## 3 Path Tracking Control Based on States Estimation
### The Process Noise  
If there is no track skidding over the ground, for the driving roadheader, its heading speed \(\dot{x}\) along the \(x\) axis and turning angular speed \(\dot{\varphi}\) would be the following when the vehicle is turning to the left.  
Figure 7: Path tracking results using optimized BP Neutral Network. (**a**) The differences between the planned positions along path #1 and the real positions during tracking; (**b**) The differences between the planned positions along path #2 and the real positions during tracking.  
\[\dot{x}=\frac{1}{2}(r\omega_{L}+r\omega_{R}),\ \dot{\varphi}=\frac{r}{b}(\omega_{R}- \omega_{L}) \tag{6}\]  
where \(\omega_{L}\) and \(\omega_{R}\) are the rotating speed of the driving wheels for the left and right track, respectively; \(r\) is the radius of the driving wheel; \(r\omega_{L}\), \(r\omega_{R}\) are the line speed of the two tracks; \(b\) denotes the width between the left and right tracks.  
Track skidding is highly related to the material and moving conditions of the track, as well as the geological characteristics of the ground. 'Skidding ratio' is usually used to characterize the effect of skidding on the line speed of the track, and it is often expressed numerically by a ratio, such as \(i_{L},i_{R}\) for the left and right track, respectively. When skidding is considered, the \(\dot{x}\) and \(\dot{\varphi}\) in Equation (6) would change to  
\[\begin{array}{l}\dot{x}=\frac{r}{2}[\omega_{R}(1-i_{R})+\omega_{L}(1-i_{L}) ]\\ \dot{\varphi}=\frac{r}{b}[\omega_{R}(1-i_{R})-\omega_{L}(1-i_{L})]\end{array} \tag{7}\]  
Consequently, to achieve the equivalent displacement along the \(x\) axis, the recommended \(\omega_{L}\) and \(\omega_{R}\) are  
\[\begin{array}{l}\omega_{R}=\frac{1}{r(1-i_{R})}(v+0.5b\omega)\\ \omega_{L}=\frac{1}{r(1-i_{L})}(v-0.5b\omega)\end{array} \tag{8}\]  
and \(i_{L}\), \(i_{R}\) are able to be calculated for records as  
\[i_{L}=\frac{r\omega_{L}-(\dot{x}-b\dot{\varphi})}{r\omega_{L}}\quad,\quad i_{ R}=\frac{r\omega_{R}-(\dot{x}+b\dot{\varphi})}{r\omega_{R}} \tag{9}\]  
The track skidding is treated as process noise in this paper and is dealt with in a statistical way, instead of introducing the mechanism analysis on the formulation of the skidding into the path tracking model.  
According to the Bekker theory [13], the formulation of driving force relates to the width of the track, the length of the track touching the ground, the skidding ratio, the shear displacement, as well as the shear stress, which is determined by the characteristics of the soil, such as the shear strength, the pressure-settlement curve, the cohesive deformation, the frictional deformation modulus, and so on. Based on that formula, [PERSON] [13] analyzed the relationship between the traction force and skidding ratio of the tracks on a deep-sea tracked miner, and proposed that for certain ground situations, it is possible to estimate the value of skidding ratio \(i\) for a single track when the maximum traction force happens. This estimated skidding ratio is not the true value, and the real skidding ratio would be variable when the track is moving. However, it is valuable to reflect a general skidding level under that ground situation.  
Inspired by this, [PERSON] [16] determined a group of approximate skidding ratios for several different soil conditions similar to those underground on roadways. The obtained skidding ratios are used in this paper to determine the distribution of the skidding ratio, that is, the skidding ratio varies, obeying the uniform distribution with boundary values at [0.1, 0.35]. Therefore, it is possible to treat the skidding effect as process noise in the path tracking control through states estimation.