When I picked out all the hardware during the design phase, I didn’t really explain any of my calculations. Here are a few numbers of interest:
Encoder calculations
I decided I wanted encoders with a minimum resolution of 1000 pulses/rev. With quadrature counting, that gives me 4000 counts/rev. For the pendulum, this makes the resolution 0.09 degrees/count (or 0.00157 radians/count for those that care). For position measurement, the 1.25″ diameter gear on the encoder gives a position resolution of 0.00098 inches/count or 0.0249 mm/count. These resolutions should give adequate feedback even when the speeds are fairly slow.
Motor calculations
IG32P Motor Performance Chart
I started selecting a motor by figuring out what the power limitations were. From the beginning I planned on using the NI 9505 module, which has a limit of 30V and 5A. I decided to use a 24V motor since that is a standard voltage and there are a large number of robot motors available that use 24V. The motor I picked is 24V with a stall current of 5A. The motor (without gearbox) has a maximum speed of 6000 RPM and a stall torque of about 1900 g-cm (0.186 N-m) according to the chart above. There were several gearbox combinations available, so I picked the 19:1 ratio that will give me fairly high stall torque of 3.54 N-m. With an approximate total cart and pendulum mass of 1.4 kg and a moment arm of 1″ (gear radius), the max acceleration of the cart will be 99 m/s^2. However, this drops off quickly with speed. Other factors (cable carrier, support platform) will probably limit the acceleration to 50 m/s^2 or so. With the 2″ gear, the maximum speed is 33 inches/sec (84 cm/sec) and will be fast enough to travel the length of the track in about 1.5 seconds. If the max speed isn’t high enough, I will look into getting the 14:1 gearbox, which would reduce the torque a little bit and increase the speed.