# Model Predictive Control of Running Biped Robot
## 5 Performance Validation
### Running on Flat Terrain  
Simulations of the biped robot running on a flat terrain were performed. The duration of the support phase, \(T_{s}\), was initially set at 0.1 s and it was gradually decreased to 0.07, 0.065, and 0.060 s as the robot sped up. The duration of the flight phase, \(T_{f}\), remained constant at 0.1 s in all runs. Before the robot started running, it bent its knees such that making the COM located 0.4469 m from the ground, which was used as \(Z^{ref}\). Table 3 shows how \(T_{s}\) and \(\lambda\) changed in a simulation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Symbols** & **Values** & **Unit** & **Symbols** & **Values** & **Unit** \\ \hline \(k\) & 2000 & kN/m & \(\mu_{d}\) & 0.8 & - \\ \(c\) & 1.0 & kNs/m & \(\mu_{s}\) & 1.0 & - \\ \(m_{1}\) & 1.3 & - & \(\
u_{d}\) & 0.15 & m/s \\ \(m_{2}\) & 1.0 & - & \(\
u_{s}\) & 0.1 & m/s \\ \(m_{3}\) & 2.0 & - & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Friction parameters. \begin{table}
\begin{tabular}{c c c} \hline \hline \(T_{s}\) & **Simulation Time** & \(\lambda\) \\ \hline
0.100 s & 1.10 s\(\sim\)3.00 s & 6.69 \\
0.070 s & 3.00 s\(\sim\)4.02 s & 7.35 \\
0.065 s & 4.02 s\(\sim\)5.34 s & 7.51 \\
0.060 s & 5.34 s\(\sim\)10.00 s & 7.69 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Duration of support phase. The control parameters used in the simulations, such as the weights and parameters for constraints of MPC and momentum control, were listed in Table 4.  
To measure the running speed relative to the size of the robot, Froude number (\(F_{r}\)), a dimensionless number defined by  
\[F_{r}=\frac{v^{2}}{g!},\]  
was often used (e.g., Refs. [43; 44; 45]), where \(v\) is the speed of the robot and \(l\) is the leg length. Five simulations were carried out, each with a different target speed: \(2.5\) m/s (\(F_{r}=1.0194\)), \(3.5\) m/s (\(F_{r}=1.9980\)), \(4.5\) m/s (\(F_{r}=3.3028\)), \(5.5\) m/s (\(F_{r}=4.9337\)) and \(6.5\) m/s (\(F_{r}=6.8909\)). Figure 7a shows how the foot placement relative to the hip changed when the target speed was \(4.5\) m/s. Companying this with Figure 7b, it is clear that the size of the footstep gradually increased as the speed of the robot increased in \(1.8\leq t\leq 4\), and finally settled around \(0.15\) m. The speed of the robot closely tracked the desired at \(4.5\) m/s as shown in Figure 7b. \begin{table}
\begin{tabular}{c c c} \hline
**Symbols** & **Description** & **Value** \\ \hline \multirow{3}{*}{\(\alpha_{\text{COM}}^{X}\)} & Weight of MPC for control & \\  & input in the horizontal & \(1.0\) \\  & direction & \\ \(\beta_{\text{COM}}^{X}\) & Weight of MPC for speed in & \\  & the horizontal direction & \(2.9\times 10^{7}\) \\ \(\gamma_{\text{COM}}^{X}\) & Weight of MPC for ZMP & \(1.1\times 10^{11}\) \\ \(\alpha_{\text{COM}}^{Z}\) & Weight of MPC for control & \(1.0\) \\ \(\beta_{\text{COM}}^{Z}\) & Weight of MPC for speed in & \(3.9\times 10^{7}\) \\ \(\gamma_{\text{COM}}^{Z}\) & Weight of MPC for \(\lambda\) & \(1.2\times 10^{11}\) \\ \(\mu\) & Friction coefficient for friction & \(0.65\) \\  & cone of MPC & \\ \(\alpha_{B}^{L}\) & Weight of momentum control & \(1.0\) \\  & for control input & \(0.0\) \\ \(\beta_{B,l}^{L}\) & Weight of momentum control & \(4.7\times 10^{7}\) \\ \(\gamma_{B,l}^{L}\) & for linear momentum & \(7.6\times 10^{7}\) \\ \(\beta_{B,a}^{L}\) & Weight of momentum control & \(2.2\times 10^{4}\) \\ \(\gamma_{B,a}^{L}\) & for angular momentum & \(0.1\) \\  & Scale factor of variable weight & \(0.1\) \\ \(\rho_{l}\) & for linear momentum & \(0.1\) \\  & Scale factor of variable weight & \(0.1\) \\ \(\rho_{a}\) & for angular momentum & \(0.1\) \\ \hline \end{tabular}
\end{table} 
Table 4: Control parameters. The angular motion of the upper body was generated based on momentum control. Figure (a)a shows how the pitch angle of the hip changed. During 1.8 s \(<t<\) 4 s, the robot's upper body leaned backward 5 degrees due to the body acceleration.