# A Review of Foot-Terrain Interaction Mechanics for Heavy-Duty Legged Robots
## 3 Dynamics Analysis of Robot
### Models of Pressure-Sinkage for Mobile Robot
#### 3.1.2 Models for Pressure-Sinkage at Non-Zero Slip Conditions  
When the robot's foot slides on the terrain, a tangential force perpendicular to the normal pressure direction is generated. The tangential force can lead to tangential deformation, causing lateral soil loss and resulting in slip sinkage. Therefore, adding very little cohesive moist sandy soil to hard soil can also reduce slip sinkage. [PERSON] discovered that the sinkage occurring when the robot is functioning on soil with non-zero slip can be described as a combination of static sinkage and sliding sinkage. The soil deformation model is shown in Equation (21). Then  
\[z=z_{o}+z_{j} \tag{21}\]  
where \(z_{o}\) represents static sinkage, and \(z_{j}\) represents dynamic sinkage.  
The model for predicting soil sliding was proposed by [PERSON] [86]. The [PERSON] prediction model is shown in Equation (22). Then  
\[z=z_{\text{o}}+h_{\text{gr}}i/(1-i) \tag{22}\]  
where \(h_{\text{gr}}\) is the grouser height, and \(i\) is the slip ratio.  
A new non-zero slip model was studied by [PERSON]. This model can be used to represent the relationship between pressure and settlement [96]. The [PERSON] model is shown in Equation (23). Then  
\[z=z_{\text{o}}+iH_{\text{p}} \tag{23}\]  
where \(H_{\text{p}}\) is the propagation depth of soil deformation that can only be evaluated through experiments.  
[PERSON] [97] considered the phenomenon of foot sinkage in the normal direction due to rotation. Through experimental verification of the hemispherical foot of the planetary exploration robot CREX, a semi-empirical formula for sinkage, rotation direction angle, and normal stress was established. The outdoor experiment of CREX and the rotating sinkage  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Name** & **Model Parameters** & **Equation Number** & **References** \\ \hline [PERSON] & \(k\), \(n\) & (5) & [80,81] \\ [PERSON] & \(k_{c},k_{\phi},b,n\) & (6) & [82,83,84] \\ [PERSON] & \(k_{c},k_{\phi},k_{1},k_{2},b,n,c,\gamma\) & (7), (8) & [86] \\ [PERSON] & \(C_{m},s_{0},s_{m}\) & (10) & [87] \\ [PERSON] & \(k_{\text{S}},A_{\text{N}}\) & (11) & [88] \\ [PERSON]–Crossley & \(\delta,n_{1},n_{2},m,k_{\text{N}},k_{\text{FNN}},k_{\text{TN}}\) & (12), (13) & [89] \\ [PERSON] & \(k_{1},k_{2},b,n,\alpha,\beta\) & (14) & [91] \\ [PERSON] & \(K_{\text{N}},C_{\text{N}},n_{1},n_{2},m\) & (17) & [93] \\ \hline \hline \end{tabular}
\end{table}
Table 4: Zero slip pressure–sinkage models provided in the literature.  
experimental device are shown in Figure 34. The lateral sinkage characteristics can be well described by Equation (24). Then  
\[h_{\text{lateral}}=A(1-\exp(B*slip)) \tag{24}\]  
The rotational sinkage behavior can be described by Equation (25). Then  
\[\text{Sinkage}=K*\text{Stress}*\text{footradius}^{2}\Big{(}1-\text{e}^{B\theta} \Big{)} \tag{25}\]  
Overall, compared to the wheels of wheeled robots, the foot of the legged robot does not produce a significant rotation angle. Compared with zero slip, the pressure and sinkage caused by non-zero slip are smaller. When analyzing the normal force of heavy-duty legged robots, it is generally believed that the slip amount of foot sinkage is relatively small.