Good references below, but the situation at hand is both complicated,
and probably temporary.

We are having trouble with an electromechanical system, and we are
trying to hack together a simplistic instrument that will help us
diagnose the true source of a problem.

The system has two main parts, BOTH driven by the same variable-speed motor.

Part A is a Sliding Stage that moves back and forth. It is driven off of
the main motor shaft using a wheel and eccentric rod. (Think "steam
locomotive piston".) This part is working OK.

Part B is driven off of the sliding stage. There is a 4" gear at the
bottom of a shaft that rotates around a vertical axis. The shaft is
actually a lead-screw. Every time the sliding stage moves forward, a
pawl (little spring loaded arm) pushes the gear around a few degrees.
That comprises one "cycle" of the system.  That rotation of the gear
(and leadscrew) will raise a platform a very tiny amount. (50 microns
per cycle. yes, one two-hundredth of a millimeter.) On the backstroke,
the pawl just slides backward, sliding over the gear teeth but not
moving the gear. BTW, this is all occurring at -30 degrees F, which
makes mechanical contact against a rubber mouse ball (say) kinda
unworkable and probably inconsistent.  Rubber gets hard and loses its
grip in that environment.

The core of the problem is that the platform is not rising by a uniform
amount. There is conjecture that sometimes the gear actually moves
-backward-, as though the pawl arm is dragging it back. My goal is to
prove or disprove that conjecture, by measuring the travel distance of
the gear, over many cycles. We -expect- a uniform forward movement, but
any backward movement would cause the problem we see. There may be
periodic elements involved, like the vertical shaft sticking every
revolution. (About 30 cycles.)

SO! Right now, we want a simple method to measure the travel of the
gear. My thought is to rig an optical mouse above the gear, near the
outer edge. For one cycle, the outer edge of the gear moves about an
inch. (well within the resolution range of an optical mouse.) This mouse
is not the UI mouse, but a second one hooked in just to be a measurement
instrument.

If this was a perfect and long-term arrangement, a Resolver sounds like
a fine idea. We are looking for a fast, cheap, easy way to make that
measurement, in a way that can be sampled and logged by a computer. AN
optical mouse offers enough resolution, we have it on hand, all we gotta
do is intercept the data stream.