position, acceleration is a change in the change
reference system as a nonrotating coordinate
of position. However, before any change can have
frame. It can be either stationary or moving
meaning, it must include the unit of time.
linearly at a uniform speed, in which there are no
Therefore, you can define a change per unit of
inherent forces such as gravity.
time as a rate of change. Thus, a rate of change
You can make a simple test of whether you
of displacement is velocity. A rate of change of
are in a true inertial system by releasing an
velocity is acceleration. A rate of change of a rate
object and observing its motion. If you release
of change of displacement is acceleration.
the object without imparting any acceleration to
Differentiation is the process of investigating
it, the object remains in its position relative to
or comparing how one physical property varies
you. If you throw the object, it continues on an
with respect to another. Integration, the reverse
undeviating path at a constant speed. Such a
of differentiation, is the process of summing all
system can exist only in empty space, far from
rate of changes that occur within the limits under
any mass, for all masses contain gravitational
investigation.
forces. A reference system attached to the earth
The inertial navigation system is not a
can closely approximate an inertial system. For
differentiating system; it is an integrating system.
this system to work, you must balance the
However, before integration can be done, it must
gravitational force on a body by a second force.
first have a rate of change. Therefore, the
For example, an object sliding on a flat,
inertial navigation system, when stripped to its
frictionless plane on the earth's surface would
barest essentials, is a detector and an integrator.
move in a NEARLY straight line. The object will
It first detects changes of motion. It then
have a NEARLY constant speed, as you saw in
integrates these changes of motion with time to
the earth's coordinate system.
arrive at velocity, and again with time to arrive
at displacement.
NOTE: The word nearly is emphasized
because the object will deviate slightly from its
Fundamentals of Integration
straight-line motion. The cause of this deviation
is the earth's rotation about its axis.
Since an INS performs integration, the follow-
ing is a review of integrating principles.
Newton's second law of motion shares
The equations for the integrals of acceleration
importance with his first law in the inertial
navigation system because the inertial navigation
and velocity are --
system works on Newton's second law. Newton's
second law of motion states "Acceleration is
proportional to the resultant force and is in the
same direction as this force." Thus, the second
law is written
The physical quality in the above equation that
When acceleration (a) is integrated
over
pertains to the inertial navigation system is
acceleration. You can derive velocity and
a specific period of time (dt), the result is velocity
When velocity (v) is integrated
displacement from acceleration. For example,
over a specific period of time (dt), the result is
consider this fact: Before an object can change
displacement (s). Therefore, when acceleration (a)
its state of rest or state of motion, it must first
is integrated twice
over a specific period of
experience an acceleration. Since acceleration is
time (dt ), the result is displacement (s).
a change in velocity and velocity is a change in
2
