# Dynamic Modeling of the Dissipative Contact and Friction Forces of a Passive Biped-Walking Robot
## 2 Biped Model
### Normal and Friction Force of the Model  
The normal and friction forces appear on the feet and ground only when it is in contact. This contact (normal) and friction (tangential) force acts on the point when the penetration appears. Since the leg impacts the ground, a small penetration appears between the feet and the ground, as shown in Figure 2. The contact force (and friction) can only be applied when a relative penetration appears in the feet. This relative penetration can be obtained with geometric equations of the interception between the sphere (feet) and plane (ground):  
\[\delta_{i}=r_{o}^{normal}-r \tag{6}\]  
where \(r_{o}^{normal}\) is the normal distance between the ground (plane) and the center of the feet (O), r is the radius of the feet and \(\delta_{i}\) is the penetration of the feet i. The penetration will only appear for positive values of \(\delta_{i}\). Additionally, the penetration can occur for one foot, both or none if there is bounce. The distance between the lowest point of the feet and the ground is defined in Equation (6). Figure 2: The contact model of the biped feet with small penetrations. Most models of the literature use [PERSON]'s contact model, which is restricted to frictionless surfaces and perfectly elastic solids and does not dissipate energy [28]. To describe the possible dissipation of energy, several models of dissipative contact force (a modified Hertz contact law) have been compared: the [PERSON] [29] and [PERSON] et al. [30], [PERSON], Hunt & Crossley, [PERSON], and [PERSON] [22]. We came to the conclusion that the viscoelastic contact model between the ground and the feet that works best for this model is the Dissipative Nonlinear Flores Contact Force Model (hysteresis damping parameter - energy dissipation):  
\[F_{N}=k\delta^{3/2}+X\delta^{3/2}\dot{\delta}. \tag{7}\]  
where F\({}_{\text{N}}\) is the contact normal force, \(\delta\) is the relative penetration and \(\dot{\delta}\) denotes the relative normal contact velocity. K and X are the stiffness parameter and the hysteresis damping factor, respectively, which are dependent on the radius of the feet and the material properties of the feet and floor. In this research, the dissipation of energy due to the impacts has been considered. The contact and penetration between the feet and the ground is obtained with geometric equations. When a foot of one leg is in contact with the ground (applying a penetration), that support leg swings on its contact. The other leg, which is in the air, must be able to perform the whole step without scratching the floor. To avoid this, in the experiments, a special track was built to avoid the oscillating leg from scratching the floor. To apply the same to the simulations, the condition that the contact will occur only when \(\delta>0\) and \(\omega>0\) has been implemented. Thanks to that, the model avoids scratching the feet like in the [PERSON] model [31]. The prototype of passive walking with the [PERSON] robot is shown in Figure 3.  
For the friction model, the majority of passive biped models in the literature uses Coulomb's dry friction law: a friction model without the Stribeck effect. However, we consider that the Stribeck effect is very important for a model with a dissipative viscoelastic contact, and for this reason several friction forces have been analyzed in this study, obtaining that the friction force with the Stribeck effect that works better for this model is Bengisu and Akay. The Bengisu and Akay [32] friction force graph and its two equations are shown in Figure 4.  
Figure 3: The prototype of PASSIVE BIPED WALKING. This model has the great advantage that for very small velocity values, close to zero, the friction force is low and realistic, although there has been slippage. This is ideal for this model, which always has low velocities close to sliding [33]. To make these results realistic, the increase in time in the simulation must be small. For this passive biped, \(\xi=1000\) and \(\mathrm{v}_{0}=0.0001\) m/s.