# Dynamic Modeling of the Dissipative Contact and Friction Forces of a Passive Biped-Walking Robot
## 5 Conclusions  
The supporting foot slippage and the viscoelastic dissipative contact force of the passive walking biped has been studied and modeled. The smooth forward dynamics model for the whole walking is presented and verified. Several contact forces and friction forces are compared, demonstrating that the ones that work best in a passive biped model are the Flores contact/impact normal force and Bengisu friction.  
Knowing that the model works properly, a sensitivity analysis has been carried out, obtaining interesting results with different numerical methods to solve the equation of motion (standard, augmented, baumgarte, penalty, the index-1 and direct) and different integrator schemes (ode integrator scheme, Euler integrator scheme and the Runge-Kutta methods). These results have shown their consistency. With these results, it can be concluded that, for this model, the augmented method or baumgarte method, combined the Runge-Kutta integrator, is a good solution, keeping the constraint violations under an intimate value with a low computational cost.  
This model can be implemented in parametric programs, rendering the opportunity to carry out simulations of different mechanisms, bipeds, robots and machines possible. We believe that this is an important contribution to the scientific community. The results show the robustness and reliability of this dynamic model with impact and friction. Thus, this same method could be used to develop other models with impact and friction and obtain reliable simulations.  
Some of the most interesting results that have been obtained with the simulations of the model are: In this research, it is easy to see that there is a proportional relationship between mechanical work (gravitational potential energy) and metabolic/passive cost (impacts). We can conclude that a lower contact stiffness and sufficiently high coefficient of restitution correspond to soft feet and the biped walking more steadily. The damping factor does not change the stability of the passive biped walking; it influences the bounce between the feet and the ground. With a very low and very high constant of damping, the biped bounces too much and becomes unstable. The friction coefficient is only related to the slippage. For a very small coefficient of friction, the walking biped slips. The changes of the stiffness value modify the dissipation energy and the time-period (the walking length is larger).  
Figure 17: Number of steps for different coefficients of restitution and friction coefficients.  
**Author Contributions:** Investigation, Conceptualization and Software: E.C.; Software: M.J.G.G.; Methodology: J.M. resources and formal analysis: C.C.; Writing--review & editing: R.G. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was financially supported by the Spanish Government through the MCYT project \"RETOS2015: sistema de monitorzacion integral de conjuntos mecanicos criticos para la mejora del mantenimiento en el transporte-maqstatus.\".  
**Acknowledgments:** The authors would like to thank the collaboration between the University of Minho and the University Carlos III of Madrid, with which it has been possible to carry out this research.  
**Conflicts of Interest:** The authors declare that they have no conflict of interest.