## 2 Materials and Methods
### 3D Modeling of Rowing Forces
#### 2.3.7 Control of Rope Tension Forces during Recovery  
In the presented rowing models, the resistance forces resulting from air friction during recovery were not explicitly modeled. Instead, the resistance was provided with the residual forces acting at the oarlock, which results from tensioning the ropes of the tendon based parallel robot to control the translational motion of the end-effector, i.e., shortened oar. A technical challenge regarding the use of a tendon based parallel robot is that rendering of high rowing forces at the end-effector due to rapid oar movements can result in unwanted transversal vibrations on the ropes and resonance on the robot. Since the vibrations mainly occur when the ropes are flexed, the stiffness control of the robot plays an important role in the haptic rendering of the rowing models on the simulator. Therefore, the natural (eigen) frequency of the system needs to be determined to understand the vibrational characteristics and assure the stiffness of the robot. Assuming that the ropes are modeled as taut strings in our robot, the wave equation that governs the natural transversal vibration of the rope can be described as follows [38]:  
\[\omega_{n}=\frac{n\pi}{l_{i}}\sqrt{\frac{\tau_{i}}{\zeta}} \tag{27}\]where \(\omega_{n}^{i}\) is the eigen frequency of the \(i\)th rope, \(n\) is \(n\)th frequency, \(l_{i}\) is the length of the \(i\)th rope, \(\tau_{i}\) is the tension force on the \(i\)th rope, and \(\zeta\) is the mass per unit rope length. According to Equation (27), the natural frequency for transversal vibration of the rope is proportional to the square root of the rope tension and inversely dependent on the length of the rope. Confirming the relationship presented in Equation (27), three out of five ropes of the tendon based parallel robot are lengthened during catch-to-drive transition in the simulator, which reduces \(\omega_{n}\) and thus results in increased transversal vibrations on the ropes. However, the vibration on the ropes can be suppressed if the eigenfrequency \(\omega_{n}\) is increased. Although increasing the tension on the ropes would yield a stiffer tendon-based parallel robot, increased stiffness would also yield higher friction forces at the oarlock, which may cause unrealistically high resistive forces during recovery. Therefore, we could not use fixed, high tension forces on the ropes during whole rowing cycle to counteract the vibrations resulting from high rowing forces during catch and drive. In the previous studies on the rowing simulator, the vibration problem could be suppressed by setting the minimum rope tension force (\(\tau_{rope}^{min}\)) to a fixed value of 50 N, since the rendered forces at the end-effector were low due to the fact that body-arm rowing was used as the rowing task [24; 25; 26; 27]. However, this minimum rope tension of 50 N yielded neither realistic recovery forces nor eliminated the transversal vibration issue for the rapid, full-body rowing movement. Therefore, we implemented a position-dependent \(\tau_{rope}^{min}\) adjustment in the LIAC model. Three different thresholds for \(\tau_{rope}^{min}\) were defined dependent on the vertical movement of the oar (\(\delta\)) (Figure 4): \(\tau_{rec}^{min}\) for the rope tension during recovery, \(\tau_{transit}^{min}\) during catch and release, and \(\tau_{drive}^{min}\) during drive. The linear adjustment of the rope tension forces were defined as follows:  
\[\tau_{rope}^{min}=\left\{ \begin{array}{ll}\tau_{rec}^{min}&\text{for }\delta_{ Transit}<\delta\\ \tau_{rec}^{min}+\left(\tau_{transit}^{min}-\tau_{rec}^{min}\right)\left( \frac{\delta_{Transit}-\delta}{\delta_{Transit}-\delta}\right)&\text{for }\delta_{Surface}<\delta\leq \delta_{Transit}\\ \tau_{transit}^{min}+p_{imm}\left(\tau_{drive}^{min}-\tau_{transit}^{min} \right)&\text{for }\delta\leq\delta_{Surface}\end{array} \right. \tag{28}\]  
where \(\delta_{Transit}\) is an empirically set value (\(1^{\circ}\)) for the vertical oar angle above the simulated water surface.