## 2 Materials and Methods
### Calculations  
#### 2.5.1 Inertia Tensors  
The electric and conventional bicycles are modelled according to the work of [PERSON] et al. [PERSON] et al. (2018). The bicycle and cyclist are simplified to geometric shapes fully determined by skeleton points 1 to 33. Point 1 to 31 are defined in the work of [PERSON] et al. Point 32 is the centre of gravity of the lower mid-tube and point 33 is the centre of gravity of the luggage rack, which is necessary to define the position of the battery unit. All the skeleton points are presented in Figure 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Wheel Motor** & **Mid Drive Motor** & \
\begin{tabular}{c} **Battery Pack in** \\ **Lower Mid Tube** \\ \end{tabular} &
\begin{tabular}{c} **Battery Pack in** \\ **Luggage Rack** \\ \end{tabular} \\ \hline Weight & 2.9 kg & 3.1 kg & 2.5 kg & 2.5 kg \\ Dimensions [x,y,z] & [160, 100, 160] mm & [200, 180, 150] mm & [330, 90, 90] mm & [370, 120, 80] mm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Standardized dimensions and masses of different motor and battery units. \begin{table}
\begin{tabular}{c c c c} \hline \hline
\begin{tabular}{c} **Value** \\ **Conventional** \\ \end{tabular} & **Electric** & **Bicycle Geometrics** & **Value** \\ \hline Stem length & 190 mm & 270 mm & Steer tube radius & 20 mm \\ Front frame mass & 3.20 kg & 4.02 kg & Lower mid-tube radius & 27.5 mm \\ Rear framework mass & 8.03 kg & Seat tube radius & 16 mm \\ Front wheel mass & 3.82 kg & Seat post radius & 14 mm \\ Rear wheel mass & 3.80 kg & Rear fork radius & 10 mm \\ Rear hub width & 130 mm & Front fork radius & 14 mm \\ Seat post length & 210 mm & Bottom bracket height & 290 mm \\ Seat tube angle & 69\({}^{\circ}\) & Trail & 100 mm \\ Seat tube length & 490 mm & Head tube angle & 68\({}^{\circ}\) \\ Wheel base & 1120 mm & Steer axis tilt & 22\({}^{\circ}\) \\ Rear wheel radius & 345 mm & Chain stay length & 445 mm \\ Front wheel radius & 345 mm & Fork length & 470 mm \\ Handlebar radius & 11 mm & Front hub width & 100 mm \\ Steer stem radius & 14 mm & Handlebar length & 100 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Prototypical dimensions and masses of the seven bicycle configurations. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
\begin{tabular}{c} **Value** \\ **Conventional** \\ \end{tabular} & **Electric** & **Parameter** & **Value** \\ \hline Forward lean angle & 65\({}^{\circ}\) & 70\({}^{\circ}\) & Upper leg mass & 9.71 kg \\ Weight & 64 kg & Upper arm mass & 0.99 kg \\ Length & 170 mm & Torso mass & 30.93 kg \\ Hip joint to hip joint & 250 mm & Lower leg mass & 3.11 kg \\ Head circumference & 540 mm & Lower arm mass & 1.13 kg \\ Lower arm circumference & 220 mm & Head mass & 3.11 kg \\ Lower arm length & 290 mm & Upper leg length & 460 mm \\ Lower leg circumference & 290 mm & Upper leg circumference & 500 mm \\ Lower leg length & 460 mm & Upper arm length & 260 mm \\ Torso length & 420 mm & Upper arm circumference & 240 mm \\ Chest depth & 150 mm & Shoulder to shoulder & 390 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Anthropometrics of a standardized woman on the electric and conventional bicycle. The inertial tensors of the shapes are calculated in their centres of gravity and are rotated according to their individual position around the \(y\)-axis and \(x\)-axis, given by Equation (11). \[I_{l}=R(ra)\ I_{cg}\ R(ra)^{T} \tag{11}\]  
where \(ra\) is the rotation angle around the axis and \(R(ra)\) is the rotation matrix around the axis. The rotation matrix around the \(y\)-axis is given by Equation (12) and the rotation matrix around the \(x\)-axis is given by Equation (13). \[R(ra)y = \left[ \begin{array}{ccc}\cos(ra)_{i}&0&\sin(ra)_{i}\\ 0&1&0\\ -\sin(ra)_{i}&0&\cos(ra)_{i}\end{array} \right] \tag{12}\]  
\[R(ra)_{X} = \left[ \begin{array}{ccc}1&0&0\\ 0&\cos(ra)_{i}&-\sin(ra)_{i}0\\ 0&\sin(ra)_{i}&\cos(ra)_{i}\end{array}