# Effects of Torso Pitch Motion on Energy Efficiency of Biped Robot Walking
## 5 Simulation Results and Discussion
### Simulation Results of TPM  
The following shows the details of TPM when the step length is 0.3 m and the gait period is 1.0 s.  
Figure 4 shows the angle trajectory of each joint, and Figure 5 shows the angular velocity trajectory of each joint. It can be seen that the joint angle is smooth and the joint angular velocity is continuous. However, the angular velocity is not smooth when switching from DSP to SSP. This means that the angular acceleration is not continuous at this moment, because the ankle of the swing leg starts to lift at that time, and its acceleration is not continuous. Similarly, the joint angular acceleration is also discontinuous when the SSP is switched to the next DSP because the acceleration of the swing ankle is discontinuous at that moment.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mass (kg)** & **Length (m)** & **CoM 1 (m)** & **Inertia (kg.m2)** \\ \hline Torso & 31.047 & 0.657 & 0.309 & 1.405 \\ Thigh & 3.305 & 0.35 & 0.187 & 0.0275 \\ Shank & 1.385 & 0.282 & 0.117 & 0.0134 \\ \hline \hline \end{tabular}  
* It denotes the distance between the center of mass (CoM) of each link and its lower joint.  
\end{table}
Table 1: Physical parameters of the biped model.  
Figure 6 shows the trajectories of joint torques. In DSP, because the system is over-actuated, the number of independent actuators is greater than the degrees of freedom; therefore, the driving torque of the two joints is released, and only \(\tau_{2},\tau_{4},\tau_{5}\) are actuated. In SSP, we can see that the driving torque of the knee joint of the support leg is much larger than that of the other joints. It plays the role of support the whole upper body and swing leg.  
Figure 4: Angle trajectories.  
Figure 5: Trajectories of angular velocities.  
According to the planned gait and the dynamic equation, the ground reaction force on the robot foot can be obtained, as shown in Figure 7, where \(\left[F_{1x};F_{1x}\right]\) is the ground reaction force (GRF) on support foot, \(\left[F_{2x};F_{2x}\right]\) is the GRF on swing foot. It is observed from Figure 7 that (1) the swing foot is only subjected to the ground reaction during the DSP; (2) the normal component of the GRF is positive, which ensures the unilateral constraint characteristics of the GRF; (3) the tangential component of the GRF is much smaller than the normal component of GRF, which ensures no sliding between the foot and the ground; and (4) during SSP, the normal component of the GRF applied on the support foot is approximately equal to the gravity of the robot (about 400 N).  
Figure 8 shows the ZMP trajectory and the upper and lower boundaries of the ZMP in one step. The stability margin of ZMP is \(l_{af}\) = 0.1 m, \(l_{ab}\) = 0.05 m, while the horizontal distance from ankle to toe is 0.13 m and the horizontal distance from ankle to heel is 0.07 m. It can be seen from the figure that ZMP is within the stability margin, which ensures the stability of the walking.  
Figure 6: Torque trajectories.  
Figure 7: Trajectories of ground reaction force (GRF).  
Figure 9 shows the stick diagram of the five-link robot walking on the horizontal ground, in which the dashed line represents the DSP, and the solid line represents the SSP. From the stick diagram, we can observe the overall movement of the biped robot, i.e., that the posture at the end of each step is the same as that at the beginning of each step.  
Generally, the gait pattern designed by the proposed method is simple and easy to implement. Although the contact constraint between the ground and the foot is not considered in the gait generation, the simulation results verify that the friction force is very small relative to the support force during walking. Therefore, the biped robot can walk without sliding even if the friction coefficient between the foot and the ground is small.