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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.

Can somebody please help me?

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This sounds exactly like cam design. When designing a cam profile, you typically want the cam follower to be at certain displacements at certain times, and you can usually choose how it gets there.

For example:

enter image description here

As you point out, there are many curve options. Curves are chosen to eliminate discontinuities in position, velocity, and acceleration. And when possible, minimizing acceleration.

When I was a cam designer, this book: Cam Design by Clyde H. Moon was our bible. It is very old, but still pretty useful.

In cam design, a velocity or acceleration discontinuity will make the follower jump off the cam or cause excess wear. You don't really have that issue with a robot arm, but I still believe that having a smooth acceleration profile is desirable.

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