# A Review of Foot-Terrain Interaction Mechanics for Heavy-Duty Legged Robots
## 3 Dynamics Analysis of Robot
### Models of Pressure-Sinkage for Mobile Robot
#### 3.1.1 Models for Pressure-Sinkage at Zero Slip Conditions
\[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]4 NASA’s ATHELTE robot [90]: (**a**) outdoor experiment; (**b**) reaction force and deformation. [PERSON] from Jilin University designed and manufactured four typical structures of feet, namely hemispherical feet, semi-cylindrical feet, rectangular feet, and circular feet. Research has been conducted on how the size, shape, and density of quartz sand particles affect the matrix's physical characteristics and the mechanical performance of foot penetration. On the three types of quartz sand, the intrusion resistance and pressure of the hemispherical feet are lower than those of the other three mechanical feet. It was found that as the particle size of quartz sand increases, the invasion resistance of the mechanical foot first increases and then decreases. The revised model has been obtained [92]. The corrected integral equations are shown in \[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]5 The mechanical feet and intrusion testing equipment are shown in \[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]6  
Furthermore, [PERSON] posited that in the normal direction, the interaction force typically resembles the force exerted on a spring and damper located at the foot. The spring damping model shows the interaction between the robot's foot and the terrain in the normal direction [93]. The foot-terrain interaction in the normal direction is shown in \[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]7 The mathematical model of spring damping can be shown in Equation (15). Then  
\[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]8  
where \(F_{\text{TN}}\) is the normal force exerted by the terrain on the foot. \(m_{\text{F}}\) is the mass of the foot. \(\delta_{\text{T}}\) and \(\delta_{\text{F}}\) represent the deformation of the terrain and feet, and \(\delta\) is the sum of them. \begin{table}
\begin{tabular}{c c} \hline \hline
**Plate Shape** & \(\beta\) \\ \hline Circular & 4 \\ Square & 4 \\ Rectangular & \(2(a+b)/a\) \\ Elliptical & \(\left\{\begin{array}{ll}2(a+b)/a,&\text{Max}\\ 4\sqrt{\frac{1}{2}(a^{2}+b^{2})}/a,&\text{Min}\end{array}\right.\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometric parameters of the flat plate. Figure 31: Mechanical feet and intrusion testing equipment [92]. If the damping is ignored and the spring is linear, the mathematical model of the normal force can be shown in Equation (16). Then  
\[F_{\text{N}}=k_{\text{FN}}\delta_{\text{F}}=k_{\text{TN}}\delta_{\text{T}} \tag{16}\]  
where \(k_{\text{FN}}\) is the normal stiffness coefficient of the foot. \(k_{\text{TN}}\) is the normal stiffness coefficient of the terrain. When \(\delta=\delta_{\text{F}}+\delta_{\text{T}}=F_{\text{N}}/k_{\text{FN}}+F_{\text {N}}/k_{\text{TN}}\), the equivalent equation for the mechanics of foot-terrain contact can be rewritten as follows:  
\[F_{\text{N}}=\frac{\delta}{1/k_{\text{FN}}+1/k_{\text{TN}}} \tag{17}\]  
When a legged robot is in motion, the normal load on the foot is not constant. During the process of contact between the foot and the terrain, the deformation of the terrain and the speed of the foot are also in a state of change. [PERSON] believes that during the foot-terrain interaction, the foot tip may shift multiple times and come into contact with the terrain again, including before contact, contact, departure, and recontact [94]. The first contact process can be represented by the Hunt-Crossley model, while the second contact process can be represented by an improved model. The improved model of normal force can be obtained and shown in Equation (18). [PERSON] and [PERSON]'s research lays the foundation for improving the terrain stability of the heavy-duty hexapod robot ELSpider.