## 2 Materials and Methods
### Parameter Definition and Calculation  
Performance parameters were calculated from raw position data for both measurement systems (PosEst and dGNSS). Filtering and parameter calculations were conducted in the same manner for both systems. The parameter definitions and calculations are addressed in this section for the different phases of a ski jump. The forces acting on the ski jumper in the in-run are illustrated in Figure 3.  
During the in-run, the normal force will have a substantial increase when the ski jumper is entering the curved section due to the centrifugal force and the normal force can be formulated as  
\[F_{N}=m(g\cos(\varphi)+\frac{\
u^{2}}{r}), \tag{1}\]  
where \(m\) is the skier's mass, \(g\) the gravitational acceleration, \(\
u\) the relative velocity and \(r\) the radius of the curve. The aerodynamic forces are defined as acting parallel (drag) and perpendicular (lift) to the direction of relative motion [44], and are defined as  
\[F_{D}=\frac{1}{2}\rho\
u^{2}C_{D}A \tag{2}\]  
and  
\[F_{L}=\frac{1}{2}\rho\
u^{2}C_{L}A, \tag{3}\]  
where \(\rho\) is the air density, \(\
u\) the relative velocity, \(A\) the frontal area of the object and \(C_{D}\) and \(C_{L}\) the drag and lift coefficients. The friction force will also act parallel to the motion and is defined as  
\[F_{f}=\mu F_{N}, \tag{4}\]  
Figure 3: Gravitational (\(mg\)), normal (\(F_{N}\)), friction (\(F_{f}\)), drag (\(F_{D}\)) and lift (\(F_{L}\)) forces acting on a ski jumper in an in-run, together with the coordinate system and angle of the hill (\(\varphi\)). where \(\mu\) is the coefficient of friction. The total force acting parallel to the direction of motion, i.e., braking the ski jumper, can be summed up as the braking force where \(F_{b}=F_{D}+F_{f}\). In the same way, the total force acting perpendicular to the motion can be summed up as \(F_{p}=F_{N}+F_{L}\). Due to the nature of the ski jump, the direction of the ski jumper may change rapidly at particular moments. The angle between the direction of motion and the global coordinate system will be  
\[\varphi=\arctan\biggl{(}\frac{\varphi_{y}}{v_{x}}\biggr{)}, \tag{5}\]  
where \(\varphi<0\). Hence, the local coordinate system of the ski jumper can be found by rotating the coordinate system as follows  
\[ \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix} = \begin{bmatrix}\cos(\varphi)&\sin(\varphi)\\ -\sin(\varphi)&\cos(\varphi)\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} . \tag{6}\]  
In this local coordinate system, \(F_{x^{\prime}}=F_{b}\) and \(F_{y^{\prime}}=F_{p}\). Thus, the braking and perpendicular forces acting on the ski jumper can be found from  
\[F_{b}=m\bar{x}\cos(\varphi)+m(\bar{y}+g)\sin(\varphi) \tag{7}\]  
and  
\[F_{p}=-m\bar{x}\sin(\varphi)+m(\bar{y}+g)\cos(\varphi). \tag{8}\]  
The dGNSS cannot estimate \(F_{f}\), \(F_{D}\), \(F_{N}\) and \(F_{L}\) separately but only as the sum of the forces acting parallel and perpendicular to the ski jumper, hence \(F_{b}\) and \(F_{p}\) are analysed in the in-run. The in-run in ski jumping has a low friction and a high speed, thus the main component of \(F_{b}\) is assumed to be the drag force [6]. The main component of \(F_{p}\) is assumed to be \(F_{N}\), both because this will increase through the curved section and because \(F_{L}\) is assumed to be small [45]. As soon as the ski jumper is air-borne both ski friction and normal force vanish and \(F_{b}=F_{D}\) and \(F_{p}=F_{L}\). Hence drag and lift are analysed for the flight phase. The lift-to-drag (\(F_{L}/F_{D}\)) ratio is also presented as this is seen as an important parameter to describe the quality of the flight phase [46; 47]. The knee angle (\(\theta\)), hip angle (\(\gamma\)) and body angle of attack (\(\psi\)), defined in Figure 4, together with their angular velocities, were derived from the PosEst systems landmarks of ankle, knee, hip and shoulder. Figure 4: Definition of parameters calculated from the PosEst data: (**a**) shows the knee angle (\(\theta\)) and the hip angle (\(\gamma\)).