Second order motion profile generator

The generated second order motion profile can be used as a smoothend reference signal (step function) for control systems.

The motion profile consists of three phases of acceleration, constant velocity, and deceleration. The start and end position and velocity and maximum velocity and rate of change of speed while accelerating and decelerating are given to the function. Three unknowns which are the duration of each phase of the motion should satisfy two requirements of the desired distance and the velocity subject to the constraint of total time of the task.