## 2 Materials and Methods
### 3D Modeling of Rowing Forces
#### 2.3.5 3D Rowing Model with Linear Blade Immersion and Adjusted Drag/Lift Coefficients (LIAC)  
In 2019, we developed a new 3D rowing model (linear blade immersion, adjusted drag/lift coefficients or LIAC), which was essentially based on Equations (1) and (2). In the model, \(F_{Dmg}\) was defined in the opposite direction of the oar blade velocity while \(F_{Lift}\) was perpendicular to drag. During a continuous movement of the oar blade in the water, the magnitude and the direction of \(F_{Dmg}\) and \(F_{Lift}\) change based on the values of drag (\(C_{D}\)) and lift (\(C_{L}\)) force coefficients, respectively. Furthermore, \(C_{D}\) and \(C_{L}\) were dependent on the instantaneous angle of attack. For the horizontal plane (\(\mathcal{P}_{xy}\)), we used a look-up table based on [34], which delivered the values of \(C_{D}\) and \(C_{L}\) depending on the calculated instantaneous angle of attack. Extending the model to the vertical plane (\(\mathcal{P}_{yz}\)), we employed a separate instantaneous angle of attack (Figure 3). As a first step, the relative oar-blade velocity vector was defined in the boat-fixed coordinate system (\(\chi_{Bout}\)) as \(\vv{V}_{xyz}=\left[ \begin{array}{cc}V_{O\alpha r_{x}}&V_{O\alpha r_{y}}&V_{ O\alpha r_{z}}\end{array} \right]^{T}\). To calculate the angles of attack for the oar blade motion in the horizontal (\(\mathcal{P}_{xy}\)) and vertical (\(\mathcal{P}_{yz}\)) planes, the projection of \(\vv{V}_{xyz}\) vector onto these planes were described as \(\vv{V}_{xy}=\left[ \begin{array}{cc}V_{O\alpha r_{x}}&V_{O\alpha r_{y}}\end{array}  \right]^{T}\) for the horizontal and \(\vv{V}_{yz}=\left[ \begin{array}{cc}V_{O\alpha r_{y}}&V_{O\alpha r_{z}}\end{array}  \right]^{T}\) for the vertical planes. According to the model presented in [34], the angle of attack used for the horizontal plane (\(\mathcal{P}_{xy}\)) was defined between the velocity of the oar blade (\(\vv{V}_{xy}\)) and the longitudinal (transverse) axis of the oar blade, yielding \(\alpha_{xy}\). Following a similar approach, the angle of attack used for the vertical plane (\(\mathcal{P}_{yz}\)) was defined between the \(\vv{V}_{yz}\) component of the oar blade velocity and the cross section (vertical axis) of the blade, resulting in \(\alpha_{yz}\). For the implementation of \(\alpha_{xy}\) and \(\alpha_{yz}\) calculations in a Matlab/Simulink(r) model, the following relationships were used (Figure 3):  
\[\alpha_{xy}=\beta-\pi-\theta \tag{12}\]  
\[\alpha_{yz}=\gamma-\frac{\pi}{2}+\varphi, \tag{13}\]  
where \(\beta\) is an angle that is defined between \(\vv{V}_{xy}\) and \(S_{x}\) in the horizontal plane (CCW around \(S_{z}\)), and \(\gamma\) is the angle between \(\vv{V}_{yz}\) and the \(S_{y}\) (CCW around \(S_{x}\)). Figure 3: Coordinate systems and forces acting on the oar blade in the LIAC (linear blade immersion, adjusted drag/lift coefficients) model. Blade forces on the horizontal plane are shown in the left pane (top view). Blade forces on the vertical plane are shown in the right pane (side view of the boat). In the right pane, the enlarged circle shows the cross-section of the blade. Subsequently, drag and lift coefficients for both the horizontal and vertical forces were defined as \(C_{D}^{xy}=f_{D}\big{(}\alpha_{xy}\big{)}\), \(C_{L}^{xy}=f_{L}\big{(}\alpha_{xy}\big{)}\), \(C_{D}^{yz}=f_{D}\big{(}\alpha_{yz}\big{)}\), and \(C_{L}^{yz}=f_{L}\big{(}\alpha_{yz}\big{)}\), where the functions \(f_{D}\) and \(f_{L}\) were implemented as a look-up table based on the presented experimental values [34]. In the final step, the projected areas of the oar blade in the horizontal and vertical planes were calculated as follows:  
\[A^{xy}=A_{b}p_{imm}cos\varphi \tag{14}\]  
\[A^{yz}=A_{b}p_{imm} \tag{15}\]  
where \(A_{b}=0.13\) m\({}^{2}\) was chosen for a big blade used in sweep rowing. \(p_{imm}\) is the proportion of the oar blade in the water, which was calculated as  
\[p_{imm}=\left\{