integrator produces an output. The second

Look at the velocity curve for the time

integrator ceases to produce an output when the

first integrator (velocity) ceases to produce an

obtained when acceleration is integrated over the

output. The velocity integrator continues to

same interval. The velocity curve is the output of

produce until receiving an acceleration that

interval t o to t1, velocity is changing in an

balances out the initial acceleration. At this point,

it produces a net acceleration of zero. The readout

increasing or positive direction. This means that

device stops at the point where the net accelera-

a positive acceleration is taking place. Velocity is

tion is zero. Until reaching this condition, the

constant during interval tl to t2, which means

readout device shows a continuous change in

acceleration is zero. At time t2, velocity begins

to decrease. This says that an acceleration is again

displacement.

The return trip is described as follows. The

taking place. In this case the acceleration is

train is at point B during time interval t3 to t4;

negative. At time t3, both acceleration and

it begins traveling backwards to point A at time

velocity are zero.

t4. The acceleration detector senses an accelera-

tion a2, which is negative and slightly less than

The purpose of an INS is to keep track of

the previous acceleration, a1. At time t5 i t

position and not total distance traveled. To do

reaches a steady velocity, and acceleration goes

this it integrates all values of acceleration (positive

to zero at this point. Note that velocity is now

and negative) detected over the interval. There-

negative since the direction of travel is reversed.

fore, it is the net value of acceleration that

Since the size of acceleration a2 is less than that

interests the INS. For instance, in the interval to

of al, maximum velocity on the return trip is less.

to t3, all accelerations that occur over the

Therefore, the time required to return to point

interval are summed, giving a net value at time

A is greater. Interval to to t7, greater than time

interval to to t3 reflects this fact. The train begins

7-28, view A) is the process of summing the area

to stop within a short distance of point A. This

bounded by the acceleration curve and the time

happens at time t6, producing an acceleration of

axis. The area above the time axis is positive,

a2 as sensed by the acceleration detector. The train

and the area below the time axis is negative. Since

comes to a full stop at time t7. Here the detector

the areas above and below the time axis are equal,

senses zero acceleration. Since the net accelera-

the net value for interval to to t3 acceleration is

tion over the interval is again zero, the output of

zero. The integral of acceleration for the interval

the first integrator (velocity) is zero. The second

to to t3 is therefore zero. This means that the

integrator (displacement) output stops with the

velocity at time t3 is equal to the velocity at time

displacement readout device showing the reference

to, in this case zero.

value. This value is the same originally noted at

Integrating velocity from time tO to t3 i s

reference point A.

The simple single-axis INS just described will

the job of the second integrator. It gives

detect and compute all changes in displacement.

B units of displacement on the displacement axis

However, the acceleration detector (accelerometer)

at time t3. The displacement readout device

must **retain **its straight-line orientation. Also, all

changes continuously as long as the second

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