So this is an odd question and more of a discussion but I am interested in trying to model the equations for the movement of the geneva mechanism. Below is a picture of what I think is one of the clearer graphs I could find:
As you can see it forms a nice continuous function on the displacement and velocity, but is discontinuous on the acceleration and jerk. I have tried to copy these functions as best as I could in desmos and you can see that here. It is clear to me that the equation must look something like: $$j(t)=\delta(t)-\delta(t-1)+f(t)$$ However I am struggling to find a nice equation which fits this shape. I tried using Lagrange polynomial interpolation but the peaks are far higher than I would have liked, a rough equation was: $$f(t)=\frac{0.1(t-0.5)(t-1)(t-0.3)(t-0.7)}{(-0.5)(-1)(-0.3)(-0.7)}+\frac{-1t(t-1)(t-0.3)(t-0.7)}{(0.5)(0.5-1)(0.5-0.3)(0.5-0.7)}+\frac{0.1t(t-0.5)(t-0.3)(t-0.7)}{1(1-0.5)(1-0.3)(1-0.7)}$$ If anyone else has any ideas to put forward or a way of actually deriving the proper equation that would be amazing. Thank you :)