# A Three-Dimensional Parametric Biomechanical
## 2 Multibody Model
### Constraints  
The multibody model consists of 84 unconstrained degrees of freedom (DOF), as there are 14 three-dimensional rigid bodies. The following points are introduced to define the configuration of the human body on the bike: the saddle point, the handgrip points (2\(\times\)), and the footpeg points (2\(\times\)). The hands are fixed to the handlebar in the handgrip points, defined on the hands, and the feet are fixed to the chassis in the footpeg points, defined on the feet and, so, 24 DOF are constrained. The head center is fixed and aligned with the trunk's longitudinal axis, at a given distance and, so, 6 DOF are constrained. All these fixed constraints are enforced by the following standard equations [68; 69; 70; 71; 72]:  
\[(\mathbf{P}_{i}-\mathbf{P}_{j})\cdot\mathbf{x}_{i} = 0, \tag{1}\] \[(\mathbf{P}_{i}-\mathbf{P}_{j})\cdot\mathbf{y}_{i} = 0,\] (2) \[(\mathbf{P}_{i}-\mathbf{P}_{j})\cdot\mathbf{z}_{i} = 0,\] (3) \[\mathbf{x}_{i}\cdot\mathbf{y}_{j} = 0,\] (4) \[\mathbf{z}_{i}\cdot\mathbf{x}_{j} = 0,\] (5) \[\mathbf{z}_{i}\cdot\mathbf{y}_{j} = 0, \tag{6}\]  
where \(\mathbf{P}_{i}\) is the connection point on body \(i\) to which a frame with axes \(\mathbf{x}_{i}\), \(\mathbf{y}_{i}\), \(\mathbf{z}_{i}\) is attached, while \(\mathbf{P}_{j}\) is the connection point on body \(j\) to which a frame with axes \(\mathbf{x}_{j}\), \(\mathbf{y}_{j}\), \(\mathbf{z}_{j}\) is attached. Spherical joints constrain the trunk to the arms on RSJC/LSJC and the forearms to the hands on RWJC/LWJC, constraining 12 DOF; see Figure 1b. Similarly, spherical joints constrain the thighs to the trunk on RHJC/LHJC and the shank to the feet on RAJC/LAJC, also constraining 12 DOF; see Figure 1b. The trunk is attached to the saddle point through a spherical joint, constraining 3 DOF. The equations related to all these 9 spherical joints are given as (1)-(3). The arms are connected to the forearms through revolute joints centerd in REJC/LEJC, with the rotation axis perpendicular to the plane defined by SJC, EJC, and WJC, constraining 10 DOF. Similarly, the thighs are connected to the shanks through revolute joints centerd in RKJC/LKJC with the rotation axis perpendicular to the plane defined by HJC, KJC, and AJC, constraining 10 DOF. The equations related to these four revolute joints are given as (1)-(3), (5), and (6). The remaining 7 DOF are the inputs for the numerical analyses (together with human height and mass) and can be related to the rotation of the trunk along the horizontal (roll), vertical (yaw), and transverse (pitch) axes through the saddle point; the distance between elbows and the distance between knees; and the distance between the right elbow and the sagittal plane and the distance between the right knee and the sagittal plane. In more detail, the trunk axes are constrained to have a fixed orientation with respect to the saddle point (and related frame), as follows:  
\[\mathbf{x}_{i}\cdot\mathbf{z}_{j}=\cos\left(\frac{\pi}{2}+\alpha\right), \tag{7}\]  
\[\mathbf{x}_{i}\cdot\mathbf{y}_{j}=\cos\left(\frac{\pi}{2}-\beta\right), \tag{8}\]  
\[\mathbf{y}_{i}\cdot\mathbf{z}_{j}=\cos\left(\frac{\pi}{2}+\gamma\right), \tag{9}\]  
where the indices \(i\) and \(j\) refer to the trunk and saddle frames, respectively, while \(\alpha\), \(\beta\), and \(\gamma\) are the rider's longitudinal lean (or pitch) angle, yaw angle, and roll angle, respectively, with respect to the saddle.