# Dynamic Modeling of the Dissipative Contact and Friction Forces of a Passive Biped-Walking Robot
## 3 Verification of the Model  
In order to verify the model of this paper, the results obtained with the same initial conditions as in the real experiment were contrasted. [PERSON] et al. [31] built a passive biped and a special ground, as shown in Figure 3. They used the step sounds to determine the time between steps (period). The sound recording is shown in Figure 5. The two significant high peaks occurred because the microphone was close to those impacts. Figure 4: Representation of the [PERSON] and Akay model. These results have been used and verified by more authors: [PERSON] et al. [27] found a stable period of the passive dynamic walker's gait in which the leg does not bounce or slip, and he checked his results by using [PERSON]'s experiments. In order to verify the rationality of the impact/contact and the feasibility of the model in this document, the initial parameters and conditions of [27] and [31] are used in the simulation, and the results of the simulation are compared with them. [PERSON] obtained the time between steps for several experiments. Figure 6 shows the time period of every step of [PERSON]'s experiments compared to the time period of our model under the same initial conditions. The error between the model and the average of the experiments of every step are: 8.02%, 0.42%, 1.69%, 2.56%, 7.69%, 0.00% and 4.00%. From these values, a maximum detected error of 8% is obtained, as well as an average error of less than 2%. With this we can verify that, for the same initial conditions, the experimental results coincided with the simulation. It can be seen in Figure 7 that with these initial conditions the biped robot is walking with a stable and passive gait (constant stride and period) during the slop. These figures show one completed gait of the biped model: a set of pictures of the motion produced by the passive biped-walking robot in different phases of the gait. It can been noted that the walking is stable and passive along the slop. Figure 5: Step sound recording of the experiments. Figure 6: Comparison of the time period. The penetration can be seen in Figure 8, and the normal force can be observed in Figure 9. It can be observed how the period of these graphs is the same as the period obtained from the experiment. In both of these figures, it can be observed that, as there is a great initial impact when the foot hits the ground, this causes a large penetration, for an instant of time, and after that initial impact the values are stabilized. This impact/impulse is transmitted from one foot to the other one, in some cases causing bounces, instability or chaos. It is also observed that when the foot is in the air, it does not suffer a normal force or cause penetration. These graphs are one foot, and the graphs of the other foot would be symmetrical. Figure 7: Passive model of the 2D/3D gait. Figure 11 displays the position constraints violation for all the methods, and Figure 12 shows all the methods except the standard one. The scale in Figure 11 is different from the scale in Figure 10. In this analysis, all the methods named above (standard, baumgarte, augmented, penalty, index1, index1 aug and described) are utilized to solve the dynamics system. Figure 11: All the position constraints violations. Figure 10: Leg angle for different slopes. It can be seen that the constraints violations are getting out of control with the standard method. Nevertheless, with the other methods (baumgarte, augmented, penalty, index1, index1 aug and described) the constraint violations remain under control indefinitely. It can be observed that the constraints violation of the augmented lagrangian formulation, baumgarte stabilization approach and the direct correction approach have the same order of magnitude and that the results are similar. With these results, the robustness of the model has been demonstrated. It works properly with all the methods. Figure 13 compares the computational time for different solving methods. The standard method (standard lagrange multiplier) is the most efficient.