Kinematic Joints -- Composition of Rotations
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Demonstrations on overhead projector:
1.) Composition of many 2D rotations around the same point: ==>
This is obviously just a single rotation around the same point.
There is just one degree of freedom!
2.) Composition of many arbitrary 2D movements: ==> Can be expressed as a single rotation around some fixed point.
It is easy to see that this can be expressed as a translation
(e.g., on the center of gravity) plus a single rotation (for alignment).
But how do we find the center of a single pure rotation that achieves the same thing?
If we assume that point A gets transformed into point A' by traversing
across some circular arc, then the rotation point must lie on the
bisector of this arc. If this process is repeated for several
point pairs: B, B'; C, C' ... then all the bisector lines should
intersect in a single point: the center of rotation, R.
3.) Composition of many 3D rotations around the same point:
==> Can be expressed as a single rotation around the same point.
see: Euler_rotation_theorem
Demonstrations with two colored paper cones fixed at their tips at the origin.
Easy
to see that (colorless) cone shapes could be brought into coincidence
with a single rotation that aligns the two axis directions.
But how do we get the azimuth directions to line up as well?
==> Swing one cone axis around a conical surface that provides the needed azimuth orientation;
e.g., swinging through a half-cone will change azimuth by 180 degrees.
Consequences:
-- Any rotation-state can be achieved with a single rotation from some given starting state.
-- The specification of a certain orientation using this "axis+angle"
representation is unambiguous and more continuous than using Euler
angles.
-- The "axis+angle" representation is preferred in animation. ==> Quaternions.
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