on the kinematics of three-link spatial cam mechanisms · 2013. 6. 4. · on the kinematics of...

Meccanica33: 349–361, 1998.c© 1998Kluwer Academic Publishers. Printed in the Netherlands.

On the Kinematics of Three-Link Spatial Cam Mechanisms

AHMED RAMAHI 1 and YILMAZ TOKAD2

1Eastern Mediterranean University, Mechanical Engineering Department, G. Magusa, Mersin 10, Turkey2Eastern Mediterranean University, Electric and Electronic Engineering Department, G. Magusa, Mersin 10,Turkey

(Received: 29 April 1997; accepted in revised form: 27 November 1997)

Abstract. In this paper, using only the linear algebra, compact expressions for the description of the relativeinstantaneous screw motion of two rigid bodies observed from a fixed reference frame is derived. These expres-sions are then used for the study of the generation of contact surfaces of three-link spatial, spherical and planarcam mechanisms. The surfaces of the cam and the follower are generated by the sweeping action of the relativeinstantaneous screw axis between the two bodies.

Sommario. In questo lavoro, usando soltanto metodi algebrici, si ricavano espressioni compatte per la de-scrizione dell’atto di moto relativo elicoidale di due corpi rigidi usati in un sistema di riferimento fisso. Questeespressioni sono poi usate per la generazione di superfici coniugate di meccanismi a camma, generici, sfericie piani, a 3 membri. Le superfici della camma e del cedente sono generati dall’inviluppo dell’asse del motoistantaneo elicoidale visto dai due corpi.

Key words: Instantaneous screws, Camms, Indexing, Mechanisms, Mechanics of machines.

1. Introduction

The instantaneous motion of a rigid body in space can be represented by a simultaneous rotationabout and a sliding along a unique line called the instantaneous screw axis (I). The ratio ofthe magnitudes of the linear (the sliding) and the angular (the rotation) velocities is known asthe pitch of the instantaneous screw [1]. This motion of the rigid body is usually referred to asinstantaneous screw motion. However, when two bodies,R1, andR2, are arbitrarily moving withrespect to a reference frame60, then one considers three different instantaneous screw axes: Twoabsolute I10, I20 and one relative I21 screw axes. According to the Aronhold-Kennedy theoremthese three axes share a common perpendicular [2]. Note that the relative instantaneous screw axisI21 is the set of all points common to both bodies, with relative velocity of minimum magnitude[3]. This implies that a higher-pair with I21 being its contact line is a pair with minimumpower losses. By making use of this fact, Xiao and Yang [4] have studied the kinematics ofspatial gearing. However, their study was based on the concept of velocity screws representedby dual number matrices. A seemingly related study of spatial cams has also been consideredin the literature by Raven [5] and Dhande [6]. However, three-link spatial cam mechanismswith minimum power losses were originally tackled by Gonzalez-Palacios and Angeles [7]. Bymaking use of the Aronhold-Kennedy theorem with the aid of vector algebra they have obtainedthe contact surfaces of both the cam and the follower using the condition that the relative velocityat contact point be of minimum magnitude. Various forms of these surfaces are then used inspatial, spherical and planar indexing cam mechanisms. Later, Gonzalez-Palacios and Angeles[8] extended their approach to spherical indexing cam mechanisms with a roller between the

350 Ahmed Ramahi et al.

cam and the follower. In fact, Gonzalez-Palacios and Angeles [9] have established a completeframework for the synthesis of various types of cam mechanisms with minimum power losses. Intheir work, spatial, spherical and planar cam mechanisms are integrated into a single formulation.They also discussed the pressure angle distributions. However, their formulation is based on amapping of the motion from the Euclidean space to the dual space. Hence, in this study againthe dual-number algebra was the basic tool for cam synthesis.

In this paper, as an alternate to the use of dual-number algebra, a complete formulation fordescribing the instantaneous screw motion of two rigid bodies arbitrarily moving with respect to afixed frame, is given using simply the ordinary linear algebra. The results are then used to generatecontact surfaces of three-link spatial, spherical and planar cam mechanisms with minimum powerlosses, i.e., with minimum sliding velocity along the contact line. A few examples are given toillustrate the approach.

2. Fundamental Background

A spatial vectora, in a fixed Cartesian coordinate system60, is represented by a 3× 1 columnmatrix a or equivalently by a 3× 3 skew symmetric matrixA. Therefore, the scalar productl = a · b and the vector productc = a × b of the two vectorsa andb are denoted in60 by therespective matrix productsl = aT b andc = Ab. The use of such representation of the vectorsusually provides some simplifications in the establishment of complicated vectorial relations[10, 11].

When a Cartesian coordinate system61 is attached rigidly to a moving rigid body withrespect to a fixed coordinate system60, then the position vectorsrA

1 andrA0 of a point A of the

rigid body with respect to61 and60 are related as [12][rA

01

]=

[0T1

0ξ1

0 1

] [rA

11

](1)

where the coefficient matrix is called the ‘homogeneous transformation’ matrix and indicatedsimply by0H1, and0T1 is the coordinate transformation matrix which describes the orientationof61 with respect to60, while0ξ1 is the translation vector which gives the position of the originof 61 with respect to60. Since the rigid body is moving andrA

1 is a constant vector in61, toobtain an expression for the velocity of the point A, we differentiate Equation (1) with respect totime. Hence, the velocityvA

0 of the point A measured in60 which may be augmented to expressin the homogeneous form[

vA01

]=

[0�1 −0�1

0ξ1 + 0v1

0 1

] [rA

01

], (2)

where0�1 is the skew symmetric representation of the angular velocity vector0ω1 and

0v1 = d/dt0ξ1.

However, in finite kinematics, Chasles theorem states in space that the position of a point Aon the displaced rigid body may be described (in60) as the combination of a finite rotation(angleθ ) about and a finite translation (λ) along a screw axis of the body. This property may beexpressed mathematically as [13, 14][ ∗rA

01

]=

[0T1 (I − 0T1) ζ + λu

0 1

] [rA

01

], (3)

On the Kinematics of Three-Link Spatial Cam Mechanisms351

where∗rA0 is the position vector of point A after a finite displacement of the rigid body, andζ

is the position vector of any point on the screw axis. Note that the coordinate transformationmaxtrix 0T1 may be given by [15, 16, 17]

0T1 = I + (1 − cosθ)U2 + sinθU,

where, according to Euler’s theorem [18],u is the direction cosine vector of the rotation axis,U is its skew-symmetric matrix representation, andθ is the rotation angle about this axis.

When the displacement becomes infinitesimal (see also [2]) we haveθ → dθ, λ → dλ andEquation (3) reduces to[

rA ′01

]=

[I + dθU −dθUζ + dλu

0 1

] [rA

01

],

whererA ′0 is the position vector of point A after the rigid body (61) being displaced infinitesi-

mally. Hence the infinitesimal displacement dr = rA ′0 − rA

0 is given by[dr

0

]=

[[I + dθU −dθUζ + dλu

0 1

]−

[I 00 1

]] [rA01

].

Dividing both sides by dt , we obtain[νA

00

]=

[ωU −ωUζ + vu

0 0

] [rA

01

],

whereω = dθ/dt andv = dλ/dt are, respectively, the magnitudes of the rotational velocityabout and the sliding velocity along the screw axis at a given instant. SinceωU = � representsthe skew symmetric representation of the angular velocity vector0v1, then with this notationthe last equation may be augmented and written in the following homogeneous form[

vA01

]=

[0�1 −0�1ζ + p0v1

0 1

] [rA

01

]. (4)

Equation (4) expresses that the instantaneous motion of a rigid body in space can be describedas a screw motion about an instantaneous screw axis I with a pitchp. From Equation (4), onecan write

vA0 = 0�1(r

A0 − ζ )+ p0v1.

The last equation shows clearly that the points lying on the screw axis I have velocities withminimum-magnitude since in this case we either have(rA

0 − ζ ) = 0 or (rA0 − ζ )‖0v1.

Equation (4) allows us logically to represent this screw motion in a more compact form. Infact, if we letn = ωu = v andm = −Nζ = −�ζ , wherem is the moment vector ofn withrespect to the origin of60, further if we letm = m + pn, then Equation (4) takes on the form[

vA01

]=

[N m

0 1

] [rA

01

]. (5)

This equation suggests that the velocity distribution may be determined by a pair of orderedvectorsτ = (n; m), called the instantaneous or instant (for short) screw whilen andm arecalled the Ball vectors of the instant screw [19]. Comparing Equations (2) and (4), the instantscrewτ may be expressed as

τ = (n,m) = (0v1; 0v1 − 0V10ξ1).


Note that if the pairτ = (n,m) is given, then the following four physical properties of theinstant screw can be determined [20]:

1. The unit directionu of its axis is given by

u = n

|n| . (6)

2. The magnitude of its angular velocity is given by

v = |n|. (7)

3. The position vectorζ⊥ of the point on its axis nearest to the origin of60 is given by

ζ⊥ = N m

nT n= 1

n2N m. (8)

4. The magnitude of the transitional velocity along its axis is calculated from

ν = ωP, (9)

where

p = nTm

nT n.

Note also that if the body is rotating only, then the pitch of the screw is zero and we haveτ = (n; m). However, if it is translating only, thenτ = (0; νu).

3. Kinematics of Two Arbitrarily Moving Rigid Bodies

Consider two rigid bodiesR1 andR2, represented respectively by the attached body coordinatesystems61 and62, moving in space arbitrarily with respect to the fixed reference frame60 asshown in Figure 1. Letτ10,τ20andτ21be the absolute and relative instantaneous screws associatedwith these rigid bodies. We would like to obtain the physical properties (u21, ω21, ζ

⊥21, ν21) of τ21

(read 2 with respect to 1) when those ofτ10 andτ20 are given. To establish these properties (seealso [21]); let the pointA1 fixed in61 and the pointA2 fixed in62 be two momentaril coincidentpoint with a position vectorrA

0 measured in60. Then, the relative velocityνA2/A1, of the pointA2 relative to pointA1, measured in60 can be determined by making use of Equation (5);

vA2/A1 = vA2 − vA1

or [v

A201

]−

[v

A101

]=

[vA2/A1

0

]=

[[N20 m200 1

]−

[N10 m100 1

]] [rA

01

]

which implies that the Ball vectors(n21; m21) of the relative instant screwτ21 are given by

t21 = (n21; m21) = (n20; m20)− (n10; m10) = (n20 − n10; m20 − m10). (10)

According to the Aronhold-Kennedy Theorem, the I21 of τ21 intersects orthogonally thecommon perpendicular of I10 and I20. This theorem may be demonstrated as follows. Let(k; k◦)be the Plucker coordinates of the common perpendicular of the absolute screw axes I10 and I20,then we can write the conditions [21]

kT n10 = 0 and kT n20 = 0 (orthogonality) (11)


Figure 1. Two rigid bodies moving arbitrarily with respect to a fixed flame60.

kTm10 + nT10k◦ = 0 and kTm20 + nT20k

◦ = 0 (intersection). (12)

Now, by making use of Equation (10), (11) and (12), it follows that

kT n21 = 0,

kTm21 + nT21k◦ = 0.

This proves that I21 is a perpendicular intersector of the common perpendicular of I10 and I20.Now, since the instant screwsτ10 andτ20 are given in the formsτ10 = (n10; m10 + p10n10)

andτ20 = (n20; m20 + p20n20), then the physical properties ofτ21 are obtained directly fromEquations (6)–(9) as follows

(v21)60 = n21 = n20 − n10,

ω221 = |n21|2 = |n20 − n10|2,

(ζ⊥21)620 = 1

|n20 − n10|2(N20 − N10)(N10ζ10 − N20ζ20 + p20n20 − p10n10)

and

ν21 = 1

|n20 − n10|(n20 − n10)

T (N10ζ10 − N20ζ20 + p20n20 − p10n10), (13)

where the position vectorsζ10 andζ20 are indicated in Figure 1. Equation (13) constitutes thecomplete expressions to describe the relative instantaneous screw motion of two arbitrarilymoving rigid bodies in space with respect to a fixed frame60.

From the practical point of view and without the loss of generality, we may choose thecoordinate system used by Gonzalez-Palacios and Angeles [7] (shown in Figure 2) in which thetwo vectorsζ10 andζ20 represent, respectively, the position of I10 and I20. From Figure 2 withthe selected sense of rotation angle one can write

n10 = v10 = ω10

0

01

, n20 = v20 = −ω20

0

sinαcosα

, ζ10 =

0

00

, ζ20 =

a

00

,


Figure 2. A special selection of the coordinate system.

P10 = |v10||v10| = ν10

ω10and P20 = |v20|

|v20| = ν20

ω20.

Hence, the physical properties ofτ21 given in Equation (13) take the following explicit expres-sions:

(v21)60 = − 0

ω20 sinαω20 cosα + ω10

,

|v21| =√ω2

20 + 2ω10ω20 cosα + ω210,

(ζ⊥21)60 = 1

|v21|2

(ω2

20 + ω10ω20 cosα)a + (ω20ν10 − ω10ν20) sinα00

and

ν21 = ν20ω20 + ν10ω10 + (ω20ν10 + ω10ν20) cosα + aω10ω20 sinα

|ω21| . (14)

4. Generation of Contact Surfaces of Three-link Spatial Cam Mechanisms

In this section we derive general expressions for the point-coordinates of the contact surfaces ofboth the cam and the follower of a three-link spatial cam mechanism. These contact surfaces areruled surfaces generated by the sweeping action of I21 onto each of the cam and the follower.

4.1. Transformation of Line Coordinates

The usual description of a line in space is the Plucker representation which may be given bya pair of column matrices L= (k; k0), wherek andk0, respectively, represent the directioncosine vector of the line and its moment vector, moment being taken with respect to the originof 60. However, since in our case the aim is to plot the ruled surfaces generated by I21, thenwe are interested in relating the initial direction of I21 and the position of a point on I21 to its


final direction and position of that point. Note that the calculation of the moment is unnecessary.Therefore, we will use a relation different from that used in [15] or [22]. Hence, if we describeI21 with its unit direction and the position vector of the point on it nearest to the origin, i.e.(I21)60 = (u21; ζ⊥

21)60, then this relation can be obtained as follows u∗

i

ζ ∗i

1

6(0)i

=[

0T(0)i 00 0H

(0)i

]−1 u21

ζ⊥211

60

(15)

with i = 1, 2. Equation (15) transforms the line coordinates of I21 from60 to6(0)i , where6(0)

i

is the zero time (initial) position of the coordinate system6i , and0H(0)i represents the position

(both the orientation and the location) of6(0)i with respect to60. Similarly, we have

uiζi1

6(0)i

=[

Ti (t) 00 Hi (t)

] u∗

i

ζ ∗i

1

6(0)i

. (16)

Equation (16) relates the unit direction of I21 and the position vector of a point on it before themotion just started, i.e. at zero time,(u∗

i ; ζ ∗i )6(0)i

, to its direction and the position vector of that

point after the motion, i.e. at any time,(ui; ζi)6(0)i , both expressed in the same fixed coordinate

system6(0)i . Note thatHi (t) defines the position (both the orientation and the location) of6i

with respect to its zero time position6(0)i .

4.2. Cam-Follower Contact Surfaces

As mentioned earlier, three-link spatial cam mechanism with I21 being the contact line betweenthe cam and the follower is a mechanism with minimum power losses of this higher pair. Hence,in the design of such cam mechanism, the surface of the camR1 and that of the followerR2

must be generated by the sweeping action of I21 onto each of the two bodies. However, since themotion ofR2 relative toR1 is opposite to that ofR1 relative toR2, then the sweeping action ofI21 to generate the surface ofRi(i = 1, 2), which is assumed stationary, is performed by rigidlydisplacing I21 about the shaft ofRi in the opposite direction of the motion both in rotation andtranslation. This can mathematically be expressed by the use of Equation (16) as

u∗∗i

ζ ∗∗i

1

6(0)i

=[

Ti (t) 00 Hi (t)

]−1 u∗

i

ζ ∗i

1

6(0)i

. (17)

Hence, the point coordinates of the cam’s and the follower’s ruled surfaces described, respec-tively, in6(0)

1 and6(0)2 are given by

(ri(t, µ))6(0)i= ζ ∗∗

1 + µu∗∗1 (18)

and

(r2(t, µ))6(0)2= ζ ∗∗

2 + µu∗∗2 , (19)

whereµ is a real parameter represents the used width of the contacting surfaces.


5. Revolute–Higher-Revolute Cam Mechanisms

A cam mechanism in which both the input and the output are of revolute pairs and with thecontact line coinciding with I21 is called a revolute–higher-revolute (RHR) mechanism [9]. Letthe input–output function of the mechanism be expressed as

φ = φ(ψ)

whereω10 = dψ/dt andω20 = dφ/dt . Then for such an input–output pairs we haveν10 =0, ν20 = 0, andω20 = φ′ω10. Whereφ′ = dφ/dψ . Let us assume that the layout of a (RHR)mechanism is as in Figure 2 with I10 being the shaft of the cam, then from Equation (14) weobtain

(u21)60 = v21

|ω21| = −1

η

0

φ′ sinαφ′ cosα + 1

, (20)

(ζ⊥21)60 = a

η2

(φ′)2 + φ′ cosα

00

, (21)

ν21 = ω10aφ′ sinα

η, (22)

whereη =√(φ′)2 + 2φ′ cosα + 1. Again, based on Figure 2, we have

0H(0)1 = [I ], 0H

(0)2 =

1 0 0 a

0 cosα sinα 00 − sinα cosα 00 0 0 1

, H1 (t) =

cosψ − sinψ 0 0sinψ cosψ 0 0

0 0 1 00 0 0 1

and

H2 (t) =

cosφ sinφ 0 0− sinφ cosφ 0 0

0 0 1 00 0 0 1

.

Hence, by making use of Equations (15)–(21), the point coordinates of the cam and the followersurfaces are given, respectively, as

(r1(ψ,µ))6(0)1= a((φ′)2 + φ′ cosα)

η2

cosψ

− sinψ0

− µ

η

φ′ sinψ sinαφ′ cosψ sinαφ′ cosα + 1

, (23)

(r2(ψ,µ))6(0)2= a

((φ′)2 + φ′ cosα

η2− 1

) cosφ

sinφ0

− µ

η

sinφ sinα

− cosφ sinαφ′ cosα

. (24)

Note that the surfaces generated by Equations (23) and (24) may correspond to a spatial, sphericalor planar cam mechanism depending upon the values selected fora andα. Thus, we may


Figure 3. A planar indexing cam mechanism with two indexing steps (n = 2).

classify the cam mechanism as: Spatial:(a 6= 0, α 6= 0), Spherical:(a = 0, α 6= 0), Planar:(a 6= 0, α = 0). It is evident from Equation (22) that a pure rolling is achieved in the cases ofspherical and planar cam mechanism. All of these conclusions are in agreement with the resultsalready established in [9].

6. Numerical Examples

In this section we give two numerical examples: In the first example we consider the generationof the contact surfaces for three-link indexing cam mechanisms (see Appendix), while in thesecond example generation of pitch surfaces of gears with coplanar axes [23] is given. All thesurfaces are plotted by using the software Maple-V (release 3). Note that in plotting the indexingcam mechanisms for the selected indexing stepn, the samen number of different positions ofthe cam are indicated on the same figure.

Example 1: In this example we generate the contact surfaces of a planar, spherical andspatial three-link indexing cam mechanisms. Cycloidal motion is used to generate thesecontact surfaces. Figure 3 shows a planar indexing cam mechanism with two indexing steps(n = 2) where we have takenµ = 0.5 to 1.0,a = 1 (unity),1ψ = 180◦, α = 0. Fig-ure 4 indicates a spherical indexing cam mechanism with three indexing steps (n = 3) whereµ = 0.5 to 1.0,a = 0,1ψ = 120◦, a = 60◦. Finally, in Figure 5 a spatial indexing cam mech-anism of four indexing steps (n = 4) with µ = 0.7 to 1.0, a = 1 (unity),1ψ = 90◦, a = 45◦

is given.

Example 2:Here we show that the results obtained in Equations (23) and (24) can also be appliedto generate pitch surfaces for gears with coplanar axes, namely, bevel gears and spur or helicalgears. The input–output relation for such gear pairs is given by

φ(ψ) = mGψ,

wheremG = N2/N1 is the gear ratio, in which Ni (i = 1, 2) is the number of teeth on geari.Figure 6 shows the pitch surfaces of a pair of bevel gears withmG = 2, µ = 0.6 to 1.0, a =0, a = 60◦ while in Figure 7 the pitch surfaces of an internal pair of spur gears withmG =2, µ = 0.6 to 1.0, a = 1 (unity),a = 180◦ is given.


Figure 4. A spherical indexing cam mechanism with three indexing steps (n = 3).

Figure 5. A spatial indexing cam mechanism with four indexing steps (n = 4).


Figure 6. Pitch surfaces of a pair of bevel gears withmG = 2 and 60◦ shaft angle.

Figure 7. Pitch surfaces of an internal pair of spur gears withmG = 2.


7. Conclusions

In this paper a complete formulation procedure for describing the instantaneous screw motionof two arbitary moving rigid bodies, observed from a fixed reference frame, is presented. Theformulation is simple and based on the ordinary linear algebra which results in compact expres-sions useful to implement on a digital computer. The results so obtained are particularly used togenerate the contact surfaces of three-link spatial, spherical and planar cam mechanisms. Thecontact surfaces are ruled surfaces generated by the sweeping action of the relative instantaneousscrew axis for the cam and the follower. The sweeping action is performed by using 7× 7 linecoordinates transformation matrix to bypass the calculation of the moment. In the examplesection, various 3D plots are presented to demonstrate the approach.

Appendix A: Input–Output Function of Indexing Cam Mechanisms

In the indexing mechanisms used for intermittent motion a full rotation of the cam is dividedinto two intervals of lengths1ψ and 2π − 1ψ . In the second interval, the follower angleφremains constant; however, in the first intervalφ is assumed to change according to the relation

φ(ψ) = 2π

nε

(ψ

1ψ

)

whereε = ε(x), (ψ/1ψ), is a normal input–output function with the property 06 ε6 1,06 x 6 1, andn is the number of indexing steps for one full rotation of the follower. Forthe cam-follower mechanisms used in the example of this paperε(x) is taken as the cycloidalfunction [9].

References

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15. Bottema, O. and B. Roth, B.,Theoretical Kinematics, North-Holland, Amsterdam 1979.16. Frazer, R., Duncan, W.J. and Collar, A.R.,Elementary matrices, Cambridge University Press, 1963.17. Argyris, J., ‘An excursion into rotations, computer methods in applied mechanics and engineering’,

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on the kinematics of three-link spatial cam mechanisms · 2013. 6. 4. · on the kinematics of...

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