## 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)
\begin{array}{ll}0&\text{for }\delta_{Surface}<\delta\\ \Big{|}\frac{\delta-\delta_{Surface}}{\delta_{InWater}-\delta_{Surface}} \Big{|}&\text{for }\delta_{InWater}<\delta\leq\delta_{Surface}\\ 1&\text{for }\delta\leq\delta_{InWater}\end{array} \right. \tag{16}\]  
where \(\delta_{Surface}\) and \(\delta_{InWater}\) are the vertical oar angles at which the tip of the blade touches the water surface and the blade is fully immersed in the water, respectively. Placement of Equation (16) in Equations (14) and (15) allowed for a smooth transition of forces between the drive and recovery. For the calculation of \(A^{xy}\), the projected area was assumed to be dependent only on the longitudinal oar angle \(\varphi\) since horizontal rotation of the oar \(\theta\) was included in the calculation of horizontal angle of attack \(\alpha_{xy}\), which would update the magnitude of drag and lift coefficients according to the fluid dynamic model [34]. Moreover, for the calculation of \(A^{yz}\), we assumed that the projected area would only depend on the immersion ratio of the blade since the vertical angle of attack \(\alpha_{yz}\) was dependent on the longitudinal oar angle \(\varphi\). In addition, a further trigonometric adjustment of the \(A^{yz}\) based on the horizontal angle \(\theta\) resulted in low forces in the vertical plane during the pilot tests conducted on the simulator, which were not preferred by the experts. Therefore, we did not include any dependency of other oar angles to estimate \(A^{yz}\). Based on the previous calculations, the drag and lift forces of the model (LIAC) that resulted from the horizontal, vertical, and longitudinal rotation of the oar blade were calculated as follows:  
\[F_{Drag}^{xy}=\frac{1}{2}C_{D}^{xy}\rho A^{xy}\big{(}\overrightarrow{V}_{xy} \big{)}^{2} \tag{17}\]  
\[F_{Lift}^{xy}=\frac{1}{2}C_{L}^{xy}\rho A^{xy}\big{(}\overrightarrow{V}_{xy} \big{)}^{2} \tag{18}\]  
\[F_{Drag}^{yz}=\frac{1}{2}C_{D}^{yz}\rho A^{yz}\big{(}\overrightarrow{V}_{yz} \big{)}^{2} \tag{19}\]  
\[F_{Lift}^{yz}=\frac{1}{2}C_{L}^{yz}\rho A^{yz}\big{(}\overrightarrow{V}_{yz} \big{)}^{2}. \tag{20}\]  
To calculate the emerging horizontal (\(F_{H}^{xy}\)) and vertical (\(F_{V}^{xy}\)) plane forces acting on the oar blade in the boat-fixed coordinate system (\(\chi_{Bont}\)), the drag and lift forces resulting from Equations (17)-(20) were transformed in the horizontal and vertical planes:  
\[\overrightarrow{F}_{H}^{xy}={}^{Bont}\mathcal{R}_{Stram}(-\pi)^{Stream} \mathcal{R}_{xy}(\beta)\left[ \begin{array}{cc}F_{Drag}^{xy}&F_{Lift}^{xy} &0\end{array} \right]^{T} \tag{21}\]  
\[\overrightarrow{F}_{V}^{xy}={}^{Bont}\mathcal{R}_{Stram}(-\pi)^{Stream} \mathcal{R}_{yz}(\gamma)\left[ \begin{array}{cc}0&F_{Drag}^{yz}&F_{Lift}^{yz} \end{array} \right]^{T} \tag{22}\]  
where \({}^{Bont}\mathcal{R}_{Stram}(\theta_{rot}=-\pi)\) was used to define the \(\chi_{Stram}\) in the boat-fixed coordinate system (\(\chi_{Bont}\)), and \({}^{Stram}\mathcal{R}_{xy}(\beta)\) and \({}^{Stram}\mathcal{R}_{yz}(\gamma)\) were the transformation matrices in the horizontal and vertical planes, respectively. According to the model presented by [34], the resulting plane forces of \(\overset{\rightarrow}{F}_{H}^{Xb}\) and \(\overset{\rightarrow}{F}_{V}^{Xb}\) define two distinct 2D models in the horizontal (\(\mathcal{P}_{xy}\)) and vertical (\(\mathcal{P}_{yz}\)) planes, respectively. In order to generate a 3D force (\(F_{Our}\)) from two 2D models (\(\overset{\rightarrow}{F}_{H}^{Xb}\) and \(\overset{\rightarrow}{F}_{V}^{Xb}\)), we employed a combination factor, which would satisfy the following constraints:  
\[F_{Our}=f\big{(}V_{xyz}^{2}\big{)}C_{1}f\big{(}V_{xy}^{2}\big{)}+C_{2}f\big{(}V _{xy}^{2}\big{)},\text{ where }C_{1},\text{ }C_{2}\leq 1\text{ and }C_{1}+C_{2}=1. \tag{23}\]  
\(C_{1}\) and \(C_{2}\) are the combination factors for the horizontal and vertical planes, respectively.