\(
\DeclareMathOperator*{\oiint}{{\normalsize{\subset\!\supset}}
\llap{\iint}} \)The Klein
Bottle
as the Configuration Space of a Mechanism
Consider a mechanism with a Klein bottle as its configuration space
(c-space).
Klein bottle
As I'm thinking about a sort-of 3-link, 2D robot arm mechanism that
would have a Klein bottle as its c-space, Link 1 should be ground. Link
2 should just be an extruded long and slender ellipse that is a crank.
Link 3 should be a square with corners labeled labeled A (upper left),
B (lower left), C (lower right), and D (upper right) CCW. Link 3 should
start rotating around a pin joint through point D for
270\(^{\circ}\) CCW, so that we now have the configuration A (upper
left), D
(lower left), C (lower right), and B (upper right). Then the pin joint
of Link 3 should be able to slide on up to the top right corner A, then
to the right to B, so the B is on the top in the upper right corner of
the square with configuration A (upper left), D (lower left), C (lower
right), and B (upper right) still and B as the new rotation point. It
is important that the pin joint at D cannot slide until the mechanism
has rotated 270\(^{\circ}\) CCW and that the pin joint at D cannot
rotate
as it is sliding, for else the mechanism would have mobility M = 3 and
could not have a 2D Klein bottle as its c-space but would instead have
to have a 3-manifold as its c-space. Any further motion of Link 3
repeats this pattern (B rotates 270\(^{\circ}\) CCW, then slides to C
and to D, D rotates 270\(^{\circ}\) CCW, then slides to A and to B,
etc.). This should force motion for Link 3 to recreate
the non-orientable loop in the above graphic.