I'm currently following a course on coursera on quadrotors, and i'm currently focusing on the trajectory planning mechanism.

In the course, it's said that one of the criterias when planning a trajectory, is the smoothness of it, and they define this mathematically as minimizing an "input" that can vary and depends on the system itself (here the quadrotor).

I somehow understand that in order to have a smooth trajectory, we need to reduce a specific quantity that i define as "trajectory noise" and cause troubles during the flight and mathematically they define this as follow:

1- We want to reduce this quantity (which i qualify as a trajectory noise) along all the trajectory, and thus we take the integral, basically that how i understood it, correct me if i'm wrong.

2- Later in the course, they take several example of inputs and they reduce their square, for example, they consider:

• The velocity, and thus they optimize the square of the velocity.
• The acceleration, and thus they optimize the square of the velocity
• The 3rd derivative (Jerk)
• The forth derivative (Snap)

And then at the end, they prove with some mathematical processes that the Snap (4rd derivative) is proportional to the rotation force applied to the quad-rotor (due to the inertia) and therefore they consider the quad-rotor as a 4th degree system, which means the best trajectory is the trajectory that is computed when optimizing the snap of each degree of freedom (x, y, z, roll, pitch, yaw) and they show videos of autonomous quad-rotors flying autonomously based on a snap optimized trajectory and indeed it works pretty well.

My questions are:

1- Why is the optimized function depending only on either velocity, or acceleration or jerk or snap, i mean why it can't some a combination of these with custom functions ?

2- Why specifically optimizing the square of an input ?

3- Why snap is the best input for having the smoothest trajectory ? i mean being proportional to the rotation force doesn't seem to me as an obvious critaria to be used for such purpose.

Please correct me if i'm wrong in any of my declarations.