# A Review of Foot-Terrain Interaction Mechanics for Heavy-Duty Legged Robots
## 3 Dynamics Analysis of Robot
### Tangential Force Models
Then  
\[F_{\rm r}=-\frac{\exp(s/K^{\prime})-\exp(-s/K^{\prime})}{\exp(s/K^{\prime})+ \exp(-s/K^{\prime})}\mu_{f}F_{\rm N}K^{\prime}=1.5K \tag{30}\]  
where \(s\) is the shearing displacement. \(K\) is the shear displacement modulus, affected by the physical characteristics of the feet and terrain. \(\mu_{f}\) is the friction coefficient between hard  
Figure 35: Three-dimensional interaction mechanical model on tangential plane [93]. Figure 36: Foot–terrain interaction in the tangent direction [94]. terrain and foot materials. For deformable terrain, \(\mu_{f}\) is related to the characteristics of the soil and feet. However, when applying the new three-dimensional mechanical model to practical simulations, there is a lack of consideration for the damping effect between the feet and the terrain. The model is unstable when the feet interact dynamically with the terrain. Consequently, the force of interaction along the tangent direction is altered when the foot proceeds in a unidirectional movement. The interaction force in the tangent direction is shown in Equation (31). Then  
\[\left\{ \begin{array}{l}F_{\text{r}}=\frac{\exp(s/K^{\prime})-\exp(-s/K^{ \prime})}{\exp(s/K)+\exp(-s/K^{\prime})}\mu_{f}F_{\text{N}}-c_{T}\hat{s}\sqrt{| s|}\\ 0<s<s_{\text{max}}-\kappa\end{array} \right. \tag{31}\]  
where \(\mu_{f}\) is the friction coefficient. \(c_{T}\) is the tangential damping coefficient. The suggested models are developed across three distinct categories: a flexible foot interacting with rigid terrain, a rigid foot on pliable terrain, and a flexible foot engaging with deformable terrain. The model proposes different mechanical models based on the geometric characteristics of different feet, which can accurately characterize the mechanical conditions of the feet in practical applications. The foot-terrain mechanics model parameters are shown in Table 5. An SVM method is proposed that uses two specific tactile movements of the heavy-duty legged robots to extract physical information features for effective terrain classification. Through the normal compression and tangential friction motion of legged robots, the representative interactive data is obtained to characterize the terrain features [104], which plays an important role in improving the accuracy of the model. [PERSON] fully considers the shape characteristics of the contact surface and slip surface under the assumptions of some classical soil mechanics for the limit-bearing theory. The normal and the tangential mechanical models of flat foot and sand, horizontal strip foot and sand, circular and sand, and rectangular flat foot and sand, as well as the interaction model of curved foot and sand, have been established [105]. It has a positive effect on calculating the foot mechanics of robots with different shapes. The tangential force models are shown in Table 6.  
The terrain mechanics models can establish a close connection between the parameters in soil mechanics and the foot contact mechanics of the robots. It is convenient to conduct a mechanical analysis of the foot-terrain interaction of the heavy-duty legged robots. However, there is still a long way to go in the research of foot-terrain mechanics for heavy-duty legged robots. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model Name** & **Model Parameters** & **Equation Number** & **References** \\ \hline [PERSON] & \(\mu\) & (26) & [99; 100] \\ [PERSON]–[PERSON] & \(f_{c}\,C_{t}\) & (27) & [89] \\ [PERSON] & \(\beta_{\text{F}}\) & (28) & [93] \\ [PERSON]–[PERSON] & \(s\), \(K^{\prime}\), \(\mu_{f}\,s_{\text{max}}\,\kappa\) & (30), (31) & [94] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The tangential force models provided in the literature.