Research on Path Planning and Control Method for Secondary Autonomous Cutting of Cantilever Roadheader in a Large-Section Coal Roadway

Abstract

A cantilever roadheader is the main tunneling equipment for underground coal mine roadways. The key to the safe, efficient and intelligent development of coal enterprises is to achieve the autonomous cutting and intelligent control of the cantilever roadheader. In order to realize the automatic cutting shaping control of a large-section coal roadway, the path planning and control method of secondary automatic cutting of a cantilever roadheader were studied. The Wangjialing 12307 belt roadway was used as the engineering background, the vertical displacement law of the roadway roof under different cutting paths was simulated with the FLAC 3D software, the reasonable cutting path was determined according to the actual situation, and the underground industrial test was carried out. The simplified model and spatial position and attitude coordinate system of the roadheader were established, the kinematics of the roadheader was analyzed, and the position and attitude expression of the cutting head center in the roadway coordinate system was obtained. The simplified model of the cutting head was established, the position expression of the pick in the roadway coordinate system was derived, the position coordinate of the inflection point and the cutting step distance were determined according to the relationship between the cutting head and the roadway boundary, and the cutting path control flow was designed. Finally, the reliability of the cutting path control method was verified with a MATLAB simulation. The research works provide a theoretical foundation for path planning and control to realize “secondary autonomous cutting of cantilever roadheader”.

1. Introduction

Coal is an important energy source for the development of most countries in the world, and China’s annual demand for coal remains high. It can be said that fossil energy such as coal will remain the main energy source in China for a long time to come [1,2,3]. Coal mining mainly adopts the underground mining method, and a large number of roadways need to be excavated to ensure the efficient production of coal, so it can be said that roadway tunneling is the premise of coal mining [4,5]. The main corollary equipment for roadway excavation in coal mines in China is an anchor drilling machine and cantilever roadheader, and the cantilever roadheader integrates the functions of cutting, walking, transportation and dust reduction [6]. However, the tunneling of domestic underground roadways is mainly realized by the manual operation of the driver, and the forming accuracy of the roadway section and the efficiency of the excavation will inevitably be affected by the factors of the roadheader driver. In addition, with the increasing depth of the roadway, the working environment under the shaft is becoming worse and worse, and the personal safety of the workers during tunneling cannot be fully guaranteed. Therefore, the development direction of coal mining enterprises is to realize the automation and intelligence of tunneling equipment, and the autonomous cutting and intelligent control of the roadheader are the key technical problems that need to be solved urgently for the development of coal mining enterprises [7,8,9].

Therefore, many scholars at home and abroad have made numerous explorations on the autonomous cutting and intelligent control methods of roadheaders. Yang et al. took the cutting motor current of the roadheader during cutting as the basis for the change in cutting load, and adopted PID control and adjustment of the expansion and contraction speed of the hydraulic cylinder to propose an intelligent cutting control method for the roadheader cutting section [10]. Zhao et al. combined principal component analysis and neural network to predict the cutting efficiency of a roadheader, which improved the cutting efficiency of the roadheader [11]. Dolipski et al. studied the relationship between cutting head speed and cutting energy consumption of a roadheader [12]. Tuncer et al. used an improved genetic algorithm for the dynamic path planning of mobile robots [13]. Ebrahimabadi et al. and Jasiulek et al. used a GA-BP neural network to enable the roadheader to identify the hardness of coal or rock in front of it when cutting, so as to improve the cutting performance [14,15]. Su et al. established the cutting model of a trapezium roadway interface, determined the inflexion point position of cutting with an approximate iteration method, and simulated it with the MATLAB software, greatly improving the cutting shaping effect of the roadway [16]. Hu et al. simulated the change law of stress and displacement of the surrounding rock of a roadway under different cutting paths by using FLAC 3D, and determined the optimal cutting path in combination with the actual situation in the mine [17]. Xu et al. established the spatial pose model of a cantilever roadheader by using the D-H parameter method, deduced the spatial pose matrix of roadheader and cutting head, determined the coordinates of the inflection point of the cutting path, and finally simulated it with MATLAB [18]. Zhang et al. realized the adjustment of the pitching angle of the roadheader during tunneling by combining the particle swarm optimization algorithm and PID [19]. Yang et al. used a genetic algorithm and a mutation particle swarm optimization algorithm to optimize the path target point set, effectively improving the roadheader’s attitude planning capability [20].

Although predecessors have conducted a lot of research, it was mainly based on the single cutting of a roadheader. For large-section roadways, it is difficult for the current mainstream cantilever roadheader to cut the roadway into shape at one time, and roadways needed to be formed by secondary cutting, which required research on the control method of the roadheader’s secondary autonomous cutting roadways.

Therefore, based on the research background of the Wangjialing 12307 belt roadway, this paper studied the path planning and control method of the secondary autonomous cutting of a cantilever roadheader. Based on the geological conditions of Wangjialing, the roadway tunneling model was established using FALC 3D, the vertical variation law of the surrounding rock of the roadway under different cutting paths was simulated, the reasonable cutting path was determined in combination with the actual situation in the mine, and the underground industrial experiments were carried out. Then, the space pose model of the cantilever roadheader was established, the space pose transformation matrix of the roadheader cutting head was derived using the D-H parameter method, and the limit cutting area of the cutting head was simulated using MATLAB. Finally, the inflexion coordinates of the reasonable cutting path were determined, the cutting path control method was proposed, and the cutting path control process was simulated with MATLAB. This research provide a theoretical foundation for path planning and control to realize the “secondary autonomous cutting of cantilever roadheader”.

2. Cutting Path Planning

The roadway section is formed after being cut by the cutting head according to a certain cutting path, and different cutting paths will have different effects on the front coal wall and the roof of the empty roof area, thus affecting the actual efficiency of the roadheader and the personal safety of the workers. Therefore, determining a reasonable cutting path is an important means to ensure the high efficiency and safety of roadway excavation. This chapter took the Wangjialing 12307 belt roadway as the engineering background, and used the FLAC 3D software to simulate and analyze the vertical displacement of the surrounding rock of the roadway under different cutting paths.

2.1. Project Overview

The section size of the Wangjialing 12307 belt roadway is 5.6 m × 3.55 m, belonging to a large-section roadway. As the cantilever roadheader cannot shape the roadway after one cutting, the method of secondary cutting is adopted. First cut the roadway section on the left, then move the roadheader to a suitable place, and then cut the roadway section on the right. The schematic diagram of the secondary cutting roadway is shown in Figure 1.

2.2. Model Establishment

According to Saint Venant’s principle, the influence intensity of roadway excavation will decrease with an increase in the distance from the excavation point. It is generally believed that the influence range is 3–5 times the excavation radius of the roadway [21]. Accordingly, a roadway model with a size of 50 m × 10 m × 40 m was established, and the grid density near the coal seam was properly densified to improve the accuracy of the simulation. The model is shown in Figure 2.

According to the physical and mechanical test report of the Wangjialing 12307 belt roadway, the geological parameters of each coal bed were assigned. The numerical simulation geological parameters are shown in

Table 1. Geological parameters of the numerical simulation.

Number of Layers	Thickness /m	Poisson Ratio	Elastic Modulus /GPa	Internal Friction Angle/°	Tensile Strength /Mpa	Density kg/m3	Cohesion /MPa	Rock Character
1	3.80	0.20	38.74	34	2.10	2400	9.25	Siltstone
2	4.17	0.29	12.65	32	1.02	2460	3.50	Sandy mudstone
3	5.99	0.25	32.74	31	1.62	2400	8.16	Fine-grained sandstone
4	3.18	0.29	12.65	32	1.02	2460	3.50	Sandy mudstone
5	6.02	0.35	3.96	40	0.51	1400	1.60	Coal
6	1.05	0.29	12.65	32	1.02	2460	3.50	Sandy mudstone
7	1.09	0.25	32.74	31	1.62	2400	8.16	Fine-grained sandstone
8	14.70	0.20	38.74	34	2.10	2400	9.25	Siltstone

. A vertical stress of 7.35 MPa on the top boundary of the model was applied, and the Mohr Coulomb yield criterion was used for the simulation.

2.3. Simulation Analysis of Cutting Path

The current mainstream cutting paths of cantilever roadheaders include “loop” and “snake”. In order to reduce the idling time of the cutting head, four cutting paths were designed as follows, as shown in Figure 3.

2.3.1. “Snake” Cutting Path Simulation

Cutting path 1 and cutting path 2 were simulated, and the vertical displacement of roadway roof under the two cutting paths was analyzed, as shown in Figure 4 and Figure 5.

(1)

Simulation analysis of cutting path 1

As shown in Figure 4, the maximum vertical displacement of the roadway roof increases with the increase in cutting times. When the roadway is shaped, the roadway roof displacement reaches the maximum value of about 126 mm. This is because with the increase in cutting times, the two sides of the roadway are gradually formed, and the displacement of the two sides increases, leading to an increase in the roof displacement. In addition, with the gradual shaping of the two sides of the roadway, the maximum vertical displacement amplitude of the roadway roof gradually slows down, and the coal body gradually becomes stable.

(2)

Simulation analysis of cutting path 2

The cutting process under cutting path 1 was similar to that under cutting path 2, only the cutting path was opposite. Therefore, in order to reduce the space occupied by the simulation results, the roof vertical displacement nephogram after the fifth and tenth cutting was selected for analysis.

As shown in Figure 5, the maximum vertical displacement of the roadway roof increases with the increase in cutting times. When the roadway is shaped, the maximum displacement of the roadway roof reaches about 123 mm.

2.3.2. “Loop” Cutting Path Simulation

Cutting paths 3 and 4 were simulated, and the vertical displacement of the roadway roof under the two cutting paths was analyzed, as shown in Figure 6 and Figure 7.

(1)

Simulation analysis of cutting path 3

According to an analysis of Figure 6, the maximum displacement of the roadway roof after the first cut is 96 mm. After the second cut, the maximum displacement of roadway roof reaches 99 mm. After the third cut, the roadway floor is formed, the maximum displacement of the roof is 105 mm, and the maximum displacement of the floor is 40 mm. The left side is formed after the fourth cut, and the maximum displacement of the roadway roof is 107 mm, with a small change. With the increase in cutting times, the displacement of the roadway roof increases continuously, and the maximum displacement of the roadway roof after cutting is 126 mm.

(2)

Simulation analysis of cutting path 4

Similarly, as the cutting process under cutting path 4 was similar to that under cutting path 3, in order to reduce space, the roof vertical displacement nephogram with representative cutting times was selected for analysis.

According to an analysis of Figure 7, after the first cut, the roadway floor is formed and the floor displacement is small. After the third cut, the roof is formed, and the roof displacement is 95 mm. The left side is formed after the fourth cut, and the drift roof displacement is 98 mm, with a small change. With the increase in cutting times, the displacement of the roadway roof also increases correspondingly, but the displacement variation amplitude of the roadway roof is smaller than that under working conditions 1~3. The maximum displacement of the roadway roof after cutting is 120 mm.

2.4. Determination of Cutting Path

According to the statistical analysis and numerical simulation Figure 4, Figure 5, Figure 6 and Figure 7, the maximum displacement and displacement change amplitude of the roadway roof of cutting path 4 were minimal, and the roof was easy to control during roadway cutting. Combined with the actual situation of Wangjialing and the numerical simulation results, the cutting path is determined as shown in Figure 8 below.

In Figure 8, the dotted line represents the motion track of the cutting head, the black arrow represents the cutting direction of the cutting head, and the numerical sequence represents the sequence of the central position of the cutting head.

The cutting head is drilled at the left side of the roadway. After the cutting head has fully entered, the cutting arm swings from left to right to 3300 mm from the left side of the roadway, then the cutting arm swings from bottom to top and cuts along the “reverse loop” cutting path. The cantilever roadheader adjusts its position for the second cutting. The cutting head is drilled 1600 mm from the right side of the roadway. After the cutting head is fully entered, the cutting arm cuts along the “reverse loop” cutting route.

2.5. Industrial Test of Cutting Path

The industrial test of the cutting path was conducted in the belt roadway of the Wangjialing 12307 working face. According to the field measurement, after the roadway was cut and formed, the cutting section was flat and the forming effect was good. The roadheader could remain stable during the cutting process and was convenient for the roadheader driver to operate. The roadway cutting section is shown in Figure 9.

3. Kinematics Analysis of Cantilever Roadheader

The kinematics of the roadheader can be divided into two parts: cutting mechanism kinematics and body kinematics. The kinematics of the cutting mechanism mainly describes the spatial position and attitude relationship between the cutting head and the body during the cutting process, while the kinematics of the body mainly describes the spatial position and attitude relationship between the body and the roadway during the cutting process of the roadheader. The cutting mechanism and body kinematics analysis are the theoretic foundation for the roadheader to realize autonomous cutting and control, which has a significant influence on the cutting and shaping effect of the roadway section.

3.1. Establishment of Space Position and Attitude Coordinate System

The cutting head, rotary table, cantilever, hydraulic cylinder and traveling mechanism of a roadheader can be regarded as consisting of a series of translated or rotated joints and connecting rods [22]. Based on this, a simplified model of the robotization cantilever roadheader and its spatial position and attitude coordinate system were established [23], as shown in Figure 10.

In the figure: O_h-X_hY_hZ_h is the roadway coordinate system, the O_hZ_h axis points vertically to the roadway roof, the O_hX_h axis points to the heading direction of the roadheader, and the O_hY_h axis is parallel to the roadway floor.

O₀-X₀Y₀Z₀ is the center of the gravity coordinate system of the roadheader body, which is convenient for establishing the position and attitude transformation relationship among the cutting head, body and roadway coordinate system.

0_i-X_iY_iZ_i (i = 1, 2, 3, 4) is the coordinate system of the cutting mechanism of the roadheader, which is established at each link joint. 0₁-X₁Y₁Z₁ is the rotary table coordinate system, 0₂-X₂Y₂Z₂ is the cantilever coordinate system, 0₃-X₃Y₃Z₃ is the cutting head expansion part coordinate system, and 0₄-X₄Y₄Z₄ is the cutting head coordinate system.

In addition, a, d, θ and α are a joint variable. a_i is the distance of translation along the O_iX_i axis so that Z_i and Z_i+1 coincide, θ_i is the angle of rotation around the O_iZ_i axis to make the X_i−1 axis and X_i axis collinear, d_i is the distance of translation along the O_iZ_i axis to make the X_i−1 axis and X_i axis parallel, α_i is the angle of rotation around the X_i axis to make Z_i and Z_i+1 collinear, and d is the retraction stroke of the cutting head.

3.2. Kinematics Analysis of Cutting Mechanism

3.2.1. Forward Kinematics Solution of Cutting Mechanism

The forward kinematics problem of the cutting mechanism of the roadheader is to solve the spatial position and posture of the cutting head in different coordinate systems when the geometric parameters and joint variables of each member of the cutting mechanism is known. The D-H parameter method can be used to solve the forward kinematics problem, to realize the spatial position and orientation of the cutting head.

Based on the principle of the D-H parameter method [24], the coordinate vector {xⁱ, yⁱ, zⁱ, 1}^T of any point in the coordinate system O_i-X_iY_iZ_i and the coordinate vector {xⁱ⁻¹, yⁱ⁻¹, zⁱ⁻¹,1}^T in the relative coordinate system O_i-1-X_i-1Y_i-1Z_i-1 have the following relationship:

$${\left\{{\mathrm{x}}^{\mathrm{i}-1},{\mathrm{y}}^{\mathrm{i}-1},{\mathrm{z}}^{\mathrm{i}-1},1\right\}}^{\mathrm{T}}={}_{\mathrm{i}}{}^{\mathrm{i}-1}\mathrm{T}{\left\{{\mathrm{x}}^{\mathrm{i}},{\mathrm{y}}^{\mathrm{i}},{\mathrm{z}}^{\mathrm{i}},1\right\}}^{\mathrm{T}}$$

(1)

where ${}_{\mathrm{i}}{}^{\mathrm{i}-1}\mathrm{T}$ is the position and attitude transformation matrix of the coordinate system O_i-X_iY_iZ_i relative to the coordinate system O_i-1-X_i-1Y_i-1Z_i-1.

According to Formula (1), the coordinate vector {x⁴, y⁴, z⁴,1}^T of any point in the coordinate system O₄-X₄Y₄Z₄ of the cutting head and the coordinate vector {x⁰, y⁰, z⁰,1}^T in the coordinate system O₀-X₀Y₀Z₀ of the fuselage center of gravity have the following relationship:

$${\left\{{\mathrm{x}}^0,{\mathrm{y}}^0,{\mathrm{z}}^0,1\right\}}^{\mathrm{T}}={}_1{}^0\mathrm{T}·{}_2{}^1\mathrm{T}·{}_3{}^2\mathrm{T}·{}_4{}^3\mathrm{T}{\left\{{\mathrm{x}}^4,{\mathrm{y}}^4,{\mathrm{z}}^4,1\right\}}^{\mathrm{T}}={}_4{}^0\mathrm{T}{\left\{{\mathrm{x}}^4,{\mathrm{y}}^4,{\mathrm{z}}^4,1\right\}}^{\mathrm{T}}$$

(2)

According to Figure 10, the linkage parameters of the cutting mechanism were established, as shown in

Table 2. Connecting rod parameters of cutting mechanism.

Connecting Rod i	ai-1	αi-1	di	θi
1	a0	0°	d1	θ1
2	a1	90°	0	θ2
3	a2	90°	d3+d	0
4	0	0°	d4	0

.

According to the parameter table of the linkage of the cutting mechanism and Formula (2), the position and attitude transformation matrix of the cutting head relative to the gravity center coordinate system ${}_4{}^0\mathrm{T}$ of the fuselage was obtained:

$$\begin{gathered} {}_4{}^0\mathrm{T}={}_1{}^0\mathrm{T}·{}_2{}^1\mathrm{T}·{}_3{}^2\mathrm{T}·{}_4{}^3\mathrm{T}= \\ \left[\begin{array}{cccc}c{\mathsf{\theta }}_{1}c{\mathsf{\theta }}_2& s{\mathsf{\theta }}_1& c{\mathsf{\theta }}_{1}s{\mathsf{\theta }}_2& a_0+a_{1}c{\mathsf{\theta }}_1+\left(d+d_3\right)c{\mathsf{\theta }}_{1}s{\mathsf{\theta }}_2+a_{2}c{\mathsf{\theta }}_{1}c{\mathsf{\theta }}_2+d_{4}c{\mathsf{\theta }}_{1}s{\mathsf{\theta }}_2\\ c{\mathsf{\theta }}_{2}s{\mathsf{\theta }}_1& -c{\mathsf{\theta }}_1& s{\mathsf{\theta }}_{1}s{\mathsf{\theta }}_2& a_{1}s{\mathsf{\theta }}_1+\left(d+d_3\right)s{\mathsf{\theta }}_{1}s{\mathsf{\theta }}_2+a_{2}c{\mathsf{\theta }}_{2}s{\mathsf{\theta }}_1+d_{4}s{\mathsf{\theta }}_{1}s{\mathsf{\theta }}_2\\ s{\mathsf{\theta }}_2& 0& -c{\mathsf{\theta }}_2& d_1-\left(d+d_3\right)c{\mathsf{\theta }}_2-d_{4}c{\mathsf{\theta }}_2+a_{2}s{\mathsf{\theta }}_2\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$

(3)

where c represents cos, s represents sin, ${}_{\mathrm{i}}{}^{\mathrm{i}-1}\mathrm{T}\left(\mathrm{i}=1, 2, 3, 4\right)$ is the posture transformation matrix of the cutting mechanism coordinate system i relative to the cutting mechanism coordinate system i−1, and

$$\begin{gathered} {}_1{}^0\mathrm{T}=\left[\begin{array}{cccc}{\cos \mathsf{\theta }}_1& -{\sin \mathsf{\theta }}_1& 0& {\mathrm{a}}_0\\ {\sin \mathsf{\theta }}_1& {\cos \mathsf{\theta }}_1& 0& 0\\ 0& 0& 1& {\mathrm{d}}_1\\ 0& 0& 0& 1\end{array}\right],{}_2{}^1\mathrm{T}=\left[\begin{array}{cccc}{\cos \mathsf{\theta }}_2& -{\sin \mathsf{\theta }}_2& 0& {\mathrm{a}}_1\\ 0& 0& -1& 0\\ {\sin \mathsf{\theta }}_2& {\cos \mathsf{\theta }}_2& 0& 0\\ 0& 0& 0& 1\end{array}\right] \\ {}_3{}^2\mathrm{T}=\left[\begin{array}{cccc}1& 0& 0& {\mathrm{a}}_2\\ 0& 0& -1& -\left({\mathrm{d}}_3+\mathrm{d}\right)\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right],{}_4{}^3\mathrm{T}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& {\mathrm{d}}_4\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$

Equation (3) can be substituted into Equation (2) to obtain the coordinates {x⁰, y⁰, z⁰} of the cutting head center O₄ (at this time, x⁴ = y⁴ = z⁴ = 0,) in the fuselage gravity center coordinate system O₀-X₀Y₀Z₀ as follows:

$$\left\{\begin{array}{c}{\mathrm{x}}^0={\mathrm{a}}_0+{\mathrm{a}}_{1}cos{\mathsf{\theta }}_1+cos{\mathsf{\theta }}_{1}sin{\mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)+{\mathrm{a}}_{2}cos{\mathsf{\theta }}_{1}cos{\mathsf{\theta }}_2+{\mathrm{d}}_{4}cos{\mathsf{\theta }}_{1}sin{\mathsf{\theta }}_2\\ {\mathrm{y}}^0={\mathrm{a}}_{1}sin{\mathsf{\theta }}_1+\left(\mathrm{d}+{\mathrm{d}}_3\right)sin{\mathsf{\theta }}_{1}sin{\mathsf{\theta }}_2+{\mathrm{a}}_{2}cos{\mathsf{\theta }}_{2}sin{\mathsf{\theta }}_1+{\mathrm{d}}_{4}sin{\mathsf{\theta }}_{1}sin{\mathsf{\theta }}_2\\ {\mathrm{z}}^0={\mathrm{d}}_1-cos{\mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)-{\mathrm{d}}_{4}cos{\mathsf{\theta }}_2+{\mathrm{a}}_{2}sin{\mathsf{\theta }}_2\end{array}\right.$$

(4)

3.2.2. Inverse Kinematics Solution of Cutting Mechanism

The inverse kinematics problem of the cutting mechanism of the roadheader is to know the geometric parameters of each link of the cutting mechanism and solve the joint variables of each link when the cutting head is positioned relative to different coordinate systems. The inverse kinematics solution of the cutting mechanism is the basis for the planning and precise control of the cutting head’s moving track.

The inverse kinematics of the cutting mechanism can be solved with the inverse transformation method [23]. The position and attitude transformation matrix ${}_4{}^0\mathrm{T}$ of the cutting head relative to the gravity center coordinate system of the fuselage could also be expressed as:

$${}_4{}^0\mathrm{T}=\left[\begin{array}{cccc}{\mathrm{n}}_{\mathrm{x}}& {\mathrm{o}}_{\mathrm{x}}& {\mathrm{a}}_{\mathrm{x}}& {\mathrm{p}}_{\mathrm{x}}\\ {\mathrm{n}}_{\mathrm{y}}& {\mathrm{o}}_{\mathrm{y}}& {\mathrm{a}}_{\mathrm{y}}& {\mathrm{p}}_{\mathrm{y}}\\ {\mathrm{n}}_{\mathrm{z}}& {\mathrm{o}}_{\mathrm{z}}& {\mathrm{a}}_{\mathrm{z}}& {\mathrm{p}}_{\mathrm{z}}\\ 0& 0& 0& 1\end{array}\right]=\left[\begin{array}{cccc}\mathrm{n}& \mathrm{o}& \mathrm{a}& \mathrm{p}\\ 0& 0& 0& 1\end{array}\right]$$

(5)

where a is the approach vector, o is the azimuth vector, n is the normal vector, and p is the position vector. Three orthogonal unit column vectors (n, a and o) describe the position and orientation of the cutting head.

It could be seen from the analysis of Equation (5) that each variable in the pose transformation matrix of Equation (3) should be equal to the corresponding element in the matrix composed of joint variables of Equation (5). From these equations, the relationship between joint variables and n, o, a, p vectors could be obtained.

From the equality of elements (1,2) and (2,2) in Equation (3) and Equation (5), it could be obtained that ${\mathsf{\theta }}_1=-\arctan \frac{{\mathrm{o}}_{\mathrm{x}}}{{\mathrm{o}}_{\mathrm{y}}}$ or ${\mathsf{\theta }}_1={\mathsf{\theta }}_1+180°$.

According to the equality of (3,1) and (3,3) elements in Equation (3) and Equation (5), we could obtain ${\mathsf{\theta }}_2=-\arctan \frac{{\mathrm{n}}_{\mathrm{z}}}{{\mathrm{a}}_{\mathrm{z}}}$ or ${\mathsf{\theta }}_2={\mathsf{\theta }}_2+180°$.

According to the corresponding equality of (1,4), (2,4) and (3,4) elements of Equation (4) and Equation (9), we could obtain $\mathrm{d}=\frac{{\mathrm{p}}_{\mathrm{x}}-{\mathrm{a}}_0+\left({\mathrm{a}}_1+{\mathrm{a}}_2\right){\mathrm{o}}_{\mathrm{y}}-{\mathrm{a}}_2{\mathrm{n}}_{\mathrm{x}}}{{\mathrm{a}}_{\mathrm{x}}}-{\mathrm{d}}_3-{\mathrm{d}}_4$ or $\mathrm{d}=\frac{{\mathrm{p}}_{\mathrm{y}}-{\mathrm{a}}_1{\mathrm{o}}_{\mathrm{x}}-{\mathrm{a}}_2{\mathrm{n}}_{\mathrm{y}}}{{\mathrm{a}}_{\mathrm{y}}}-{\mathrm{d}}_3-{\mathrm{d}}_4$ or $\mathrm{d}=\frac{{\mathrm{a}}_2{\mathrm{n}}_{\mathrm{z}}+{\mathrm{p}}_{\mathrm{z}}-{\mathrm{d}}_1}{{\mathrm{a}}_{\mathrm{z}}}-{\mathrm{d}}_3-{\mathrm{d}}_4$. Although there were three formulas to solve d, the formula whose denominator was not 0 could be used to solve parameter d.

3.3. Kinematic Analysis of Roadheader Body

The correct expression of the position and posture of the roadheader body is the basis for the roadheader to achieve accurate and autonomous cutting. The method of describing the position and attitude of aircraft, ships and other vehicles during navigation was used for reference. The “fuselage yaw angle α”, “fuselage pitch angle β”, “fuselage rolling angle γ” and “fuselage offset L” were adopted to describe the position and attitude of the roadheader when tunneling in the roadway coordinate system [25,26]. The design centerline direction of the roadway is defined as the X_h axis; the direction vertically pointing to the roadway roof is taken as the axis Z_h; the Y_h axis is parallel to the roadway floor; the angle of rotation about the Z_h axis is called the yaw angle α; the angle of rotation about the Y_h axis is called the pitch angle β; the angle of rotation about the X_h axis is called the roll angle γ; the three-dimensional space offset distance between the fuselage center line and the roadway center line is called offset L, as shown in Figure 11.

According to the coordinate transformation theory, the fuselage center of gravity coordinate system O₀-X₀Y₀Z₀ can be obtained from the roadway coordinate system Oh X_hY_hZ_h through the following transformation: first rotate around the O_hZ_h axis α angle, then rotate around the O_hY_h axis β angle, then rotate around the O_hX_h axis γ angle, and finally move the vector L (△X, △Y, △Z) along the O_hO_h axis. From this, the position and attitude transformation matrix of the fuselage gravity center coordinate system relative to the roadway coordinate system was obtained as follows:

$$\begin{gathered} {}_0{}^{\mathrm{h}}\mathrm{T}=\mathrm{Rot}\left({\mathrm{Z}}_{\mathrm{h}},\mathsf{\alpha }\right)·\mathrm{Rot}\left({\mathrm{Y}}_{\mathrm{h}},\mathsf{\beta }\right)·\mathrm{Rot}\left({\mathrm{X}}_{\mathrm{h}},\mathsf{\gamma }\right)·\mathrm{Trans}\left(△\mathrm{X}, △\mathrm{Y}, △\mathrm{Z}\right)= \\ \left[\begin{array}{cccc}\mathrm{c}\mathsf{\alpha }\mathrm{c}\mathsf{\beta }& -\mathrm{s}\mathsf{\alpha }\mathrm{c}\mathsf{\gamma }+\mathrm{c}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{s}\mathsf{\gamma }& \mathrm{s}\mathsf{\alpha }\mathrm{s}\mathsf{\gamma }+\mathrm{c}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{c}\mathsf{\gamma }& \mathrm{c}\mathsf{\alpha }\mathrm{c}\mathsf{\beta }△\mathrm{X}+\left(\mathrm{c}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{s}\mathsf{\gamma }-\mathrm{s}\mathsf{\alpha }\mathrm{c}\mathsf{\gamma }\right)△\mathrm{Y}+\left(\mathrm{s}\mathsf{\alpha }\mathrm{s}\mathsf{\gamma }+\mathrm{c}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{c}\mathsf{\gamma }\right)△\mathrm{Z}\\ \mathrm{s}\mathsf{\alpha }\mathrm{c}\mathsf{\beta }& \mathrm{c}\mathsf{\alpha }\mathrm{c}\mathsf{\gamma }+\mathrm{s}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{s}\mathsf{\gamma }& -\mathrm{c}\mathsf{\alpha }\mathrm{s}\mathsf{\gamma }+\mathrm{s}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{c}\mathsf{\gamma }& \mathrm{s}\mathsf{\alpha }\mathrm{c}\mathsf{\beta }△\mathrm{X}+\left(\mathrm{c}\mathsf{\alpha }\mathrm{c}\mathsf{\gamma }+\mathrm{s}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{s}\mathsf{\gamma }\right)△\mathrm{Y}+\left(\mathrm{s}\mathsf{\alpha }\mathrm{s}\mathsf{\beta }\mathrm{c}\mathsf{\gamma }-\mathrm{c}\mathsf{\alpha }\mathrm{s}\mathsf{\gamma }\right)△\mathrm{Z}\\ -\mathrm{s}\mathsf{\beta }& \mathrm{c}\mathsf{\beta }\mathrm{s}\mathsf{\gamma }& \mathrm{c}\mathsf{\beta }\mathrm{c}\mathsf{\gamma }& -\mathrm{s}\mathsf{\beta }△\mathrm{X}+\mathrm{c}\mathsf{\beta }\mathrm{s}\mathsf{\gamma }△\mathrm{Y}+\mathrm{c}\mathsf{\beta }\mathrm{c}\mathsf{\gamma }△\mathrm{Z}\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$

(6)

where c is cos and s is sin.

3.4. Kinematics Analysis of Roadheader

In order to realize the transformation from the cutting head coordinate system to the roadway coordinate system, the kinematics analysis of the roadheader was carried out. The pose transformation matrix of the cutting head coordinate system relative to the roadway coordinate system was denoted as ${}_4{}^{\mathrm{h}}\mathrm{T}$, then:

$${}_4{}^{\mathrm{h}}\mathrm{T}={}_0{}^{\mathrm{h}}\mathrm{T}·{}_4{}^0\mathrm{T}$$

(7)

Then, the coordinate vector {x^h, y^h, z^h, 1}^T of the cutting head center O₄ in the roadway coordinate system O_h-X_hY_hZ_h was:

$${\left\{{\mathrm{x}}^{\mathrm{h}},{\mathrm{y}}^{\mathrm{h}},{\mathrm{z}}^{\mathrm{h}},1\right\}}^{\mathrm{T}}={}_4{}^{\mathrm{h}}\mathrm{T}·{\left\{0,0,0,1\right\}}^{\mathrm{T}}={}_0{}^{\mathrm{h}}\mathrm{T}·{}_4{}^0\mathrm{T}·{\left\{0,0,0,1\right\}}^{\mathrm{T}}$$

(8)

In order to simplify the calculation and simulation analysis, the following assumptions were made: the roadheader kept the body stable during the secondary tunneling process, without any deflection, and only moved along the O_hY_h axis; the center of gravity line of the body of the roadheader at the first tunneling section was set as the center line of the roadway, that is, the gravity center line of the roadheader body coincided with the center line of the roadway during the first tunneling, as shown in Figure 12.

According to the above assumptions, $\mathsf{\alpha }=\mathsf{\beta }=\mathsf{\gamma }=0°$ and $△\mathrm{X}=△\mathrm{Y}=△\mathrm{Z}=0$ for the first tunneling of the roadheader, and $\mathsf{\alpha }=\mathsf{\beta }=\mathsf{\gamma }=0°$, $△\mathrm{X}=△\mathrm{Z}=0$, and $△\mathrm{Y}=2800$ during the second tunneling of the roadheader.

Therefore, ${}_0{}^{\mathrm{h}}\mathrm{T}$ could be simplified as:

$${}_0{}^{\mathrm{h}}\mathrm{T}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& △\mathrm{Y}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(9)

Equation (9) and Equation (3) can be substituted into Equation (8) to obtain the coordinate expression of the cutting head center O₄ in the roadway coordinate system:

$${\mathrm{x}}^{\mathrm{h}}={\mathrm{a}}_0+{\mathrm{a}}_1{\cos \mathsf{\theta }}_1+{\cos \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)+{\mathrm{a}}_2{\cos \mathsf{\theta }}_1{\cos \mathsf{\theta }}_2+{\mathrm{d}}_4{\cos \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2$$

(10a)

$${\mathrm{y}}^{\mathrm{h}}=△\mathrm{Y}+{\mathrm{a}}_1{\sin \mathsf{\theta }}_1+\left(\mathrm{d}+{\mathrm{d}}_3\right){\sin \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2+{\mathrm{a}}_2{\cos \mathsf{\theta }}_2{\sin \mathsf{\theta }}_1+{\mathrm{d}}_4{\sin \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2$$

(10b)

$${\mathrm{z}}^{\mathrm{h}}={\mathrm{d}}_1-cos{\mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)-{\mathrm{d}}_{4}cos{\mathsf{\theta }}_2+{\mathrm{a}}_{2}sin{\mathsf{\theta }}_2$$

(10c)

4. Cutting Path Control

4.1. Pick Coordinate Determination

In fact, the roadway section is formed by the continuous rotation of the cutting head to drive the pick installed on the cutting head to continuously cut the coal wall. Therefore, the boundary of the roadway is formed when the rotation curve of the pick is tangent to the boundary of the roadway section. In order to obtain the curve of pick rotation, the cutting head was regarded as the envelope surface formed by a parabola rotating around the cutting head axis. The simplified model of the cutting head is shown in Figure 13.

In Figure 13, i is any point on the parabola (can be regarded as a pick), and r is the distance from point i to the cutting head axis. Let the equation of parabola be Z = aX² + b; the trajectory equation of pick i rotation then is:

$$\{\begin{array}{c}\mathrm{x}={\mathrm{r}}_{\mathrm{i}}\cos \mathsf{\phi }\\ \mathrm{y}={\mathrm{r}}_{\mathrm{i}}\sin \mathsf{\phi }\\ \mathrm{z}=\mathrm{a}{\mathrm{r}}_{\mathrm{i}}^2+\mathrm{b}\end{array}$$

(11)

Let the coordinate vector of the pick i in the cutting head coordinate system O₄-X₄Y₄Z₄ be:

$${\mathrm{I}}^4=\left\{{\mathrm{x}}_{\mathrm{i}}^4,{\mathrm{y}}_{\mathrm{i}}^4,{\mathrm{z}}_{\mathrm{i}}^4,1\right\}$$

(12)

Combining Formula (11), we could obtain:

$$\{\begin{array}{c}{\mathrm{x}}_{\mathrm{i}}^4={\mathrm{r}}_{\mathrm{i}}\cos \mathsf{\phi }\\ {\mathrm{y}}_{\mathrm{i}}^4={\mathrm{r}}_{\mathrm{i}}\sin \mathsf{\phi }\\ {\mathrm{z}}_{\mathrm{i}}^4=\mathrm{a}{\mathrm{r}}_{\mathrm{i}}^2+\mathrm{b}\end{array}$$

(13)

Therefore, when the cutting head shaft length, radius and cutting depth are known, r_i can be determined according to Equation (13). As the type and cutting depth of the roadheader were designed in advance during cutting, r_i could be regarded as a known quantity in the subsequent calculation.

According to the D-H parameter method, the coordinate vector $\left\{{\mathrm{x}}_{\mathrm{i}}^{\mathrm{h}},{\mathrm{y}}_{\mathrm{i}}^{\mathrm{h}},{\mathrm{z}}_{\mathrm{i}}^{\mathrm{h}},1\right\}$ of pick i in the roadway coordinate system O_h X_hY_hZ_h was:

$$\left\{{\mathrm{x}}_{\mathrm{i}}^{\mathrm{h}},{\mathrm{y}}_{\mathrm{i}}^{\mathrm{h}},{\mathrm{z}}_{\mathrm{i}}^{\mathrm{h}},1\right\}={}_0{}^{\mathrm{h}}\mathrm{T}·{}_4{}^0\mathrm{T}·{\mathrm{I}}^4$$

(14)

The coordinate expression of pick i in the roadway coordinate system O_h-X_hY_hZ_h was obtained by substituting Formulas (3), (9) and (13) into Formula (14):

$$\begin{gathered} {\mathrm{x}}_{\mathrm{i}}^{\mathrm{h}}={\cos \mathsf{\theta }}_1{\cos \mathsf{\theta }}_2{\mathrm{r}}_{\mathrm{i}}\cos \mathsf{\phi }+{\sin \mathsf{\theta }}_1{\mathrm{r}}_{\mathrm{i}}\sin \mathsf{\phi }+{\cos \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2\left({\mathrm{ar}}_{\mathrm{i}}^2+\mathrm{b}\right)+{\mathrm{a}}_0+{\mathrm{a}}_1{\cos \mathsf{\theta }}_1 \\ +{\cos \mathsf{\theta }}_1{\cos \mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)+{\mathrm{a}}_2{\cos \mathsf{\theta }}_1{\cos \mathsf{\theta }}_2+{\mathrm{d}}_4{\cos \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2 \end{gathered}$$

(15a)

$$\begin{gathered} {\mathrm{y}}_{\mathrm{i}}^{\mathrm{h}}=△\mathrm{Y}+{\cos \mathsf{\theta }}_2{\sin \mathsf{\theta }}_1{\mathrm{r}}_{\mathrm{i}}\cos \mathsf{\phi }-{\cos \mathsf{\theta }}_1{\mathrm{r}}_{\mathrm{i}}\sin \mathsf{\phi }+{\sin \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2\left({\mathrm{ar}}_{\mathrm{i}}^2+\mathrm{b}\right)+{\mathrm{a}}_1{\sin \mathsf{\theta }}_1 \\ +\left(\mathrm{d}+{\mathrm{d}}_3\right){\sin \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2+{\mathrm{a}}_2{\cos \mathsf{\theta }}_2{\sin \mathsf{\theta }}_1+{\mathrm{d}}_4{\sin \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2 \end{gathered}$$

(15b)

$${\mathrm{z}}_{\mathrm{i}}^{\mathrm{h}}={\sin \mathsf{\theta }}_2{\mathrm{r}}_{\mathrm{i}}\cos \mathsf{\phi }-{\cos \mathsf{\theta }}_2\left({\mathrm{ar}}_{\mathrm{i}}^2+\mathrm{b}\right)+{\mathrm{d}}_1-{\cos \mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)-{\mathrm{d}}_4{\cos \mathsf{\theta }}_2+{\mathrm{a}}_2{\sin \mathsf{\theta }}_{2 }$$

(15c)

4.2. Determination of Inflection Point of Cutting Path

Research shows that the inflection point position of the cutting path will have a significant impact on the forming quality of the cutting section [27]. The inflection point position of the path was determined according to the planned cutting path (Figure 8) so as to achieve accurate planning and control of the cutting path. The schematic diagram of the inflection point of the cutting path is shown in Figure 14. In the figure, L is the set cutting step, h₁^h_upper and h₁^h_lower are the vertical coordinates of the upper and lower boundary of the left half of the roadway, z₁^h_upper and z₁^h_lower are the vertical coordinates of the cutting head center when cutting the upper and lower boundary of the left half of the roadway, and b₁^h_right is the horizontal coordinates of the bottom of the right boundary of the left half of the roadway.

4.2.1. Determination of Inflection Point Coordinates of Boundary Contour

When the cutting head was cutting the upper boundary of the left half of roadway, by substituting φ = 0 into Equation (15c) the following was obtained:

$${\mathrm{z}}_{\mathrm{i}}^{\mathrm{h}}={\sin \mathsf{\theta }}_2{\mathrm{r}}_{\mathrm{i}}-{\cos \mathsf{\theta }}_2\left({\mathrm{ar}}_{\mathrm{i}}^2+\mathrm{b}\right)+{\mathrm{d}}_1-{\cos \mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)-{\mathrm{d}}_4{\cos \mathsf{\theta }}_2+{\mathrm{a}}_2{\sin \mathsf{\theta }}_2$$

(16)

When the cutting head was cutting the lower boundary of the left half of the roadway, by substituting φ=π into Equation (15c) the following was obtained:

$${\mathrm{z}}_{\mathrm{i}}^{\mathrm{h}}=-{\sin \mathsf{\theta }}_2{\mathrm{r}}_{\mathrm{i}}-{\cos \mathsf{\theta }}_2\left({\mathrm{ar}}_{\mathrm{i}}^2+\mathrm{c}\right)+{\mathrm{d}}_1-{\cos \mathsf{\theta }}_2\left(\mathrm{d}+{\mathrm{d}}_3\right)-{\mathrm{d}}_4{\cos \mathsf{\theta }}_2+{\mathrm{a}}_2{\sin \mathsf{\theta }}_2$$

(17)

The upper ordinate h₁^h_upper = 3550 and the lower ordinate h₁^h_lower = 0 were substituted into Equation (16) and Equation (17) to calculate the θ₂, and θ₂ was then substituted into Formula (10c) to calculate the corresponding cutting head center coordinates z₁^h_upper and z₁^h_lower, and the vertical coordinates of inflection points 1~4 could be obtained.

In the left half of the roadway, the abscissa of the inflection point was symmetrical; therefore, only the abscissa of one inflection point needed be calculated. When the cutting head was cutting the lower right boundary of the left half of the roadway, $△\mathrm{Y}=0$ and $\mathsf{\phi }=\frac{\pi }{2}$ were substituted into Formula (15b) to obtain:

$${\mathrm{y}}_{\mathrm{i}}^{\mathrm{h}}={\cos \mathsf{\theta }}_1{\mathrm{r}}_{\mathrm{i}}+{\sin \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2\left({\mathrm{ar}}_{\mathrm{i}}^2+\mathrm{c}\right)+{\mathrm{a}}_1{\sin \mathsf{\theta }}_1+\left(\mathrm{d}+{\mathrm{d}}_3+{\mathrm{d}}_4\right){\sin \mathsf{\theta }}_1{\sin \mathsf{\theta }}_2+{\mathrm{a}}_2{\cos \mathsf{\theta }}_2{\sin \mathsf{\theta }}_{1 }$$

(18)

The corresponding θ₂ was substituted when the cutting head cut the lower boundary of the left half of the roadway and the abscissa b₁^h_right = 1825 into Formula (18); θ₁ could then be obtained, and then the abscissa of inflection points 2 and 3 could be obtained from Formula (10b). The abscissa of inflection points 1 and 4 could be obtained according to the symmetry. At this time, the coordinates of inflection points 1–4 in the left half of the roadway were calculated.

Similarly, the coordinates of the inflection point on the right half of the roadway could be obtained, but $△\mathrm{Y}$ = 2800 needed to be substituted into the calculation.

4.2.2. Determination of Cutting Step

When the roadheader cuts, if the cutting step distance L is set too small, although the cutting accuracy of the roadway section can be improved, the cutting steps required for roadway cutting and forming will increase, resulting in an increased cutting time. If the cutting step L is set too large, although it will reduce the cutting time, the cutting accuracy of the roadway section will decline, resulting in uneven roadway sides. In order to determine the cutting step distance L, the model diagram of the error between the cutting step distance and the roadway boundary was established, as shown in Figure 15.

It could be seen from the Figure 15 that the cutting error a, the cutting step L and the cutting radius r_i had the following relations:

$$\mathrm{a}={\mathrm{r}}_{\mathrm{i}}-\sqrt{{\mathrm{r}}_{\mathrm{i}}{}^2-{(\frac{\mathrm{L}}{2})}^2}$$

(19)

Formula (19) showed that when the cutting radius r_i remained unchanged, the cutting error a increased with an increase in the cutting step L. The cutting step L should be greater than r_i but not greater than 2r_i. In addition, the cutting radius r_i could be taken as a known quantity, so that when the maximum cutting error a was specified, the range of cutting step L could be determined.

4.2.3. Determination of Internal Inflection Point Coordinates

In order to form a complete and smooth cutting roadway section, the four inflection points around the roadway were considered as the benchmark, a distance of step L was gradually cut inward. Then, the coordinates of inflection point 5 were (x₁, y₁+L), inflection point 6 was (x₂−L, y₂+L), inflection point 7 was (x₃−L, y₃−L), inflection point 8 was (x₄+L, y₄−L), inflection point 9 was (x₈, y₈−L), and inflection point 10 was (x₉+L, y₉).

Similarly, the coordinates of the remaining inflection points in the right half of the roadway were as follows: the coordinates of inflection point 15 were (x₁₁, y₁₁+L), the coordinates of inflection point 16 were ($\frac{x_{11}+x_{12}}{2}$, y₁₂+L), and the coordinates of inflection point 17 were ($\frac{x_{11}+x_{12}}{2}$, y₁₃−L).

4.3. Cutting Path Control

Based on the determined inflection point coordinates of the cutting path, the cutting path control was carried out to realize the automation of the cutting process. When the cutting head reached the specified position in the horizontal direction, the rotary oil cylinder stopped working, and the lifting oil cylinder was opened to enable the cantilever to conduct a vertical cutting movement. When the cutting head reached the specified position in the vertical direction, the lifting oil cylinder stopped working, and the rotary oil cylinder began to make the cantilever perform horizontal cutting. The control program flow is shown in Figure 16.

5. Results and Discussion

5.1. Simulation Analysis of Cutting Head’s Limit Cutting Area

The relevant dimensions of the EBZ-200 cantilever roadheader were substituted into Formula (10), and the MATLAB software was used to simulate the limit cutting area of the cutting head. The simulation results are shown in Figure 17.

The simulation results in Figure 17 show the cutting process of the cutting head. Whether it was single cutting or secondary cutting is considered, the surroundings of the roadway were flat and smooth after the cutting formation, and the shape of the limit cutting section was that of a waist drum. The single cutting limit cutting area was 4100 mm × 3900 mm, the limit cutting area of the secondary cutting was 5650 mm × 3900 mm, and the ability to achieve a 5.6 m × 3.55 m-large roadway through cutting formation proved the validity of the theory. In addition, it can be seen from Figure 17d that there were some overlapping cutting areas, which decreased with increases in the moving distance ($△\mathrm{Y}$) of the roadheader, and the corresponding limit area of the secondary cutting also increased.

5.2. Simulation of Cutting Path Control

The MATLAB software was used to conduct the cutting path control simulation, as shown in Figure 18. In the simulation, the shaft length of the cutting head was set as 1000 mm, the maximum radius was 500 mm, the cutting depth was 600 mm, and the cutting step was 700 mm.

It can be seen from Figure 18 that the simulated cutting path was consistent with the set cutting path, and the cutting error around the roadway was controlled within 1 mm, which improved the cutting accuracy and realized the planning and high-precision control of the cutting path during secondary cutting.

The cutting path planning and control method proposed in this paper for the secondary cutting of a cantilever roadheader is still applicable to the cutting path planning and control of “snake” and other large-section roadways, although it was an example of a “loop” cutting path.

In addition, when it was found that there was a fault in front of the coal wall or the dip angle of the coal seam changed, the yaw angle α, pitch angle β, roll angle γ and offset L of the roadheader body were adjusted in time. Through the adjustment of the position and attitude parameters of the roadheader body, the tunneling direction of the roadheader was changed. When the position and attitude parameters of the roadheader body changed, the position coordinates of the calculated inflection point also changed accordingly, thereby changing the cutting path.

6. Conclusions

(1)

Based on the process of secondary cutting into the roadway, the Wangjialing 12307 belt roadway was used as the engineering background, the vertical displacement law of the roadway roof under different cutting paths was simulated with the FLAC 3D software, and the reasonable cutting path of the secondary cutting roadway was determined in combination with the actual situation in the mine. The effectiveness of the cutting path was also verified through industrial tests.

(2)

According to the robot principle and the D-H parameter method, the spatial pose model and coordinate system of the roadheader were established, which provided a mathematical model and a global coordinate system for the kinematics analysis of the cutting mechanism, the body and the roadheader.

(3)

The cutting structure kinematics, body kinematics and roadheader kinematics were analyzed and solved, and the pose transformation matrix of the cutting head coordinate system relative to the roadway coordinate system and the pose expression of the cutting head center in the roadway coordinate system were obtained, which provided a kinematic theoretical basis for the roadheader’s secondary autonomous cutting control system.

(4)

The simplified model of the cutting head was established, the position and pose expression of the pick on the cutting head in the roadway coordinate system was derived, the position coordinates of the inflection point and the cutting step distance were determined according to the relationship between the cutting head and the roadway boundary. The cutting path control flow was also designed, which provided the cutting path control method for the secondary autonomous cutting control system of the roadheader.

(5)

The MATLAB software was used to simulate the limit cutting area and cutting path control process of the cutting head during the second cutting. The simulation results showed that the limit cutting area was 5650 mm × 3900 mm, and the cutting error around the roadway was controlled within 1 mm, which verified the effectiveness and reliability of the automatic cutting control method. This research provides a theoretical basis for path planning and control to realize “secondary autonomous cutting of cantilever roadheader”.