# What coordinate system was used for the Apollo missions?

This is specifically about the Apollo missions, which are not touched in this related question.

Did they use one coordinate system throughout the entire mission, or did they switch depending on the phase (Earth, Earth-Moon, Moon)?

And in the context of Earth-Moon navigation, how did they deal with the fact that the moon is moving, and orbiting the Earth at an angle?

• They may have used several coordinate systems together. The computer could transform coordinates from one system to another. Distance to Earth as well as to the Moon would be important to know. – Uwe Nov 23 '18 at 11:11
• The CSM AGC was using earth-centric coordinates, the LM AGC used lunar-centric coordinates. They had to convert the latter to the former on Apollo 13, when powering up the CSM again. I am not certain which exact sets of coordinates were used. I am not clear what the last sentence is supposed to mean? Both cartesian orbital state vectors and keplerian elements can be converted between two reference frames with simple algebra. – Polygnome Nov 23 '18 at 13:19
• Here is a comprehensive overview over coordinate system used for apollo: ibiblio.org/apollo/Documents/19700076120.pdf – Polygnome Nov 23 '18 at 13:23
• But for the rendezvous maneuver between CSM and LM it might be easier when both AGC's used the same lunar-centric coordinate system. – Uwe Nov 23 '18 at 17:03

The command/service module and lunar module each used a local coordinate system.

• Each spacecraft had a main engine, which by definition was pointed in the $$-X$$ direction, producing thrust in the $$+X$$ direction.
• The right (starboard) side of the spacecraft was defined as the $$+Y$$ direction, with left (port) as $$-Y$$.
• The remaining axis was the $$Z$$ axis. For the CSM, $$-Z$$ was above the astronauts' heads, and $$+Z$$ was beneath their feet. In the LM, the astronauts faced forward in the $$+Z$$ direction, with $$-Z$$ aft.
• This meant that each individual engine (main and RCS) had a fixed position and orientation in the local coordinate system, simplifying the calculations for engine burns.

Pitch, yaw, and roll were relative to the orientation of an astronaut piloting the craft:

• For the CSM, pitch was around the $$Y$$ axis, yaw around the $$Z$$ axis, and roll around the $$X$$ axis.
• For the LM, pitch was around the $$Y$$ axis, yaw around the $$X$$ axis, and roll around the $$Z$$ axis.

The guidance computer could also convert between the local coordinate system and several other coordinate systems:

• A global coordinate system called the "Basic Reference Coordinate system", that could be centered on either the Earth or the moon. Most calculations used this as an intermediate coordinate system. The matrix used to convert between this global coordinate system and the vehicle's local coordinate system was called REFSMMAT. The motion of the Earth and moon is accounted for by entering the liftoff date and time using Verb 70.

The Basic Reference Coordinate system (BRC) is an orthogonal inertial coordinate system whose origin is located either at the earth or moon center of mass (Figure 1). The orientation of this coordinate system is defined by the line of intersection of the mean earth equatorial plane and the mean ecliptic at the beginning of the Besselian year which starts January 1, 1971. The X axis ($$u_{XI}$$) is defined by the inter­section of the earth's equatorial plane and the ecliptic in the direction of the ascending node. The Z axis ($$u_{ZI}$$) is along the mean earth north pole, and Y axis ($$u_{YI}$$) com­pletes the triad.

• The guidance computer could translate from eight additional coordinate systems: Earth launch site, passive thermal control (the "barbecue roll"), preferred lunar orbital insertion, lunar landing site, preferred lunar orbital plane change, LM ascent, preferred trans-Earth injection, and Earth entry. See pages LB-58 to LB-60 of the above reference (equations included!).

• "The guidance computers of both craft radioed each other in real time with their position and orientation." Was this done by direct digital communication between both guidance computers using bidirectional radio links? – Uwe Dec 2 '18 at 21:18
• @Uwe: Correct. This was particularly important when the CSM was the active vehicle, as the rendezvous radar was on the LM. (see image above) – DrSheldon Dec 2 '18 at 21:30
• The rendezvous radar was on the LM, but it used a radar transponder at the CSM as shown in the figure. The transponder enabled low power long distance radar measurement by replacing r^4 of the radar equation by r^2 for both directions to and from the CSM. – Uwe Dec 2 '18 at 21:47
• @DrSheldon Do you have a reference for the transmission of attitude and position data between the vehicles? I'd like to read more about that - it's news to me. – Organic Marble Dec 2 '18 at 21:52
• @DrSheldon I believe you are incorrect about the transmission of attitude and position data between the vehicles. I'd love to be proved wrong though. This very detailed writeup on lunar rendevous methods makes no mention of it. history.nasa.gov/afj/loressay.html – Organic Marble Dec 2 '18 at 22:03