## 2 Robot Modelling
### The Foot Force Distribution  
When analysing the foot force distribution, the slippage between the foot endpoint and the ground is neglected [13]. For the sagittal motions, the lateral forces of the foot endpoints can be ignored, and the ground reaction forces of each leg can be divided into a normal and a tangential component.  
The ground reaction force can cause many negative effects like structural damages and control difficulties; therefore, the value of the ground reaction force must be minimised. Here, the minimization of norm of the foot force method is used to minimize the norm solution of the foot force and to reduce the compact between the ground and the robot [14].  
A coordinate frame \(\{O_{b}\}\) is established at the geometric center of the body. The \(x\) axis points to the direction of the motion. The direction of the \(z\) axis is vertical to the ground and points upwards. The \(y\) axis is defined by the right-hand rule. The static force diagram is illustrated in Figure 3. Here, we assume the RF and LH (left-hand) legs are in the stance phase. \(\mathbf{F_{G}}=[f_{RF},f_{LH}]^{T}\) is the ground reaction force vector which contains the ground reaction force of the RF and LH legs, where \(f_{RF}=[f_{RF,x},f_{RF,z}]^{T}\) and \(f_{LH}=[f_{LH,x},f_{LH,z}]^{T}\). The vector \(\mathbf{V}=[F_{x},F_{z},T_{y}]^{T}\) contains the forces and moment acting on the robot COM in sagittal plane. Under these conditions, the forces and moment balance equations can be written as follows:  
\[F_{x}=f_{RF,x}+f_{LH,x} \tag{17}\]  
\[F_{z}=f_{RF,z}+f_{LH,z} \tag{18}\]  
\[T_{y}=f_{RF,x}z_{RF}-f_{RF,z}x_{RF}+f_{LH,x}z_{LH}-f_{LH,z}x_{LH} \tag{19}\]  
Figure 3: The static force diagram of SCalf.  
The Equations (17) to (19) can be written in a matrix form as follows:  
\[\mathbf{B}\cdot\mathbf{F_{G}}=\mathbf{V} \tag{20}\]  
where  
\[\mathbf{B}=\left[\begin{array}{cccc}1&0&1&0\\ 0&1&0&1\\ z_{RF}&-x_{RF}&z_{LH}&-x_{LH}\end{array}\right] \tag{21}\]  
In Equation (21), \(\{x_{RF},z_{RF}\}\) and \(\{x_{LH},z_{LH}\}\) are the coordinates of the support foot endpoints in \(\{O_{b}\}\).  
In Equation (20), there are 4 unknowns but only 3 equations. Here, we use the least squared method to realize the minimization of norm of the foot force method. The result is shown as follows.  
\[\mathbf{F_{G}}=\mathbf{B}^{T}(\mathbf{B}\mathbf{B}^{T})^{-1}\mathbf{V} \tag{22}\]  
In Equation (22), the matrix \(\mathbf{B}^{+}=\mathbf{B}^{T}(\mathbf{B}\mathbf{B}^{T})^{-1}\) is also the pseudoinverse matrix of \(\mathbf{B}\). As the third component of \(\mathbf{V}\) (\(T_{y}\)) is zero, the parameters in the last column of \(\mathbf{B}^{+}\) can be neglected. The value of \(\mathbf{B}^{+}\) is shown as follows.  
\[\mathbf{B}^{+}=\left[\begin{array}{cccc}\frac{b_{11}^{+}}{2\Lambda}&\frac{b_{12 }^{+}}{2\Lambda}&\times\\ \frac{b_{21}^{+}}{2\Lambda}&\frac{b_{22}^{+}}{2\Lambda}&\times\\ \frac{b_{31}^{+}}{2\Lambda}&\frac{b_{32}^{+}}{2\Lambda}&\times\\ \frac{b_{41}^{+}}{2\Lambda}&\frac{b_{42}^{+}}{2\Lambda}&\times\end{array}\right] \tag{23}\]  
where  
\[b_{11}^{+}=(x_{RF}-x_{LH})^{2}+2z_{LH}^{2}-2z_{LH}z_{RF} \tag{24}\]  
\[b_{12}^{+}=(x_{RF}+x_{LH})(z_{RF}-z_{LH}) \tag{25}\]  
\[b_{21}^{+}=(x_{RF}-x_{LH})(z_{RF}+z_{LH}) \tag{26}\]  
\[b_{22}^{+}=(z_{RF}-z_{LH})^{2}+2x_{LH}^{2}-2x_{LH}x_{RF} \tag{27}\]  
\[b_{31}^{+}=(x_{RF}-x_{LH})^{2}+2z_{RF}^{2}-2z_{LH}z_{RF} \tag{28}\]  
\[b_{32}^{+}=(x_{RF}+x_{LH})(z_{LH}-z_{RF}) \tag{29}\]  
\[b_{41}^{+}=(x_{LH}-x_{RF})(z_{RF}+z_{LH}) \tag{30}\]  
\[b_{42}^{+}=(z_{RF}-z_{LH})^{2}+2x_{RF}^{2}-2x_{LH}x_{RF} \tag{31}\]  
\[\Lambda=(x_{RF}-x_{LH})^{2}+(z_{RF}-z_{LH})^{2} \tag{32}\]  
The Jacobian matrix in Equation (16) can be written as follows:  
\[\mathbf{J_{G}}=\left[\begin{array}{cc}\mathbf{J_{F}}&\mathbf{0}\\ \mathbf{0}&\mathbf{J_{H}}\end{array}\right] \tag{33}\]  
In Equation (33), \(\mathbf{J_{F}}\) and \(\mathbf{J_{H}}\) are the Jacobian of the RF and LH legs respectively.