## 1 Introduction  
Signals from motion analysis systems are contaminated with noise or error, resulting from electrical interference in the system, skin motion, and inaccurate data digitization. The noise has features that are different from the actual signal: low amplitude, nondeterministic, and diverse frequency range [1]. Therefore, raw signals or data must be smoothed or filtered to eradicate such noise while preserving the original signal traits. Traditionally, biomechanics data, specifically displacement data, were smoothed to obtain velocity and acceleration using several methods such as polynomial, splines, and Fourier series, as well as digital filtering. Cubic spline has been proven to be a better smoothing technique than polynomial as a second derivative; that is, the acceleration of displacement data is well-fitted by using cubic spline rather than orthogonal polynomial [2] and Chebyshev polynomial [3]. Polynomial produces an oversmoothed acceleration curve, which provides unrealistic acceleration values in running events [3], and an oversmoothed angular acceleration curve, which attenuates the peaks and falsifies the time historiesof single segment motion [4]. However, a higher-order spline, specifically quintic spline, is found to be superior over cubic spline. A comparison between the third derivatives of cubic and quintic spline methods in vertical jump data from [PERSON] and [PERSON] [5] indicates that quintic spline is a better smoothing technique than cubic spline, and the presence of a discontinuity in the third derivative of cubic spline supports the argument [6]. Quintic spline also produced a well-fitted raw angular displacement film data and its second derivative, as shown in the research of [PERSON] et al. [4]. Cubic spline tends to be zero at endpoints of the second derivative of displacement data [3]. This occurrence is known as endpoint error [7], while some identify this term as an endpoint problem [8; 9], a boundary effect [10], or an edge effect [11]. [PERSON] et al. [2] managed to reduce zero second derivative at endpoints with cubic spline by adding three extra data points at the beginning and end of each dataset. [PERSON] et al. [3] showed that the cubic spline function produced zero acceleration at endpoints for a weight-dropping experiment. Similarly, the vertical acceleration of a dropping golf ball with cubic spline emphasized the presence of zero acceleration at endpoints, which supports quintic spline as a better smoothing method than cubic spline [9]. The superiority of quintic spline is found not only over cubic spline but also over digital filter and Fourier series. [PERSON] and [PERSON] [7] compared four popular smoothing methods, Butterworth digital filter, Fourier series, cubic spline, and quintic spline, in terms of root mean squared (RMS) residual errors of acceleration in endpoint regions using [PERSON]'s [12] modification of [PERSON] et al. 's [4] raw angular displacement data. Quintic spline produced the most accurate acceleration values, which was close to the endpoints of the modified [PERSON] et al. [4] dataset compared to the other three methods, although none of the methods completely eradicated the endpoint errors [7]. Quintic spline is also the most accurate smoothing method compared to stepwise polynomial regression and simple polynomial regression, although stepwise polynomial regression can be used as an alternative to quintic spline [13]. Nevertheless, all previous methods of smoothing and differentiating raw displacement data mentioned above are in the discrete data form. Analysis using discrete data leads to the discarding of essential features in the data [14; 15]. Hence, functional data analysis (FDA) is introduced as a rising alternative statistical method that approaches data analysis from a functional perspective. FDA can transform discrete data into a functional form before proceeding to any form of analysis [16].