Well, how are you going to get around corners? Perhaps the simplest solution,
used by many mice, is a wheelchair drive where a pair of wheels are driven
independently on either side of the mouse. Turn one of them faster than
the other and your mouse will execute a turn. Turns of any radius are
possible and the mouse will be able to turn about its own centre. Bear
in mind that the part of the mouse that lives in the maze below wall height
(5cm) had best not hit the walls when you spin on the spot.

NOTE – about trikes. Sharp turns need inside wheel going backwards. two
motors at rear can sort this out.

Tricycle mice have a single steering wheel which may or may not also
be the driving wheel. If the steering wheel is not also driving then you
will need some kind of differential on the driven wheels if you are to
prevent them from skidding on turns. With a non-driven steering wheel,
you are limited in terms of the minimum turning circle. Clearly, it will
not be possible to turn the steering wheel through 90 degrees and then
start the driving wheels. Turns will have to be entered gently. With a
driven steering wheel, turns can be as small as your mouse footprint.
It is probably better not to try and drive backwards with a tricycle mouse
as the steering problem is much harder even though the turning circles
are smaller. Another problem you will face with the tricycle is the speed
at which you can turn the steering wheel. If you are tempted to use a
standard RC servo, remember that these are not that fast. Even a quick
servo will take about 100ms to turn through 60 degrees. In a competitive
mouse, an entire right angle turn will need to be complete in under 250ms.
I am not sure it will be possible with RC servos. You could make your
own steering servo. Steal the potentiometer, motor and control board from
a servo and make a new gearbox with a more suitable gear ratio. There
are likely to be problems because the control circuits is probably optimised
for the dynamics of the servo as made. You may have settling, overshoot
or even oscillation problems to fix. In either case, where are you going
to get data about distance travelled in a tricycle mouse. A couple of
tricycle mice exist and run very successfully.

There are a number of four (and more) wheel mice about. These are mechanically
complex and require small (expensive) components and good engineering
skills. Why bother? To be honest I am not sure but here are some thoughts.
With a wheel at each corner, steering has both wheels on the same side
electronically or mechanically coupled. Turn the wheels so that their
axles all pass through an imaginary radius drawn through the centre of
a single circle and you have excellent, smooth cornering. Both inside
wheels travel at the same speed as do both outside wheels. Odometry is
a relatively simple problem. Put encoders an all wheels and you can use
ABS and traction control to get the highest performance from you motors.
Reduce the mechanical complexity a little by putting a pair of driving
wheels in the middle for a six-wheel mouse. I expect the driving wheels
would need to be independently mounted so that you can get sufficient
traction without lifting any other wheels off the ground on bumpy circuits.
Just using the front wheels to steer is surprisingly tricky. For that
you need to create or emulate Ackerman steering. The inner wheel needs
to point in a slightly different direction to the outer wheel. When turning,
the rear wheels describe a slightly different arc so you can’t cut your
corners too close. Reversing is a nightmare. For the same reason, steering
with the rear wheels is no fun either. If you have access to the components
and appropriate skills, I expect a well-built four or six wheeled mouse
could be very competitive.

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