Generation S Curve speed profile

[#1]

Hi,

I am trying to control a Industrial AC Servo motor using my XE166 device.

The controller interfaces with the servo controller using the PULSE and DIRECTION control.

To achieve a jerk-free motion I have been trying to create a S Curve motion profile (motor speed v/s time).

Calculating instantaneous speed is no problem as I know the distance moved by the motor per pulse, and the pulse duration.

I need to understand how to arrive at a mathematical equation that I could use, that would tell me what should be the nth pulses duration to have the speed profile as an S-Curve.

Since these must be a common requirement in any domain requiring motion control (Robotics, CNC, industrial) there must be some standard reference to do it.

With anticipation

[#2]

Your motion has up to 5 phases, limited by the jerk, max acceleration and max speed.

The 3 phases have the following distance (s) formulae:
s = v * t
s = 1/2 a t²
s = 1/6 j t³

You might remember the first 2 from your physics education in school.

You have a slowdown and a brake phase, which should be symmetrical. You probably can take it from there.