[Psychology] Mechanical Assemblies Phần 2 ppsx

38 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES

FIGURE 3-5. Schematic Diagram of Matrix Transforms Applied to the Stapler. Left: The parts of the stapler have been

replaced by blobs. Right: Straight-line arrows have been added to relate frames on the same part. Curved arrows have been

added linking the coordinate frames of assembly features on different parts to indicate which ones are to be joined in order to

assemble the parts. Double curved lines indicate the KCs that were identified in Chapter 1.

FIGURE 3-6. Schematic Representation of a Transform.

The transform T contains a translational part represented by

vector p and a rotational part represented by matrix R. Vec￾tor p is expressed in the coordinates of frame 1. Matrix R

rotates frame 1 into frame 2.

vectors are assumed to be column vectors, so a transposed

vector is a row vector.) On a component-by-component

basis, transform T is

where vector p is expressed in the coordinates of the orig￾inal frame and r,;

are the direction cosines of axis i in

frame 1 to axis j in frame 2.

Transform T can be used to calculate the coordinates

of a point in the second coordinate frame in terms of

the first coordinate frame. The coordinates of a point are

given by

3.C. MATRIX TRANSFORMATIONS 39

Then, in general, if q is a vector in the second frame,

its coordinates in the first frame are given by q'\

This says that q' is obtained by rotating q by R and

then adding p.

Suppose a transform T consists only of matrix R, and

suppose that we want to find the coordinates of the end of

a unit vector along the z axis of the rotated second frame

in terms of the unrotated first frame. The calculation is

This result shows that the columns of matrix R tell

where the coordinate axes have rotated. That is, the first

column tells where the x axis went, and so on. The ele￾ments of each column are the cosines, respectively, of the

x,y, and z components of the new axis expressed in the

original frame.

Matrix R can be generated a number of ways. One way

is to rotate once about each coordinate axis. This will gen￾erate one elemental rotation matrix. Matrix R can then be

created by multiplying the elemental matrices into one an￾other. The elemental matrices, as discussed in [Paul], are

The order in which T's and R's are multiplied is impor￾tant, and different sequences will create different results.

For example,

rotates vector u into a new orientation w by first rotating

90° about the z axis in the frame in which u is measured,

then 90° about _y in the same frame. However,

rotates vector u into a new orientation w' by first rotating

about the y axis and then about the z axis. Equation (3-9)

can also be interpreted as saying, Rotate u 90° about its

original y axis, then 90° about its new z axis. Similarly,

Equation (3-10) can be interpreted as saying: first rotate

u 90° about its original z axis and then rotate it 90° about

its new y axis.

A transform that simply repositions a frame without

reorienting it is

A transform T that comprises a translation px along x

followed by a rotation of 90° about the new (translated) z

could then be written

We can also compute the inverse of a transform. In

words, the inverse of T should undo what T did. If

or, equivalently, if

then

The transform in Equation (3-16) is the inverse of the

transform in Equation (3-15). Embedded in these relation￾ships is the fact that, for rotation matrices,

3.C.2.b. Examples

Here are some examples that illustrate the rules for using

trans and rot, including the effects of doing so in different

sequences.

then

40 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES

Equation (3-18) reminds us of the rule regarding se￾quence of application of a transform. It contains the trans￾forms that we will use in the examples here.

<— use original axes

trans(px,Q,Q)rvt(z,90) (3-18)

use new axes

We will compare this combined transform with one that

contains the same matrices but does something completely

different:

We will calculate the effects in both cases, applying the

transforms from left to right and from right to left. First,

Equation (3-18) is expanded in Equation (3-20). The ac￾tions are performed in both sequences in Figure 3-7. It is

seen that both sequences result in the same new frame.

FIGURE 3-7. Illustration of Two Ways of Interpreting

Equation (3-20). Left: Performing the operations from right

to left requires that the original XYZ axes be used through￾out the action. Hence, we first rotate 90° about Z and then

translate a distance px along the original X axis. Right: Per￾forming the operations from left to right requires that the new

axes be used throughout the action. (For the first operation,

new and original have the same orientation.) Hence, we first

translate a distance px along the original/new X axis and then

rotate 90° about the new (translated) frame's Z axis.

Second, we will perform the actions of Equation (3-19)

in both sequences. This is illustrated in Figure 3-8. First,

Equation (3-19) is expanded in Equation (3-21). Again,

we see that the same final frame results. Of course, it is

different from the frame that results from the operations

in Equation (3-20).

= trans(0,px,Q)rvt(z,90)

3.C.2.C. Composition of Transforms

The main use of transforms is to permit chaining a series

of them together so that we can locate a distant frame by

means of several intermediate frames. This is done merely

by multiplying one transform by another, as shown in

Figure 3-9.

The following forms are equivalent:

The first thing to notice about the matrix in the fifth

equation is that it follows the form of the general trans￾form: a rotation matrix in the upper left, a position vector

at the right, and a row of three zeroes and a one along

the bottom. Thus the composition of two transforms is

another transform. This means that we can continue to

chain transforms in this way, obtaining another transform

each time. The second thing to notice is that we can say

3.C. MATRIX TRANSFORMATIONS 41

FIGURE 3-8. Illustrating Three Ways to Interpret Equation (3-21). Left: Performing the operations (rotz, 90) trans(px,0,0)

right to left requires using the original axes, including honoring the location of the origin when performing the rotation

about (original) Z. Middle: Performing the operations frans(0, px,0)rot(z, 90) left to right requires using the new axes, again

including honoring the location of the origin when performing the rotation about (new) Z. Right: Performing operations

trans(0, pXlO)rot(z,90) right to left requires rotating first 90° about Z and then translating a distance px along (original) Y.

These and other interpretations of Equation (3-21) give the same result.

FIGURE 3-9. Illustrating the

Composition of Two Trans￾forms. 7~12 locates frame 2 in

frame 1 coordinates. IQI locates

frame 1 in frame 0 coordinates.

7"o2 locates frame 2 in frame 0

coordinates.

Example rotation transform function Rz = rotz(theta)

% creates rotation matrix about axis Z

% input in radians

ct = cos(theta)

st = sin(theta)

Rz = [ct -st 0 0; st ct 0 0; 0 0 1 0; 0 0 0 1]

Conversion from degrees

to radians

Translation transform

function degtorad = dtr(theta)

% converts degrees to radians

degtorad = theta*pi/180

function Tr = trans(;c, y, z)

% creates translation matrix

Note: Function Rz is an example of a rotation operation. Similar functions for rotat￾ing about the other axes are easy to write using Equation (3-6) and Equation (3-8).

in words what the composite transform does: It translates

along PQI, then rotates by 7?oi, then translates along p\i,

and finally rotates again about R\2. The third thing to

notice is that the composite transform 7o2 accomplishes

in one leap what TQ\ followed by T\2 do one step at a time.

When we write a transform, say TQ\, we are able to

convert any vector expressed in frame 1 coordinates into

frame 0 coordinates. We can also convert any transform ex￾pressed in frame 1 coordinates so that its effect appears in

frame 0 coordinates. Such a transform might be called T\I.

If frame 2 is rotated in some complex way from frame 0,

it may be easier to express the effect (a translation or a

rotation) that we want in frame 2 coordinates and then

calculate the effect in frame 0 coordinates by writing

The order in which we multiply transforms is impor￾tant. If T\ and TI are transforms, then

This fact is used in constructing Equation (3-23), which is

the basic equation of matrix transforms, as well as in the

examples in Equation (3-20) and Equation (3-21). When

we multiply a transform TQ\ from the right by another

transform T\2, we use TQ\ as the base, effectively adding a

coordinate frame T\2 to a chain of frames that begins at the

left end of the chain with a base frame whose transform

is /, the identity transform.

Table 3-1 gives some useful MATLAB4

functions for

working with transforms.

If we are careful about how we choose the subscripts of

transforms, we can easily read them as a recipe for walking

from frame to frame: 7};

takes us from frame / to frame j.

When we compose two transforms, as in T^ = 7}^ Tkj, we

can say that subscript k is "used up" when 7}* and T^ are

chained together to form 7}y. This means that frame k no

longer needs to be represented explicitly because its effect

has been absorbed in T(j. Tfj then carries us directly from

frame / to frame j. Careful subscripting is very important

in debugging complex chains of frames, especially when

they are used for variation analysis.

4

MATLAB is a trademark of The Math Works, Inc.

TABLE 3-1. Three Useful MATLAB Functions

for Operating on Transforms

42 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES

We can also express small changes in a transform using

a transform. This is highly convenient because it means

that we can use the same mathematics to express both the

nominal location and the varied location of a frame, and

hence of a part or a feature on a part. This is how we will

perform variation analyses in Chapter 6.

The kinds of variations that we can express this way

are errors in rotation or translation, that is, errors in R or

in p. These may be written as follows:

FIGURE 3-10. Properties of the Error Transform. If DT

is an error in 7", then the erroneous T' is expressed as

r = T DT.

The upper left 3x 3 submatrix 8R is a differential ro￾tation matrix. Its elements correspond to a small error in

rotation of 80X about *, 89y about y, and 80Z about z. The

vector dp contains small differential translations dx, dy,

and dz. We may write the differential rotation matrix as

shown because, if the rotations are small enough, we may

consider them to be in the form of a vector like a rotation

rate vector, and the order in which they are accomplished

does not matter.5

The properties of the differential transform are illus￾trated in Figure 3-10. If there is an error DT in a transform

T, then the varied transform is expressed as

Next, we will show how to use chains of transforms to

represent assemblies of parts joined by features.

3.D. ASSEMBLY FEATURES AND FEATURE-BASED DESIGN

This section takes up the topic of features in assembly.

First we give some history, then we define manufacturing

features and assembly features, and finally we show how

to use transforms to locate features on parts and chain parts

together via feature frames to create a connective assem￾bly model. This will equip us to use the same mathematical

framework to model assemblies linked by features having

either nominal or varied locations.

5To prove this, form rotation matrices mt(x, 89X), rot(y, 89y), and

mt(z, 89Z), multiply them together, substitute 89 for sin 80 and 1 for

cos 89, and eliminate all terms in powers of 9 above 1.

Examples that use the methods in this section are given

in Section 3.E.4.a.

3.C.3. Variation Transforms

Here, again, the order is important. We accomplish

transform T and then we apply the error DT. If the error

occurs before transform T is applied, that is, if it occurs

in the untransformed frame, then

For completeness, we introduce the equivalent notation

where

Multiplying these together creates the error trans￾form DT: