## 3 Turning Gait Planning in the Non-Inertial Reference (IRF)  
### Planning of the COM Trajectory  
In order to make the description more intutive, we first define three directions related to the robot. The forward direction is called the \(y\) direction. The leftward and rightward direction is called the \(x\) direction. The vertical direction is called the \(z\) direction. As the basic model of the robot walking, the LIP model simplifies the complex dynamic model of the robot. As is shown in Figure 4, the trajectory of the COM can represent the robot motion when the robot only moves in translation. \[\dot{x}_{\rm{com}}=\omega^{2}(x_{\rm{com}}-x_{\rm{zmp}}). \tag{5}\]  
Figure 3: Different radii of the robot turning walk. Here, \(\omega=\sqrt{\frac{g}{h_{\rm com}}}\). \(h_{\rm com}\) is the height of the COM. \(x_{\rm zmp}\) is the position of the ZMP which can be planned ahead. \(x_{\rm com}\) is the position of the COM. The input of the LIP model is the \(x_{\rm zmp}\) and the output is the \(x_{\rm com}\). This model is applicable when the robot does not rotate around the z-axis. When compared with the straight-line walk, the system for the turning walk is more complex. For the turning walk, because of the existence of \(f_{a_{\rm r}},f_{a_{\rm r}},f_{rin}\), and \(f_{\omega}\). we must improve the traditional LIP model. As we can know from Section II, for the \(y\) direction in the IRF, the resultant of the inertial force is zero, so the traditional LIP is still applicable. However, for the \(x\) direction in the IRF, the resultant force is \(f_{sum}\), so we can get the model of the turning walk from Figure 5. In this section, we call the \(f_{\rm sum}\) force another name, \(f_{ct}\), in order to simplify the subsequent derivation. Because of force \(f_{ct}\), the ZMP moves by a short distance. Based on this new model, we can then obtain the new dynamics. \[x_{\rm com}^{\rm IRF}=\omega^{2}[x_{\rm com}^{\rm IRF}-(x_{\rm zmp}+\frac{y_{ \rm com}^{\rm IRF}}{\omega^{2}r_{\rm com}})], \tag{6}\]  
where \(r_{c}\) is the turn radius of the COM. \(x_{\rm zmp}^{\rm IRF}\) is the ZMP position in the IRF, and \(x_{\rm com}^{\rm IRF}\) is the COM position in the IRF. The \(x_{\rm zmp}^{\rm IRF}\) position of the new ZMP has changed compared to that in the straight-line walk, as shown in Formula (7):  
Figure 4: Linear inverted pendulum of the robot. Figure 5: Model of the linear inverted pendulum (LIP) of the robot’s turning walk in the \(x\) direction. \[x_{\rm zmp}^{\rm IRF}=x_{\rm zmp}+\frac{y_{\rm com}^{\rm IRF}}{\omega^{2}\tau_{ \rm com}}. \tag{7}\]  
So, the new LIP model can be expressed as follows:  
\[\dot{x}_{\rm com}^{\rm IRF}=\omega^{2}(x_{\rm com}^{\rm IRF}-x_{\rm zmp}^{\rm IRF }), \tag{8}\]  
We can, therefore, treat turning walk as straight walk when we add the ZMP deviation: The new state space of the LIP system can then be expressed as \(x^{\rm IRF}=[x_{\rm com}^{\rm IRF},\dot{x}_{\rm com}^{\rm IRF},\dot{x}_{\rm zmp }^{\rm IRF}]^{T}\). The input of the system is the velocity of the ZMP:  
\[\dot{x}^{\rm IRF}=\left[ \begin{array}{ccc}0&1&0\\ \omega^{2}&0&-\omega^{2}\\ 0&0&0\end{array} \right]x^{\rm IRF}+\left[ \begin{array}{c}0\\ 0\\ 1\end{array} \right]\dot{x}_{\rm zmp}^{\rm IRF}, \tag{9}\]  
After doing the above simplification, turning walk can be converted into straight-line walk in the IRF when the straight-line walk is applied to a force (\(f_{ct}\)), as shown in Figure 6. Because the velocity of the COM in the \(y\) direction is not constant, the force \(f_{ct}({\rm t})\) will also change all the time. According to Formula (7), we know that the \(\Delta x_{\rm zmp}({\rm t})\) is also not constant:  
\[f_{ct}({\rm t})={\rm m}\omega^{2}(x_{\rm zmp}^{\rm IRF}({\rm t})-x_{\rm zmp}( {\rm t}))={\rm m}\omega^{2}\Delta x_{\rm zmp}({\rm t}), \tag{10}\]  
where \(m\) is the mass of the robot. Through changes in the coordinates, this model can be divided into two components [7; 25]. One is the stable mode and the other is the unstable mode, as is shown in Formula (11):  
\[