Where to Attach the Faces

ensure that the parasitic engagement is out of the range of normal operation of the device.

7.4 Where to Attach the Faces

We describe the pair of faces in a library entry with respect to a local coordinate system included in that entry. To attach the faces to actual components, we must rst transform the coordinates of the faces to match those of the components. Fortunately, we can precompute the necessary coordinate transformations and include them in the library. Hence, this task is performed as part of constructing the library.

We illustrate this task by deriving expressions to transform the coordinates of the faces in Figure 7-2 to match those of an arbitrary pair of components. Our goal is a set of templates (such as Figure 7-15) that show where to attach the faces.

We start by placing the library frame (the coordinate frame of the faces) in the global coordinate frame as shown in Figure 7-11. The origin of the library frame is located at the point (

X

^L;

Y

^L) in the global frame and is inclined by angle L from the horizontal.

(X Y_L, _L)

x ΩL

XL

Y_L

θ

X_G YG

Figure 7-11: The location of the library coordinate frame with respect to the global coordinate frame.

(We use bold serif fonts, like

X

^L, to refer to the origin of a coordinate frame. We use sans serif fonts, like^XR, to refer to an axis of a coordinate frame.)

Next, we place the rotor frame (the coordinate frame of the component to which we attach the rotating face) in the global coordinate frame as shown in Figure 7-12.

The origin of the rotor frame (i.e., the rotor's pivot) is located at the point (

X

^R;

Y

^R) in the global frame. R is the rotation angle of the rotor.

Similarly, we place the slider frame (the coordinate frame of the component to which we attach the translating face) in the global coordinate frame as shown in

(X Y_R, _R)

XR

Y_R

X_G YG

θR

Figure 7-12: The location of the rotor coordinate frame with respect to the global coordinate frame.

XS

XG

Y_S YG

xS

ΩSG

(XSG,Y_SG)

Figure 7-13: The location of the slider coordinate frame with respect to the global coordinate frame.

7.4. WHERE TO ATTACH THE FACES 129 Figure 7-13. The slider frame translates along a xed guide inclined an angle SG

from horizontal. Theguide's origin is located at the point (

X

^SG;

Y

^SG) in the global frame. ^xS is the displacement of the slider along the guide.

The rotating face rotates about the origin of the library frame, and the rotor rotates about the origin of the rotor frame. Because the face is rigidly attached to the rotor, the face and the rotor must rotate about the same point. Hence, the two origins must actually be the same point:

X

^L =

X

^R

Y

^L =

Y

^R

Similarly, the translating face and the slider must translate in the same direction:

L= SG

Figure 7-14: (a) The location of the rotating face in the library frame. (b) The location of the rotating face in the rotor frame.

Figure 7-14a shows a new version of the library frame in which we have included our new knowledge about the location of the origin and the inclination from horizon-tal. The gure also shows the absolute rotation angle, , of the rotating face. We can express this angle in terms of the other angles in the gure:

= SG+

Figure 7-14b shows another view of the rotor frame in which we have rigidly attached the rotating face to the rotor body. The face is attached to the rotor at some unknown angular oset, , which is a free design parameter. The gure also

shows the absolute rotation angle, , of the rotating face. We can derive a new expression for in terms of the angles in the gure:

=R+

If we combinethese two expressions for, we obtain the desired expression relating

R to :

R=+ SG^,

For our convenience, we eliminate in favor of a new parameter off. The new parameter is related to the old one by:

off = SG^,

Using this new parameter, our expression relating R to takes the simple form:

R=+off (7^:36)

Similarly, we eliminate from Figure 7-14b to obtain Figure 7-15. We use this new gure as our template for attaching the rotating face to a rotating body.

XR

YR

-θoff

ΩSG

Figure 7-15: Template indicating where to attach the rotating face to the rotor.

Now we turn our attention to the translating face. Figure 7-16 shows the slider guide, slider, and library frames in the global frame. Our goal is to determine the location of point

P

, the bottom end of the translating face, with respect to the slider frame.

We begin by expressing

P

in the library frame:

P

^L= ^x

h

!

where the superscript \L" indicates that

P

is expressed in the library frame.

Next, we express

P

in the frame of the slider guide:

7.4. WHERE TO ATTACH THE FACES 131

P

^SG =

P

^L+ ^{X Y}

!

where the point (^X, ^Y) is the location of the library frame with respect to slider guide frame. (Here again, the superscript \SG" indicates the coordinates are ex-pressed in the slider guide frame.) We use a standard rotation matrix (rotating by

,SG) to express this location in terms of the absolute locations (i.e., the locations with respect to the global frame) of the slider guide frame and the library frame:

^X

Figure 7-16: The slider frame and the library frame in the global frame.

Because, the slider frame is displaced from the slider guide frame by a distance

xS, we can use our expression for

P

^SG to obtain an expression for

P

in the slider frame:

P

^S =

P

^SG+ ^,x0^S

!

Combining all of the previous expressions we obtain:

This gives us the location of

P

^S in terms of several design parameters (SG,

X

^R,

Y

^R,

X

^SG, and

Y

^SG) and two position parameters: ^x and ^xS. Because the face is rigidly attached to the slider, the position of

P

in the slider frame must be a constant.

Hence, we must be able to eliminate the two position parameters (which are variables) from our expression for

P

^S.

To this end, we note that ^x and ^xS measure the positions of the face and the slider (respectively) with respect to references that are xed in the global frame. Also, because the face is rigidly attached to the slider, if the face undergoes a displacement, the slider must undergo the identical displacement. These two facts combined require that ^xS and ^x dier by at most a constant. Calling that constant ^xoff we obtain:

xS =^x+^xoff (7^:37)

Substituting this expression into our previous expression for

P

^S we obtain the desired result: With this expression, we now know where to attach the face to the slider. We simply attach the face so that its lower end is located at the point

P

^S, as shown in Figure 7-17.

As a second example, we consider the faces in Figure 7-2 for the case in which we measure^x(the position of the translating face) positive to the left rather than to the right. As we discussed previously, dening ^xpositive to the left turns this geometry into an implementation for qcs-curves of type F in Figure 7-1.

In this case the position of the rotor and the position of the rotating face are still related by Equation 7.36, and we still use Figure 7-15 as our template for attaching the face to the rotor. Similarly, the position of the slider and the position of the translating face are still related by Equation 7.37. However, we must derive a new template for attaching the translating face to the slider.

In the previous case, when we measured ^x positive to the right, we placed the library frame and the slider frame into the global frame such that the axes of the library and slider frames pointed in the same direction. (That is, ^XL pointed in the same direction as ^XS, and ^YL pointed in the same direction^YS as shown in Figure 7-16.) However, when we measure^xpositive to the left rather than to the right, we place the two frames in the global frame such that their axes point in opposite directions.

7.4. WHERE TO ATTACH THE FACES 133 Doing this, we derive a new expression for the location of

P

in the slider frame:

P

^S =

"

cos(SG) sin(SG)

,sin(SG) cos(SG)

#

X

^R,

X

^SG

Y

^R,

Y

^SG

!

+ ^,xoff

,h

!

(7^:39) The dierence between this expression and Equation 7.38 is the sign in front of

\^h." We also derive a new template, shown in Figure 7-18 indicating where to attach the face to the slider. In our previous template, the face was located in the rst quadrant of the coordinate system. Here, because the axes of the slider frame are rotated 180 degrees from the previous case, the face is in the third quadrant.

X_S Y_S

P^S w ψ

Figure 7-17: Template indicating where the translating face is attached to the slider:

The lower end of the translating face is attached to the slider at the point

P

^S.

XS

Y_S

P^S ψ ^w

Figure 7-18: A template for attaching the translating face to the slider when the position, ^x, of the translating face is measured positive to the left relative to the library frame.

Dans le document MASSACHUSETTS INSTITUTE OF TECHNOLOGY ARTIFICIAL INTELLIGENCE LABORATORY A.I. Technical Report No.
 March,
 (Page 127-135)