# 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
In the N2M model, distinct asymptotes in both the elastic and plastic sections differentiate the soil's elastic and plastic properties. The sinkage equipment for experimental equipment is shown in Figure 28. Then  
\[p=\bigg{(}\frac{C_{m}}{A^{m}}+\frac{s_{m}}{A^{1-m}}z\bigg{)}\bigg{(}1-\exp \bigg{(}-\frac{s_{0}}{C_{m}}\frac{z}{A^{1-m}}\bigg{)}\bigg{)} \tag{10}\]  
where \(A\) is the diameter of the contact surface, \(C_{m}\), and \(s_{0}\), \(s_{m}\), and \(m\) are model parameters. The sinkage index \(N\) serves as a variable to represent the impact of terrain characteristics and various other factors. [PERSON] formulated a model capable of mirroring the impact of normal load, the size of the plate or transmission, and slip on the relationship between pressure and sinkage [88]. The plate sinkage experiment is shown in Figure 29a. The pressure-sinkage relationship is shown in Figure 29b. The general form of the model that takes into account the influencing factors is shown in Equation (11). Then  
\[p=k_{\text{S}}z\lambda_{\text{N}} \tag{11}\]  
where \(k_{\text{S}}\) is the stiffness modulus of the terrain, which plays a leading role in determining the load-bearing performance of the terrain, in Pa/m. \(\lambda_{\text{N}}\) is a dimensionless function that reflects other key factors. Figure 28: Sinkage equipment for experimental equipment [87]. Figure 29: Sinkage experiment and pressure–sinkage relationship [88]: (**a**) plate sinkage experiment; (**b**) pressure–sinkage relationship. A Theoretical Exploration of the Wheel-Legged Composite Robots  
[PERSON] and [PERSON] proposed the Hunt-Crossley model. It describes the relationship between the equivalent stiffness and damping of the contact between the object and the terrain. The physical characteristics and the boundary conditions are fully revealed during the contact process [89]. The Hunt-Crossley model was applied by NASA to the study of the lunar hexapod robot ATHELETE. The wheels are modeled as three-dimensional springs to calculate reaction force and deformations [90]. The research results indicate that the model can accurately predict the sinking phenomenon of the robot and has sufficient accuracy for gait planning and execution. Future work can further improve the model to enhance prediction accuracy and apply it to more types of robots. NASA's ATHELETE robot is shown in Figure 30a. The reaction force and deformation are shown in Figure 30b. The Hunt-Crossley model is shown in Equation (12). Then  
\[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]0  
where \(\delta\) is the sum of foot and terrain deformations. \(C_{\text{N}}\) is the damping coefficient. \(n_{1}\) and \(n_{2}\) are the indicators of the stiffness terms. \(m\) is the exponent of the damping term, which can be set to 0 (the linear spring damping model) or \(n_{1}\) (the simplified Hunt-Crossley model). \(k_{\text{N}}\) is the equivalent stiffness coefficient. Then, the equivalent stiffness \(k_{\text{N}}\) can be obtained by Equation (13). Then  
\[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]1  
A Theoretical Exploration of the Legged Robots  
[PERSON] and [PERSON] conducted comprehensive research on sandy soil and clay by integrating the bearing capacity model introduced by [PERSON] and [PERSON] with the pressure-sinkage models of [PERSON] and [PERSON] [91]. Considering the influence of the size and shape of the contact object, different parameters are provided for the shape of the contact surface, such as circular, square, rectangular, and elliptical. The geometric parameters of the flat plate are shown in \[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]2 Through experimental verification, a new pressure-sinkage model is proposed, as shown in Equation (14). Then  
\[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]3  
where \(k_{1}\) and \(k_{2}\) are the soil shear strength values. \(\alpha\) and \(\beta\) are dimensionless geometric constants.