Path Tracking of Underground Mining Boom Roadheader Combining BP Neural Network and State Estimation

Abstract

This paper proposes a path correction scheduling strategy for the underground mining boom roadheader by ably combining a back propagation (BP) neural network and state estimation. First, a pose deviation-based tracking model is designed for the roadheader, and it is then further studied and optimized by incorporating the benefits of BP neural networks into the model adaptation. Considering the fact that there is skidding between tracks on the ground and errors during the instant pose detection of the roadheader underground, singular value decomposition (SVD)–Unscented Kalman filtering is applied to estimate the real pose deviation, based on the summarized distribution regularities of the track skidding ratios and the pose detection errors, instead of complicated analysis mechanisms. The BP neural network and states estimation are well combined in structure, enabling this scheduling strategy to update the control law and revise the control instruction simultaneously in the procedure. The proposed path tracking model for the roadheader is simple and clear, without adding extra devices or massive algorithms, which is attractive in terms of industrial use. The path tracking simulations show that this proposed strategy achieves path tracking well in different scenarios and is of high adaptability when facing complex trajectory while still giving stable control instructions for the roadheader.

1. Introduction

Tunneling is an important part of many civil engineering works, such as mining [1], subways and railways [2], the large hadron collider [3] and so on. Longitudinal shaft cantilever-type roadheaders (or boom roadheaders, hereinafter referred to as “roadheaders”) can be considered core equipment for underground coal mining. The roadheader is mainly used for roadway forming before comprehensive mining [1]. The roadheader is usually composed of a cutting mechanism, shovel plate mechanism, frame, conveyor, walking mechanism, hydraulic pressure and control system [4], as shown in Figure 1. In the process of roadway tunneling, the idealized basic actions of the roadheader are walking along the center line of the planned roadway, slotting and then cutting. However, in actual work, geological hindrance, maneuverability errors, unclear underground space posture and other factors cause the roadheader to deviate from the planned centerline gradually, leading to overbreak or underbreak interface of the roadway and thus lowering the quality of the roadway and decreasing the tunneling efficiency. Autonomous walking scheduling and path tracking, i.e., path correction, for the roadheader is necessary to achieve robotized tunneling based on the roadheader [5].

The research on path correction control of tracked vehicles can be generally divided into three categories according to different research focuses. The first group refers to the closed-loop optimization control strategy, based on a model that is used in wheeled vehicle trajectory planning and tracking and that makes appropriate improvements to ensure the model description is as close as possible to the actual walking characteristics of tracked mobile robots. Model predictive control (MPC) is a typical multi-objective rolling optimization model based on feedback information. Chen and Danielson [6] presented a rate-based MPC for controlling a multi-motor electric vehicle (EV) with good tracking performance and passenger comfort. Sanchez [7] introduced a unified MPC strategy to solve the path-following and trajectory-tracking problems of constrained vehicles.

The second group focuses on improving the dynamic model of the tracked vehicle and embeds the maneuvering error caused by the mechanical structure of the tracked vehicle into the dynamic model to enhance the control effect. For example, Zhao Huanyu [8] studied the dynamic model of articulated tracked vehicles and the dynamic model of induction motors, incorporating ground friction and track skidding force analysis into the dynamic models. Based on this, the electromechanical coupling model of articulated tracked vehicles was established. For articulated tracked vehicles, fuzzy PID control is established, with distance deviation and heading angle deviation as control objectives. Some researchers have ignored the influence of friction between track and ground and improved the model by estimating the skidding rate. For example, Kang Yiting [9] established the dynamic model of the tracked vehicle according to the sphere–surface contact principle and improved the model by estimating the skidding rate based on steady-state turning experiments.

The third group focuses on the kinematics model of the tracked vehicle. Wang Shuai [10] proposed an improved pure tracking algorithm by establishing the kinematics model of the tracked vehicle to address the problem of the vehicle failing to rectify the lateral deviation quickly during path tracking. M.A. Subari [11] proposed a Sup-controller based on the kinematics model of the tracked vehicle, which combined the PID controller and skidding mode controller and used the particle swarm optimization algorithm. These improved kinematics model–based controllers usually apply complex algorithms, which can bring challenges to the performance and calculation speed of the control core board. Another point is that the effects caused by track skidding are rarely considered, which can be substantial for tracked vehicles, especially when the ground conditions are complicated. According to the kinematic model of tracked vehicles, Zou Ting [12] proposed a modified PID control method based on a back-stepping derivation, and used bi-axial accelerometers to estimate the skidding degree to realize the path tracking. Han Qingjue [13] designed a skidding control system for a deep-sea tracked vehicle over certain types of soft ground to determine the most likely skidding and make use of the driving force.

The above-mentioned studies provide references for the path tracking control of tracked vehicles from different viewpoints. However, research specifically referring to the systematic scheduling and tracking control strategy for the roadheader underground is not abundant. A few examples are as follows. Ma Hongwei [14] integrated vision and inertial navigation to measure the posture of the roadheader and used fuzzy PID and neural network PID to correct the pose deviation. Wu Miao [15] proposed a pose detection system for the roadheader underground using the SINS together with laser sensing, and Zhang Minjun [16] then proposed an autonomous rectification drive control strategy based on a neural network PID algorithm, according to the skidding rate under different driving environments. Yang Jianjian [17] also introduced general autonomous sensing and control technologies that involved intelligent rapid heading in coal mining. These works focus more on practical inertial posture detection and navigation for the roadheader as well as the basic control model for this bulky tracked vehicle.

We should point out that the inevitable errors in the execution of the pose detection and vehicle adjustment would decrease the effect of path tracking control of the roadheader. To deal with errors, most scholars use filtering to estimate the current pose and predict the next possible pose of the vehicle. Wu Hongyun [18] introduced Kalman filtering and adaptive Kalman filtering to reduce environmental interference, thereby improving the tracking performance of deep-sea tracked mining vehicles. Ge Jingwei [19] and Taeklim Kim [20] also improved the Kalman filter according to the physical characteristics of noise and the sensory characteristics, respectively, to improve the pose detection accuracy. B. Venkata Krishnaveni [21] and Sun Mingxu [22] adopted UKF and EKF, respectively, to achieve high positioning accuracy within UWB pose detection systems.

To summarize, there is a need for the improvement of the path correction strategy in terms of dynamic adaptability as well as the ability to deal with possible maneuver and measurement errors in the process of roadheader scheduling and control. This paper proposes a path tracking control based on the pose deviation model for the roadheader, combining BP neural networks and state estimation. It improves the control model using BP neural networks while considering the inevitable pose measurement error and process error using state estimation. Thus, the control law is designed to update in a timely fashion and the control instruction is revised at the same time by implying the SVD–Unscented Kalman filtering. This strategy is believed to make the path tracking control of the roadheader more robust when facing uncertainty as a result of pose detection errors and track skidding, without adding complications due to extra devices or algorithms, which ensures ease of use in industrial coal mining applications.

2. Scheduling Requirements and Modeling Foundations of Roadheaders

Roadways are tunneled by a series of basic movements, such as direction and speed adjustment, drilling and cutting of the coal wall, section shaping and fixing, coal and rock transportation, and so on. Bolt support is a temporary support used before the arch hydraulic support is implemented when a section is completed; the roadheader needs to withdraw appropriately to provide working space for bolting. When the pose of the roadheader is unpredictable, path planning and tracking must be executed, which is referred to as “roadheader scheduling” in this paper.

For many wheeled robots or vehicles, path planning and tracking can be carried out synchronously by optimal control. Since the expected roadway for coal mining is usually a straight line inside the coal bed, it is not necessary to replan a path for the roadheader after each section is formed. Instead, it is better to pre-plan a path for tracking than to use dynamic path planning. In other words, this paper only focuses on the tracking strategy under a planned path.

A pre-planned path for the roadheader to travel back onto the target roadway includes well-planned turning speed ($\dot{\phi }$) and forward velocity ($\dot{x}$) for the roadheader according to certain operational requirements, as introduced in the literature [23].

To achieve certain turning speed and forward velocity, the motion of the tracked vehicle is considered similar to that of wheeled ones: when the driving wheels on both sides rotate at the same angular speed and in the same direction, the vehicle moves forward or backward in a straight line; when they rotate at same angular speed but in opposite directions, the vehicle would turn in place; when they rotate at different angular speeds that are well designed, the vehicle would turn along a certain expected curve.

Nevertheless, the roadheader scheduling is different from that of wheeled vehicles or small tracked vehicles on the ground, based on the following considerations. For example, the turning speed, forward velocity, and their gradients for the roadheader are usually limited for safety or by a restricted working space; the track skidding between the tracks and ground is difficult to formulate exactly; the tolerable position derivation is small compared with the size of the vehicle, and therefore a slight movement might result in over adjustment; with a certain level of pose detection rate and accuracy, the accumulated measurement errors should be taken into account.

2.1. Basic Path Tracking Control for the Roadheader

The tracking control of the roadheader was carried out based on a pose deviation model [24], which is demonstrated in Figure 2. An isosceles triangle, which takes the centroid of the roadheader as its apex and the ends of the two tracks as its bottom points, is used to display the pose (position of the centroid, orientation) of the roadheader; the dotted triangle in front represents the scheduled next pose of the roadheader.

$(x,y)$ and $(X,Y)$ are the coordinates of the roadheader in the vehicle coordinate system $xoy$ and geodetic coordinate system $XOY$, respectively. The letter $\mathrm{b}$ presents the distance between the two tracks; $v$ presents the instant heading velocity; $\phi$ presents the turning angle of the roadheader; $\omega$ presents the turning angular speed, which is also $\dot{\phi }$; $\theta$ presents the orientation of the roadheader.

Taking $\dot{x}$ and $\dot{y}$ as the instant velocity of the vehicle along the $x$ axis and $y$ axis of the vehicle coordinate system, respectively, the instant heading velocity $v$ is actually composited by $\dot{x}$, $\dot{y}$ and the linear velocity caused by the steering motion of the vehicle. Considering that the turning angular speed is usually very small, the corresponding linear velocity and $\dot{y}$ are ignorable compared with the $\dot{x}$. Hence, the value of $v$ is very close to $\dot{x}$, namely, $v\approx \dot{x}$.

We take the subscript c as the current state of a certain parameter, d as the destined state, and e as their difference. For instance, we denote the current pose of the centroid as ${{P}}_c={(X_c,Y_c,{\theta }_c)}^T$, and the destined pose as ${{P}}_d={(X_d,Y_d,{\theta }_d)}^T$; thus, their difference in the vehicle coordinate system is

$$p_e=\left[\begin{array}{c}{x}_e\\ y_e\\ {\theta }_e\end{array}\right]=\left[\begin{array}{ccc}\cos {\theta }_c& \sin {\theta }_c& 0\\ -\sin {\theta }_c& \cos {\theta }_c& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{X}_d-X_c\\ Y_d-Y_c\\ {\theta }_d-{\theta }_c\end{array}\right]$$

(1)

The differential of $P_e$ is deduced to be

$${\dot{P}}_e=\left[\begin{array}{c}{\dot{x}}_e\\ {\dot{y}}_e\\ {\dot{\theta }}_e\end{array}\right]=\left[\begin{array}{c}{v}_d\cos {\theta }_e-v_c+y_e{\omega }_c\\ -x_e{\omega }_c+v_{d}sin{\theta }_e\\ {\dot{\theta }}_d-{\dot{\theta }}_c\end{array}\right]=\left[\begin{array}{c}{v}_d\cos {\theta }_e-v_c+y_e{\omega }_c\\ -x_e{\omega }_c+v_{d}sin{\theta }_e\\ {\omega }_d-{\omega }_c\end{array}\right]$$

(2)

Based on Equations (1) and (2), a control law for updating current heading velocity $v_c$ and turning angular speed ${\omega }_c$ is designed as Equation (3) to realize $P_e\to {\left(0,0,0\right)}^T$ within limited steps, which is path tracking.

$$\{\begin{array}{c}{v}_c=k_1{x}_e+v_d\cos {\theta }_e+y_e{\omega }_c\\ (1+v_d x_e){\omega }_c={\omega }_d+v_d{}^{2}sin{\theta }_e+k_2{\theta }_e+k_2{v}_d y_e\end{array}$$

(3)

Equation (3) is obtained by defining the switching function and using the back-stepping approach, and it is simplified as Equation (4), considering that (1) ${\omega }_c$ is very small compared with $x_e$; (2) $\left|v_d{x}_e\right|\ll 1$; (3) ${\dot{v}}_d\approx 0$; (4) $\sin {\theta }_e\approx {\theta }_e$.

$$\mathrm{u}=\left[\begin{array}{c}{v}_c\\ {\omega }_c\end{array}\right]=\left[\begin{array}{c}{k}_x{x}_e+v_d\cos {\theta }_e\\ {\omega }_d+v_d{k}_y{y}_e+k_{\theta }\sin {\theta }_e\end{array}\right]$$

(4)

where $k_x , k_y,$ and $k_{\theta }$ are random positive coefficients of the pose errors $x_e$, $y_e$ and θ_e, respectively.

According to the Lyapunov stability criterion, if a suitable Lyapunov function $V(x_e,y_e,{\theta }_e)\ge 0$ is determined and it fulfills $\dot{V}(x_e,y_e,{\theta }_e)\le 0$ based on Equations (1), (2) and (4), then $p_e$ is limited and $\underset{n\to \infty }{\lim }{p}_e=0$.

We take the Lyapunov function as $V=\frac{1}{2}{x}_e{}^2+\frac{1}{2}{y}_e{}^2+2k{\sin }^2\frac{{\theta }_e}{2}$, where $k$ is a random positive number, and consider the differential of $V$:

$$\begin{array}{ll}\dot{V}& =x_e{\dot{x}}_e+y_e{\dot{y}}_e+2k{\dot{\theta }}_e\sin \frac{{\theta }_e}{2}\cos \frac{{\theta }_e}{2}\\ & =x_e(v_d\cos {\theta }_e-v_c+y_e{\omega }_c)+y_e(-x_e{\omega }_c+v_{d}sin{\theta }_e)+k({\omega }_d-{\omega }_c)sin{\theta }_e\\ & =-x_e{v}_c+v_d{x}_e\cos {\theta }_e+v_d{y}_{e}sin{\theta }_e+k({\omega }_d-{\omega }_c)sin{\theta }_e\end{array}$$

We substitute Equation (4) into $\dot{V}$ and obtain

$$\dot{V}=-k_x{x}_e{}^2+v_d{y}_{e}sin{\theta }_e-k{k}_y{v}_d{y}_{e}sin{\theta }_e-k{k}_{\theta }{sin}^2{\theta }_e$$

(5)

From Equation (5), it is obvious that if $k{k}_y=1$, then $\dot{V} =-k_x{x}_e{}^2-k_y{}^{-1}{k}_{\theta }{\sin }^2{\theta }_e \le 0$, namely, $\underset{n\to \infty }{\lim }{p}_e=0$.

2.2. Optimization of Weights Based on BP Neural Network

According to the deduction of Equation (4), as long as $k_x , k_y,$ and $k_{\theta }$ are positive, the position and orientation errors would converge to 0 within a limited adjustment period. To verify this, two groups of simple pre-designed trajectories, i.e., two series of planned positions together with orientations are given for the roadheader at certain initial deviation situations to the track. Since the centreline of the roadway is usually straight, the two target centrelines of the roadway should be along the lines $Y=2$ and $X=2$, respectively, in the geodetic coordinate $XOY$. Correspondingly, the initial poses of the roadheader are $(0,1,\pi /6)$ and $(1,0,\pi /3)$, respectively. Here, $(0,1)$ and $(1,0)$ are coordinates, and $\pi /6$ and $\pi /3$ are orientations.

As shown in Figure 3a, path #1 leads the roadheader to move from $(0,1,\pi /6)$ onto the line $Y=2$, while path #2 leads the roadheader onto the line $X=2$ from $(1,0,\pi /3)$. The expected orientations for the roadheader along the paths are also presented in Figure 3b, and their variances, i.e., the steering angular velocities, are shown in Figure 3c.

To simulate the path tracking process using Equation (4), we used the parameters as shown in

Table 1. The data used in simulations.

Parameters	Value or Description	Parameters	Value or Description
Radius of driving wheel r	0.5 m	Random positive coefficients	$k_x$	0.6
Distance between the tracks b	1 m		$k_y$	0.7
Width of track B	320 mm		$k_{\theta }$	0.2
Length of track where touching the ground l	3 m	Maximum rotating speed of the driving wheel	4 rad/s
Maximum travelling speed $v_{\max }$	$v_{\max }$ = 0.8 m/s	Range of angular acceleration	(0, 0.2)
Referred average speed	$v_{ref}$ = 0.6 m/s

. Note that the values of $k_x , k_y, k_{\theta }$ are randomly taken to be 0.6, 0.7, and 0.2. The simulation results are shown in Figure 4, and for brief illustration, only the coincidences between the planned and tracked paths are shown here.

It can be seen that both sets of adjustments achieve the purpose of path tracking, i.e., the roadheader moves close to the planned path and finally moves onto the target centerline within a certain number of adjustment steps in the two cases of simulation. However, it is obvious that the tracking effect following path #2 is not as good as the case of path #1 in the early stage of tracking.

Commonly, changing the values of $k_x , k_y, k_{\theta }$ for simulation is carried out repeatedly for trials. A set of values at 0.1, 0.5, and 0.5 for $k_x , k_y, k_{\theta }$, respectively, results in more satisfying tracking performances for both cases, as shown in Figure 5.

Incidentally, some other sets of $k_x , k_y, k_{\theta }$, for instance, 0.5, 0.2 and 1.8, also give good tracking results. To summarize, first, the path control using Equation (4) will not fail if the values of $k_x , k_y, k_{\theta }$ are suitable. Second, different sets of $k_x , k_y, k_{\theta }$ would give different tracking effects in the aspects of convergence speed of the position and orientation deviations, or the overlap ratio of the tracked path over the pre-planned path. Lastly, $k_x , k_y, k_{\theta }$ giving satisfying tracking results is not unique in terms of value, and these different sets of values apparently show no patterns. The BP neural network might be appropriate for the situation presented above, to obtain ideal values for $k_x , k_y, k_{\theta }$ automatically while eliminating the pose deviation effectively in the process of tracking. Therefore, as shown in Figure 6, the nonlinear optimization of the BP neural network is applied to the tracking control based on the pose deviation model.

It is simple to apply the BP neural network within the path tracking control based on the pose deviation model:

• The used BP neural network has 3 layers and is 9-9-3. The 9 inputs are the desired posture ${{P}}_d={(X_d,Y_d,{\theta }_d)}^T$, the current posture ${{P}}_c={(X_c,Y_c,{\theta }_c)}^T$, and the position and orientation errors $p_e={(x_e,y_e,{\theta }_e)}^T$; the 3 neurons on the output layer are the parameters $k_x , k_y, k_{\theta }$.
• The activation function for the hidden layer is the positive–negative symmetric Sigmoid function: $f(x)=\tanh (x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}$; the activation function for the output layer is the non-negative Sigmoid function: $g(x)=\frac{1}{2}(1+\tanh (x))=\frac{e^x}{e^x+e^{-x}}$; and the cost function is taken as $E(x)=\frac{1}{2}{(P_d-P_c)}^2=\frac{1}{2}{p}_e{}^2$.
• There is no clear physical concept between the inputs and outputs, and previous training for a final set of optimized $k_x , k_y, k_{\theta }$ before the path tracking simulation is not necessary. Instead, the initial values of $k_x , k_y, k_{\theta }$ are set randomly and are updated iteratively based on a back-propagation algorithm. As the pose deviation converges to zero, the values of $k_x , k_y, k_{\theta }$ become steady.
• In each adjustment, the driving commands for the roadheader, i.e., the heading velocity $v$ and the turning angular speed, and the values of ${[v_c,{\omega }_c]}^T$ are calculated using Equation (4) with the $x_e,y_e,{\theta }_e,{\omega }_d,v_d$, etc., which are obtained by instant measurements and predictions from the last adjustment. Then, the position and orientation errors $p_e={(x_e,y_e,{\theta }_e)}^T$ would be updated by $p_e\to p_e+{\dot{p}}_e\Delta t$, followed by the updating of the cost $E(x)$. The variance of the cost guides fixing of the connection weights $w_{ij}^{(1)},w_{jm}^{(2)}$ by a negative gradient direction searching (the superscript (1) means from the input layer to the hidden layer; (2) means from the hidden layer to the output layer. The subscripts i, j, k refer to the i-th, j-th, k-th neuron from the three layers, respectively). The outputs $k_x , k_y, k_{\theta }$ are renewable until the error ${\mathrm{p}}_e$ converges to zero.

Using the structure shown by Figure 6, the path tracking simulation following path #1 and path #2 is carried out using the same conditions as introduced previously. The results are shown in Figure 7. Notice that since the initial position of the roadheader is identical with the start point of the planned path; this means the initial position derivation is zero, and thus the first command given to the ${[v_c,{\omega }_c]}^T$ is to maintain. In other words, the first iteration produces nothing and is wasted, while the total number of iterations is set equal to the number of points along the planned path; therefore, as seen in Figure 7, there is always around one step between the aimed and local position. However, the tracked paths overlap the planned paths quite well, indicating that the position and orientation deviations converged around zero at their early steps.

In addition, recall that the values of $k_x , k_y, k_{\theta }$ in each adjustment are obtained by instant optimization and are not unique, even for a repetition of the same simulation. The basic path tracking control model using the BP Neutral Network introduced above is effective for the road header as long as the driving commands were carried out with no errors.

3. Path Tracking Control Based on States Estimation

It is imaginable that the real displacement of the roadheader would fail to be exactly as expected by executing the command $u={(v_c,{\omega }_c)}^T$. The possible internal reason focuses on whether the dynamic model of the driving system is correct; the possible external reason focuses on the following issues. One reason is that there is skidding between the tracks and the ground. The other fact is that there are errors in the instant measurements of the pose of the roadheader. This section will discuss these two external aspects, which are treated as process noise and measurement noise, respectively, in the basic path tracking control based on the pose deviation model.

3.1. 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 $\dot{\phi }$ would be the following when the vehicle is turning to the left.

$$\dot{x}=\frac{1}{2}(r{\omega }_L+r{\omega }_R), \dot{\phi }=\frac{r}{b}({\omega }_R-{\omega }_L)$$

(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 $\dot{\phi }$ in Equation (6) would change to

$$\begin{array}{l}\dot{x}=\frac{r}{2}[{\omega }_R(1-i_R)+{\omega }_L(1-i_L)]\\ \dot{\phi }=\frac{r}{b}[{\omega }_R(1-i_R)-{\omega }_L(1-i_L)]\end{array}$$

(7)

Consequently, to achieve the equivalent displacement along the x axis, the recommended ${\omega }_L$ and ${\omega }_R$ are

$$\begin{array}{l}{\omega }_R=\frac{1}{r(1-i_R)}(v+0.5b\omega )\\ {\omega }_L=\frac{1}{r(1-i_L)}(v-0.5b\omega )\end{array}$$

(8)

and $i_L$, $i_R$ are able to be calculated for records as

$$i_L=\frac{r{\omega }_L-(\dot{x}-b\dot{\phi })}{r{\omega }_L} , i_R=\frac{r{\omega }_R-(\dot{x}+b\dot{\phi })}{r{\omega }_R}$$

(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, Han Qingjue [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, Zhang [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.

3.2. The Measurement Noise

In the terms of position and attitude measurement of the roadheader, Fu Shichen [25] designed and built an ultra-wideband (UWB) pose detection system, which consists of four UWB positioning base stations and three local positioning nodes on the roadheader. Using the UWB P440 modules from the company “Time domain”, together with the micromechanical gyroscope and shaft encoder developed by the company “Action”, a prototype of the roadheader pose detection system was built, as shown in Figure 8. Using 12 sets of distance measurements, a time-of-arrival (TOA) positioning model [26] was established to calculate the 3D coordinates of three positioning nodes on the roadheader fuselage, and the three attitude angles (heading, pitching, and rolling angles) of the roadheader were then solvable according to the concept of inertial navigation system (INS).

The roadheader pose detection experiments were carried out in a narrow tunnel. In the experiments, the base stations were immobile while the roadheader was moving slowly, and the pose measurement accuracy was different as the distance between the base stations and nodes changed in scale.

Figure 9 presents the error frequency histogram of the ranging involved in the pose detection system based on ultra-wideband (UWB) within different distance scales, i.e., 20 m, 35 m, 57 m, and 90 m. It shows that the distance errors comply with the normal distribution and have a standard deviation less than 10 mm and a mean value between ±5 mm. The 3D coordinates and heading angle of the roadheader were calculated using the measured distances, and according to the experimental results, the mean error and standard deviation of the calculated coordinates are below 2 cm and of the heading angle are below 2.5°.

Recall that the pose of the roadheader is used for a subtraction with the desired ones along the pre-designed path to form the pose deviation $p_e={(x_e,y_e,{\theta }_e)}^T$ in the basic path tracking control model in Section 2.1. It is reasonable to assume that the measurement errors of the coordinate deviation $x_e$, $y_e$ obey a normal distribution with the mean value of 0 and a standard deviation of 4 cm, and the direction angle deviation ${\theta }_e$ obeys a normal distribution with a mean value of 0 and a standard deviation of 5°.

3.3. Patch Tracking Control Based on State Estimation

We define the state of the roadheader as $S=(x_e,y_e,{\theta }_e, {\omega }_L, {\omega }_R)$, where $(x_e,y_e,{\theta }_e)$ is the pose deviation, which is obtained according to the predesigned path and the pose measurements in real time by the UWB pose detection system on-board. The scheduled rotation rates of the left and right driving wheels ${({\omega }_L, {\omega }_R)}^T$ then are calculated through Equation (8), where the skidding ratio $i_L$ and $i_R$ are uncertain. Notice that the state $S$ varies as the recommended heading velocity $v_c$ and turning angular speed ${\omega }_c$ change as in Equation (4), with the aim that $\underset{n\to \infty }{\lim }{p}_e=0$. Apparently, the path tracking control of the roadheader based on the pose deviation model, which is expressed by Equations (1), (2), (4) and (8), is represented by a typical nonlinear system:

$$\begin{array}{l}{x}_{k+1}=S_{k+1}=S_k+{\dot{S}}_k\cdot \Delta t+{\epsilon }_k=\mathrm{f}(S_k, u_k)+{\epsilon }_k\\ y_{k+1}=\mathrm{h}(S_k)+{\delta }_{k+1}\end{array}$$

(10)

where ${\epsilon }_k$ is the process noise and ${\delta }_{k+1}$ is the observation error, as introduced in Section 3.1 and Section 3.2, respectively. Since the skidding between the tracks and ground results in part of the errors between the real pose of the roadheader and the expected pose, ${\epsilon }_k$ is equivalent to the effort of skidding; the observation item, i.e., the pose deviation, is obtained based on measurements; therefore, ${\delta }_{k+1}$ is equivalent to measurement noise.

We are able to treat the nonlinear system represented by Equation (10) as a joint estimation problem, to which the nonlinear filtering method is widely used for solution. There are many nonlinear filtering methods for state estimation, including the Kalman filter, which is used to estimate linear system and has great applicational value in nonlinear systems; however, the error introduced by linear approximation would lead to divergence [27]. The SVD–Unscented Kalman Filtering is able to suppress divergence while having better estimation accuracy and matrix operations during iterations; therefore, it is applied in this work.

The main idea in Unscented Kalman Filtering (UKF) for state estimation is to approximate the probability distribution of the concerned parameters directly by nonlinear transformation, instead of approximating the linearized model of the system [28]. For Unscented Kalman Filtering (UKF), to realize states estimation without derivative operations, the Unscented Transform (UT) is commonly used to calculate the statistical properties of random variables that are formulated by nonlinear expressions. A brief introduction on UT is given in the following section.

For a random vector $y$, which is the nonlinear transformation of an N-dimensional independent variable $x$ by $f(x)$, we take $P_x{|}_{N\times N}$ as the covariance matrix of $x$, and $\overline{x}$ as the mean of $x$; to obtain the statistics of $y$, a sigma-point vector matrix $\chi$ is required and is formed by using sigma-point sampling method:

$$\chi =[\overline{x} |{ }_{ i=0} \overline{x}+{(\sqrt{(N+\lambda )P_x})}_i|{ }_{i= 1,\cdots ,N } \overline{x}-{(\sqrt{(N+\lambda )P_x})}_i{|}_{i=N+ 1,\cdots ,2N} ]$$

(11)

where $\lambda ={\alpha }^2(N+\kappa )-N$ is the scale parameter, in which $\alpha$ decides the diffusion range of the sigma-points and is a small positive constant; $\kappa$ is a sub-scale parameter taking a value of 0 in general state estimation and 3-N in parameter estimation; ${(\sqrt{(N+\lambda )P_x})}_i$ is the i-th column of the weighted square root matrix $\sqrt{(N+\lambda )P_x}$.

In normal cases, $P_x$ is a positive definite symmetric matrix, and to figure out $\sqrt{(N+\lambda )P_x}$, the eigenvalue decomposition of $P_x$ is taken. Suppose $P_x=Q\Lambda Q^{-1}$ (where $Q$ is a N × N matrix with the eigenvectors of $P_x$ as its columns; $\Lambda$ is the diagonal matrix with the corresponding eigenvalues as its diagonal elements, $P_x{}^{1/2}=Q{\Lambda }^{1/2}{Q}^{-1}$). On the other hand, for a positive definite symmetric matrix, its eigenvalue decomposition is equivalent to its SVD decomposition. Suppose $P_x=U\Lambda V^T$ (where $U=V=Q , Q^{T}Q=I$, and $\Lambda$ is the same as just introduced above), therefore $P_x{}^{1/2}=U{\Lambda }^{1/2}{V}^T$.

Notice that $\sqrt{(N+\lambda )P_x}$ stands with the precondition that $P_x$ is a positive definite symmetric matrix. However, a morbid covariance matrix is often encountered in the implementation of UKF, i.e., the covariance matrix fails to be positive symmetric during filtering. To solve this, SVD decomposition is always taken instead of eigenvalue decomposition in the Unscented Transform, which is called SVD–Unscented Kalman Filtering.

The procedure of state estimation based on SVD–Unscented Kalman Filtering is as follows:

• Initialization

${\overline{x}}_0=\mathrm{E}[x_0]; P_0=\mathrm{E}[(x_0-{\overline{x}}_0){(x_0-{\overline{x}}_0)}^T]$

${\overline{x}}_0$ is the initial mean value of $x$, while $P_0$ is the initial covariance matrix of $x$. We construct the augmented matrix about ${\overline{x}}_0$ with the mean value of the process noise and measurement noise, which are both zero initially, and the augmented matrix about $P_0$:

$$\begin{array}{l}{\widehat{x}}^a{}_0=({\overline{x}}_0{}^T 0 0) { }^T \\ P_0{}^a=diag(P_0, P_{\epsilon }, P_{\delta }) \end{array}$$

where the superscript ‘a’ means ‘augmented’. $P_{\epsilon }$ and $P_{\delta }$ are the covariance of process noise $\epsilon$ and measurement noise $\delta$, respectively.

• We take a set of sigma-points and form the vector matrix ${\chi }_k$ by using Equation (11) and $P_x{}^{1/2}=U{\Lambda }^{1/2}{V}^T$.

• Time updating

We produce samples $x^{*}{}_{k+1|k}=f({\chi }_k,u_k)$ through function $f(x)$, obtaining the new augmented matrices:

$$\begin{array}{l}{\widehat{x}}^a{}_{k+1|k}\approx {\displaystyle \sum _{i=0}^{2N}{W}_i^{(m)}{x}^{*}{}_{k+1|k}(i)} \\ P_{k+1|k}^a\approx {\displaystyle \sum _{i=0}^{2N}{W}_i^{(c)}\{x^{*}{}_{k+1|k}(i)-{\widehat{x}}^a{}_{k+1|k}\}{\{x^{*}{}_{k+1|k}(i)-{\widehat{x}}^a{}_{k+1|k}\}}^T}\end{array}$$

(12)

where weights $W_0^{(m)}=\lambda /(N+\lambda )$, $W_0^{(c)}=1-{\alpha }^2+\beta$ and $W_i^{(m)}=W_i^{(c)}=\frac{1}{2(N+\lambda )}$; $\beta$ depends on the prior distribution of $x$, and usually $\beta =2$ in the case of Gassain.

We take a set of sigma-points to form the vector matrix ${\chi }_{k+1|k}$ by Equation (11) using ${\widehat{x}}^a{}_{k+1|k}$ and $P_{k+1|k}^a$. We substitute ${\chi }_{k+1|k}$ into the observation function of the nonlinear system:

$$\begin{array}{l}{y}_{k+1|k}=\mathrm{h}({\chi }_{k+1|k}) \\ {\widehat{y}}_{k+1|k}\approx {\displaystyle \sum _{i=0}^{2N}{W}_i^{(m)}{y}_{k+1|k}(i)} \end{array}$$

(13)

• State updating

$$\begin{array}{l}{P}_{yy}={\displaystyle \sum _{i=0}^{2N}{W}_i^{(c)}\{y_{k+1|k}(i)-{\widehat{y}}_{k+1|k}\}{\{y_{k+1|k}(i)-{\widehat{y}}_{k+1|k}\}}^T}\\ P_{xy}={\displaystyle \sum _{i=0}^{2N}{W}_i^{(c)}\{{\chi }_{k+1|k}(i)-{\widehat{x}}^a{}_{k+1|k}\}{\{y_{k+1|k}(i)-{\widehat{y}}_{k+1|k}\}}^T}\end{array}$$

$$\begin{array}{l}{K}_k=P_{xy}{P}_{yy}{}^{-1}\\ {\widehat{x}}^a{}_{k+1}={\widehat{x}}^a{}_{k+1|k}+K_k(y_{k+1}-{\widehat{y}}_{k+1|k})\\ P_{{}_{k+1}}^a=P_{k+1|k}^a-K_k{P}_{yy}{K}_k{}^T\end{array}$$

(14)

We take the state $S=(x_e,y_e,{\theta }_e, {\omega }_L, {\omega }_R)$ as the N-dimensional independent variable $x$; take the pose of the roadheader in the XOY coordination, $(X_c,Y_c,X_c)$, as the observation vector $y$ in the procedure; with the process noise ${\epsilon }_k$ being relevant with the skidding ratio, which varies obeying the uniform distribution with boundary values at [0.1, 0.35]; the measurement noise ${\delta }_{k+1}$, which obeys a normal distribution with the mean value of 0 and a standard deviation of 4 cm; and the direction angle deviation ${\theta }_e$ obeys a normal distribution with a mean value of 0 and a standard deviation of 5°.

Given an initial state $S_0$ and a set of pre-designed poses along the expected path, using SVD-Unscented Kalman filtering, the ${\widehat{x}}^a{}_{k+1}$ and ${\widehat{y}}_{k+1|k}$ from Equation (14) would be obtained as the estimated values of the state $Q=(x_e,y_e,{\theta }_e, {\omega }_L, {\omega }_R)$ and the $(X_c,Y_c,X_c)$ at the moment k + 1, respectively. Then, $(x_e,y_e,{\theta }_e)$ is fed back into Equation (4) for the recommended $v_c$ and ${\omega }_c$, i.e., the heading speed $\dot{x}$ and turning angular speed $\dot{\phi }$, respectively.

4. Path Tracking Control of Roadheader Combining BP Neural Network and State Estimation

Comprehensively considering the optimization of the coefficient $k_x , k_y, k_{\theta }$ as introduced in Section 2.2, the effect of skidding, which is equivalent to a sort of process noise, and the possible pose measurement noise during the tracking control, this paper improves the path tracking control based on pose deviation model by combining the BP neural network and state estimation. The framework of the proposed model is shown in Figure 10. The state estimation is implemented through SVD–Unscented Kalman filtering during the path tracking in which the used control low is renewed in a timely manner in the matter of coefficients.

Breakdowns are not considered in the framework, as shown in Figure 10. If this happens, for safety, the power would be cut off according to the general protection procedures involved in the control system of the roadheader; then, the path correction would break immediately.

5. Simulations of the Path Tracking Control for the Roadheader

To evaluate the performance of the proposed path tracking model, a group of tracking simulations are carried out in this section. A transition path heading to the aimed roadway was designed previously for the yawed roadheader, which is shown in Figure 11a with a series of positions. The corresponding expected orientation of the roadheader during tracking is shown in Figure 11b, and the required steering angular velocity in the process is shown in Figure 11c. Other relevant parameters used in the simulations are specified in Table 1, but not the randomly chosen $k_x , k_y, k_{\theta }$, as BP neural network optimization is applied. Instead, they would update regularly until the pose deviations become zero.

To explain how the strategy of path tracking combining state estimation and BP neural network is carried out, the tracking simulations consider three scenarios for comparison. The first scenario is as specified in Section 2.2: $k_x , k_y, k_{\theta }$ is not optimized and the noises are not considered. The second scenario is to obtain a set of $k_x , k_y, k_{\theta }$ through the BP neural network but without using state estimation when noises are considered. The third scenario is to use both $k_x , k_y, k_{\theta }$ optimization and state estimation as the proposed path tracking, combining state estimation and BP neural network to deal with the uncertainty during the real path tracking for the roadheader.

The path tracking under the first scenario is successful when using the basic tracking control based on the pose derivation model. The results are close to the situation presented by Figure 5, which is achieved by choosing suitable values for $k_x , k_y, k_{\theta }$. Although this is rarely the case in a real situation, it is another example to verify that the basic path tracking control based on the pose derivation model is theoretically reasonable for the roadheader.

The simulation results of path tracking under the second and third scenarios are shown in Figure 12 and Figure 13, respectively. Figure 12a shows the coincidence between the planned path and centroid trajectory during the scheduling process. Figure 12b shows how the position and orientation errors change in the process; these errors are expected to converge to zero as soon as possible. Figure 12c shows the variance of the angular speeds of the driving wheels for the tracks on the left and right sides, respectively. Similar explanations are suitable for the illustrations in Figure 13a–c.

Comparing Figure 12a and Figure 13a, it can be seen that the simulated tracking path and expected path are generally coincident in both scenarios. However, the situation in Figure 13a applying states estimation in addition is better, namely, the differences between expected and tracking paths are smaller, and the obtained whole tracking path is visually smoother.

Comparing Figure 12b and Figure 13b, it can be seen that

(1) The position and orientation errors $y_e$ and ${\theta }_e$ of the roadheader converge to a small range around 0 within a limited number of tracking steps in both scenarios. The convergence rate in Figure 13b is faster than that in Figure 12b.

(2) The position error $x_e$ of the roadheader also converges as tracking takes place in both scenarios; however, it does not end at zero but at an average slightly higher than zero.

According to the illustration in Figure 1 and Equation (1), it is acceptable that $x_e$ is larger than $y_e$, as long as the errors are not absolutely zero. A reasonable explanation is as follows: recall that $x_e$ is the summed projection of $X_d-X_c$ and $Y_d-Y_c$ along the x-axis of vehicle coordinate system $xoy$ from the geodetic coordinate system $XOY$, while $y_e$ is the summed projection of $X_d-X_c$ and $Y_d-Y_c$ along the y-axis of $xoy$. When the orientation ${\theta }_c$ of the roadheader under the geodetic coordinate system $XOY$ is close to 0°, the summed projection along the x-axis is larger than that along the y-axis since $\cos {\theta }_c$ is always larger than $\sin {\theta }_c$. On the contrary, if ${\theta }_c$ is inclined to 90°, $y_e$ would be larger than $x_e$ as long as $X_d-X_c$ or $Y_d-Y_c$ $XOY$ is not zero. Since the differences between the expected and real local pose of the roadheader are not likely to be exactly zero, if the designed path follows the X-axis more likely than the Y-axis of $XOY$, the position error $x_e$ would be slightly above zero and $y_e$ as well. This fits the situations shown in Figure 12b and Figure 13b.

To summarize based on (1) and (2), under certain levels of processing and measurement noise, such as those specified Section 3.1 and Section 3.2, the tracking control proposed in this paper is able to achieve path tracking regardless of whether states estimation is adopted. However, the tracking performance variance between the expected and adjusted paths, the pose error convergence rate, is improved notably by taking the benefits of states estimation by implanting SVD–Unscented Kalman filtering in the process of path tracking control using BP Neural network optimization.

It is observed by comparing Figure 12c and Figure 13c that the angular speeds of the two driving wheels are more stable by applying states estimation. The sharp jumps along the curve in Figure 13c correspond to the abrupt change points of the required steering angle speed, which are normal scheduling adjustments.

In conclusion, the path tracking simulations of different scenarios show that the proposed scheduling control strategy combining the BP neural network and state estimation is simple in structure and easy to carry out. Compared with the basic pose deviation-based tracking control strategy, the tracking effect of the proposed method is better and more stable. It is of great potential to be applied in practical roadheader scheduling underground in mining and to be included in the intelligence framework for the collaborative control between roadheaders and other equipment in coal mining.

6. Conclusions

This paper proposed a path correction scheduling strategy, namely, a path tracking control model for the underground mining boom roadheader by combining the BP neural network and state estimation ably. It refers to the pose deviation model of general vehicle motion as the basis and designs a path tracking control model to drive the pose deviation of the roadheader. The target trajectory converges to zero by recommending angular velocities for the left- and right-side driving wheels according to the local pose detection of the roadheader. In order to improve the engineering applicability of this basic model, the control law is simplified by introducing a set of coefficients, $k_x , k_y, k_{\theta }$, which are random positive values corresponding to the position and orientation errors $x_e,y_e,{\theta }_e$, respectively.

In view of the phenomenon that different allocation of coefficients $k_x , k_y, k_{\theta }$ in the control law leads to different tracking effects, the control law is dynamically optimized by updating coefficients $k_x , k_y, k_{\theta }$ using the BP neural network, which is beneficial for dealing with complicated situations, such as devious trajectory for tracking or disadvantageous initial pose deviation.

Additionally, considering that there is skidding between tracks upon the ground and that there are errors in the instant pose measurements of the walking roadheader in practice, this paper treats the unpredictable effects of track skidding as the process noise and the pose detection errors as the observation noise. Thus, we solve the uncertainty in the control model by applying the statistical method instead of analyzing the complex mechanism of track skidding or pose detection of a working roadheader. With the summarized distribution regularities of the track skidding ratios and the pose detection errors of certain positioning systems, SVD–Unscented Kalman filtering is applied to estimate the real pose deviation to the tracked path and to renew the scheduling control instruction for the roadheader.

The novelty of this scheduling strategy is that by combining the BP neural network and state estimation into the pose deviation-based path tracking control model for the roadheader, the involved control law is enabled to update dynamically according to the local tracking performance; simultaneously, the control instructions have a chance to be further revised by considering errors during the procedure of tracking. This is believed to make the path tracking control of the roadheader more robust when facing uncertainty resulting from pose detection errors and track skidding.

The path tracking simulations also show that our proposed strategy is able to achieve path tracking well within a few operations and is of high adaptability when facing complex trajectory while still giving stable control instructions for the driving wheels. The structure of the scheduling strategy is simple and clear, without adding extra devices or massive algorithms; this is attractive in terms of industrial applications in coal mining. Our research goal is to fully study the feasibility and performance of this proposed scheduling strategy for path correction in practical applications of roadway forming using roadheaders in coal mining.