Generalized evaluation for the transmission performance of mechanisms

Ut,ck. Mack.Tk~wyVoL29, No. 4, PlP.60"t--6IS.1994 Copyrisht© IW4Eb,.~ $cim: Lal Prinad in Gnat Ogitain.All ~ 00~-U4X/g¢$6.00+0.00

GENERALIZED EVALUATION FOR THE TRANSMISSION PERFORMANCE OF MECHANISMS MING-JUNE TSAI and HONG-WEN LEE Depaxtment of Mechanical Engineering. National Cheng Ktmg University,Tainan, Taiwan 70101, g.o.c. (Received 15 June 1992"receivedfor publication3 June 1993)

Abstract--l• this paper the conceptof igeneralizodtransmission wrenchgrew (GTWS)that ¢har•cterizes the transmission properties of mechanisms is introduced. Basedon this concept, measurements of the transmimvity and the manipulability are d~nned. Both of tham have the expressions similar to the manipulability index, which is • quantitative me•sure of the closeness of • robot confilluration to sinllularity. The variable lead ~rew transmission mechanism provides an example for illustrating the evaluation of transmission performance.

INTRODUCTION The evaluation of transmission performance plays an important role in the design of mechanisms. It offers valuable criteria for designers to choose better design parameters and to construct mechanisms with high quality of transmission. Transmission performance of a mechanism is a quantitative measure of the effectiveness of power transmission from the input member to the output member. Therefore, overall transmission performance should consider both the transmissivity to the output and the manipulability from the input [I]. For a given mechanism, the measure of overall transmission performance is also associated with the problem of the closeness to singular configurations. It can be assured that the singularity of the mechanism must correspond to the zero measure of transmissivity or manipulability. Traditional transmission criteria, such as transmission angle, pressure angle and transmission factor, all only dealt with the quality of transmission to the output. The transmission angle, first introduced by Air [2], has been widely accepted as the evaluation of transmissivity for planar ,I-bar linkages. The application of transmission angle to the synthesis of planar 4-bar linkages can refer to some works [3-10]. Similarly, the pressure angle [I !-!7] is used to evaluate the transmissivity of cam mechanisms. In spatial cam mechanisms, however, the pressure angle is merely applicable to the cases of point contact [18], in which only pure force is transmitted. In the case of line (curve) or surface contact, there is no suitable definition to evaluate the transmission performance since, at the contact pair, not only force but also torque may be transmitted. And they have received little attentions. Transmission properties of spatial linkages can be clearly illustrated by the transmission wrench screw ('INVS)[19]. Bali's virtual coemcient [20] between the TWS and output velocity screws had been taken as the transmission factor by Yuan et aL [21]. It is generally applicable to evaluate the transmissivity of spatial linkages. Normalizing the transmission factor, a geometrically-based transmission index between the values of 0 and I was developed by Sutherland and Roth [19]. Further, a modified definition of the transmission index is proposed by Tsai and Lee [!]. However, the use of these transmission criteria requires that the TWS of the linkage is uniquely determined. It is very difficult to identify the TWS for some spatial mechanisms such as overconstrained linkages and complex cam mechanisms. These criteria are only useful for those mechanisms whose floating joints (intermediate connecting joints) are possessed of five degrees of freedom, in which one reciprocal screw can be found. In this paper, the reciprocity relationship of a twist screw and a wrench screw is treated as the orthognnal condition in six-dimensional vector space. For spatial linkages with less than five 607

608

Mr~o-Jta~ Ts~ and HoNo-W~ I.~

floating joint screws, a particular TWS is determined using the Gram-Schmidt orthogonal process, and termed as the generalized transmission wrench screw (GTWS). The measurements of transmissivity and manipulability are thus developed. Both of their expressions are similar to the measure of mainpulability for robot manipulators [22-24]. In fact, the measurement of manipulability for mechanisms is based on the same idea as that for robots or manipulators. The process of evaluating the transmissivity and manipulability are illustrated by a variable lead screw transmission mechanism, which is composed of a planar crank-slider linkage and a variable lead screw mechanism (VLSM). The crank-slider linkage is a simple planar mechanism, and can be taken as a good example to illustrate the measure of transmission performance. The VLSM is of a sophisticated design [25-28] that the traditional concept of pressure angle can not readily be applied. It is because that the contact geometry of the lead screw with the mesh element is usually not a point but a line or a space curve. The measurements of transmissivity and manipulability developed in this paper can be successfully used to evaluate the transmission performance of this complex cam mechanism. GENERALIZED TRANSMISSION

WRENCH SCREW

The motion transmission from the input member to the output member in a mechanism is always accomplished either with a higher pair (direct contact of a cam-follower mechanism) or a series of lower-pair joints (a linkage). In simple cam-follower mechanisms, it is usually only a pure force transmitted at the point of contact. The pure force is thus the TWS of the cam mechanism. In spatial linkages, not only force but also torque can be transmitted from one link to another through their connecting joints. As pointed out in Ref. [19], the TWS can be determined only when the floating joints of a spatial linkage exactly have five degrees of freedom. For the linkages with less than five independent floating joint screws, there are infinite number of screws that can be used as TWS. This becomes a static indeterminant problem. The nature of TWS is thus ambiguous. In this section, a particular TWS for the spatial linkages with less than five degrees of freedom in their floating joints is introduced. In general, a screw can represent a twist motion of a rigid body and, dually, a wrench exerted on the rigid body. The instantaneous motion of a rigid body relative to a reference frame can be represented by a twist screw. In Pliicker ray coordinates, a twist screw can be written by a 6 x I vector as $--co$--[L

M

N

P*

Q*

R*] T,

(I)

where oJ is the amplitude of $, and $ is the normalized screw or called the unit screw. [L M N] represents the rotatinal velocity of the rigid body, and [P* Q* R*] represents the translational velocity of a point in the body which is instantaneously coincident with the origin of the reference frame. On the contrary, a wrench screw expressed in Plficker axis coordinates is given by ~--fS=[P'*

Q'*

R'*

L'

M'

N]T,

(2)

where f is the intensity of ~, and S is the normalized or unit wrench screw. [P'* Q'* R'*] represents the resultant moment with respect to the origin ofthe reference frame, and [L' M' N'] represents the force exerted on the rigid body. The transformation of screw coordinates from ray (axis) coordinates to axis (ray) coordinates is called the elliptic polarity of the screw and is one-to-one corresponding [29, 30]. At twist $ is said to be reciprocal to a wrench ~ if and only if their rate of work is vanishing. It leads to the reciprocity condition: ~T~ = LP'* + MQ'* + NR'* + P*L' + Q*M' + R*N' = 0.

(3)

The operation in equation (3) is nothing but the inner product of $ and ~. Therefore, a pair of reciprocal screws composed of a twist and a wrench can be treated as a pair of orthogonal vectors in six-dimensional space. It is important to notice that the orthogonality implies the reciprocal condition, and it is different from that defined by Sugimoto and Duffy [31], which encounters the problems: (a) dimensional inconsistency, (b) dependence on the choice of units and (c) dependence on the choice of the origin of coordinates [32].

Tmo.

609

I~'d'ormL~ of SD

$i

Fig. I. A single-loop linkage whose input twist is $,, output twist is $0, and floating joint screws are $*, gz . . . . . and $,.

Figure I shows a single-degree-of-freedom closed-loop linkage, in which the number of floating joint screws is n (n ~ 5). The input motion of the linkage is a twist about $,, and the output motion is a twist about $0. In general, the floating joint screws denoted as $1, [;2. . . . . and $, are linearly independent, and define an n-order screw system (n-system). The space of this n-system spanned by the floating joint screws is expressed by v = {$1, $2 . . . . $ . }

(4)

O(V) = n.

(5)

and its order is Since the linkages has one degree of freedom, all the joint screws of the linkage must be linearly dependent [33]. Except for special configurations, the input screw or the output screw is always the linearly independence of the floating joint screws. Therefore, the mobile space of the linkage defined by all the joint screws can be given as u = {$,, $,. $2 . . . . . $.}

or

{$0.S,,$2 . . . . . $.}

O(U) = n + 1.

(6)

(7)

Based on screw theory, a wrench space that is reciprocal to the twist space, V, should be of order (6-n). Since the reciprocity relationship has been treated as the orthogonal condition previously, it is adequate to define the reciprocal space as v • = {s,, s, .....

s+_,},

O(V "~) = 6-n,

(s)

(9)

where S,, $2 . . . . . $+., are a basis of V ~ and all reciprocal to the twist screws in V. It is known that the TWS of the linkage is reciprocal to all the floating joint screws, i.e. the TWS is a linear combination of the basis of V ~. If the order of V ~ is I (n = 5), the linkage has only one TWS at that configuration. However, there are also a number of mechanisms whose order O(V x) are more than i (n < 5), e.g. the overconstrained linkages and some spatial cam mechanisms. There exist a total of oo(~') wrench srews in V x, in which the TWS can lie. The TWS can not be uniquely determined due to static indeterminacy. This static redundancy in wrench space is similar to the kinematic redundancy in twist space. In order to determine the GTWS for those mechanisms, we outline the following method. Firstly, we image that a number of (5-n) single-degree-of-freedom joitns are, in serial, hypothetically added to the linkage between $, and $o as shown in Fig. 2. The twist screws of those joints, denoted by $,+z to $s, must be linearly independent. The additional screws should not belong to U, and then they are all linearly independent to the original joint screws of the linkage.

610

Mmo-Jt,~ Ts~J and HOt~-WEN LU

• a*lQ

~

2

, . . . . . . .

.

IIIIIII Fig. 2. The imaginary floating joints and their screws of the linkage.

Assuming the imaginary joints are all inactive in the linkage. This assumption guarantees that the instantaneous motion of each twist about any of additional screws does not affect the instantaneous motion of any twist about the screws in U. It is followed that a wrench transmitted in the linkage should do no work against the imaginary joints. Clearly, a total of oo(s'~ sets of those additional screws can be selected to satisfy this assumption. A reasonable way to choose those additional screws is assuming their coordinates orthogonal to the basis of U. Therefore, the orthogonal space of U is spanned by those additional screws and given by u"

=

{S.+,, S.+2 ..... $,},

O(U ~) = 5-..

(I0)

(I !)

Each screw in U ~ should have no projective component into the screws in U. Moreover, the space defined by the original and additional floating joint screws can be expressed by w = {S, ..... S., S.+, ..... S,},

(12)

o(w) = 5. (13) The GTWS of the linkage is defined as a wrench screw reciprocal to all the floating joint screws, $, to Ss. Since the order of W is 5, the GTWS can be uniquely determined. A screw orthogonal to the basis of U ~ must belong to the space U. [t results that the GTWS is lying in the mobile space U, and orthogonal to $,,$2 . . . . and $,. Using the Gram-Schmidt orthogonal process [3 i, 34], the GTWS of the linkage can be formulated by two ways:

$, $,~$, S~S,

$, S~$, S~S,

$2 $T$~

"'" "'"

$. S,%

$~S~

""

$,*S.

S.S,

$.$:

...

S~S.

(14)

and

$i0 ==

$o S, $**So $,~$, SI$o S~$, $1So

$,~S,

S: "'" $. ST$~ "'" $T$. $~S~ "'" $I$. S~$:

""

$,~$,

(15)

Tmon

l~"rfomuu~ of meOnni~s

611

If the linkage is not in the degenerated special configuration, normalizing ~ and ~a, the same GTW$, denoted as S , is found. The GTWS obtained from the orthogonal process has the meaning of minimum norm solution similar to the pseudo-inverse of the matrix. The minimum norm solution is the most efficient way of force transmission for such a linkage. Similarly, in kinematics, it is also the most efficient motion for a redundant linkage. It is important to note that, if O(V) = 5, the GTWS found by equations (14) or (15) is the only TWS of the linkage, which is geometrical invariant under the coordinate transformation. The linkage is at a stationary configuration when either the input or the output screw are linearly dependent to the floating joint screws [33]. In such cases, either ~,~ or ~ becomes a null screw (a zero vector in six-dimensional space). The null ~, corresponds to the stationary configuration with respect to the output, and the null ~ corresponds to the stationary configuration w.r.t, the input. If the floating joint screws are further linearly dependent, uncertainty configuration [33] of the linkage occurs. The GTWS of the linkage obtained either by equation (14) or (15) becomes to a zero vector. Those special configurations are closely related to the measure of transmission performance, and will be described in next section. THE M E A S U R E OF T R A N S M I S S I O N P E R F O R M A N C E A generalized approach for the measure of transmission performance is developed. The measure of overall transmission performance of a linkage has two parts. One is the measurement of transmissivity to evaluate the effectiveness of power transmission between the GTWS and the output screw. The other is the measurement of manipulability to indicate the effectiveness of power transmission between the input screw and the GTWS. In the following, the measure of transmission performance is accomplished by two ways. One is using the traditional rate of work (or virtual coefficient) method. Another systematic approach is then developed and called the method of Jacobian matrix.

Rate of work method Consider the spatial linkage shown in Fig. I, the condition for the linkage to be movable is all the joint screws are linearly dependent. Therefore, we have ¢o,$~+ ca, $1 + ¢o25~+ ' "

+ co,$, + mo$0 = 0.

(16)

Since the GTWS is orthogonal to all the floating joint screws, the inner products of unit GTWS (S,) with $1, $2 . . . . . and $, must be all zeros. It leads to the relationship between the input and the output speeds: ($TS,~, + ($~S,)to0 = O.

(17)

For convenience, the Bali's virtual coefficient is referred to as the rate of work between the unit GTWS and the output screw, and has been taken as the transmissivity factor (TF) in Ref. [19]. Similarly, the rate of work between unit GTWS and input screw is recognized as the manipulability factor (MF) in Ref. [21]. Consequently, we have TF MF

I$~S,I, = I$Ts, I.

=

(18)

09)

The measure of transmission performance is thus characterized by the GTWS. It is observed that TF is determined by the coefficient of the output speed too, and MF is determined by the coefficient of the input speed to, in equation (17). The rate of work method, which only depends on the joint geometry of the linkage, requires the unit GTWS to be determined. The advantage is able to normalize TF and MF for some linkages [20, 21]. However, for some complex mechanisms such as the spatial cam mechanisms with line (curve) or surface contact, it is difficult to obtain the normalized values of TF and MF. There are no suitable normalized indices for global transmissivity and manipulability to suit all the mechanisms. Moreover, the determination of GTWS is computational intensive and time consuming. As mentioned before, the GTWS is only a particular TWS of the linkage. There is no guarantee that the transmission actually occurs through the GTWS. Since the non-normalized transmission

612

Mmo-Ju~ Ts~ and Ho~3-W~ L~

factors are frame variants and their values may change by changing the unit or dimenuon, the GTWS should be carefully appfied to evaluate the transmission performance for the mechanisms without such concerns. Jacobian matrix method The GTWS of a single-loop linkage can be formulated as ~,, in equation (14) or ~ in equation (15). If the unit GTWS (St) in equation (17) is replaced by ~, and ~ respectively, we can obtain:

det(J~J~)cu~+ det(J~oJ,)cuo= 0,

(20)

det(J~J0)cu, + det(JoTJ0)cuo= 0,

(21)

whereJ~=[$, $1 $2 ... $,] and J0=[$o $1 $2 .-" $ , ] a r e b o t h 6 x ( n + l ) matrices, and called the input Jacobian matrix and the output Jacobian matrix respectively. Since the square matrix JoTJ,is the transpose of J~Jo, the determinants of both them are equal. Therefore, equations (20) and (21) can be combined and further reduced to ~ ¢ o ~

± ~/det(JoTJ0)oJ0= O.

(22)

Equation (22) provides another expression for the relationship between the input and the output speeds of the linkage. The coefficients of the input and the output speeds in equation (22) should be proportional to those in equation (i 7), respectively. Similar to the definition of TF in equation (18), the coefficient of the output speed in equation (22) can be taken as an indicator for transmissivity. The measurement of transmissivity of the linkage then is defined as ~/o= dv/~JoTJ0) •

(23)

The measurement of manipulability, which is proportional to MF in equation (17), can also be defined by the coefficient of the input speed in equation (22) as ~/,= ~

.

(24)

Equation (24) is virtually the similar form of manipulability defined for robot manipulators [22]. As the meaning of manipulability for robots, the scalar value ~/, then provides a quantitative measure of the closeness of the linkage to singularity w.r.t, the output. Likewiscly, the scalar value ~0 is a quantitative measure of the closeness to singularity w.r.t, the input. Notice that the input and output Jacobian matrices are configurational-depcndenc It is concluded that the measure of transmission performance also indicate the degree of how a mechanism is near to a singularity configuration. The scalar values, ~0 and ~/,, are both dimension, unit, and frame variants. They are not in a normalized sense. In order to avoid misleading, it is suggested that each of them should be calculated based on the same frame with the same unit of dimension, and should be used carefully. The Jacobian matrix method is more generalized, and does not need an GTWS explicit. Both of ~/oand ~/, can be easily determined from the Jacobian matrices which are just composed of the joint coordinates of a given linkage. They provide relative indices of transmission performance for the linkage. It is worthy to note that all the special configurations of the linkage should correspond to zero values of r/0 or r/,. Generally speaking, the Jacobian matrix method is more straight forward and efficient to obtain transmission performance of a complex mechanism. EXAMPLE A variable lead screw transmission mechanism is given in Fig. 3, in which the input member is link 2 (crank) and the output is link 6 (variable lead screw). This compound mechanism consists of a planar crank-slider linkage and a VLSM. The crank-slider linkage serves as the input device for the VLSM. The VI~M has been used in some industrial applications, such as the looms [25, 26] and the propulsion of vehicles [27]. In the VLSM, the motion is accomplished through the direct contact of the meshing element (link 5) and the lead screw. To evaluate the transmission performance of the whole mechanism, therefore, is to measure the effectiveness of power transmitted from the crank to the slider, and that from the slider to the lead screw. Both of them are discussed below.

Tnmm~ion perfomum~ of mechaaima

613

Fi8. 3. The variant lead screw transmission mechanism.

The crank-slider linkage

As shown in Fig. 4, the joint screws of the crank-slider linkage in coordinate system {XYZ} can be written as $,=[0

0

i

0

0

$~=[0

0

I

aSe

$2=[0

0

I

0

0] T

aCe -:

o]"

0] T

$o=[0 o 0 1 0 0]T

(25)

where z = ( b C ~ - aCO). a and b are the link lengths of the linkage, and symbols $0 and C0 stand for sin 0 and cos 0 or, similarly, for other corresponding angles. The unit GTWS of the linkage solved by normalizing ~,a and ~m is

S,--[O 0 - z S ~

C~

-S~

0]T,

(26)

where S~ = S0/t, and ~ = b/a is called the link length ratio. Due to the pitch of S, is zero, the unit GTWS of the linkage is a unit force, F = [C~ - S~ 0]T, as shown in Fig. 4. Therefore, the transmission factor and manipulability factor obtained by equations (18) and (19) are TF = IC~ l,

(27)

MF = I-zS,f I.

(28)

Y

$

Vi1~

Fig. 4. Joint screws of the crank.slider linkage.

614

M~J~

T~u ~ d Ho~G-WI~ LEE

R i.dl~'l i ~ I,Ir~pI I~ -

Z =

Fig. 5. A coaxial PHR linkage.

Moreover, the measurements of transmissivity and manipulability that depend on the joint screws of the linkage are solved by equations (23) and (24) as ~lo = [z + aCOI -- JbC~l,

(29)

tl, = Jaz S0I = [bz S~ J.

(30)

Clearly, the values of t/0 and ~/i are proportional to the TF and MF, respectively, when t/0 is zero, i.e. C ~ = 0, the crank-slider linkage is in a stationary configuration w.r.t, the input. On the other hand, the linkage is in a stationary configuration w.r.t, the output when r/; is zero (S~ ---0 or S0 = 0). According to the definition of pressure angle, the quality of transmissivity is indicated by the angle ~ between F and Vo. Similarly, the quality of manipulability is indicated by the angle between F and V~as given in Fig. 4. The transmissivity index (TI) and manipulability index (MI) respectively obtained by normalizing TF and MF [I] can be written as TI

=Ic~l=Jl ~°,

(31)

I

(32) In equation (31), it is clear that if a larger link length ratio is chosen, a higher tranmissivity performance is obtained. Since the crank is rotating continuously through 360 °, the M[ should vary between 0 and I. The manipulability performance, therefore, is not an important appraisal in such a linkage. It is concluded that manipulability performance is negligible for the planar #-bar linkages with a fully rotating driving crank. The V L S M

It is well known that a lead screw mechanism is generally used to change linear motion to angular motion or vice versa. In kinematics, a lead screw mechanism can be simplified as a coaxial PHR linkage schematically shown in Fig. 5. Suppose that the input linear velocity of the P-joint is ~, and the angular velocity of the R-joint is ~. The instantaneous motion of the H-joint can be represented by a twist screw, gh, whose pitch satisfies dz h = --

(33)

d#~"

In general, if the ratio of the linear velocity and the angular velocity is linear, the pitch of $, is constant. However, for some special applications, it requires that the pitch of $, is variable. In such case, the coaxial PHR linkage becomes the so-called VLSM, and geometric design of the H-joint is very complex. A simple VLSM is given in Fig. 6, in which a conical meshing element (link 5) is added between the slider (link 4) and the cylindrical lead screw (link 6). The conical meshing element is connected to the slider with a revolute joint in order to reduce the friction between them. This rotational freedom does not affect the input-output relation of the VLSM. Therefore, it is reasonable to assume that the meshing element is fixed in the slider. It is necessary for evaluating the generalized transmission performance of the mechanism to identify the joint screws. The joint screws of the simple VLSM in coordinate system {XoYoZo} given in Fig. 6 can be conveniently expressed by $i:[0

0 0

!

0 0]T,

$h--[l

0 0 h

0 0]T,

$o=[1

0 0 0 0 0]T.

(34)

Transmission ix'rfomumce of mechanisms

615

The measurements of transmissivity and manipulability solved by equations (23) and (24) are found to be

no = Ihl,

(35)

~/, = 1.

(36)

The measurement of transmissivity ~/o is proportional to the pitch of Sh. However, it is not reasonable for the manipulability in equation (36) being unit. It is because the input screw of the P-joint is an infinite-pitch screw which is normalized at its dual part. To measure the manipulability, it is necessary to change the normalization of the intermediate twist screw $~. Similar to $,, the twist screw $h normalized at its dual part is given by

0 o,

o o]

(37)

Thus, the measurement of manipulability is modified to be

It follows that the measurement of manipulability is of inverse proportion to the variable pitch of $,. The singularity of the VLSM is corresponding to w/0= 0 or q~ = 0, i.e. the pitch, h, is zero or infinity. When h = 0, the H-joint becomes an R-joint instantaneously. There is no input can be transmitted into the mechanism. When h = ~ , the H-joint becomes a P-joint instantaneously. For any input, there is not output generated. The conical meshing element in Fig. 6 is usually curve-contact with the cylindrical lead screw. There is not suitable definition to evaluate the transmission performance of such case. Using the concept of pressure angle, we can only obtain the transmission index at a specific point of contact. The relative coordinate system and the surface parameters of the meshing element are shown in Fig. 7, in which the fixed coordinate system is {XoYoZo}. The coordinate system {X,Y,Z,} is attached to the cylindrical lead screw, and {XaY2Z2} attached to the conical meshing element. In the fixed coordinate system, the position vector and the unit normal vector at any point on the surface of the conical meshing element are given by p=[(r+6tan~,)C0'+z

(r+6tana)S0'-d

n = [C0'C~,

S0'C~,

(p+~)]t

(39)

- S~]T.

(40)

The absolute velocity of contact point at the meshing element is always the linear velocity along the Xo-axis, and expressed by v,=~[I

0 o]t.

(40

So

Fig. 6. A simple variable lead screw mechanism with a conical meshing clement.

M~c,-JuNz Ts~u and HONC,-WZNI.~

616

Z2 Z I~ °

~ O '

Yo

Y2

Yl

X2

Fig. 7. The coordinate systems and surface parameters of meshing element.

Since the angular velocity of the lead screw is ~ along the Xo-axis, the absolute velocity of the corresponding contact point at the lead screw is found to be V0--~[0

(p+6)

-{(r+6tan~)S0'-d}]

=-z [0 0t + 6 ) h

z

--{(r + 6 tan~)SO'-d}] T.

(42)

As proposed by Chakraborty and Dhande[18], the conditions of conjugate contact can be expressed by

nT(Vs- Vo) =

ESO' + F C e "

+ G = 0,

(43)

where E = (r + 6 tan ~)S~, - (p + 6)C0c, F= h C~, G = - dS..

Assume Vo makes an angle 7 with the Ye-axis, equation (42) can be written as

Vo=~A[0

C~, S~]T

(44)

where

,4 -- ~/(p + 6) 2 + {(r + 6 tan ,,)so' - d} 2, ,= tan_.(.-{(, +6 tan ~)SO'-d}/

(p+6)

Hence, the conjugate condition in equation (44) is reduced to

(hC~)CO" -- (A CyC~)SO' + A $7S~ = O.

(45)

The pressure angle at a given point of contact is the angle, ~, between n and V0. Therefore, the transmission index is given by TI = lC~, = ,(C3,C~)S0' - $3,S~, = I~- C~C0' 1.

(46)

The manipulability index may be evaluated from the angle ¢ between n and VI, and expressed by MI = [C~[ = [C~C0', = [h (CTCr,S0' - $7S~)I.

(47)

T m n

performance of meehani~lns

617

For a specific contact point, the TI and MI can be obtained. It is shown that TI is proportional to h, and MI inversely proportional to h. These results agree well with the generalized measure of the transmission performance. Both of the TI and MI are varied at different locations along the contact curve. It is conscious that, no matter what the contact geometry (a point, a line, or other) of the mechanism is, there is only one global performance for tranmissivity or manipulability at a specific configuration. However, the normalized values of the global transmissivity or manipulability of the VLSM with curve-contact situation can not be easily obtained. The quantifies of transmissivity and manipulability of the PHR linkage can be normalized when the pitch of Sh is constant. Suppose that the lead screw in Fig. 6 is replaced by a trapezoidalthreaded screw with single constant thread, and the conical meshing element is replaced by a spherical meshing element. Thus, the meshing element is of point contact with the lead screw. Figure 8 shows that the trapezoidal-threaded screw is unrolled one turn at that contact point, in which the transmission force (contact force) is F, the lead of the trapezoidal-threaded screw is l, and the distance from the axis of screw to the contact point is p. Since there always exist an inclined angle p in the trapezoidal-threaded screw, it is only the force component, FCp, that can be effectively transmitted at the contact point. Hence, the pressure angle of this VLSM is 0 between FCp and V o. Vo is the tangential velocity of contact point at the trapezoidal-threaded screw. Similarly, the angle 2 between FC# and V~(the linear velocity of the meshing element) can be used to indicate the quality of manipulability. The angle 0 is complementary to the angle 2. The transmission index and manipulability index can be given by TI -- C/~$2,

(48)

MI = COC2.

(49)

From the geometry shown in Fig. 8, the angle 2 satisfies i

h

tan 2 = 2np -- p '

(50)

where h is the constant ratio of the linear velocity to the angular velocity in the trapezoidal-threaded screw. By equation (50), both of TI and MI can be rewritten as TI -

hC#

MX=

pc# + h"

(52)

If the meshing element is surface-contact (for more practical case) with the trapezoidal-threaded screw, p is usually taken as the mean radius. In case of line or curve contact using a cylindrical or conical roller meshing element, p can use the nominal radius of the lead screw, which is usually the average of the inner radius and outer radius. It is concluded that the normalized transmissivity

Axis

~

Fig. 8. Force transmission in trapezoidal-threaded screw.

Vi

2x 13

618

Mf~,-Jtn~ Tx4a and HoN~-W~ L ~

and manipulability indices of a trapezoidal-threaded screw device are dependent on the pitch h, the radius p, and the inclined angle p. For the cases that the radius p is an average value, such as curve or surface contact, the normalized value of transmission performance is only an approximate measurement. CONCLUSION

Traditional evaluation of transmission performance requires the TWS to be identified. For those mechanisms with less than five floating joint screws, the number of transmission wrench screws is infinite. In this paper, a particular transmission wrench screw formulated by the Gram-Schmidt orthogonal process is proposed. It is shown that the evaluation for transmission performance of mechanisms is generalized by the quantitative measure of the closeness of mechanisms to singularity. Using this concept, the transmission performance of the VLSM has been shown strongly dependent on the pitch of a twist screw, i.e. the transmission ratio of linear velocity to angular velocity. Although the generalized measurements of transmissivity and manipulability are not normalized, it provides an efficient way to understand the transmission properties of such mechanisms. Acknowledgements--The authors are thankful to the National Science Council of Republic of China, for supporting this research under Grant NSC 80-0422-E006-06. This project is directed by Drs Hong-Sen Yan and gong-Shen Lee. REFERENCES I. M. J. Tsai and H. W. Lee, ASME 1992 Design Technical Conf, Tempe 13-16 September, De-Vol, 47, pp. 295-303 (1992). Also accepted by ASME J Mech Des. (in press). 2. H. Air, Werkstattstechnik 26, 61 (1932). 3. F. Freudenstein and M. S. Chew. ASME JI Mech. Des. 101, 51 (1979). 4. tl. Funabashi and F. Freudenstein, ASME JI Mech. Des. 101, 20 (1979). 5. K. C. Gupta. ASM£ JI Engng Ind. 99, 360 (1977). 6. A. K. Khare and R. K. Dave, Mech. Mach. Theory 14, 319 (1979). 7. A. K. Khare and R. K. Dave, Mech. Mach. Theory 15, 77 (1980). 8. S. V. Kulkarrnia nd R. A. Khan, Proc. 6th World Congr. Theory Mach. Mech., 15-20 December, New Delhi. India. p. 94 (1983). 9. J. T. Pugh, ASME JI. Mech. Transm. Automn Des. 106, 437 (1984). 10. J. W. H. Sun and K. J. Waldron. Mech. Mach. Theory 16, 385 (1981). II. !. I. Arthovolevskii, Theory of Mechanisms, p. 566. Science Publishers. Moscow (1965). 12. F. Y. Chen, Mechanics and Design of Cam Mechanisms. Pergamon Press, Oxford (1982). 13. F. Dyson, Principle of Mechanism, p. 106 OUP, Oxford (1928). 14. W. G. Green, Theory of Machines, p. 772. Biackie & Son, Glasgow (1955). 15. D. Lent, Analysis and Design of Mechanisms, p. 181. Prentice-Hall, Englewood Clifs, N.J. (1961). 16. H. H. Mabie and C. F. Reinholtz, Mechanisms and Dynamics of Machinery, p. 73. Wiley, New York (1987). 17. J. E. Shigley and J. J. Uicker Jr, Theory of Machines and Mechanisms. McGraw-Hill, New York (1980). 18. J. Chakraborty and S. G. Dhande, Kinematicsand Geometry of Planar and Spatial Cam Mechanisms. Wiley, New York (1977). 19. G. Sutherland and B. Roth, ASME JI Engng Ind. 95, 589 (1973). 20. R. S. Ball, A Treatise on the Theory of Screws, Cambridge University Press (1990). 21. M. S. C. Yuan, F. Freudenstein, and L. S. Woo. ASME JI Engng Ind. 93, 67 (1971). 22. T. Yoshikawa, Ist Int. Syrup. Robotics Res., p. 735 MIT Press, New York (1984). 23. T. Yoshikawa, Int. J. Robot. Res. 4, 3 (1985). 24. M. J. Tsai and Y. H. Chiou, Mech. Mac& Theory 25, 575 (1990). 25. L. Pezzoli, United States Patent No. 4624288, 25 November (1986). 26. G. Genini, United States Patent No. 4052906, !1 October (1977). 27. R. F. Gillie, Mech. Mac& Theory 13, 523 (1978). 28. J. Y. Liu. Ph.D. Dissertation, National Cheng Kung University, Tainan, Taiwan, R.O.C. (1992). 29. H. IApkin and J. Duffy, ASM£ .ll Mech. Transm. Automn Des. 107, 377 (1985). 30. H. Lipkin and J. Daffy, AgM[ J! Mech. Transm. Automn Des. II0, 138 (1988). 31. K. Sugimoto and J. Duffy, Mech. Mac& Theory 17, 73 (1982). 32. J. Duffy. J. Robot. Sys. 7, 139 (1990). 33. K. H. Hunt, Kinematic Geometry of Mechanisms. Clarendon Press, Oxford (1978). 34. D. R. Kerr and D. J. Sanger. Mech. Mac& Theory 24, 87 (1989).