Last revised 12 February 2009.

At the heart of all guidance and navigation computations by the Apollo
Guidance Computers (AGC) onboard the CSM and LM was the State Vector
(SV): a set of six numbers giving the spacecraft's location and
velocity in an appropriate inertial, orthogonal coordinate
system. In essence, an inertial coordinate system has three axes
with each pointing in a fixed direction relative to the stars. In an
orthogonal system the three axes are mutually perpendicular.

From time to time, Houston could update the onboard SV using tracking
and other available date; but, generally, it was the AGC's job to
update the SV at about two second intervals During coasting
flight, with the engines off, the AGC used the Coasting Integration
Routine (Section
5.2 in the GSOP) to update the SV. Although Earth was the
dominant gravitational influence in earth orbit and, likewise the Moon
in lunar orbit, the effects of both had to be included to update the SV
with sufficient accuracy. The spacecraft orbit was similar to a
Keplerian ellipse, but not quite identical. The Coasting
Integration Routine made use of a venerable, 19th-century perturbation
technique called Encke's Method.

During powered flight, the Average-G Routine was used in solving the
equations
of motion using (1)integrations of three-component dynamic
acceleration provided by IMU Pulsed Integrating Pendulous
Accelerometers (PIPAs) and (2) three-component values of gravitational
acceleration.. A detailed description of the AGC can be found at Ron Burkey's
comprehensive AGC site, especially the Guidance System Operations
Plan (GSOP). A detailed discussion of the Average-G Routine used
to update the SV during powered flight can be found in GSOP
section 5.3.2.

Here is a brief description of the routine.

Let's imagine that State Vector was last updated to time, **t**,
and had a three-component spacecraft location vector
denoted **r(t)**
and a velocity vector, **v(t)**.
The underscores indicate that **r(t)** and **v(t)** are vector
quantities. In addition, the computer had a value of
gravitational acceleration vector, **g(t)**, calculated at the
spacecraft location, **r(t)**.

Since the last update, a time interval Dt has passed and the PIPAs have
accumulated a dynamic, vector velocity change, **Dv(Dt)**. Note that this
velocity change does not include any effects of gravity, because the
accelerometers and the spacecraft experience virtually the same
gravitational pull. To
begin the update, the Average-G Routine updates the location vector

The terms enclosed in square brackets can be thought of as the velocity updated halfway, to the time

So far, the routine has been using the gravitational acceleration it had a time,

Next, the velocity vector is updated

where
the gravity term uses the average of the values at the end of the
interval, **g(t+****Dt****), **and the value at
the beginning, **g(t****)** . Hence the
name of the routine.