while still maintaining a specified c

02 obtained from the solution of (24) (see the fifth of (21)).

4. Conclusions

The work contained here establishes the requirement for general conjugate motion between two

bodies that are in direct contact. Essentially, the work is a spatial equivalent of the EulerSavaryequation for envelopes. This is a study of an implication of Dooner s Third Law of Gearing [1] forsingle point contact that combines screw theory, the well-known differential geometry of surfaces,

and a novel application of higher-order reciprocity. This is done with the sole purpose of de-termining the curvature properties of a single curve that represents the travel of the point of

contact on the output gear tooth surface. Requirements to determine this include knowledge ofthe input gear tooth surface at the point of contact as well as the relative motion between the two

gears.It has been shown that when the direction for the path of travel of the point of contact in the

input gear is specified, the sum (c02 ) can be determined of the rate-of-change (c02) of geodesic

curvature and of the second rate-of-change (a002) for the direction of the path of travel in the outputgear (see (21) and (24)). This means that, at least to second order motion and second order re-ciprocity, one has a free choice to change geodesic curvature in order to steer the output path in

some desired way, provided that the sum, the fifth of (21), remains the same.Further work is required to see if there are any additional relationships that can be used or that

must be satisfied in order to determine a second, independent curve on the output gear tooth

surface. That second curve together with the first shown here would completely determine the

output gear tooth surface. Also, further work could concentrate on determining actual direction(a1) of point path travel in the input gear instead of accepting it as a given.

Appendix A

This section proves Theorems 1 and 2. Consider that the 2D surface r ru; v is a function ofcurvilinear coordinates u and v and that a curve is traced in it. Then, say that the coordinates u

and v are functions of arc length s of that curve. It is well known that normal curvature is (see[15]):

k Lu02 2Mu0v0 Nv02; A:1where u0 du

dsand v0 dv

dsand where L, M, and N are functions of u and v

L n o2r

ou2; M n o

2r

ouov; and N n o

2r

ov2A:2

and where n is the surface normal at the point. Also, it is well known that

1 Eu02 2Fu0v0 Gv02; A:3where E or

ou2, F or

ou or

ov, and G or

ov2. This with (A.1) means



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k Lk2 2Mk N

Ek2 2Fk G A:4

where k

u0

v0 tan

a

, where a is the spin angle that establishes a tangent direction. Equation(A.4) demonstrates a well known result that normal curvature is known when the point is known(i.e. u and v determines E, F, G, L, M, and N) and when a tangent direction is specified (i.e. a or k).

In other words, it is not explicitly dependent on first order changes u0 and v0, but rather it isdependent on their ratio k.

Next, consider that the arc-length-rate-of-change of normal curvature is given by the derivative

k0 L1u02 2M1u0v0 N1v02u0 L2u02 2M2u0v0 N2v02v0 2Lu0 2Mv0u00 2Mu0 2Nv0v00; A:5

where L1 oL=ou, M1 oM=ou, N1 oN=ou, L2 oL=ov, M2 oM=ov, and N2 oN=ov arefunctions of u and v. The objective is to demonstrate that k0 is not explicitly dependent on thesecond-order changes u00 and v00. Now, since the derivative of (A.3) is equal to zero, (k times) thederivative of (A.3) can be subtracted from (A.5), and the result can be expressed in the form

k0 L1 kE1u02 2M1 kF1u0v0 N1 kG1v02u0 L2 kE2u02 2M2 kF2u0v0 N2 kG2v02v0

2L kEu0 M kFv0 u00

u0v00

v0

; A:6

where (A.1) and (A.3) have been used again for substitutions and where for example E1 oE=ou.Multiplying the last part of (A.6) by v02=v02 and then dividing all again by (A.3) and then both topand bottom by v02 yields

k0 X1k2 2Y1k Z1u0 X2k2 2Y2k Z2v0

Ek2 2Fk G 2Xk Yk0

Ek2 2Fk G; A:7

where k0 is the arc-length-rate-of-change ofk and Xi Li kEi, Yi Mi kFi, and Zi Ni kGi, itaking on the values of nil, 1 and 2. The final desired result is obtained when the first term of (A.7)is divided by

ffiffiffiffiffiffiffiffiffiffiffiA:3p and subsequently divided top and bottom by v0, which yields:k0 X1k

2 2Y1k Z1k X2k2 2Y2k Z2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEk2 2Fk G3

p 2Xk Yk0Ek

2 2Fk G: A:8

This shows a new result that k0 is known when the point is known (i.e. u and v determines E, F, G,

L, M, N, E1, F1, G1, L1, M1, N1, and E2, F2, G2, L2, M2, N2) and when a tangent and its arc-length-rate-of-change are specified (i.e. k and k0). Now, since the expression for geodesic torsion (t) isanalogous (see [15]) to (A.1), the expression for arc-length-rate-of-change (t0) is analogous to (A.8)(so it is proved by analogy).

References

[1] D. Dooner, A. Seireg, The Kinematic Geometry of Gearing, John Wiley, New York, 1995.

[2] E.A. Dijksman, Motion Geometry of Mechanisms, Cambridge University Press, Cambridge UK, 1976.



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[3] M. Griffis, Proving Dooners Third Law of Gearing: a study of curvature, in: Proceedings of Ball 2000, 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 911, 2000, University of Cambridge, 2000.

[4] F. Litvin, Gear Geometry and Applied Theory, Prentice-Hall, New Jersey, 1994.

[5] D. Wu, J. Luo, A Geometric Theory of Conjugate Tooth Surfaces, World Scientific Publishing, Singapore, 1992.

[6] A. Dyson, A General Theory of the Kinematics and Geometry of Gears in Three Dimensions, Clarendon Press,

Oxford, 1969.

[7] E. Rimon, J. Burdick, in: A Configuration Space Analysis of Bodies in Contact-II: 2nd Order Mobility,

Mechanisms and Machine Theory, vol. 30(6), Elsevier Science Ltd, Pergamon, Great Britain, 1995.

[8] C. Cai, B. Roth, On the spatial motion of a rigid body with point contact, in: Proceedings of the 1987 IEEE

International Conference on Robotics and Automation, Raleigh, North Carolina, 1987.

[9] D. Montana, The kinematics of contact and grasp, International Journal of Robotics Research 7 (3) (1998).

[10] S.R.S. Ball, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge UK, 1900.

[11] E.H. Bokelberg, K.H. Hunt, P.R. Ridley, in: Spatial Motion-I: Points of Inflection and the Differential Geometry

of Screws, Mechanism and Machine Theory, vol 27(1), Pergamon Press, New York, 1992, pp. 115.

[12] Louis Brand, Vector and Tensor Analysis, John Wiley, New York, NY, 1947.

[13] Felix Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, MacMillan Company, NewYork, 1939.

[14] O. Bottema, B. Roth, Theoretical Kinematics, North Holland Publishing, Amsterdam, 1979 (reprinted in 1990,

Dover Publications, Mineola NY).

[15] C.E. Weatherburn, in: Differential Geometry of Three Dimensions, vol. 1, Cambridge University Press,

Cambridge, UK, 1961.


Documents

A Study of Curvature for Single Point Contact Griffis