# Design and Research of a Flexible Foot for a Multi-Foot Bionic Robot
## 3 Analysis of Mechanical Properties of the Foot End  
### Foot Force Analysis  
As shown in Figure 3, in order to facilitate the analysis of the foot-end force, three RFP (Yubo Intelligent Technology Co., Ltd. in Hangzhou, China.) film pressure sensors were placed at the contact plane between the foot end and the pressure plate. The main function of the three sensors was to perform data calculation and parameter identification. As shown in Figure 3a, each of the three sensors was at an angle of 120\({}^{\circ}\). The force model is shown in Figure 3b. The fixed surface was linked to the FSS pressure sensor through the push-pull rod. The non-fixed surface contacted the ground \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Symbol** \\ \hline Spring wire diameter & d \\ Outer diameter & D \\ inside diameter & D\({}_{2}\) \\ Free height & H\({}_{0}\) \\ Pitch & t \\ Spiral angle & \(\alpha\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spring parameter table. Figure 2: Spring structure. to form a deformation. This was equivalent to a spring with a spring constant of \(\mathrm{k_{2}}\). A fixed surface coordinate system O and a non-fixed surface coordinate system O\({}_{1}\) were established as shown in the figure. When the foot was perpendicular to the ground, the two coordinate systems were in the same direction, and the length of the flexible material in the natural state was \(\mathrm{h}\). When the force was applied in the vertical direction, we assumed the projection positions of the three RFP film pressure sensors on the fixed surface were \((x_{1},y_{1})\), \((x_{2},y_{2})\), and \((x_{3},y_{3})\), the fixed surface forces are \(\mathrm{F_{1}}\), \(\mathrm{F_{2}}\), and \(\mathrm{F_{3}}\), and the coordinates of the three points corresponding to the non-fixed surface were A \((x_{1},y_{1},\mathrm{h}-\mathrm{F_{1}}/\mathrm{k_{2}})\), \(\mathrm{B}\) (\(x_{2},y_{2},\mathrm{h}-\mathrm{F_{2}}/\mathrm{k_{2}}\)), and \(\mathrm{C}\) (\(x_{3},y_{3},\mathrm{h}-\mathrm{F_{3}}/\mathrm{k_{2}}\)). The torque of \(\mathrm{F_{1}}\) to O\({}_{1}\) is:  
\[\mathrm{M_{1}}=\overset{\rightarrow}{r}\times\overset{\rightarrow}{F}= \left| \begin{array}{ccc}\mathrm{i}&\mathrm{j}&\mathrm{k}\\ x_{1}&y_{1}&z_{1}\\ \mathrm{F_{1x}}&\mathrm{F_{1y}}&\mathrm{F_{1z}}\end{array} \right| \tag{10}\]  
where \(z_{1}=0\), \(\mathrm{F_{1x}}+\mathrm{F_{1y}}=0\), and \(\mathrm{F_{1z}}=\mathrm{F_{1}}\), so  
\[\mathrm{M_{1}}=\left( \begin{array}{ccc}x_{1}\mathrm{F_{1}}\\ -y_{1}\mathrm{F_{1}}\\ 0\end{array} \right)\left( \begin{array}{ccc}\mathrm{i}&\mathrm{j}&\mathrm{k} \end{array} \right). \tag{11}\]  
Similarly, the combined torque of the three RFP film pressure sensors is  
\[\mathrm{M_{1}}=\left| \begin{array}{c}\mathrm{M_{x}}\\ \mathrm{M_{y}}\\ \mathrm{M_{z}}\end{array} \right|=\left| \begin{array}{c}x_{1}\mathrm{F_{1}} +x_{2}\mathrm{F_{2}}+x_{3}\mathrm{F_{3}}\\ -y_{1}\mathrm{F_{1}}-y_{2}\mathrm{F_{2}}-y_{3}\mathrm{F}\\ 0\end{array} \right|. \tag{12}\]  
The combined torque is  
\[|\mathrm{M}|=\sqrt{\mathrm{M_{x}^{2}}+\mathrm{M_{y}^{2}}+\mathrm{M_{z}^{2}}}= \sqrt{\mathrm{M_{x}^{2}}+\mathrm{M_{y}^{2}}}. \tag{13}\]  
The angle between the direction of the force and the \(x\)-axis is  
\[\alpha=\arccos(\mathrm{M_{x}}/|\mathrm{M}|). \tag{14}\]  
The joint force is  
\[\mathrm{F}=\mathrm{F_{1}}+\mathrm{F_{2}}+\mathrm{F_{3}}. \tag{15}\]  
Figure 3: Mechanical analysis model of foot tip: (**a**) device model; (**b**) force model. Take three non-collinear vectors in a non-fixed plane:  
\[\left\{ \begin{array}{l}\stackrel{{\rightarrow}}{{\rm AB}}=(x_{2}-x_{1 },\ y_{2}-y_{1},\ ({\rm h-F_{2}/k_{2}})-({\rm h-F_{1}/k_{2}}))\\ \stackrel{{\rightarrow}}{{\rm AC}}=(x_{3}-x_{1},\ y_{3}-y_{1},\ ({\rm h-F_{3}/k_{2}})-({\rm h-F_{1}/k_{2}}))\\ \stackrel{{\rightarrow}}{{\rm BC}}=(x_{3}-x_{2},\ y_{3}-y_{2},\ ({\rm h-F_{3}/k_{2}})-({\rm h-F_{2}/k_{2}}))\end{array} \right\} \tag{16}\]  
Let the normal vector of the non-fixed surface be  
\[{\rm n}=(\alpha,\beta,\gamma). \tag{17}\]  
Thus:  
\[\left\{