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Let's imagine we've got a uniformly sampled curved trajectory of a moving ball in 3D space without gravity. Let's assume that some dynamic force is applied to the moving ball to update its trajectory between samples. Is it possible to estimate minimal efforts required to update trajectory of the moving ball between samples?

For example: Two examples of moving balls

In case A there was no force applied between t and t+1 because the ball moved with constant speed.

In case B some dynamic force changed position of the ball from P(t) to P(t+1)

I would like to know the way to estimate minimal efforts required to move the ball from P(t) to P(t+1) in general case.

Edit:

One possible interpretation of "effort" might be the time integral of the magnitude of the force applied. This would quantify the total "scalar impulse" applied on the ball. This is also the relevant quantity when you want to perform a maneuver in a spacecraft by consuming the least amount of fuel (assuming the change in the mass of the spacecraft is negligible)

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  • $\begingroup$ Are you given that no force is applied before $t$ and after $t+1$ in case B? Also, the minimal work required would be zero if the separation between P(t-1) and P(t), and between P(t+1) and P(t+2) remain the same as in case A. This would mean the speeds (and hence the kinetic energies) at P(t) and P(t+1) are the same, meaning no net work would be required. $\endgroup$
    – Puk
    Commented Jul 30, 2019 at 8:30
  • $\begingroup$ @Puk the force might be applied at any time between samples $\endgroup$
    – Maksym Ganenko
    Commented Jul 30, 2019 at 8:31
  • $\begingroup$ @Puk Kinetic energy of the ball at t-1 and t+1 might be the same, but some force had been applied to move from P(t) to P(t+1) to accelerate and decelerate the ball. That required non-zero work. $\endgroup$
    – Maksym Ganenko
    Commented Jul 30, 2019 at 8:40
  • $\begingroup$ The force may do no work at all if it is always perpendicular to the velocity of the ball. If it isn't always perpendicular, it does some positive work and some negative work, for a total of zero work. $\endgroup$
    – Puk
    Commented Jul 30, 2019 at 8:44
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    $\begingroup$ I think a natural quantity to minimize would be the $\int F(t) dt$, i.e. the time integral of the magnitude of the force applied. This would quantify the total "scalar impulse" applied on the ball. This is also the relevant quantity when you want to perform a maneuver in a spacecraft by consuming the least amount of fuel (assuming the change in the mass of the spacecraft is negligible). I don't know what you mean by the positions being band limited. It seems to me that requiring the ball to pass through the points $P(t)$ and $P(t+1)$ at times $t$ and $t+1$ already enforces this condition. $\endgroup$
    – Puk
    Commented Jul 30, 2019 at 9:28

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You can numerically estimate the acceleration of the path, by interpolating the data using a cubic spline or some other method in order to estimate the time derivatives of the positions. Then do it again to estimate the acceleration needed to follow the path. Finally, if you know the mass of the object, you can $\vec{F} = m \vec{a}$ to estimate force needed.

Below is an example using a single coordinate X as input (blue), and evaluating the speed Xp (red) and acceleration Xpp (green) in Excel.

screen

I am using my own custom array function Derivative(x_range,y_range) to do the job.

Depending on what programming environment you have access to, try to find interpolating functions. They exist for MATLAB, C, Fortran, Python, Java, C# and all other major programming languages. You can try porting the C code from NRBOOK in section 3.3 relating to Cubic Splines.

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    $\begingroup$ Thank you for posting the answer, I came to the same conclusion at the end of the day, but I decided to interpolate using Lanczos kernel instead of cubic splines. $\endgroup$
    – Maksym Ganenko
    Commented Jul 31, 2019 at 7:39

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