# Kinematic Determination of the Aerial Phase in Ski Jumping
## 3 Results and Discussion
### Determination of \(\tau_{opt}\)  
Average values of \(LD\)-ratios and rate-of-change for HS106 are shown in Figure 3.  
The standard deviation in rate-of-change in the mid-region of the aerial phase p\({}_{\text{m}}\) was \(\sim\)0.005 s\({}^{-1}\); thus, \(\tau_{0}\) was set to 0.005 s\({}^{-1}\) and a sensitivity analysis was performed with \(\tau\) ranging from 0.005 s\({}^{-1}\) to 0.015 s\({}^{-1}\). Figure 4 shows examples where \(\tau\) was set to 0.005 s\({}^{-1}\), 0.01 s\({}^{-1}\), and 0.015 s\({}^{-1}\) to find the start and end of the steady glide phase for one jump on HS106.  
As shown in Figure 3(a), parts of the aerial phase where the \(LD\)-ratio is steady will not be defined as steady glide, if the \(\tau\) thresholds are set too small. This effect will vary from jump-to-jump and, consequently, the standard deviation of the defined start and end point for all jumps in HS106 will be high for small threshold values and become smaller as the threshold is increased. The length of the steady glide phase will increase with increasing thresholds. If the \(\tau\) threshold is too large, the algorithm will define parts of the early flight  
Figure 3: Average values of \(LD\)-ratio, shown in (**a**) and rate-of-change of the \(LD\)-ratio in (**b**) with \(n=38\). Gray shaded error bands show the standard deviation.  
and landing preparation (Phase 3.1 and 3.1) as steady glide. In these regions, the \(LD\)-ratio is rapidly decreasing; thus, the variation in HS106 of the \(LD\)-ratio at the defined start and end of all jumps will be high. The standard deviation of the defined start and end point (SD\({}_{1,2}\)) and the \(LD\)-ratio (SD\({}_{3,4}\)) for \(\tau\) between 0.005 s\({}^{-1}\) and 0.015 s\({}^{-1}\) of HS106 are displayed in Figure 5.  
Low threshold values cause large variations in the start and end points, while high thresholds cause large variations in the \(LD\)-ratio. The cost function, based on nSD\({}_{1-4,\tau}\)  
Figure 4: Example of a jump for which the steady glide phase was defined with the three rate-of-change thresholds (**a**) \(\tau_{0}=0.005\) s\({}^{-1}\), (**b**) \(\tau=0.01\) s\({}^{-1}\), and (**c**) \(\tau=0.015\) s\({}^{-1}\). In the lower graphs, the horizontal black dashed lines show the rate-of-change threshold and the vertical red dashed lines indicate the start and end of the steady glide phase, where rate-of-change exceeded \(\tau\).  
Figure 5: Sensitivity analysis of the standard deviation of the starting point, end point, and \(LD\)-ratio at these points for different thresholds. (**a**) displays the standard deviation of the start and end point (SD\({}_{1,2}\)) of the steady glide phase. (**b**) displays the standard deviation of the \(LD\)-ratio at the start and end point (SD\({}_{3,4}\)) for all \(\tau\) tested.  
from the SD\({}_{1-4}\) in Figure 5, is shown in Figure 6, and is used to find the optimal threshold value \(\tau_{opt}\); i.e., the \(\tau\) with the lowest relative standard deviation.  
From this analysis, the optimal threshold value was determined to be \(\tau_{opt}\) = 0.01 s\({}^{-1}\).