# A Three-Dimensional Parametric Biomechanical
## 4 Sensitivity Analysis  
The multibody model was employed to investigate the effect of the main model parameters--namely knee distance, elbow distance, lean angle, and rider height--on the rider's inertial properties. The rider's yaw and roll (\(\beta\) and \(\gamma\) in (8) and (9)) were assumed to be zero, while the elbows and knees were assumed to be symmetric with respect to the sagittal plane (\(d_{er}=d_{e}/2\) and \(d_{kr}=d_{k}/2\)). The analysis was performed over three different types of motorcycle: sport (tucked position), touring, and a scooter (upright position). A 74 kg rider was assumed, whose posture was defined with the parameters in Table 2. Four data sets were considered. Sensitivity with respect to the rider mass was not considered, as the center of mass does not change with it, while the gyration radii are linearly proportional to it. Indeed, the position of the whole center of mass \(\mathbf{G}\) can be computed from the positions of the center of mass of its \(i\) components \(\mathbf{P}_{i}\):  
\[\mathbf{G}=\frac{\sum m_{i}\mathbf{P}_{i}}{\sum m_{i}}=\frac{\sum\alpha_{i}m\mathbf{P}_{i} }{m}=\sum\alpha_{i}\mathbf{T}_{i} \begin{bmatrix}\beta_{i}^{x}l_{i}\\ \beta_{i}^{y}l_{i}\\ \beta_{i}^{z}l_{i}\\ 1\end{bmatrix} =\sum\alpha_{i}\mathbf{T}_{i} \begin{bmatrix}\beta_{i}^{x}\gamma_{i} l_{i}\\ \beta_{i}^{y}\gamma_{i}h\\ \beta_{i}^{z}\gamma_{i}h\\ 1\end{bmatrix} . \tag{15}\]  
Figure 3: Horizontal (**a**) and vertical (**b**) position of \(\mathbf{G}\): numerical vs. experimental. The mass \(m_{i}\) of each body segment is proportional to the whole-body mass \(m\) through the coefficient \(\alpha_{i}\) (Tables A1-A3 and A5), while the position of the center of mass of each body segment is proportional to the body segment length \(l_{i}\) through the coefficients \(\beta_{i}^{z}\), \(\beta_{i}^{x}\), and \(\beta_{i}^{y}\) (Tables A1-A3 and A5), when expressed in the local reference frame. The projection onto the global reference frame is obtained through a \(4\times 4\) roto-translation matrix \(\mathcal{T}_{i}\), which has the \(3\times 3\) rotation matrix \(\mathcal{R}_{i}\) of the local reference frame in its upper-left corner (whose columns consist of the unit vectors of the local reference frame, expressed in the global reference frame) and the location of the origin of the reference frame \(\mathcal{O}_{i}\) in its fourth column (again expressed in the global frame); the (4,4) value is always one [1,70]. Finally, the length of the body segment \(l_{i}\) is proportional to the body height \(h\) through the coefficient \(\gamma_{i}\) (Tables A1-A3 and A5). Equation (15) indicates that the position of the center of mass of the whole body only depends on the height (and, of course, on the data set selected for the coefficients \(\alpha_{i},\beta_{i},\) and \(\gamma_{i}\)). Similarly, the whole-body inertia matrix, \(\mathcal{J}_{G}\), can be obtained from the parallel axis theorem as:  
\[\mathcal{J}_{G} = \sum\left(\mathcal{R}_{i} \begin{bmatrix}m_{i}\rho_{x}^{2}&m_{i} \rho_{xy}^{2}&m_{i}\rho_{xz}^{2}\\ m_{i}\rho_{xy}^{2}&m_{i}\rho_{y}^{2}&m_{i}\rho_{yz}^{2}\\ m_{i}\rho_{xz}^{2}&m_{i}\rho_{yz}^{2}&m_{i}\rho_{z}^{2}\end{bmatrix} \right) \mathcal{R}_{i}^{T}+m_{i}(\mathbf{r}_{i}^{T}\mathbf{r}_{i}\mathbf{I}-\mathbf{r}_{i}\mathbf{r}_{i}^ {T}) \tag{16}\] \[= m\sum\left(\mathcal{R}_{i} \begin{bmatrix}\alpha_{i}\eta_{x}^{2} l_{i}^{2}&\alpha_{i}\eta_{xy}^{2}l_{i}^{2}&\alpha_{i}\eta_{xz}^{2}l_{i}^{2}\\ \alpha_{i}\eta_{xy}^{2}l_{i}^{2}&\alpha_{i}\eta_{yz}^{2}l_{i}^{2}&\alpha_{i} \eta_{xz}^{2}l_{i}^{2}\\ \alpha_{i}\eta_{xz}^{2}l_{i}^{2}&\alpha_{i}\eta_{yz}^{2}l_{i}^{2}&\alpha_{i} \eta_{xz}^{2}l_{i}^{2}\end{bmatrix} \right)\mathcal{R}_{i}^{T}+\alpha_{i}(\mathbf{r }_{i}^{T}\mathbf{r}_{i}\mathbf{I}-\mathbf{r}_{i}\mathbf{r}_{i}^{T})\] \[= m\sum\left(h^{2}\alpha_{i}\mathcal{R}_{i}