Endpoint error in smoothing and differentiating raw kinematic data: An evaluation of four popular methods

'Endpoint error' describes the erratic behavior at the beginning and end of the computed acceleration data which is commonly observed after smoothing and differentiating raw displacement data. To evaluate endpoint error produced by four popular smoothing and differentiating techniques, Lanshammar's modification of the Pezzack et al. raw angular displacement data set was truncated at three different locations corresponding to the major peaks in the criterion acceleration curve. Also, for each data subset, three padding conditions were applied. Each data subset was smoothed and differentiated using the Butterworth digital filter, cubic spline, quintic spline, and Fourier series to obtain acceleration values. RMS residual errors were calculated between the computed and criterion accelerations in the endpoint regions. Although no method completely eliminated endpoint error, the results demonstrated clear superiority of the quintic spline over the other three methods in producing accurate acceleration values close to the endpoints of the modified Pezzack el al. data set. In fact, the quintic spline performed best with non-padded data (cumulative error = 48.0 rad s^-2). Conversely, when applied to non-padded data, the Butterworth digital filter produced wildly deviating values beginning more than the 10 points from the terminal data point (cumulative error = 226.6 rad s^-2). Each of the four methods performed better when applied to data subsets padded by linear extrapolation (average cumulative error = 68.8 rad s^-2) than when applied to analogous subsets padded by reflection (average cumulative error = 86.1 rad s^-2).