[Ether]

Originally Posted by martinrand

Potentially, I could generate an sigmoid function to represent position

Which particular sigmoid function did you have in mind?

Assuming you meant the Logistic Function:

x(t) = L/(1+exp(-k*(t-t_m))),

where:

t_m = the t-value of the sigmoid's midpoint

L = the curve's maximum value

k = the steepness of the curve

...that would be a poor choice.

To see why, suppose you want to go from x=0 and v=0 and a=0 at t=0 to x=5ft and v=0 and a=0 at t=3seconds.

Then L and t_m would be L=5 t_m=3/2.

The only free parameter left is k; try k=2, k=3, k=4, k=5.

As can be seen from the graphs, there is no acceptable value for k.

k=2: a, v, and x are non-zero at beginning and end points, and x has not reached 5 at t=3.

k=3: better, but a and v are still nonzero at endpoints

k=4: a is still not quite zero at endpoints. max accel has grown to 7.7

k=5: endpoints look good, but max accel is now 12, and max jerk is 77

Compare those graphs to the Figure on page 1 of this paper, which uses a sinusoidal function for x(t), resulting in

x=0, v=0, and a=0 at t=0, and

x=5, v=0, a=0 at t=3, and

max accel=3.5, max speed=3.3, max jerk=7.3