Brushless Motor Encoder Read

I think if the speed controller was completely off, and you connect it to the battery, it would snap to some position. However, depending on the number of poles in the motor, you might find that it's not just one of six positions, but one of a multiple of six positions. In other words, you could get it to snap into line with one of the phases, but there may be multiple points that represent that phase, and the motor will likely just seek the nearest one.

Of course, if the ESC were to become activated while you had this alternate connection in place, you'd probably smoke at least a couple of the MOSFETs, so, hazard there to worry about. As long as it's off though, I don't think the speed controller would be harmed, b/c if you think about it, those MOSFETS routinely switch the battery voltage to make approximately 3-phase AC out of it. There should be protection diodes in the 6-way bridge that will protect it from the inductive kick-back when you break your alternate connection, too.

How do I get the distance multiplier?

You measure or calculate how far your device will go for e.g. one full rotation, e.g. in cm. Then you take that length and divide it by the number of pulses per revolution the encoder delivers and you got your distance multiplier in cm/pulse.

How do I get initial position?

The encoder itself can only tell you the rotation angle relative to the "index" position. If the motor can do more than a single rotation in either direction there is no way the encoder alone could tell you the absolute position after power on.

A common solution is to have a limit switch, either mechanical, inductive, or optical (IR), at one end of the rail. Upon power up, the controller drives the motor toward the end of the rail with the limit switch until the switch detects arrival at the end. This is then the "0" position and any movement/rotation of the motor is measured via the encoder relative to that fixed position.