Calculate kinematics of body movement from the set of spatial coordinates

Question

Short intro

I have a set of 3D (x,y,z) spatial coordinates of arm movement obtained using motion capture system. The example set of such coordinates looks like this (rounded up):

 arm = c(-420.1, -419.8, -419.6, -419.4, -419.1, -418.8, -418.5, -418.2, 
-417.9, -417.5, -417.1, -416.8, -416.4, -416, -415.5, -415.2, 
-414.8, -414.3, -413.9, -413.5, -413.1, -412.6, -412.1, -411.6, 
-411.1, -410.6, -410.1, -409.5, -408.9, -408.3, -407.7, -407.1, 
-406.5, -405.8, -405.1, -404.5, -403.8, -403.1, -402.5, -401.9, 
-401.2, -400.5, -399.9, -399.2, -398.6, -397.8, -397.1, -396.3, 
-395.7, -395.2, -394.6, -394, -393.4, -392.9, -391.8, -391.7, 
-391.8, -391.6, -391.3, -390.8, -390.3, -389.6, -389.1, -389.7, 
-389.4, -387.9, -387.5, -387.9, -387, -386.7, -387.2, -387, -386.8, 
-386.6, -386.3, -386.1, -385.8, -385.8, -385.8, -385.8, -385.6, 
-385.5, -384.6, -384.5, -384.5, -384.5, -384.5, -384.5, -384.5, 
-384.6, -385.8, -386.2, -386.9, -387.2, -387.1, -387.5, -387.8, 
-388.1, -388.4, -388.9, -389.2)

where each number above stands for location of the arm on the x axis, through time. I want to obtain the kinematic markers of average velocity, peak velocity, peak acceleration, peak deceleration and jerk index. Jerk index is defined as 'magnitude of the jerk averaged over the entire movement and relating to the smoothness of movement'.

How far did I get?

From what I read on kinematics in Wikipedia and this site I know that I can calculate average velocity using this equation:

$$ v = \frac {\Delta x}{\Delta t} $$

I know the duration $t$ of the movement, but I am not certain how do I define displacement $x$ for this particular coordinate set. Could it be just subtracting minimal from maximal value $max(arm)-min(arm)$?

I don't know how to obtain peak velocity.

Calculating average acceleration seems easy using this equation: $$ a = \frac {\Delta v}{\Delta t} $$

but not sure how to get peak acceleration and peak deceleration.

No idea how I would approach calculating jerk index (magnitude of the jerk averaged over the entire movement and relating to the smoothness of movement).

If any more complex calculations would have to be involved, I work mainly in R, also in MATLAB regarding functions.

Answer 1

To be honest, most of these questions could probably be answered with a bit of reading on Wikipedia, but anyway:

• Displacement is defined as the change in position - hence the notation $\Delta x$. And the change in any quantity $X$ is defined as the final value minus the initial value, $X_f - X_i$.
• To calculate peak velocity, you'll have to determine the velocity at each time step and find the highest one. There are a couple approaches you could use to get individual velocities. You could take the naive approach of determining the differences between adjacent data points, $v_n = (x_{n+1} - x_n)/(t_{n+1} - t_n)$, which would give you an approximation for the velocity between data points. Or you could use $v_n = (x_{n+1} - x_{n-1})/(t_{n+1} - t_{n-1})$, the standard 2-point first order finite difference, which would tell you the velocity at each data point (except the first and last, of course). Or you could do some sort of interpolation on the points (which you could ask about at Scientific Computation and then find the maximum of the resulting curve. There are more complex methods which can improve the accuracy of your approximation a bit, but whichever method you use, you will still only get an approximation to the peak velocity. It is impossible to know the actual peak velocity from the data you have.
• To calculate peak acceleration, you can do the same thing except with velocities instead of positions. I'd recommend using the 3-point second order finite difference, $a_n = (x_{n-1} - 2 x_n + x_{n+1})/[(t_{n+1} - t_{n})(t_{n} - t_{n-1})]$, which corresponds to taking the numerical derivative of the velocities computed by the first method. Again, it's just an approximation; you can use fancier methods which will get you a better approximation, but you can't compute it exactly.
• There isn't any such thing as deceleration in physics, or in other words, it is just acceleration when the acceleration happens to be negative. So "peak deceleration" would probably be the most negative value of acceleration.
• For the jerk index, you could compute the jerks at each half-time step as the derivatives of acceleration and average their magnitudes. The formula for the third derivative would be $J_n = (x_{n-1} - 3 x_n + 3 x_{n+1} - x_{n+2})/[(t_{n+2} - t_{n+1})(t_{n+1} - t_{n})(t_{n} - t_{n-1})]$

Some of these computations may be implemented as functions in Matlab; it's probably worth checking the documentation. Though they would also be pretty easy to do manually.

Answer 2

You need a tool to convert the data points into a) a polynomial fit, or b) a set of cubic splines which are easily differentiable. You might need to smooth the data first to get a nicer results. I have made an VBA script for Excel to do this because I used it with measured cam follower data.

Maybe if I convert the script into Matlab I can post it for you (maybe)

PS1. My reference for cubic splines is http://www.nrbook.com/c section 3.3. PS2. Do not use finite differences (Change over Change) as the results with be very unstable.

Answer 3

If you want a point value of v or a. Use a spline interpolation of x and then derive it once for v and twice for a. There is a cubic spline function in Matlab.