# Given a set of 3d positions and timestamps, what is the best numerical method to estimate the velocity of an object at the last position?

I am working on a VR game where the player is able to grab an object and throw it. I would like to estimate the object velocity at the moment of release. I want to base this estimation on a recording of 3D positions and timestamps, and I want to use a numerical method in order to deal with signal noise.

The following picture describes my question. Given a set of points [(p1, t1), (p2, t2), ...] what is the velocity V at the last point?

Here is an example of the data I have recorded:

points_data = [[2.83043766 1.9541018  2.95477581]
[2.96888566 1.88202035 3.03757858]
[3.03159261 1.80428386 3.07880282]
[3.07919741 1.76630378 3.11723733]
[3.11075521 1.74352813 3.13725162]]

points_time = [[ 0.      ]
[-0.019358]
[-0.0394  ]
[-0.057304]
[-0.074427]]


The vector array points_data contains 3d (x, y, z) coordinates in meters, points_time contains time in seconds relative to a moment of release (t = 0).

• I have tried simple differentiation y1 - y0 / dt at the last point, but the result is rather noisy since VR data is also noisy.
• I then tried exponential smoothing to filter out the noise, but this results in a lag in the resulting velocity estimation.
• Yesterday I implemented least squares regression to estimate a cubic polynomial through the point data and use it to estimate the velocity. The result is pretty good, but I think I need a higher order estimation to make it 'feel' right to the player.
• This morning I was thinking about finite differences, and I think something like a higher order backward finite difference could work very well, but it looks like it is only applicable to a fixed time step. In VR, some frames take longer than others.

Any ideas are appreciated! Thanks!

• have you considered Kalman filters? Commented Jul 15, 2021 at 7:01

There exists recursive polynomial approximation algorithm for N points: $$\begin{array} {rl} INPUT: &[(t_0,(x,y,z))], where f(t_0)=(x,y,z)\\ PROCESS:& P([(t_0, (x,y,z))]) \rightarrow f(t)\\ OUTPUT:& P([]) = (0,0,0)\\ & P(pair : rest) = (t_0,(x,y,z))=pair,\\ & f(t)=(t-t_0)*P(rest)+(x,y,z)\\ \end{array}$$

This algorithm gives you $$f(t) : R\times R\times R$$, which is polynomial approximation $$f:(t)\rightarrow(x,y,z)$$.

Then velocity can be calculated on point $$t_0$$ using the definition of derivative once the function is available: $$f'(t) = \lim_{k\to 0} \frac{f(t+k)-f(t)}{k}$$

THEOREM: if $$f(a)=b$$, then $$f(x)-b=0$$ equation has a root $$x=a$$.

THEOREM: when equation $$f(x)=0$$ has root $$x=a$$, then $$(x-a)$$ is a component of the polynomial describing f.

THEOREM: if polynomial has $$N$$ roots $$r=[r_1,r_2,..,r_n]$$, then a polynomial in form $$f(x) = (x-r_1)*(x-r_2)*...*(x-r_n)$$ fully describes the polynomial function.

• You can only solve this problem with an explicit or implicit model of the motion. Is there gravity? Does the object experience electrical attraction and repulsion from some fixed charges? For instance, you might adopt the model of minimum force applied to the object, or that forces come in instantaneous impulses, or ... Adopt your underlying model, then express your path search as a global minimization problem. Commented Jul 10, 2021 at 20:09
• in the above algorithm, its possible to choose the N points/timestamps in such way that some kind of error function is minimized. While it can utilize all the points, if there's errors in the dataset, some obviously broken points might need to be discarded to get better results. Lower-degree polynomials might give more consistent results. Simplest solution is to pick N most recent points, but any selection is allowed for the algorithm. N=5 or N=6 is probably what I'd use.
– tp1
Commented Jul 10, 2021 at 20:44