# how do I decide what is the order of the polynomial I need to use in a polynomial iterpolation for a trajectory?

I am a student studying robotics, and I am focusing on trajectory planning for a robotic arm. In particular, I am studying polynomial interpolation for defining the trajectories, and I am having some doubts.

I have understand that ny using a polynomial interpolation for defining the timing law of a trajectory I have a smooth interpolation. Also, I have seen in many examples, that usually, if I don't need to include acceleration in my profile, I can choose a cubic polynomial and if I have to include also acceleration, I have to use a quintic polynomial (this concept is not really clear to me, I am just using it automatically).

As far as I have understood this concept is relate to the number of parameters I have to use, but is this a fixed rue?

What I mean is, how do I decide what is the order of the polynomial I need to use?

In many example I see it is said that it is sufficient a second order polynomial, or sometimes a third order polynomial, but I have not clear why.