# A Review of Foot-Terrain Interaction Mechanics for Heavy-Duty Legged Robots
## 3 Dynamics Analysis of Robot
### Tangential Force Models  
The aim of integrating tangential interaction models is to illustrate that the shear displacement of the foot on the terrain induces an effect on the tangential plane. In traditional contact models, models that separately describe tangential forces often consider normal force and tangential relative motion position as common determining factors [98]. The heavy-duty legged robots generate tangential friction when performing the tangential relative motion. In terms of the tangential mechanical models of the foot, the most classic tangential friction model is the Coulomb model [99; 100]. As a static friction theory, the Coulomb model has a small computational complexity. Its mechanical parameters are easy to identify. The Coulomb model has been widely used [101; 102]. The Coulomb model is shown in Equation (26). Then  
\[F=\left\{ \begin{array}{ll}F_{\text{C}}+(F_{\text{S}}-F_{\text{C}})e^{-|v/v_{ \text{S}}|\delta_{\text{S}}}+F_{\text{v}}&\text{if $v\
eq 0$}\\ F_{\text{e}}&\text{if $v\
eq 0$ and $|F_{\text{e}}|<F_{\text{S}}$}\\ F_{\text{S}}\text{sgn}(F_{\text{e}})&\text{otherwise}\end{array} \right. \tag{26}\]  
The Hunt-Crossley model also describes the action model of tangential force, as shown in Equation (27). Then  
\[F_{\text{T}}(t)=-f\times\text{sign}(v_{\text{T}})\times F_{\text{N}}(t)-C_{ \text{t}}\times v_{\text{T}}(t) \tag{27}\]  
In the study of terrain mechanics, the spring damping system has been widely used in foot-terrain interactions. [PERSON] [103] validated that a simple spring damping system can explain the characteristics of human walking by establishing models and conducting experiments. Conventional 3D models segment the deformation of feet or terrain into two perpendicular directions on a tangential plane, specifically the \(x\) and \(y\) axes. Significant errors are generated because the coupling effect of deformation is ignored. Figure 34: CREX robot [97]: (**a**) outdoor experiment; (**b**) rotating sinkage experimental device. In order to consider the coupling effect, [PERSON] [93] proposed a new three-dimensional mechanical model of the foot-terrain interaction in combination with a spring damping system. The three-dimensional interaction mechanical model on a tangential plane is shown in Figure 35. The three-dimensional mechanical model of the foot-terrain interaction can be shown in Equation (28). Then  
\[\left\{ \begin{array}{l}F_{z}=F_{\rm N}\\ F_{x}=F_{\rm T}\cos\beta_{\rm F}\\ F_{y}=F_{\rm T}\sin\beta_{\rm F}\end{array} \right. \tag{28}\]  
The process of tangential interaction can also be seen as a normal interaction. As the foot slides across the terrain, the tangential force escalates in correspondence with the rising shear displacement and velocity. The tangential force quickly stabilizes as the shear speed decreases to a certain value. Contrary to the previous growth process, the tangential force suddenly drops to zero when the foot moves in the opposite direction. The force in a specific tangent direction is usually represented by the Coulomb friction model. If the terrain is relatively hard, the tangential force model uses a modified form. When the direction of relative motion velocity changes, the friction force remains unchanged and does not meet the physical boundary conditions of the friction process, resulting in singular solutions in the simulation. When introducing the hyperbolic tangent function to establish a foot-terrain tangential force model, the value of \(th(\delta)\) is infinitely close to 1 or -1 to describe the saturation of tangential friction. The foot-terrain interaction in the tangent direction is shown in Figure 36. The hyperbolic tangent mathematical model is shown in Equation (29). Then  
\[th(\delta)=\frac{sh(\delta)}{ch(\delta)}=\frac{e^{\delta}-e^{-\delta}}{e^{ \delta}+e^{-\delta}} \tag{29}\]  
For deformed soil [85], a modified model based on the Janosi equation is proposed; it can be shown in Equation (30).