# Anti-Slip Gait Planning for a Humanoid Robot in Fast Walking
## 5 Simulation  
The above methods proposed in this paper are verified on the joint simulation platform composed of ADAMS 2017 and Simulink 2016b with a time step of 1 ms, which balances the calculation accuracy and speed. During the simulation, we made multiple calculations and adjustments. The finalized dynamics parameters of the simulation are listed in Table 1. These robot configuration parameters in Table 1 are chosen depending on the newly designed humanoid robot in Beijing Institute of Technology. It has a similar structure and the same number of degrees of freedom of the leg with P2, BHR-5 and HRP-5P [4; 5; 6], as shown in Figure 1. The advantage of the newly designed robot is the more powerful leg joints. At the same time, the ratio of the leg's mass to the total mass of the robot is larger. For this configuration, a walking step length of 0.44 m gives a small joint acceleration. Because of lack of stability control, long-distance walking introduces accidental factors, complicating the problem. So we set the number of walking steps to 6, which is enough to examine the effect of planning methods. In our previous experiments, the tested static friction coefficient \(\mu_{s}\) between the sole and the ground is 0.75. Since the friction coefficient \(\mu_{d}\) used for calculating the threshold moment \(T_{fcri}\) should be smaller than the static friction coefficient, we set \(\mu_{d}\) as 0.5. Under the condition of satisfying the friction constraint, reducing the double-support phase time ratio is beneficial to reduce the maximum swing velocity and acceleration. However, the swing leg switches to the support leg during the double-support phase, and the robot switches the support leg at the same time, so the time of double-support phase cannot be too short. In order to achieve a smooth trajectory as well as high stability of foot landing and lifting, the swing leg's motion in the z direction is planned by quartic interpolation, so that the speed and acceleration tend to be zero at the moment of lifting and landing the foot. However, due to the inevitability of a slight collision between the sole and the ground, it is easy to generate a large shift of the ZMP, which requires the double-support phase to provide a large support region. Therefore, it is necessary to maintain a certain double-support phase time to provide a large ZMP stability margin. After trial and error, we set the double-support phase time ratio as 0.1. Under these conditions, the resulting calculated minimum walking step cycle is 0.528 s and maximum walking speed is about 3.0 km/h. Figure 4 shows the snapshots taken from the video of biped walking in simulation which are totally planned by our proposed method (Case 1). The robot obtains a stable walking pace without any slipping at a speed of 3.0 km/h. \begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Value** & **Unit** \\ \hline trunk mass & 30 & kg \\ thigh mass & 10 & kg \\ shank (including foot) mass & 10 & kg \\ DoFs of single leg & 6 & - \\ CoM height of the inverted pendulum & 0.7 & m \\ walking step length & 0.44 & m \\ walking steps & 6 & - \\ ground friction coefficient & 0.75 & - \\ walking step cycle & 0.528 & s \\ single-support phase & 0.476 & s \\ constant walking speed & 3.0 & km/h \\ \hline \end{tabular}
\end{table} 
Table 1: Dynamics parameters in the simulation. Figure 4: Snapshots of biped walking in Case 1. The robot walks straightly along the center line all the time stably. In order to verify the superiority of our algorithm, we have also simulated three other cases. The following are the exact operations done in these four cases (see Supplementary Materials):  
Case 1: Plan the pattern exactly using the methods described in this article, namely, plan the swing leg's motion by the friction constraint method, and then compensate for the reference ZMP. Case 2: Plan the swing leg's motion by the friction constraint method, but don't compensate for the reference ZMP. The purpose of setting Case 2 is to examine the effect of ZMP compensation on walking stability.