# 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  
A Theoretical Exploration of the Wheeled Robots  
In 1913, [PERSON] proposed the relationship between the pressure applied to the soil and the sinkage after conducting relevant experiments on the relationship between pressure and sinkage [80,81]. The updated model is shown in Equation (5). Then  
\[p=kz^{n} \tag{5}\]  
where \(n\) is the model parameter, and \(n\) > 0. In the mid-19 th century, [PERSON] conducted specialized research on the plasticity of soil subsidence and driving resistance [82, 83, 84] In classical soil mechanics, when considering the [PERSON] model, the sinkage modulus \(k\) is bifurcated into two distinct components. One part represents the influence of the cohesion of the soil itself, while the other part represents the influence of the internal shear angle. The shear angle mentioned here is actually the friction angle. The Bekker model also considers the geometric shape of the contact surface. The Bekker model played an irreplaceable role in evaluating the motion performance of the wheeled robots, the tracked robots, and the legged robots [85]. The Bekker model is shown in Equation (6). Then  
\[p=\bigg{(}\frac{k_{c}}{b}+k_{\phi}\bigg{)}z^{n} \tag{6}\]  
where \(b\) is the smaller dimension of the contact patch. \(k_{c}\) is a sinkage modulus influenced by soil cohesion. \(k_{\phi}\) is a sinkage modulus influenced by the soil friction angle. [PERSON] introduced two distinct pressure-sinkage models, each tailored to specific soil conditions. The first model, as shown in Equation (7), features model parameters whose dimensions remain constant regardless of the sinkage index, which stands in contrast to the Bekker model. On the other hand, the second equation, as shown in Equation (8), employs dimensionless model parameters and is primarily designed for highly compacted soil [86]. [PERSON]'s second model is particularly suited for very dense soil, showcasing its versatility. Then  
\[p=(k_{1}+k_{2}b)(z/b)^{n} \tag{7}\]  
Then  
\[p=\big{(}ck_{c}+\gamma k_{\phi}b\big{)}(z/b)^{n} \tag{8}\]  
where \(k_{1}\) and \(k_{2}\) are model parameters. \(c\) is the soil stickiness, and \(\gamma\) is the unit weight of the soil. Notably, the Reece model proves highly effective for wheeled robots navigating frictionless clay and firm soil with minimal sinkage. It represents a substantial enhancement over the Bekker model. However, it is worth noting that the Reece model has not undergone extensive testing in softer clay soils, leaving room for further evaluation and refinement in these specific conditions. Experimental curves showing pressure and sinkage are used to define the relationship between pressure and sinkage in soil. A semi-empirical hyperbolic law is established by [PERSON] and [PERSON]. By analyzing the compressive strength of the soil, two constants that can be utilized are proposed; they are the bearing capacity \(p_{0}\) and the soil compression coefficient \(k\). The relationship between pressure and sinkage can be obtained, as shown in Equation (9). Then  
\[p=p_{0}\frac{1-\exp(-2 kz/p_{0})}{1+\exp(-2 kz/p_{0})} \tag{9}\]  
where \(p_{0}\) is the bearing capacity of the soil. [PERSON] and [PERSON] [87] selected three standard soils: a sand type is used for soils with frictional properties, a silt type for soils with cohesion, and a silty sand type for soils exhibiting both cohesive and frictional characteristics. A pressure-sinkage model N2M was proposed for the interaction between circular contact surfaces and soil. The N2M model is shown in Equation (10). It considers the mechanical behavior of the soil. Smallvertical sinkage is similar to elastic behavior. For large sinkage, they are similar to plastic behavior. The initial linear function describes the linear relationship of sinkage pressure within the elastic and plastic areas of the sinkage pressure diagram. The subsequent composite function delineates the shift from the elastic to the plastic region.