I'm a researcher in a Physical Therapy department and I came across an equation for a "roughness index" (RI) as a measure for cycling (as in a bicycle) smoothness. We have our subjects (who all have cerebral palsy) cycle on a stationary trike and we measure the angles of the crank and pedals, as well as forces in the pedals, to try to characterize how well they cycle.
The equation is presented in a paper without discussion or reference, although internet searching yields an "International Roughness Index" (IRI) for highways. While I can understand how this would be related to the RI, I have not yet been able to download any references for IRI. Regardless of suitability, I have some questions about the mathematical properties of the equation. The equation as presented is as follows:
$$RI = \sum_1^{360} \lvert dR/ds\rvert,$$
where $R$ is the instantaneous cranking speed and $s$ is the crank position. The sum is from 1 to 360 because the indicated derivative is computed (as the derivative of a polynomial that is fit to the crank speed data) for each integer degree between 1 and 360.
What confuses me is that the crank speed $R$ is the derivative of the crank position $s$ relative to time, so this equation seems essentially equivalent to:
$$RI = \sum_1^{360} \left\lvert \frac d {ds} \left( \frac {ds} {dt} \right) \right\rvert,$$
Which seems like it should simplify. However, my recollection of differential calculus is less than ideal. I want to cancel the $ds$'s, but that doesn't seem right.
In addition, it seems to me that the summation is actually an integral, so that a more general representation of RI would be:
$$RI = \int_s \left\lvert \frac d {ds} \left( \frac {ds} {dt} \right) \right\rvert ds,$$
which I want to simplify (assuming $\frac d {ds} \left( \frac {ds} {dt} \right) $ is positive or zero over $s$) to:
$$RI = \frac {ds} {dt} (t = t_2) - \frac {ds} {dt} (t = t_1),$$
Where $t_1$ is the time at $s=0$ and $t_2$ is the time at $s=360$.
However, some simple excel programming, playing with with different crank accelerations and taking derivatives numerically, does not come close to verifying the above equation.
In short, my questions are:
1) Where did I go wrong?
2) Can I simplify the original equation for RI?
Thanks in advance, -Jamie (newbie)