How to calculate velocity from X and Y positions and time?

What would be the best/most correct way of calculating velocity (total velocity, not X and Y components of velocity) from the following X and Y positions at the given timepoints? Not sure whether (and exactly how) to use a diff or a gradient function... Also, how could I get a one dimension position value, for the purpose of calculating STD at the given timepoints for several of these datasets?

Table =

20×3 table

Time PosX PosY

_______ ______ ______

0 363.79 255.3

0.00833 352.32 252.27

0.01666 342.07 236.94

0.02499 320.48 225.84

0.03332 287.69 209.97

0.04165 272.58 197.78

0.04998 261.67 193.81

0.05831 261.72 205.85

0.06664 269.66 224.84

0.07497 283.41 250.91

0.0833 290.57 250.67

0.09163 292.02 250.13

0.09996 296.21 251.24

0.10829 298.06 252.43

0.11662 306.25 265.04

0.12495 320.4 286

0.13328 334.88 315.34

0.14161 346.22 333.17

0.14994 357.84 336.76

0.15827 366.17 333.72

Answer 1

[Torsten]

@Daniela Pereira

for the purpose of calculating STD at the given timepoints for several of these datasets?

STD of what ?

[Daniela Pereira]

STD of the position at each timepoint. I only included one data set here that depicts a trajectory of a given point throughout time. But I have several of these per experiment (up to 200 or so), which I need to compare. I know how to calculate STD of X and Y positions individually but don't know how to reduce it to a single dimension...

[Torsten]

Are the movements in X and Y direction independent or correlated ?

Maybe you should calculate the correlation coefficient between deltaX and deltaY to answer this question first.

If they are independent, it does not make sense to calculate a common STD in my opinion.

If they are correlated, well, let's see.

[Daniela Pereira]

I would guess they are correlated. These are trajectories of a mouse paw pressing a lever. After resolving the problem of how to calculate velocity at each timepoint (and eventually get a one dimension value per position...), I want to know how the mean velocities of the trajectories change and how similar are these trajectories to one another.

[Torsten]

A mouse ? And correlated movements ? Why ?

[Daniela Pereira]

Well, I can check for that correlation to be sure, you're right. But geting help with the velocity calculations for now, would be great!

[Torsten]

Yes, I already asked whether the movement from A to B was on the straight line connecting the points.

[Daniela Pereira]

It is probably not a straight line, definite not between all timepoints, at least.

[Torsten]

Since we don't know better, we assume the shortest distance between two points.

This gives:

   M=[0    363.79     255.3
    0.00833    352.32    252.27
    0.01666    342.07    236.94
    0.02499    320.48    225.84
    0.03332    287.69    209.97
    0.04165    272.58    197.78
    0.04998    261.67    193.81
    0.05831    261.72    205.85
    0.06664    269.66    224.84
    0.07497    283.41    250.91
     0.0833    290.57    250.67
    0.09163    292.02    250.13
    0.09996    296.21    251.24
    0.10829    298.06    252.43
    0.11662    306.25    265.04
    0.12495     320.4       286
    0.13328    334.88    315.34
    0.14161    346.22    333.17
    0.14994    357.84    336.76
    0.15827    366.17    333.72];
   
   dt = M(2:size(M,1),1)-M(1:size(M,1)-1,1); 
   dx = M(2:size(M,1),2)-M(1:size(M,1)-1,2);
   dy = M(2:size(M,1),3)-M(1:size(M,1)-1,3);
   dist = sqrt(dx.^2+dy.^2);
   vel = dist./dt;
   velx = abs(dx)./dt;
   vely = abs(dy)./dt;
   
   plot((M(1:size(M,1)-1,1)+M(2:size(M,1),1))/2,vel,'color','b')  % absolute velocity
   %plot((M(1:size(M,1)-1,1)+M(2:size(M,1),1))/2,sqrt(velx.^2+vely.^2),'color','g') % also absolute velocity for checking
   hold on
   plot((M(1:size(M,1)-1,1)+M(2:size(M,1),1))/2,velx,'color','r') % x velocity
   hold on
   plot((M(1:size(M,1)-1,1)+M(2:size(M,1),1))/2,vely,'color','y') % y-velocity
   hold off

vel(i) is the average absolute velocity between time instant t_i and t_i+1 if we assume that the movement is on the straight line between (x(i),y(i)) and (x(i+1),y(i+1)) (1<=i <=19).

Note that the velocity arrays have one element less than the position arrays.

You may think of vel not as velocities in points, but over intervals (here: vel(i) as velocity over the interval [t_i,t_i+1)).

Answer 2

[Sam Chak]

Hi @Daniela Pereira

First, you need to obtain the distances between each point specified by the coordinates.

Then, I think you compute the velocity with this formula:

[Torsten]

@Sam Chak

Are you sure that the candidat took the shortest way between the positions ? :-)

[Daniela Pereira]

So you would use moslty the diff function to perform those calculations and not the gradient function?

[Daniela Pereira]

No, I'm not sure that the real trajectory is the shortest between each 2 positions...

[Sam Chak]

@Torsten

Ah... I overlooked "STD"... Thanks.