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Re: non-vectors + gimbal lock



At 01:35 PM 12/4/01 -0600, Dario Moreno wrote [off list]:
In fact you may
define all operations that go with vectors and if you care about
paying the cost of a shopping list, we are lead, quite naturally,
to the notion of dual space, linear functions of vectors and internal
product. This is useful when teaching about work, when you have the
product of two vectors that live in different spaces ! Do you agree?

Hmmm. I suppose you could make that work.
I'll have to come up with a better example of
a non-vector. Perhaps a list of phone numbers....

would you please explain what is a "gimbal lock" ?

Ughhh. That's a lot easier to demonstrate given a hands-on
gimbal system. But I'll _try_ to explain it.

A "free gyro" instrument is free to maintain its orientation
because it's on gimbals. As you wiggle the outer frame around,
the spatial relationship between the various gimbal rings has
to change to accomodate you. If you are sufficiently fiendishly
clever, you can maneuver it so that all three rings are coplanar.
Then if you rotate the frame in that plane, the rings can't
accomodate you, and the gyro axis is brutally disturbed.

A similar thing can happen if only two of the rings become
coplanar. This is much easier to provoke. Indeed if you
are doing radical maneuvering with a gimballed gyro, it is
just a matter of time before this happens by accident,
unless the gyro manufacturer has done something very tricky
to prevent it.

It's hard to describe just what the failure looks like. When
I was about ten years old I spent an entire day playing with
such a thing. That was a long time ago. As I (approximately)
recall, key notions include:
-- The three rings, when they are coplanar, have their
axes 90 degrees apart. If the outer ring axis is East-West,
the middle ring is North-South, and the inner ring is East-West
again. Two-ring gimbal lock occurs when the inner ring is
coplanar with the outer ring. That means that there
are two rings with colinear axes. That's fatal, because it
means that there is some plane (usually not the plane of the
rings) that contains all three axes. If you rotate the frame
in that plane, the gyro's axis will be brutally disturbed.
-- The physics of the situation is roughly analogous
to supporting a large weight with your knees straight.
If your knee-joint moves by a first-order amount, the
weight descends only a second-order amount. So it is
with gimbal lock. The two rings _can_ move so as to
accomodate what you are doing to the frame, but if they
move a first-order amount, it only accomodates you by
a second-order amount. So you've got infinite mechanical
disadvantage.

That's not a very good explanation, but it's about the best
I can do without a hands-on demonstration. Grab yourself
a set of gimbals and see for yourself. I hope I've helped
by telling you what to look for.