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ELECTRONIC DEVICE FAILURE ANALYSIS | VOLUME 18 NO. 2

In addition to the temperature effects, there is a toler-

ance on the zero position. This is a result of the switches

used to detect when an axis is at the zero position. These

switches may be mechanical, optical, or magnetic. No

matter which type of switch is used, it will have some

level of uncertainty that results in uncertainty in the zero

position.

The resolution of themachine is a function of howwell

the scale can be measured by the linear encoder. A good

encoder can provide positional resolution of 50 nm, but

the accuracy is a function of the scale.

[5]

A typical optical

scale will have a nonlinearity of 0.005 mm per meter of

travel and a thermal expansion coefficient (

) of themate-

rial towhich it ismounted.

[5]

Steel has a

of 11.5 ppm/°C,

and the

for aluminum is 23 ppm/°C. A 100-mm-long

scale will change 0.0023mm/°C. All of this is additive. The

uncertainty due to scale resolution adds to the nonlinear-

ity, zero position uncertainty, and the thermal variations

of the scale. With a typical ±2° variation in temperature

and a 100-mm-long scale, zero uncertainty of ±0.5 µm,

and the scale nonlinearity, the positional uncertainty is

±5.6 µm. This is 110 times greater than the encoder

resolution.

THE CONCEPT OF UNCERTAINTY

Uncertainty is the ultimate tolerance on any positional

move. The factors that affect uncertainty are repeatability,

accuracy, geometric factors, interaction between the axes

of movement, and environmental effects. The repeatabil-

ity, as shown in Fig. 2, will drift in position resulting from

temperature effects, and the pattern center will move as

a result of the other factors. Repeatability will always be

less than uncertainty.

THE GEOMETRY OF MACHINE

PERFORMANCE

All machines involve linear and rotational movement.

Mostmachines involvemultiple-axismovements. Thegeo-

metric relationship of the axes to each other will directly

affect the position in the other axes as movement takes

place. If the

and

axes are at 89° instead of 90°, a move

in either axis will produce a positional shift of 1.745%

of the move in the other axis. Resolution, accuracy, and

repeatability in one axis are meaningless if the axes are

not truly orthogonal. If the

axis is not normal to the

X-Y

plane, changes in

position produce changes in the tool

tip’s position referenced to the

X-Y

plane.

Determining axis uncertainty requires knowing the

geometric relationships of all three axes and the rotational

axis of the tool. This requires the manufacturer to specify

the orthogonality of the axes to each other and the nor-

mality of the

axis and spindle to the

X-Y

plane (Fig. 3).

Although the geometric variables donot directly affect the

repeatability, the interaction in position from geometric

inaccuracies makes each axis’s real position a function of

the other axes’ positions.

In addition to the geometric problems, the runout of

each axis also must be considered. The linear bearings

or linear positioner used in each axis have runout in the

perpendicular axes. A typical precision-grade profile linear

bearingwill have 2µmrunout in the vertical andhorizontal

axes.

[6]

That is, as the linear bearing ismoved, it canmove

a small amount vertically and horizontally. Even super-

precision bearings have a runout of 1.5 µm. The runout in

one axis adds directly to the positional uncertainty of the

other two axes. The runout also produces rotation about

each axis as a functionof the separationbetween the bear-

ings used. As an example, Fig. 4 shows a single axis using

four linear bearings on twoprofile rails in a square pattern:

• A 2 µmhorizontal and vertical runout in

will produce

±2 µmmovement in the

and

axes.

• The rotationpossible is a result of +2 µmonone bearing

and −2 µm on another. The pitch is then a function of

the bearing spacing. The yaw becomes a function of

the rail spacing.

• The pitch and yaw affect true position in all three axes

according to the geometric relationships and the dis-

tance from the sample to the axis plane.

All of these variables directly add to other sources

of positional uncertainty. Vertical runout on the

axis

bearings adds directly to the uncertainty of the

axis

Fig. 3

Typical

X-Y

movementmechanismwith the

axis stage

removed. There are three bearings instead of four

on each axis, although the runout and orthogonality

effects are the same as in the text.