How about sending me a fourth gimbal for Christmas?

Part of a conversation between Mission Control and Apollo 11 :

104:59:27 Garriott: Columbia, Houston. Over.

104:59:34 Collins: Columbia. Go.

104:59:35 Garriott: Columbia, Houston. We noticed you are maneuvering very close to gimbal lock. I suggest you move back away. Over.

104:59:43 Collins: Yeah. I am going around it, doing a CMC Auto maneuver to the Pad values of roll 270, pitch 101, yaw 45.

104:59:52 Garriott: Roger, Columbia. (Long Pause)

105:00:30 Collins: (Faint, joking) How about sending me a fourth gimbal for Christmas.

[Armstrong – “This is Mike at his best. We had a four-gimbal platform on Gemini.”]

I was in a conversation at the Gate Hangs High and was asked to name an Irish mathematician.  I offered Hamilton and his quaternions.

The quaternions are an an extension of our number system beyond the complex numbers, \(\mathbb C\), to \(\mathbb H\).

They are another way of representing spatial rotation and do not suffer from gimbal lock.

A number of commentators ask what is the point of number theory to which the answer is none whatsoever until it is the solution.

In order to understand quaternions, it is helpful to have some understanding of the history and proof of the four square theorem, of which more in the future.

 

A very brief explanation of the above Apollo 11 conversation :

in three-dimensional space an object (say an aircraft or spacecraft) can move in six independent directions; the six degrees of freedom.

There are three translational directions along or parallel to the three orthogonal axes of space along which an object can move forward/back, left/right or up/down :

OldTrout, Going Postal

Then there are three rotational directions around the axes which a pilot would call roll, pitch and yaw.  These correspond to the x, y and z axes respectively.

OldTrout, Going Postal

An object in space is usually moving in some combination of these directions.  Our pilots on here can tell you more than me – I just do the numbers.

The orientation of an object of an object in space can be monitored by a set of three gimbals.

OldTrout, Going Postal

If the object rotates too far in one direction, then two of the three gimbals rings may align in a single plane (gimbal lock) and the object loses a degree of freedom to rotate.  The addition of a fourth gimbal can solve the problem.

The maths with Euler angles is here : Gimbal lock
 

© OldTrout \(2019\)
 

No Audio file – Does not translate well