## 2 Materials and Methods
### 3D Modeling of Rowing Forces
#### 2.3.3 Description and Implementation of Base (1D) Models of Rowing for Blade-Water Interaction  
Both rowing models presented in this paper were originally based on the modeling of drag and lift forces acting on a blade immersed into the water, which was provided by [34] as follows:  
\[F_{Dmg}=\frac{1}{2}C_{D}\rho A\left(\vvet{V}_{o/w}\right)^{2} \tag{1}\]  
\[F_{Lift}=\frac{1}{2}C_{L}\rho A\left(\vvet{V}_{o/w}\right)^{2} \tag{2}\]  
where \(\rho=1000\frac{\text{kg}}{\text{m}^{3}}\) is the density of the water; \(\vvet{V}_{o/w}\) is the relative blade velocity between blade and water; \(A\) is the projected area of the blade (calculated perpendicular to the front of the blade); \(C_{D}\) and \(C_{L}\) are the angle of attack dependent dimensionless drag and lift force coefficients, respectively.  
Both equations were based on an experiment in which the blade was inserted into the water tank and held fixed against a one dimensional (1D) incoming flow [34]. Thus, the relationship between the forces and the velocity of the blade was only considered on the horizontal plane.  
An alternative version of the 1D rowing model described by Equations (1) and (2) was implemented for our rowing simulator by [30]. In the simplified version of the model, drag and lift forces acting on the oar blade in the horizontal plane were approximated as follows:  
\[F_{Dmg}^{H}=C_{D}(\alpha_{H})\left(\vvet{V}_{O\pi_{\rho}}\right)^{2} \tag{3}\]  
\[F_{Lift}^{H}=C_{L}(\alpha_{H})\left(\vvet{V}_{O\pi_{\rho}}\right)^{2} \tag{4}\]  
where  
\[C_{D}(\alpha_{H})=2C_{L}^{max}\textit{sin}^{2}\alpha_{H} \tag{5}\]  
Figure 2: Coordinate systems and forces acting on the blade in the BIFC (binary blade immersion, fixed drag/lift coefficients) model. Forces due to horizontal oar movement are shown in the left pane (top view). the vertical force on the blade is shown in the right pane (back view of the boat).  
\[C_{L}(\alpha_{H})=C_{L}^{max}sin2\alpha_{H} \tag{6}\]  
and the angle of attack \(\alpha_{H}\) was defined between the longitudinal axis of the oar and \(\overset{\rightarrow}{V}_{Oar\sigma}\), i.e., the oar blade velocity that depends on the 1-DoF horizontal rotation of the oar (\(\theta\)). \(C_{L}^{max}\) was calculated from the maximum value of lift coefficient \(C_{L}\) whose values were presented based on \(\alpha_{H}\) in [34].  
In principle, the force model presented in [34] with Equations (1) and (2), and the interaction forces calculated with Equations (3)-(6) are based on the assumption that the magnitude of drag and lift forces depend quadratically on the blade velocity. To our knowledge, Equations (3) and (4) were initially mentioned for drag and lift oar blade forces in [29] based on the authors' interpretation of the results from the experiments and force model presented in [35]. In [35], the authors empirically found an approximation of the drag \(C_{D}\) and lift \(C_{L}\) coefficients that depend on the angle of attack (Equations (5) and (6)), which were further considered for the modeling of rowing in [29; 36], and calculation of the efficiency of rowing oars [37].  
In [30], the oar blade was assumed to be squared (vertical with respect to water surface, i.e., \(\varphi=0^{\circ}\)), and the resultant force acting on the blade, i.e., \(F_{\theta}\), was only dependent on the horizontal (\(\theta\)) movement of the oar.  
\[F_{\theta}=\sqrt{\left(F_{Drag}^{H}\right)^{2}+\left(F_{Lift}^{H}\right)^{2}} =2C_{L}^{max}\left(\overset{\rightarrow}{V}_{Oar\sigma}\right)^{2}sin\alpha _{H} \tag{7}\]  
However, for a real rowing stroke, the resultant forces on a blade do not only depend on the horizontal rotation of the oar \(\theta\) but also on the vertical \(\delta\) and longitudinal rotation \(\varphi\). Thus, two rowing models that incorporate three rotations of the oar are presented in the following sub-sections.