Accumulated contact slip (${\ell }_s=\int |{\dot{\delta }}_s|dt$)

Accumulated contact slip measures the total distance slid at a specified contact, counting motion in either direction positively. For an absolutely continuous signed contact slip displacement in a consistently tracked local frame,It is the total variation of a function applied to the slip history. It differs from net slip ${\delta }_s(t)-{\delta }_s(0)$, which can lose information through cancellation. Contact slip displacement explains the contact-frame convention and the connection to frictional dissipation from contact slip.

For an original piecewise-linear trajectory, suppose successive slip values are $0,0.30,-0.10,0.20\mathrm{mm}$. Its net slip is $0.20\mathrm{mm}$, whileThe middle reversal contributes $0.40\mathrm{mm}$ even though it moves toward smaller signed coordinates. Recording only the first and last values would report $0.20\mathrm{mm}$ and miss eighty percent of the travelled distance in this stipulated history.

For exact samples at ordered times, the sum of absolute increments is a lower bound on the true accumulated motion; unobserved reversals can add distance between samples. The triangle inequality explains why inserting additional exact observations cannot decrease this sum. It equals the integral when the observed intervals contain no unresolved reversals and the path model is appropriate.

Measurement noise changes that conclusion about observed data. A truly stationary contact read as $+\eta ,-\eta ,+\eta$ would acquire an apparent distance $4\eta$ for positive $\eta$. More samples can reveal real reversals while also accumulating readout noise. A useful imaging study therefore reports time resolution, length measurement calibration and the treatment of noise, rather than interpreting every absolute increment as physical sliding.