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I've been reading a technical report about inertial navigation.

The image is described as "conventional mechanical gyroscope". It is then explained with "A conventional gyroscope consists of a spinning wheel mounted on two gimbals which allow it to rotate in all three axes".

I don't understand how this is supposed to rotate in all three axes? From my understanding it would need 3 gimbals to rotate in all three axes, am I wrong? (Or is an outer case considered as the third degree of freedom?)

Also how would one be able to measure orientation with the angle pick-offs? Are they rotating with the axis?

Source: https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-696.pdf

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First, to exclude possible misunderstanding, the word 'gimbal' is used not to refer to an axis, but to a plane. In the diagram the two rectangular frames are the gimbals.



With that out of the way:

The gimbal mounting depicted in the diagram is attached to some larger structure, not shown in the diagram. I will refer to that larger structure as 'the floor'.

I will refer to the two gimbals as inner gimbal and outer gimbal
The inner gimbal is what the axis of the gyro wheel is attached to, the outer gimbal is connected - through a pivot point - to the floor.

So in total we have four elements:
the floor
the outer gimbal
the inner gimbal
the gyrowheel

Four elements: that means there are three pivoting connections.

The gyrowheel can spin around any axis relative to the floor because there are three pivoting connections. Three pivoting connections: three degrees of freedom.

But yeah, the fact that the spin axis of the gyrowheel itself counts as one of the degrees of freedom is somewhat counter-intuitive.




[Later edit]

I remember seeing a youtube video of a gimmick onboard a cruise ship. There was a pool table. The gimmick was that that pool table wasn't bolted to the floor; the entire table was suspended in such a way that any rolling and pitching of the ship was counteracted by motorized movement of the suspension. The system was accurate enough so that non-moving balls would remain non-moving, and moving balls would roll in a straight line.

In order to counteract rolling and pitching it was sufficient to measure two degrees of freedom: rolling and pitching.

The floor of the room where the pool table is located is a plane. To keep the table level the system must measure the orientation of that plane relative to the direction of gravitational attraction.



A more general case is gyroscopes onboard a spacecraft. For instance, the spacecrafts for the Moon missions had mechanical gyroscopes, so that the spacecraft could be oriented correctly for, say, an orbital insertion burn.

So you have a gyro wheel spinning, suspended with respect to the spacecraft with two gimbals, and you have two pick-offs.

The spin axis of the gyro wheel keeps pointing in the same direction (if that gyroscope has motorized gimbals that compensate for bearing friction). In order to know the orientation of the spacecraft you need pick-off readings of the orientation of the spacecraft with respect to the gyrowheel. You only need two pick-offs.

The reason that you need only two pick-offs:
Compare the problem of specifying your position on Earth. It is sufficient to give two numbers: longitude and latitude.

In space: to specify where the nose of your spacecraft is pointing to you need only two numbers: the celestial counterpart of longitude and latitude.

So:
The assertion that the two gimbals "allow rotation in all three axes" is quite a murky assertion. I now believe that statement is logically flawed.

Let me go back to the story of the pool table on the cruise ship. On a big ship yaw is negligably slow. But: if the designers of that rolling/pitching counter-moving table would have gone the extra mile they would have included a third counter-moving ability: yaw-compensation.

So there can be circumstances where you do have to measure three axes. But even in the case of the gyroscopes onboard the Moon mission spacecrafts two pick-offs were sufficient for all the space navigation the astronauts needed to do.

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  • $\begingroup$ That makes sense, thank you a lot! How does one measure orientation with these angle pick-offs? $\endgroup$ Oct 25, 2020 at 19:31

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