Simple Rack and Pinion Zero Backlash (no springs)

[deemoss]

Mechanically there's no significant difference between this and a split pinion apart from the fact that you get a "gearing up" of the spring force due to a) the wide angle of the belt between the main gears and the tensioner and b) the ratio of the belt diameter to the gear diameter. This results in not needing such a beefy spring. However you can also vary the preload tension more easily than with a split pinion as the spring is external. It's rather like a spilt pinion with the pinions moved apart on the driven gear

It wouldn't be difficult to calculate the equivalent torsional preload effect at the driven gears. Yes, the green pulley removes the backlash between the big gears and the red one tensions the motor pulley.

Actually you don't need to calculate the tension force. This is because the tooth-to-tooth force is not going through a spring. The belt tensioning force need only be such that facilitate the "touching" of the teeth of the respective pinions in each direction.