#### 1. Introduction

In order to construct the mathematical model and simulation model of cutting head cutting roadway outer profile, it is necessary to study the formation mechanism of roadway surface topography, and the roof forming process of driving roadway is shown in Figure 1.

In Figure 1, hf is the residual height of the feed interval. Observing the residue of the first cut feed and the first cut feed (green filling in the picture), we can see that, after each cut, a “pit” is formed in the coal and rock and a “peak ridge” is raised between the adjacent “pits.” The shape, size/height of the “pit” and “peak ridge” are related to the cutting feed, the cutting angle, the shape of the cutting head, the radius of the cutting head, and the cone angle of the cutting head. The residual coal and rock of the adjacent cutting feed constitute the residue of the feed interval. The regular distribution of residual coal and rock in three-dimensional space forms the surface texture, and its ups and downs form a two-dimensional microprofile.

The right side of the figure shows the schematic diagram of the outer outline of the roadway without considering the cutting vibration and taking into account the cutting vibration. It can be seen from the figure that there are rules to follow in the formation of “pits” and “peak ridges” on coal and rock after cutting and feeding many times without considering vibration, and their height is basically the same. Considering the different heights of “pits” and “ridges” in the case of cutting vibration, it is necessary to analyze the vibration of the cutting head. If the vibration swing angle of the cutting head is large, the “pit” formed on the coal and rock will be deep, and if the vibration swing angle of the next feed cutting head is small, the “peak ridge” will be formed on the coal and rock.

#### 3. Vibration Analysis of Cutting Head in Cutting Process

Through the above analysis, it is concluded that the cutting load of the cutting head is the sum of the forces of all the cutters involved in the cutting. According to the principle of force line translation, the forces of all the cutters are equivalent to the rotary center of the big end face of the cutting head, and the instantaneous cutting load of the cutting head can be obtained. When the cutting head rotates to a certain position, the force acting on the cutting head is shown in Figure 2, and the radial resistance Faj, cutting resistance Fbj, and axial resistance Fcj are obtained as follows:

##### 3.2. Construction of Vertical Dynamic Model of Cutting Part

The working environment of the roadheader is bad, the working condition is complex, and the structure of the whole model of the roadheader is also relatively complex. In order to solve the problem, the dynamic model is simplified and the cutting part is regarded as a rigid body rotating around a fixed point. And, the mass of the cantilever section and the cutting head of the cutting part are concentrated in their respective centroids, and the lifting hydraulic cylinder is equivalent to a hydraulic spring with damping. Then, the dynamic model of the vertical direction of the cutting part is shown in Figure 3.

According to the relevant knowledge of mechanical vibration, Newton’s second law can be used to list the vibration equation of the cutting part as follows:

In the formula, J is the moment of inertia of the cutting head and the cantilever around the rotation point O, is the angular acceleration of the rotation of the cutting part around the rotation point, and kh is the stiffness of the equivalent hydraulic spring, which can be calculated according to

In the formula, βe is the effective volumetric elastic modulus of hydraulic oil, Ap is the average acting area of the two cavities of the hydraulic cylinder, and Vt is the average value of the equivalent total volume of the two cavities:

In the formula, S is the stroke of the hydraulic cylinder. ch is the viscous damping coefficient of the hydraulic oil in the hydraulic cylinder. x is the displacement of the hydraulic cylinder. is the moving speed of the hydraulic cylinder. θ2 is the angle between the cutting part and the horizontal plane. θ3 is the angle between the lifting cylinder and the cutting part. lr is the distance between the rotation point of the cutting part and the rotation point of the lifting cylinder. ∑M is the sum of all external moments to the rotation point.

Because the vibration swing angle θ of the cantilever section is small, it can be approximately considered that sin θ = θ; then, the displacement x of the hydraulic cylinder is

The moving speed of the lifting hydraulic cylinder is

The moment of inertia J of the cutting head and the cantilever section to the rotation point O is

In the formula, m1 and m2 are the concentrated mass of the cutting head and the cantilever section, respectively, and it is assumed that the center of mass of the cantilever section is at the middle point.

According to the above analysis, we can obtain that the vibration equation when the cantilever segment is regarded as rotating a rigid body around a fixed point is

Based on the fourth-order and five-stage Runge–Kutta variable step size algorithm, the ode45 function in Matlab is used to solve the vibration of the system numerically. The simulation parameters are shown in Table 1.

For the external load of the cutting head, the three-dimensional load calculation formula of the cutting head in the aforementioned formula (1) is substituted into the dynamic equation of the vibration of the cutting part, and the time is taken to analyze the dynamic response of the system. The results are shown in Figure 4.

Through the analysis of the vertical vibration response of the cutting part of the roadheader, the maximum positive vibration swing angle of the cutting head is 4.72° and the maximum negative vibration swing angle is −1.23°. The vertical vibration swing angle of the statistical cutting part under different cylinder pressures and different horizontal and vertical angles in the cutting space is shown in Figure 5.

The vertical vibration swing angle of the cutting part under different lifting cylinder pressure and different horizontal and vertical angles in the cutting space shows that when the vertical swing angle of the cutting part is constant and the horizontal swing angle is 0, the vertical vibration of the cutting part is the most intense. When the horizontal swing angle of the cutting part is constant and the vertical swing angle is 0, the vertical vibration of the cutting part is the smallest. The main reason is that the vibration intensity of the cutting part is affected by the cutting load. With the increase of the pressure of the lifting cylinder, the greater the stiffness of the swing vibration system of the cutting part is, the lower the vertical vibration swing angle of the cutting part is.

#### 4. Kinematic Model of Cutting Head

The coordinate system of roadheader is defined as O0X0Y0Z0, the spatial coordinate system of roadway is OXYZ and the coordinate system of each mechanism of roadheader is OiXiYiZi, where i = 1 represents the turntable coordinate system, i = 2represents the cutting part coordinate system, i = 3 represents the telescopic part coordinate system of the cutting part, and i = 4 represents the cutting head coordinate system. The relationship between the various coordinate systems is shown in Figure 6.

After the establishment of the above coordinate systems, the motion and attitude relationship of the roadheader cutting head relative to the roadway space can be determined, and the kinematics loop of the roadheader is established systematically.

If the coordinate vector set at the center of the cutting head in the cutting head coordinate system is , it can be transformed into the coordinate system of the roadway by the following formula:where , and .

#### 5. Cutting Surface Contour Model

In the process of roadway surface topography research, the establishment of accurate roadway model and cutting head motion model is the basis of simulation, and how to find out the intersection of cutting head motion model and roadway model is the key of the whole simulation work. In this section, we will construct the cutting surface contour model obtained by the intersection of the cutting head motion model and the roadway model.

This paper takes the common “ball crown + cone” cutting head of roadheader as the research object. Select the key nodes O4, Oj, C1, C2, and C3 above the cutting head, and the key node coordinates on the profile are defined as shown in Figure 7.

From the geometry of the “ball crown + cone” cutting head, it is known that the coordinate vectors of O4, Oj, C1, and C2 in the coordinate system of the cutting head are as follows:

From the kinematic model of the cutting head, we can know that the coordinate transformations of C1, and C2 relative to the roadway coordinate system are as follows:where u1 = cosθ1, u2 = −sinθ1cosθ2, u3 = −sinθ1sinθ2, u4 = −(a3 + a4 + d) sinθ1cosθ2 − a2sinθ1,  = sinθ1,  = cosθ1·cosθ2,  = −cosθ1sinθ2,  = (a3 + a4 + d) cosθ1cosθ2 + a2·cosθ1 + a1 + a0 + d0,  = 0,  = sinθ2,  = cosθ2, and  = (a3 + a4 + d) sinθ2 + b2 + b1 + b3.

The coordinates of point O4 are as follows:

Because the cutting head is a rotary body, it is considered that the outer profile of coal and rock formed by the cutting head is determined by the arc O4C1, line C1C2, and line C2C3 on the section of the cutting head, as shown in Figure 8.

The outer contour equation of the “ball crown + cone” cutting head is as follows:

Usually, the feed rate of one cutting head of the roadheader is about 550 mm, and the outer contours of the cutting head obtained by two cutting feeds will intersect at the intersection point D1 of the arc O4C1 and the line segment C1C2 on the cutting head of the previous cutting feed.

Then, the outer outline formed by the cutting head cutting coal and rock is a combination of line segments or arcs: . Then, the outer profile of coal and rock formed by the formula can be described as

#### 6. Simulation Analysis of the Characteristics of the Outer Profile of Roadway Forming

In this paper, taking the EBZ200 roadheader commonly used in coal mine as the research object, the simulation program of the outer profile of coal and rock is built in the MATLAB numerical calculation software, and the parameters in the simulation program are set by the structure size of EBZ200 roadheader as follows: a0 = 0 mm, a1 = 3000 mm, a2 = 700 mm, a3 = 2470 mm, a4 = 1775 mm, b1 = 1000 mm, b2 = 519m, b3 = 87 mm, d = 0 mm, r1 = 416 mm, α = 90°, β = 83.5, and m1 = 640 mm. The cutting feed rate d0 is set to 550 mm and the cutting lifting angle is 42°. The simulation results show that the characteristics of the formed outer profile of the roadway considering the vibration of the cutting head and without considering the vibration of the cutting head are shown in Figure 9.

After each cut, a “pit” is formed in the coal and rock, and the “peak ridge” is raised between the adjacent “pits.” The simulation results are consistent with the expected results shown in Figure 9. Considering the vibration of the cutting head, the difference between “pit” and “peak ridge” is larger. Without considering the vibration of the cutting head, the difference between the “pit” and the “peak ridge” is 105.6 mm, while in the case of considering the vibration of the cutting head, the difference between the “pit” and the “peak ridge” is 435.1 mm. It shows that the outer outline of the roadway obtained by real cutting is more rougher than that without considering the vibration of the cutting head, and it is not conducive to the accurate cutting and shaping of the roadway.

In order to quantify the characteristics of the formed outer profile of the roadway, the maximum difference between the “pit” and the uplift “peak ridge” formed on the coal and rock after cutting and the average value of the “pit” and the uplift “peak ridge” are selected as the evaluation parameters of the roadway surface topography:where Zmin is the maximum value of “pit,” Zmax is the maximum value of “peak ridge,” Zi is the value of roadway profile of each sampling point, and n is the number of sampling points.

The cutting height is set to [3600 mm, 3900 mm, 4200 mm, 4500 mm] to study the characteristics of the formed outer profile of the roadway under the condition of different roadway height and different stiffness of lifting cylinder, as shown in Figure 10.

According to formulas (13) and (14), it is calculated that when the horizontal swing angle is in the range of −45° to 45° and the vertical swing angle is in the range of 25° to 45°, the evaluation indexes of the forming outer profile characteristics of the roadway under the pressure of 18 MPa, 20 MPa, and 25 MPa, respectively, are shown in Figure 11.

The numerical simulation shows that when the cylinder pressure is 18 MPa, 20 MPa, and 25 MPa, respectively, the trend of the evaluation index of the forming outer profile of the roadway is basically the same with the increase of the cutting angle. The maximum difference between “pit” and uplift “peak ridge” on coal and rock and the average value of “pit” and uplift “peak ridge” all show an increasing trend and with the increase of cylinder pressure. The maximum Rmax of the horizontal swing angle “pit” and the uplift “peak ridge” gradually increases, mainly due to the increase of the pressure of the lifting cylinder, which is equivalent to the increase of the cutting load, the vibration of the cutting part becomes intense, and the positive and negative swing angle of the cutting head increases, resulting in the increase of the maximum Rmax. When the horizontal swing angle of the cutting part is from −45° to 45°, the maximum Rmax increases at first and then decreases, which is mainly due to the change of cutting load. The average value of horizontal swing angle “pit” and uplift “peak ridge” is almost not affected by the change of cylinder pressure, indicating that the change of cutting load mainly affects the maximum value of “pit” and uplift “peak ridge” Rmax and has little effect on the average value of the formed outer profile of the roadway.

#### 7. Experimental Study

The experimental study on the surface profile characteristics of roadway cut by roadheader is carried out by using the self-built test-bed, and the experimental results are compared with the theoretical calculation results. The test-bed is mainly composed of roadheader model prototype, simulated coal wall, and roadway surface profile measurement platform. The prototype of the roadheader model is shown in Figure 12.

The roadway surface profile measuring platform is shown in Figure 13.

The simulated coal wall after cutting is scanned by using the biaxial linear sliding table of the test platform and its high precision laser displacement sensor. The uniaxial moving range of the two-axis linear slide table is 1000 mm, the position accuracy of the linear screw slide table is 0.05 mm, the repeated positioning accuracy is 0.02 mm, and the moving speed is 100 mm/s. The detection distance of the high precision laser displacement sensor is 2000 mm, the detection accuracy is ±0.5 mm, and the resolution is 1 mm. The controller and data acquisition use Siemens S7-200 smart PLC.

The surface profile of the simulated coal wall cut by the roadheader is shown in Figure 14.

The experimental process is shown in Figure 15.

The comparison between the three-dimensional profile of the roadway surface measured by the experiment and the theoretical research results is shown in Figure 16.

The experimental results show that the errors of theoretical calculation and experimental test are mainly within ±2 mm. Generally speaking, the feasibility of the theoretical research method and the correctness of the results can be verified by experiments. In the follow-up research, the author will focus on how to optimize the geometric parameters of the cutting head and the motion parameters of the roadheader to obtain a lower roadway surface roughness and to achieve accurate autonomous cutting of the roadheader.

#### Data Availability

There are no public data yet, but it will be made public on our website (http://kczyyjy.lntu.edu.cn/) one after another.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

Zhi-xiang Liu designed the research. Miao Xie processed the corresponding data. Zhi-xiang Liu wrote the first draft of the manuscript. Shuai Wang and Chun-xue Xie helped to organize the manuscript.

#### Acknowledgments

The authors would like to thank for the financial support provided by the National Natural Science Foundation of China (nos. 51904142 and 51874158) and Liaoning Natural Science Foundation Guidance Program (no. 2019-ZD-0036).