## 2 Methods and Techniques
### Biomechanical Model Identification  
Restricting the analysis to the sagittal plane, the kinematic model of the lower limb is composed of constrained rigid elements and mechanical elements of the bicycle, used to transmit the motion to the wheel. By modeling each body segment and each mechanical element as a segment (Figure 2) it is possible to define a kinematic chain with five elements and two degrees of freedom, so the position of each member in the sagittal plane is determined by the length of each segment and two of the following angles:  
1. \(\theta_{\mathrm{c}}\), angle between the bicycle's frame and the crank;
2. \(\theta_{\mathrm{p}}\), angle between the crank and the pedal;
3. \(\theta_{\mathrm{g}}\), knee angle identified as in Figure 2;
4. \(\theta_{\mathrm{s}}\), hip-saddle angle identified as in Figure 2.  
Figure 1: Lower limb muscles that contributes to cycling involved in the model.  
The length of each segment of the model was determined by direct measurement. Once kinematic data and pedal forces are obtained--measuring \(\theta_{\mathrm{c}}\), and \(\theta_{\mathrm{p}}\) and calculating the other angles as reported in [15] (\(\theta_{\mathrm{s}}\)(\(\theta_{\mathrm{c}}\), \(\theta_{\mathrm{p}}\)) and \(\theta_{\mathrm{g}}\)(\(\theta_{\mathrm{s}}\)))--ankle, knee and hip joint moments are calculated using the inverse dynamics, using inertial parameters given by literature [34]. Afterwards, these data are used to implement the three equilibrium equations at each joint, involving the following muscles, that represent the minimum set to be involved in the model: (1) Tibialis anterior (TA); (2) Soleus (SO); (3) Gastrocnemius (GA); (4) Vastii (VA); (5) Rectus femoris (RF); (6) Short head of biceps femoris (BFs); (7) Long head of Biceps Femoris (BFI); (8) Iliacus (IL); (9) Gluteus Maximum (GLM). The relation between the muscular moments and the muscular forces at each joint j is given by the equation:  
\[\sum_{i=1}^{N_{j}}F_{i}\;\times d_{ij}=M_{j} \tag{1}\]  
where M\({}_{j}\) represents the muscular moment at the \(j\)-th joint, N\({}_{j}\) is the number of muscles acting on the \(j\)-th joint, F\({}_{i}\) is the muscular force exerted by the \(i\)-th muscle and d\({}_{ij}\) is the effective moment arm of the \(i\)-th muscle from the \(j\)-th joint. The values of muscular moment arms were calculated as a second order function of the joint angle in % of the length of the segment on which muscle belly is located, based on the equation reported in [35]. As the number of equations is not sufficient to calculate muscular force values, these were calculated by minimizing the cost function:  
\[U=\sum_{i=1}^{p}\left(\frac{F_{i}}{PCSA_{i}}\right)^{3} \tag{2}\]  
associated with the boundary conditions:  
\[0<F_{i}<F_{imax} \tag{3}\]  
here, given \(p\) total number of muscles, and being PCSA\({}_{i}\) and F\({}_{imax}\) respectively the physiological cross sectional area and the maximum force value for the \(i\)-th muscle, obtained by the literature [34]. The cubic exponent used in the equation (3), guarantees the best tradeoff between the muscular contractile force and the maximum duration of the contraction. This cost function is widely used in literature [36,37] as it relies on the co-activation of all the muscles involved in the gesture.  
Figure 2: Kinematic chain of lower limb and angle between body segment.