Replace an inductive joystick with a micro controller?

What you probably need is a potentiometer, not variable resistor.

Connect one end to +5, the other to ground and then the moving wiper will go
to your input.

To restrict the range to 1 to 4v, put fixed resistors in the two end
connections - something like GND -- 1k2 fixed --- 5k pot --- 1k2 fixed --  
+5v

Note that micros do not usually have a simple variable output voltage - you
will probably need to use PWM (Google for that). You will probably need some
sort of filtering on that PWM output to avoid confusing the drive controller

HTH
Dave

Here is what I've found out:
Note that the yellow and brown wires both control speed--the voltage on
the yellow wire ranges from 1v-4v when the joystick is moved from all
the way back to all the way forward, and the brown wire also varies
from 4v-1v for the same movement of the joystick. Similarly, the
voltages are mirrored on the blue and white wires for right and left
movement of the joystick. In order to mimic the joystick, mirrored
voltages must be applied to the pairs that control speed and direction.
For example, keeping in mind that 2.5v is neutral, if 2.5v + 1v is
applied to the yellow wire, 2.5v - 1v must be applied to the brown
wire. This is a sanity check for the joystick. If mirrored voltages
aren't present, the controller thinks the joystick is broken and goes
into a fault state.

In addition, the neutral voltage--2.5v--must be present on all wires
for three seconds when the controller is turned on as an additional
check on the state of the joystick.
Permissible control voltages actually vary from a little over 1v to
about 4.7v.

It should be possible to create the control voltages using PWM through
a low-pass filter.

Hope this helps someone else.  Power wheelchairs seem very promising as
robot bases. I found one on eBay for $75.

Dave wrote: