# Deviation Correction Path Planning Method of Full-Width Horizontal Axis Roadheader Based on Improved Particle Swarm Optimization Algorithm

Authors: Qinghua Mao; Fei Zhang; Xuhui Zhang; Xusheng Xue; Liang Wang
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

Aiming at the problem of optimal path planning for deviation correction of full-width horizontal axis roadheader in the confined roadway of the coal mine, a deviation correction path planning method of full-width horizontal axis roadheader based on improved particle swarm optimization (I-PSO) algorithm is proposed. First, according to the driving characteristics of roadheader and deviation correction strategy, the mathematical model of full-width horizontal axis roadheader is established, and the constraints based on pose analysis are determined. Then, taking the shortest total distance from the initial position to the terminal position of roadheader as the objective function of path planning, an optimization model of deviation correction path planning is established. Finally, the I-PSO algorithm is proposed to quickly and accurately solve the deviation correction path model, in order to plan a safe and shorter distance path of full-width horizontal axis roadheader. In the I-PSO algorithm, the fitness difference of particles is used to dynamically adjust the inertia weight parameters to optimize the exploration of the whole or part of activity space, and the learning factor is adjusted to speed up particle convergence and optimization of particles, and a random factor is introduced to update the position of particles to avoid local traps as much as possible. The performance test of the I-PSO algorithm shows that the I-PSO algorithm is superior to other algorithms. The EJM340/4-2 full-width horizontal axis roadheader is used as the simulation object to verify the effectiveness of the I-PSO algorithm in the deviation correction path planning of roadheader. The simulation results show that the I-PSO algorithm proposed can quickly and accurately plan a safe and shorter distance path than the WPSO algorithm and CPSO algorithm and has better convergence and stability. It has laid a good foundation for the intelligent navigation of full-width horizontal axis roadheader in coal mine.


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## Body

## 1. Introduction

On February 25, 2020, the Chinese Development and Reform Commission, Energy Administration, and other eight ministries and commissions jointly issued Guiding Opinions on Accelerating Intelligent Development of Coal Mines. At present, there is an imbalance between mining and excavation in coal mine, and the intellectualization of tunneling equipment is one of the problems that need to be solved urgently in the current coal field [1]. The full-width horizontal axis roadheader is a key equipment for rapid and intelligent tunneling at present, and its deviation correction control is one of the key technologies for realizing intelligent tunneling.Due to influence of positioning error, external environment, self-slip, and other factors, actual driving track of full-width horizontal axis roadheader will deviate from roadway center line during operation. That is, roadheader needs to be controlled independently to make it return to roadway center line, so as to ensure formation quality of roadway. At present, deviation correction path planning algorithm of roadheader and mobile robot has become a research hotspot. In view of the fact that there are many nonstructural environmental attribute problems in the background of actual path planning of roadheader roadways, Yang [2] classified the path planning problem in the nonstructural environment of roadways as a general stochastic network problem. Qu [3] formulated the body deviation correction path planning according to the traveling characteristics of the cantilever roadheader and established the position and posture deviation type of the roadheader’s caterpillar traveling. Liu [4] proposed novel best path selection approach based on hybrid improved A∗ algorithm and reinforcement learning. Liu [5] proposed a path planning method based on the particle swarm optimization trained fuzzy neural network algorithm. Ji [6] proposed a dual-surrogate assisted cooperative particle swarm optimization algorithm to seek multiple optimal solutions. The proposed algorithm can simultaneously obtain multiple highly competitive optimal solutions at a low computational cost. Zhang [7] proposed a multiobjective path planning algorithm based on particle swarm optimization for navigation path planning of coal mine rescue robot. Nazarahari [8] presented a hybrid approach for path planning of multiple mobile robots in continuous environments. It not only proposed algorithm-determined collision-free paths but also it found near optimal solution for all robots. Song [9] proposed a variable-size cooperative coevolutionary particle swarm optimization algorithm (VS-CCPSO) for FS. The experimental results show that VS-CCPSO has the capability of obtaining good feature subsets. Wang [10] proposed a path planning method combining Ferguson spline path description and improved particle swarm optimization algorithm to achieve smooth optimal path planning of microsoccer robot, which can effectively carry out collision-free path planning of robot in obstacle environment. Wang [11] proposed an improved ant colony algorithm to solve optimal path problem of mobile robots in a dynamic environment, which can effectively avoid collisions between mobile robots and dynamic obstacles. Wang [12] proposed a path planning method for mobile robots based on improved A∗ algorithm, which can effectively overcome problems of traditional A∗ algorithm, such as length of planned path is not optimal and path is not smooth enough. Hu [13] proposed an improved particle swarm optimization algorithm for robot path planning, which improves local and overall search ability. The path planning speed is faster, path distance is shorter, and robustness is stronger.On the whole, there are few reports on the research of deviation correction path planning of full-width horizontal axis roadheader in the confined roadway of coal mine. The deviation correction path planning of full-width horizontal axis roadheader is different from the path planning of cantilever roadheader and traditional path planning of mobile robot [3, 14–20]. The working space of roadheader is complex, and the environment is closed, and roadheader drives step by step according to the intercept of bolt support. The deviation position and angle adjusted in each step are small. Therefore, in order to obtain optimal deviation correction path of full-width horizontal axis roadheader, a deviation correction path planning method based on improved particle swarm optimization (I-PSO) algorithm is proposed.The main contributions of this work are as follows:(1)
The deviation correction strategy of full-width horizontal axis roadheader is proposed, and the deviation correction path model is established with the goal of the shortest total travel distance of full-width horizontal axis roadheader.(2)
The I-PSO algorithm is proposed to quickly and accurately solve the deviation correction path model, in order to plan a safe and shorter distance path of full-width horizontal axis roadheader. In the I-PSO algorithm, the fitness difference of particles is used to dynamically adjust the inertia weight parameters to optimize the exploration of the whole or part of activity space, and the learning factor is adjusted to speed up particle convergence and optimization of particles, and a random factor is introduced to update the position of particles to avoid local traps as much as possible.(3)
The rest of the paper is organized as follows: related work is summarized in Section2; Section 3 introduces the deviation correction strategy of full-width horizontal shaft roadheader; Section 4 introduces the establishment and solution of the deviation correction path planning model based on I-PSO algorithm; Section 5 introduces the simulation verification and result analysis; Section 6 compares and tests the algorithms proposed in this paper; Section 7 summarizes this paper.


## 2. Related Work

The full-width horizontal axis roadheader operates in a closed environment. During the operation of roadheader, it is necessary to ensure that roadheader has enough adjustment space in restricted roadway space and also ensure that roadheader can cut and shape roadway section according to the required size in high quality. Therefore, the position and attitude parameters of roadheader change within a certain range in the roadway of a fully mechanized tunneling face, rather than in any position and attitude state [21].As shown in Figure1, roadheader turns with its own center of gravity as benchmark. Assume that body width of roadheader is d, length is L, and drum width is D. The first rotation of roadheader takes center O of roadheader as center with OA as radius and intersects with outer wall of roadway at point B, and rotation angle of roadheader is θ. It can be seen from Figure 1 that rotation angle meets the following relationship for the following equations:(1)sinμ=d/2d/22+L/22,(2)sinβ=d/2d/22+L/22,(3)θ=μ−β,=arcsind/2d/22+d/22−arcsind/2d/22+L/22.Figure 1
Schematic diagram of steering posture of roadheader.After turningθ angle, roadheader moves forward along OD direction for s and then reaches O′ arrival point for the second rotation. In Figure 1, the gray, blue, green, and red cuboids represent the positions of roadheader before rotation, the first rotation, the advance s distance, and the second rotation, respectively. The equations (4) and (5) of the second rotation angle θ′ can be obtained from Figure 1.(4)θ′=arcsinBE+s⋅sinθO′B′−μ.Further obtained:(5)θ′=arcsind/2+s⋅d/2/d/22+L/22d/22+L/22−arcsind/2d/22+L/22.Similarly, the rotation angle of each travel in roadway can be recursively calculated. Thenth adjustment angle is presented in the following equations:(6)θn′=arcsinBE+n⋅s⋅sinθO′B′−μn−1.Further obtained:(7)θn′=arcsind/2+n⋅s⋅d/2/d/22+L/22d/22+L/22−arcsind/2+n−1⋅s⋅d/2/d/22+L/22d/22+L/22,where μn−1 is included angle between roadheader and horizontal axis after the last rotation of roadheader.By analyzing the limit angle of the full-width horizontal axis roadheader in roadway as shown in Figure2, the maximum possible deviation position and attitude of roadheader in roadway can be obtained. If CF length is x, so BC length is D−x. As shown in Figure 2, equation (8) can be obtained.(8)ll2−D−x2=dx,where l is maximum effective length of roadheader, and D is drum width.Figure 2
Schematic diagram of the maximum deflection angle of roadheader.As shown in Figure2, when the drum is cutting coal and rock, part of the pick will be embedded in the coal and rock. At this time, if the roadheader deviates from the correct track, it will cause one side of the drum pick to go too deep into the coal and rock and will suffer greater resistance. Therefore, the roadheader path planning problem is an optimization problem with constraints to avoid the separation of drum pick and coal rock. According to the mathematical model of the pose, the constraint conditions can be determined in the following equation:(9)θn′<sin−1e2x,where θn′ is the nth adjustment angle of roadheader, and e is the length of drum pick.


## 3. Deviation Correction Strategy of Full Width Horizontal Axis Roadheader

### 3.1. Tunneling Technology

When the full-width horizontal roadheader operates, it is required that roadheader travels according to the center line track of the tunneling roadway. The tunneling construction process is as follows: (1) adjust the direction angle of the body according to the current position and posture of roadheader to ensure that the travel direction of the body coincides with the center line of roadway to be excavated. (2) According to the cross-sectional shape of roadway, coal and rock are cut by a cutting mechanism, and falling coal is transported out by a scraper mechanism and conveyor in front of and behind fuselage. (3) After cutting is completed, the direction angle and speed of fuselage are adjusted to realize the forward movement of the fuselage, complete tunneling of one intercept, and enter the next construction sequence for cyclic travel. An intercept is completed every time, position and attitude detection system of roadheader equipped with fuselage will measure the coordinates of the current fuselage center position, judge whether there is any deviation from the designed roadway center line, and then plan subsequent tunneling trajectory and adjust working parameters [22, 23].


### 3.2. Deviation Correction Strategy

In the process of roadway excavation construction, the deviation correction of roadheader mainly controls left and right tracks of roadheader to make center position of roadheader coincide with center line of roadway, so as to realize the adjustment of body position [24, 25]. When roadheader is tunneling, if trajectory deviation exceeds the allowable range of error, it is necessary to replan the operation route of roadheader, so that roadheader can return to set roadway center line trajectory. The simplified schematic diagram of roadheader track deviating from roadway is presented in Figure 3. Figure 3(a) indicates that driving direction of roadheader deviates from roadway center line. Figure 3(b) indicates that the position of roadheader deviates from roadway center line, but the direction is parallel to roadway center line. Figure 3(c) indicates that both position and direction deviation of roadheader deviate from roadway center line. It is specified that the deflection angle θ of roadheader is positive if it is skewed to the left and negative if it is skewed to the right. It is stipulated that the deviation A of roadheader is positive to the left and negative to the right. The correction strategies for three cases in Figure 3 are as follows:(1)
In Figure 3(a), the direction of roadheader body only deviated from roadway center line, and the deflection angle isθ. When θ < 0, control left track of roadheader to reverse and right track to rotate forward, so that body has a certain angle of deviation, and then control left and right tracks to move a step s before turning forward at same time. Then, position and attitude detection system equipped with body will measure coordinates of current center position of body to determine whether current position of body is offset from designed roadway center line. If offset distance does not reach limit value, continue to control roadheader to make body have a certain angle of deviation and move forward a step s until center distance is less than limit value. When θ > 0, left track of roadheader is controlled to rotate forward and right track to reverse until yaw angle is less than limit value.(2)
In Figure 3(b), the center line of roadheader body deviates from center line of roadway offset distance isA, when A > 0, control left track of roadheader to rotate forward and right track to reverse, so that body has a certain angle of deviation, and then control left and right tracks to move a step s before turning forward at the same time. Then, position and attitude detection system equipped with machine body measures coordinates of current center position of machine body and judges whether there is any deviation from designed roadway center line. If deviation does not reach limit value, continue to control roadheader to make machine body have a certain deviation angle, and move forward a step s until center distance is less than limit value. When A < 0, direction is opposite.(3)
In Figure 3(c), the position and direction of center line of roadheader deviate from center line of roadway.Offset distance is A, and the deflection angle is θ. When A > 0 and θ > 0, the correction strategy is slightly the same as that in Figure (b).Figure 3
Schematic diagram of trajectory deviation of roadheader.


## 3.1. Tunneling Technology

When the full-width horizontal roadheader operates, it is required that roadheader travels according to the center line track of the tunneling roadway. The tunneling construction process is as follows: (1) adjust the direction angle of the body according to the current position and posture of roadheader to ensure that the travel direction of the body coincides with the center line of roadway to be excavated. (2) According to the cross-sectional shape of roadway, coal and rock are cut by a cutting mechanism, and falling coal is transported out by a scraper mechanism and conveyor in front of and behind fuselage. (3) After cutting is completed, the direction angle and speed of fuselage are adjusted to realize the forward movement of the fuselage, complete tunneling of one intercept, and enter the next construction sequence for cyclic travel. An intercept is completed every time, position and attitude detection system of roadheader equipped with fuselage will measure the coordinates of the current fuselage center position, judge whether there is any deviation from the designed roadway center line, and then plan subsequent tunneling trajectory and adjust working parameters [22, 23].


## 3.2. Deviation Correction Strategy

In the process of roadway excavation construction, the deviation correction of roadheader mainly controls left and right tracks of roadheader to make center position of roadheader coincide with center line of roadway, so as to realize the adjustment of body position [24, 25]. When roadheader is tunneling, if trajectory deviation exceeds the allowable range of error, it is necessary to replan the operation route of roadheader, so that roadheader can return to set roadway center line trajectory. The simplified schematic diagram of roadheader track deviating from roadway is presented in Figure 3. Figure 3(a) indicates that driving direction of roadheader deviates from roadway center line. Figure 3(b) indicates that the position of roadheader deviates from roadway center line, but the direction is parallel to roadway center line. Figure 3(c) indicates that both position and direction deviation of roadheader deviate from roadway center line. It is specified that the deflection angle θ of roadheader is positive if it is skewed to the left and negative if it is skewed to the right. It is stipulated that the deviation A of roadheader is positive to the left and negative to the right. The correction strategies for three cases in Figure 3 are as follows:(1)
In Figure 3(a), the direction of roadheader body only deviated from roadway center line, and the deflection angle isθ. When θ < 0, control left track of roadheader to reverse and right track to rotate forward, so that body has a certain angle of deviation, and then control left and right tracks to move a step s before turning forward at same time. Then, position and attitude detection system equipped with body will measure coordinates of current center position of body to determine whether current position of body is offset from designed roadway center line. If offset distance does not reach limit value, continue to control roadheader to make body have a certain angle of deviation and move forward a step s until center distance is less than limit value. When θ > 0, left track of roadheader is controlled to rotate forward and right track to reverse until yaw angle is less than limit value.(2)
In Figure 3(b), the center line of roadheader body deviates from center line of roadway offset distance isA, when A > 0, control left track of roadheader to rotate forward and right track to reverse, so that body has a certain angle of deviation, and then control left and right tracks to move a step s before turning forward at the same time. Then, position and attitude detection system equipped with machine body measures coordinates of current center position of machine body and judges whether there is any deviation from designed roadway center line. If deviation does not reach limit value, continue to control roadheader to make machine body have a certain deviation angle, and move forward a step s until center distance is less than limit value. When A < 0, direction is opposite.(3)
In Figure 3(c), the position and direction of center line of roadheader deviate from center line of roadway.Offset distance is A, and the deflection angle is θ. When A > 0 and θ > 0, the correction strategy is slightly the same as that in Figure (b).Figure 3
Schematic diagram of trajectory deviation of roadheader.


## 4. Establishment and Solution of the Deviation Correction Path Planning Model

### 4.1. Establishment of the Deviation Correction Path Planning Model

The working space of full-width horizontal axis roadheader is complex, the environment is closed, and the roadheader is driven step by step according to the intercept of bolt support. The deviation position and angle adjusted are small in each step. Therefore, according to the actual requirements of roadheader deviation correction path planning, considering the feasibility and safety of path planning comprehensively, its fitness function is presented in the following equation:(10)minf=∑i=0n−1di+de,where f represents the total distance from the initial position to the terminal position of roadheader. n represents the number of steps required to reach center line. di represents step size of each step. de represents step size of last step.In order not to affect row spacing of downhole drilling anchors,θn′ is presented in the following equation:(11)cosθn′=dldi,where θn′ is nth adjustment angle of roadheader. According to equation (9), θn′ has a constraint condition, that is, θn′<sin−1e/2x. dl represents projection distance of roadheader step size in direction of a straight roadway (i.e., row spacing of drilling anchor).


### 4.2. Improved Particle Swarm Optimization Algorithm

Particle swarm optimization (PSO) uses massless and voluminous particles as individuals and provides simple behavior rules for each particle, so that the entire particle swarm shows complex behavior characteristics. In the PSO algorithm, each particle is a feasible solution to the optimization problem, and the objective function determines a fitness value for it [26–30]. Each particle flies to a better position according to individual experience and group experience to search for the global optimal solution. According to the principle of the current optimal particle, particle Xi will change its speed and position according to the following equations:(12)vidt+1=ωvidt+c1r1pidt−xidt+c2r2pgdt−xgdt,(13)xidt+1=xidt+xidt+1,where ω is inertia weight of particle. pi is local optimal position searched by particle at time t. pg is global optimal position searched at time t. d = 1, 2, …, n. i = 1, 2, …, m. m is population size. t is current evolutionary algebra. r1 and r2 are random numbers distributed in [0, 1]. c1 and c2 are acceleration constants, called self-cognitive factors and social cognitive factors, respectively, which are used to adjust the influence intensity of pi and pg.Figure4 shows the position changes of particles in two consecutive iterations. When the maximum number of iterations or the minimum error threshold is reached, the algorithm operation process is terminated. Table 1 shows the pseudocode of the traditional particle swarm optimization algorithm.Figure 4
Schematic diagram of a particle’s position update in PSO.Table 1
The pseudocode of the conventional particle swarm optimization.
Algorithm: traditional particle swarm optimization for path planning(1) Select population size and maximum iterations(2) Define fitness function(3) Preset acceleration factors (c1, c2) and inertia weights (ω)(4) For each particle do(5)  Initialize particle speed and position(6)  Calculate the fitness value of particles(7)  Record the initial individual extreme valuepi and global extreme value pg(8) end for(9) While does not reach the maximum number of iterations or the minimum error standard do(10)  For each particle do(11)   Update particle velocity with formula (12)(12)   Update particle position with formula (13)(13)   Update particle fitness values(14)   Update individual extremepi and global extreme pg(15)  end for(16) end whileObviously, PSO has an attraction mechanism; that is, each particle is always attracted byPgd and Pid at the same time [31–39]. Specifically, the change in particle speed depends on two main factors: one is the best location found in the search history before each particle is attracted to the particle, and the other is the best location found by any particle in the search space. At the same time, the traction strength of particles in each direction depends on the distance between particles and each optimal position. The smaller the distance, the stronger the attraction. However, this algorithm also has some shortcomings, such as premature convergence of particles and falling into local optimum, so it has been constantly improved by many researchers.Because the traditional particle swarm optimization algorithm has problems of premature convergence of particles and poor quality in solving the optimal solution, an improved particle swarm optimization algorithm is proposed. The optimization strategy is proposed from three aspects: dynamically adjusting inertia weight, adjusting learning factors, and introducing random factors.


#### 4.2.1. Adjustment of Inertia Weight

Improvement ω. On the one hand, it is to improve the search efficiency, and on the other hand, it is to improve the precocity problem as much as possible [40–55]. Therefore, the difference in particle fitness is used to dynamically adjust the change of inertia weight in this paper. This function has the following advantages: (1) the function is decreasing in this interval, meeting the requirements of monotonic decreasing of inertia weight value, which is large at the beginning and small at the end; (2) the function decreases slowly at the beginning of the interval, which is conducive to the particle swarm to explore the optimal solution at a higher speed at the beginning, and it will not reduce too quickly to a smaller value and enter the local search, leading to premature; (3) this function also maintains a relatively gentle rate of change at the end of the search, which is conducive to the particle swarm to conduct local search stably at a low rate of change at the end of the search, so as to find the optimal solution. The inertia weight function designed in this paper is as follows:(14)ω=ωmin+η1−minFPit−1,FPitmaxFPit−1,FPit,(15)μ=ωmax−ωmin1−ttmax,where FPit−1 is fitness value, ωmax and ωmin represent maximum and minimum values of weights, respectively, and tmax is maximum number of iterations.I-PSO algorithm receivesωmax, searches a larger area, and quickly locates the approximate position of optimal solution along with ωmax gradually decreases, particle speed slows down, and the fine local search starts. This method makes I-PSO algorithm better control search ability, speeds up convergence speed, and improves the performance of the algorithm.


#### 4.2.2. Adjustment of Learning Factors

In the process of population search, maintaining a large value ofc1 can continuously expand the exploration area of the population, but it will lead to slow convergence of the algorithm. Keeping c2 at a relatively large value can enable particles to obtain information of neighboring particles as much as possible, thus speeding up the optimization of the algorithm, but the solution may not be optimal. Therefore, the values of c1 and c2 are very important to find the optimal solution. In order to enable particles to perform a global search as much as possible at the initial stage of the algorithm and have significant convergence speed at the later stage of the algorithm, c1 should be appropriately increased and c2 should be reduced at an early stage of the algorithm, and c1 should be appropriately reduced and c2 should be increased at the same time at the later stage of the algorithm. The updated formula design of the acceleration factor is presented in the following equations:(16)c1=c1s−c1s−c1etmaxt,(17)c2=c2s−c2s−c2etmaxt,where c1s, c1e, c2s, and c2e are the initial and final values of the particle itself and the group learning factor, respectively.


#### 4.2.3. Random Factor Introduced to Update the Position of Particles

In order to reduce the probability of falling into local traps, a random factor is introduced to increase the randomness of particle exploration when updating particle positions. The random factor is presented in the following equation:(18)Hit=Hit1+r,where Hi(t) is position of ith particle at tth time. r is a random value of [−0.1, 0.1].


### 4.3. Solution of Deviation Correction Path Planning Model Based on I-PSO Algorithm

In the deviation correction path planning of full-width horizontal axis roadheader, the population size is 2000, the maximum number of iterations is 150, and self-cognitive factorc1 and social cognitive factor c2 are both 1.5, ωmax = 0.9, ωmin = 0.1.Solution flow chart of the deviation correction path planning model based on the I-PSO algorithm is presented in Figure5.Figure 5
Solution flowchart of deviation correction path planning model based on I-PSO algorithm.The steps to solve the deviation correction path planning model based on the I-PSO algorithm are as follows:(1)
Initialize particle size, maximum iteration number, maximum inertia weight and minimum inertia weight, self-cognitive factor, and group cognitive factor.(2)
The fitness value of particles, the individual optimal value of each particle, and the global optimal value of the whole are calculated according to the objective function established in equation (14)–(15)–(16)–(17)–(18).(3)
The velocity and position of the particle are updated through equations (12) and (13), and collision test is made to adjust the collision position.(4)
According to equations (14)–(18), ω, c1, and c2 of each particle are dynamically adjusted.(5)
Calculate the fitness value of each particle after position update according to equation ((14)–(15)–(16)–(17)–(18)), and update individual optimal fitness value and global optimal fitness value.(6)
Determine whether the maximum number of iterations is reached. If so, the loop is ended to output the optimal particle. Otherwise, return to step (2).


## 4.1. Establishment of the Deviation Correction Path Planning Model

The working space of full-width horizontal axis roadheader is complex, the environment is closed, and the roadheader is driven step by step according to the intercept of bolt support. The deviation position and angle adjusted are small in each step. Therefore, according to the actual requirements of roadheader deviation correction path planning, considering the feasibility and safety of path planning comprehensively, its fitness function is presented in the following equation:(10)minf=∑i=0n−1di+de,where f represents the total distance from the initial position to the terminal position of roadheader. n represents the number of steps required to reach center line. di represents step size of each step. de represents step size of last step.In order not to affect row spacing of downhole drilling anchors,θn′ is presented in the following equation:(11)cosθn′=dldi,where θn′ is nth adjustment angle of roadheader. According to equation (9), θn′ has a constraint condition, that is, θn′<sin−1e/2x. dl represents projection distance of roadheader step size in direction of a straight roadway (i.e., row spacing of drilling anchor).


## 4.2. Improved Particle Swarm Optimization Algorithm

Particle swarm optimization (PSO) uses massless and voluminous particles as individuals and provides simple behavior rules for each particle, so that the entire particle swarm shows complex behavior characteristics. In the PSO algorithm, each particle is a feasible solution to the optimization problem, and the objective function determines a fitness value for it [26–30]. Each particle flies to a better position according to individual experience and group experience to search for the global optimal solution. According to the principle of the current optimal particle, particle Xi will change its speed and position according to the following equations:(12)vidt+1=ωvidt+c1r1pidt−xidt+c2r2pgdt−xgdt,(13)xidt+1=xidt+xidt+1,where ω is inertia weight of particle. pi is local optimal position searched by particle at time t. pg is global optimal position searched at time t. d = 1, 2, …, n. i = 1, 2, …, m. m is population size. t is current evolutionary algebra. r1 and r2 are random numbers distributed in [0, 1]. c1 and c2 are acceleration constants, called self-cognitive factors and social cognitive factors, respectively, which are used to adjust the influence intensity of pi and pg.Figure4 shows the position changes of particles in two consecutive iterations. When the maximum number of iterations or the minimum error threshold is reached, the algorithm operation process is terminated. Table 1 shows the pseudocode of the traditional particle swarm optimization algorithm.Figure 4
Schematic diagram of a particle’s position update in PSO.Table 1
The pseudocode of the conventional particle swarm optimization.
Algorithm: traditional particle swarm optimization for path planning(1) Select population size and maximum iterations(2) Define fitness function(3) Preset acceleration factors (c1, c2) and inertia weights (ω)(4) For each particle do(5)  Initialize particle speed and position(6)  Calculate the fitness value of particles(7)  Record the initial individual extreme valuepi and global extreme value pg(8) end for(9) While does not reach the maximum number of iterations or the minimum error standard do(10)  For each particle do(11)   Update particle velocity with formula (12)(12)   Update particle position with formula (13)(13)   Update particle fitness values(14)   Update individual extremepi and global extreme pg(15)  end for(16) end whileObviously, PSO has an attraction mechanism; that is, each particle is always attracted byPgd and Pid at the same time [31–39]. Specifically, the change in particle speed depends on two main factors: one is the best location found in the search history before each particle is attracted to the particle, and the other is the best location found by any particle in the search space. At the same time, the traction strength of particles in each direction depends on the distance between particles and each optimal position. The smaller the distance, the stronger the attraction. However, this algorithm also has some shortcomings, such as premature convergence of particles and falling into local optimum, so it has been constantly improved by many researchers.Because the traditional particle swarm optimization algorithm has problems of premature convergence of particles and poor quality in solving the optimal solution, an improved particle swarm optimization algorithm is proposed. The optimization strategy is proposed from three aspects: dynamically adjusting inertia weight, adjusting learning factors, and introducing random factors.


### 4.2.1. Adjustment of Inertia Weight

Improvement ω. On the one hand, it is to improve the search efficiency, and on the other hand, it is to improve the precocity problem as much as possible [40–55]. Therefore, the difference in particle fitness is used to dynamically adjust the change of inertia weight in this paper. This function has the following advantages: (1) the function is decreasing in this interval, meeting the requirements of monotonic decreasing of inertia weight value, which is large at the beginning and small at the end; (2) the function decreases slowly at the beginning of the interval, which is conducive to the particle swarm to explore the optimal solution at a higher speed at the beginning, and it will not reduce too quickly to a smaller value and enter the local search, leading to premature; (3) this function also maintains a relatively gentle rate of change at the end of the search, which is conducive to the particle swarm to conduct local search stably at a low rate of change at the end of the search, so as to find the optimal solution. The inertia weight function designed in this paper is as follows:(14)ω=ωmin+η1−minFPit−1,FPitmaxFPit−1,FPit,(15)μ=ωmax−ωmin1−ttmax,where FPit−1 is fitness value, ωmax and ωmin represent maximum and minimum values of weights, respectively, and tmax is maximum number of iterations.I-PSO algorithm receivesωmax, searches a larger area, and quickly locates the approximate position of optimal solution along with ωmax gradually decreases, particle speed slows down, and the fine local search starts. This method makes I-PSO algorithm better control search ability, speeds up convergence speed, and improves the performance of the algorithm.


### 4.2.2. Adjustment of Learning Factors

In the process of population search, maintaining a large value ofc1 can continuously expand the exploration area of the population, but it will lead to slow convergence of the algorithm. Keeping c2 at a relatively large value can enable particles to obtain information of neighboring particles as much as possible, thus speeding up the optimization of the algorithm, but the solution may not be optimal. Therefore, the values of c1 and c2 are very important to find the optimal solution. In order to enable particles to perform a global search as much as possible at the initial stage of the algorithm and have significant convergence speed at the later stage of the algorithm, c1 should be appropriately increased and c2 should be reduced at an early stage of the algorithm, and c1 should be appropriately reduced and c2 should be increased at the same time at the later stage of the algorithm. The updated formula design of the acceleration factor is presented in the following equations:(16)c1=c1s−c1s−c1etmaxt,(17)c2=c2s−c2s−c2etmaxt,where c1s, c1e, c2s, and c2e are the initial and final values of the particle itself and the group learning factor, respectively.


### 4.2.3. Random Factor Introduced to Update the Position of Particles

In order to reduce the probability of falling into local traps, a random factor is introduced to increase the randomness of particle exploration when updating particle positions. The random factor is presented in the following equation:(18)Hit=Hit1+r,where Hi(t) is position of ith particle at tth time. r is a random value of [−0.1, 0.1].


## 4.2.1. Adjustment of Inertia Weight

Improvement ω. On the one hand, it is to improve the search efficiency, and on the other hand, it is to improve the precocity problem as much as possible [40–55]. Therefore, the difference in particle fitness is used to dynamically adjust the change of inertia weight in this paper. This function has the following advantages: (1) the function is decreasing in this interval, meeting the requirements of monotonic decreasing of inertia weight value, which is large at the beginning and small at the end; (2) the function decreases slowly at the beginning of the interval, which is conducive to the particle swarm to explore the optimal solution at a higher speed at the beginning, and it will not reduce too quickly to a smaller value and enter the local search, leading to premature; (3) this function also maintains a relatively gentle rate of change at the end of the search, which is conducive to the particle swarm to conduct local search stably at a low rate of change at the end of the search, so as to find the optimal solution. The inertia weight function designed in this paper is as follows:(14)ω=ωmin+η1−minFPit−1,FPitmaxFPit−1,FPit,(15)μ=ωmax−ωmin1−ttmax,where FPit−1 is fitness value, ωmax and ωmin represent maximum and minimum values of weights, respectively, and tmax is maximum number of iterations.I-PSO algorithm receivesωmax, searches a larger area, and quickly locates the approximate position of optimal solution along with ωmax gradually decreases, particle speed slows down, and the fine local search starts. This method makes I-PSO algorithm better control search ability, speeds up convergence speed, and improves the performance of the algorithm.


## 4.2.2. Adjustment of Learning Factors

In the process of population search, maintaining a large value ofc1 can continuously expand the exploration area of the population, but it will lead to slow convergence of the algorithm. Keeping c2 at a relatively large value can enable particles to obtain information of neighboring particles as much as possible, thus speeding up the optimization of the algorithm, but the solution may not be optimal. Therefore, the values of c1 and c2 are very important to find the optimal solution. In order to enable particles to perform a global search as much as possible at the initial stage of the algorithm and have significant convergence speed at the later stage of the algorithm, c1 should be appropriately increased and c2 should be reduced at an early stage of the algorithm, and c1 should be appropriately reduced and c2 should be increased at the same time at the later stage of the algorithm. The updated formula design of the acceleration factor is presented in the following equations:(16)c1=c1s−c1s−c1etmaxt,(17)c2=c2s−c2s−c2etmaxt,where c1s, c1e, c2s, and c2e are the initial and final values of the particle itself and the group learning factor, respectively.


## 4.2.3. Random Factor Introduced to Update the Position of Particles

In order to reduce the probability of falling into local traps, a random factor is introduced to increase the randomness of particle exploration when updating particle positions. The random factor is presented in the following equation:(18)Hit=Hit1+r,where Hi(t) is position of ith particle at tth time. r is a random value of [−0.1, 0.1].


## 4.3. Solution of Deviation Correction Path Planning Model Based on I-PSO Algorithm

In the deviation correction path planning of full-width horizontal axis roadheader, the population size is 2000, the maximum number of iterations is 150, and self-cognitive factorc1 and social cognitive factor c2 are both 1.5, ωmax = 0.9, ωmin = 0.1.Solution flow chart of the deviation correction path planning model based on the I-PSO algorithm is presented in Figure5.Figure 5
Solution flowchart of deviation correction path planning model based on I-PSO algorithm.The steps to solve the deviation correction path planning model based on the I-PSO algorithm are as follows:(1)
Initialize particle size, maximum iteration number, maximum inertia weight and minimum inertia weight, self-cognitive factor, and group cognitive factor.(2)
The fitness value of particles, the individual optimal value of each particle, and the global optimal value of the whole are calculated according to the objective function established in equation (14)–(15)–(16)–(17)–(18).(3)
The velocity and position of the particle are updated through equations (12) and (13), and collision test is made to adjust the collision position.(4)
According to equations (14)–(18), ω, c1, and c2 of each particle are dynamically adjusted.(5)
Calculate the fitness value of each particle after position update according to equation ((14)–(15)–(16)–(17)–(18)), and update individual optimal fitness value and global optimal fitness value.(6)
Determine whether the maximum number of iterations is reached. If so, the loop is ended to output the optimal particle. Otherwise, return to step (2).


## 5. Simulation Verification and Results’ Analysis

In order to verify the effectiveness of the I-PSO algorithm for correcting deviation path planning of full-width horizontal axis roadheader, the EJM340/4-2 full-width horizontal axis roadheader is taken as a simulation object. Its size parameters are as follows: the length is 11.6 m, the width is 5 m, and the height is 2.6 m. The projection distance of each advance of roadheader in direction of the straight roadway is 1 m (that is, the row distance of the drill anchor is 1 m). The experimental platform is Matlab2018, and three kinds of roadheader pose state correction path planning are verified.


### 5.1. The Simulation Verification of Direction Deviation from Roadway Center Line

The deviation from the initial position of roadheader to the center line of roadway is 0 mm, and the deviation angle is 5°. As shown in Figure6, the blue curve is the path planned by the I-PSO algorithm, the red curve is the path planned by the WPSO algorithm, and the green curve is the path planned by CPSO algorithm. WPSO algorithm needs to drive 8 steps to make center point of roadheader return to roadway center line with a total driving distance of 7446.36 mm, and CPSO algorithm needs to drive 8 steps to make center point of roadheader return to roadway center line with a total driving distance of 7706.52 mm. I-PSO algorithm only needs to drive 7 steps to make center point of roadheader return to roadway center line with a total driving distance of 6825.33 mm. The number of steps and adjustment angle of roadheader based on I-PSO algorithm are presented in Table 2.Figure 6
Schematic diagram of deviation correction path planning of roadheader.Table 2
Results of deviation correction path planning based on I-PSO.
Adjust stepsn1234567Adjust angleθ (°)1.5121.4361.3981.322−1.929−1.616−1.518Steps (mm)1000.51000.31000.31000.31000.71000.5822.73The algorithm iteration diagram of I-PSO, WPSO, and CPSO is presented in Figure7. The fitness is total distance traveled by roadheader from initial position to roadway center line. In the iteration process, the fitness value of I-PSO is obviously smaller than the fitness value of WPSO and CPSO, and I-PSO algorithm achieves optimal value in 23 iterations. It can be seen that I-PSO has more advantages in convergence speed and convergence. I-PSO can quickly and accurately find an optimal path with a total length of 6825.33 mm.Figure 7
Convergence iteration diagram.


### 5.2. The Simulation Verification of Position Deviation from Roadway Center Line

The initial position of roadheader is 200 mm away from center line of roadway, and deflection angle is 0°. As shown in Figure8, blue, red, and green curves are, respectively, I-PSO planned path, CPSO planned path, and WPSO planned path. CPSO algorithm needs to drive 5 steps to make center point of roadheader return to roadway center line with a total driving distance of 4961.23 mm. WPSO algorithm needs to drive 6 steps to make center point of roadheader return to roadway center line with a total driving distance of 5302.71 mm. I-PSO algorithm also needs to drive 5 steps to make center point of roadheader return to roadway center line, but its total traveling distance is only 4756.82 mm. The number of steps and adjustment angle of roadheader based on I-PSO algorithm are presented in Table 3.Figure 8
Schematic diagram of deviation correction path planning of roadheader.Table 3
Results of deviation correction path planning based on I-PSO.
Adjust stepsn12345Adjust angleθ (°)1.4411.3731.157−1.285−1.533Steps (mm)1000.51000.31000.21000.3755.52The algorithm iteration diagram of I-PSO, WPSO, and CPSO is presented in Figure9. The fitness is total distance traveled by roadheader from initial position to roadway center line. In the iteration process, the fitness value of I-PSO is obviously smaller than the fitness value of WPSO and CPSO, and I-PSO algorithm achieves optimal value in 24 iterations. It can be seen that I-PSO has more advantages in convergence speed and convergence. I-PSO can quickly and accurately find an optimal path with a total length of 4756.82 mm.Figure 9
Convergence iteration diagram of I-PSO and PSO.


### 5.3. The Simulation Verification for Both Position and Direction Deviate from Roadway Center Line

The deviation from initial position of roadheader to center line of roadway is 220 mm, and deviation angle is 6°. As shown in Figure10, blue, red, and green curves are, respectively, I-PSO planned path, CPSO planned path, and WPSO planned path. CPSO algorithm needs to drive 4 steps to make center point of roadheader return to roadway center line with a total driving distance of 3833.47 mm. WPSO algorithm needs to drive 4 steps to make center point of roadheader return to roadway center line with a total driving distance of 3623.08 mm. I-PSO algorithm also needs to drive 4 steps to make center point of roadheader return to roadway center line, but its total traveling distance is only 3522.86 mm. The number of steps and adjustment angle of roadheader based on I-PSO algorithm are presented in Table 4.Figure 10
Schematic diagram of deviation correction path planning of roadheader.Table 4
Results of deviation correction path planning based on I-PSO.
Adjust stepsn1234Adjust angleθ (°)−1.449−1.256−1.387−1.252Steps (mm)1000.31000.21000.3522.06The algorithm iteration diagram of I-PSO, WPSO, and CPSO is presented in Figure11. The fitness is total distance traveled by roadheader from initial position to roadway center line. In iteration process, the fitness value of I-PSO is obviously smaller than the fitness value of WPSO and CPSO, and I-PSO algorithm achieves optimal value in 13 iterations. It can be seen that I-PSO has more advantages in convergence speed and convergence. I-PSO can quickly and accurately find an optimal path with a total length of 3522.86 mm.Figure 11
Convergence iteration diagram of I-PSO and PSO.The simulation verification results under above three actual working conditions indicate that I-PSO algorithm proposed can plan a safe and shorter path faster than WPSO and CPSO algorithm and has better convergence and stability.


### 5.4. Influence Rule of Different Offset Distance and Angle

In order to verify influence of different offset distances and angles on the total deviation correction distance, the total travel distances of roadheader under different offset distances and angles are simulated and analyzed. The results are shown in Figure12. The total distance of deviation correction path planning will increase with the increase of the initial position offset distance and decrease with the increase of the initial deviation angle.Figure 12
Total travel distance of roadheader under different offset distance and angle.


## 5.1. The Simulation Verification of Direction Deviation from Roadway Center Line

The deviation from the initial position of roadheader to the center line of roadway is 0 mm, and the deviation angle is 5°. As shown in Figure6, the blue curve is the path planned by the I-PSO algorithm, the red curve is the path planned by the WPSO algorithm, and the green curve is the path planned by CPSO algorithm. WPSO algorithm needs to drive 8 steps to make center point of roadheader return to roadway center line with a total driving distance of 7446.36 mm, and CPSO algorithm needs to drive 8 steps to make center point of roadheader return to roadway center line with a total driving distance of 7706.52 mm. I-PSO algorithm only needs to drive 7 steps to make center point of roadheader return to roadway center line with a total driving distance of 6825.33 mm. The number of steps and adjustment angle of roadheader based on I-PSO algorithm are presented in Table 2.Figure 6
Schematic diagram of deviation correction path planning of roadheader.Table 2
Results of deviation correction path planning based on I-PSO.
Adjust stepsn1234567Adjust angleθ (°)1.5121.4361.3981.322−1.929−1.616−1.518Steps (mm)1000.51000.31000.31000.31000.71000.5822.73The algorithm iteration diagram of I-PSO, WPSO, and CPSO is presented in Figure7. The fitness is total distance traveled by roadheader from initial position to roadway center line. In the iteration process, the fitness value of I-PSO is obviously smaller than the fitness value of WPSO and CPSO, and I-PSO algorithm achieves optimal value in 23 iterations. It can be seen that I-PSO has more advantages in convergence speed and convergence. I-PSO can quickly and accurately find an optimal path with a total length of 6825.33 mm.Figure 7
Convergence iteration diagram.


## 5.2. The Simulation Verification of Position Deviation from Roadway Center Line

The initial position of roadheader is 200 mm away from center line of roadway, and deflection angle is 0°. As shown in Figure8, blue, red, and green curves are, respectively, I-PSO planned path, CPSO planned path, and WPSO planned path. CPSO algorithm needs to drive 5 steps to make center point of roadheader return to roadway center line with a total driving distance of 4961.23 mm. WPSO algorithm needs to drive 6 steps to make center point of roadheader return to roadway center line with a total driving distance of 5302.71 mm. I-PSO algorithm also needs to drive 5 steps to make center point of roadheader return to roadway center line, but its total traveling distance is only 4756.82 mm. The number of steps and adjustment angle of roadheader based on I-PSO algorithm are presented in Table 3.Figure 8
Schematic diagram of deviation correction path planning of roadheader.Table 3
Results of deviation correction path planning based on I-PSO.
Adjust stepsn12345Adjust angleθ (°)1.4411.3731.157−1.285−1.533Steps (mm)1000.51000.31000.21000.3755.52The algorithm iteration diagram of I-PSO, WPSO, and CPSO is presented in Figure9. The fitness is total distance traveled by roadheader from initial position to roadway center line. In the iteration process, the fitness value of I-PSO is obviously smaller than the fitness value of WPSO and CPSO, and I-PSO algorithm achieves optimal value in 24 iterations. It can be seen that I-PSO has more advantages in convergence speed and convergence. I-PSO can quickly and accurately find an optimal path with a total length of 4756.82 mm.Figure 9
Convergence iteration diagram of I-PSO and PSO.


## 5.3. The Simulation Verification for Both Position and Direction Deviate from Roadway Center Line

The deviation from initial position of roadheader to center line of roadway is 220 mm, and deviation angle is 6°. As shown in Figure10, blue, red, and green curves are, respectively, I-PSO planned path, CPSO planned path, and WPSO planned path. CPSO algorithm needs to drive 4 steps to make center point of roadheader return to roadway center line with a total driving distance of 3833.47 mm. WPSO algorithm needs to drive 4 steps to make center point of roadheader return to roadway center line with a total driving distance of 3623.08 mm. I-PSO algorithm also needs to drive 4 steps to make center point of roadheader return to roadway center line, but its total traveling distance is only 3522.86 mm. The number of steps and adjustment angle of roadheader based on I-PSO algorithm are presented in Table 4.Figure 10
Schematic diagram of deviation correction path planning of roadheader.Table 4
Results of deviation correction path planning based on I-PSO.
Adjust stepsn1234Adjust angleθ (°)−1.449−1.256−1.387−1.252Steps (mm)1000.31000.21000.3522.06The algorithm iteration diagram of I-PSO, WPSO, and CPSO is presented in Figure11. The fitness is total distance traveled by roadheader from initial position to roadway center line. In iteration process, the fitness value of I-PSO is obviously smaller than the fitness value of WPSO and CPSO, and I-PSO algorithm achieves optimal value in 13 iterations. It can be seen that I-PSO has more advantages in convergence speed and convergence. I-PSO can quickly and accurately find an optimal path with a total length of 3522.86 mm.Figure 11
Convergence iteration diagram of I-PSO and PSO.The simulation verification results under above three actual working conditions indicate that I-PSO algorithm proposed can plan a safe and shorter path faster than WPSO and CPSO algorithm and has better convergence and stability.


## 5.4. Influence Rule of Different Offset Distance and Angle

In order to verify influence of different offset distances and angles on the total deviation correction distance, the total travel distances of roadheader under different offset distances and angles are simulated and analyzed. The results are shown in Figure12. The total distance of deviation correction path planning will increase with the increase of the initial position offset distance and decrease with the increase of the initial deviation angle.Figure 12
Total travel distance of roadheader under different offset distance and angle.


## 6. Discussion

In order to verify the effectiveness of I-PSO algorithm, four standard test functions are randomly selected for function optimization, and linear adaptive acceleration factor particle swarm optimization algorithm (CPSO) [46, 56], linear adaptive inertia weight factor particle swarm optimization algorithm (WPSO) [57], and I-PSO algorithm are used for performance comparison and analysis.In order to ensure the validity of the data obtained from experiment, each standard test function conducts 50 separate statistical experiments. The parameter settings of three test functions are presented in Table5. According to statistical results, optimal value, average optimal value, standard deviation, and average number of iterations are calculated to evaluate the performance of I-PSO algorithm. The experiment results are presented in Table 6.Table 5
Parameter settings of test function.
FunctionDimensionRange of valueNumber of iterationsSphere10[−100, 100]1000Rastrigin10[−5, 5]1000Rosenbrock10[−30, 30]1000Ackley10[−20, 20]1000Table 6
Result of function optimization.
FunctionIndexCPSOWPSOI-PSOSphereOptimal value3.270365 × 10−126.188575 × 10−191.675238 × 10−19Mean optimal value5.962749 × 10−44.520920 × 10−141.608277 × 10−14Standard deviation2.762931 × 10−31.611793 × 10−133.012249 × 10−14Mean iterations637601553RastriginOptimal value46.55936242.86104536.956735Mean optimal value53.93103850.30921839.9326117Standard deviation71.82217565.21717647.035322Mean iterations488459382RosenbrockOptimal value3.5349042.5973172.263560 × 10−2Mean optimal value6.7838754.9375192.609437Standard deviation3.0499512.6297742.187072Mean iterations840823599AckleyOptimal value3.227938 × 10−92.669405 × 10−94.731159 × 10−11Mean optimal value7.224737 × 10−51.363717 × 10−83.428755 × 10−9Standard deviation2.748184 × 10−41.217176 × 10−83.691472 × 10−9Mean iterations821787724It can be seen from the overall optimization results for different types of functions in Table5. The improved algorithm (I-PSO) proposed has achieved higher convergence accuracy in optimization accuracy and evolution speed and is superior to other algorithms.


## 7. Conclusion

Aiming at the problem of optimal path planning for deviation correction of full-width horizontal axis roadheader in confined roadway of coal mine, a deviation correction path planning method of full-width horizontal axis roadheader based on I-PSO algorithm is proposed. Through research and simulation analysis, the conclusions obtained are as follows:(1)
In view of driving characteristics and roadway constraints of full-width horizontal axis roadheader in coal mine restricted roadway, a mathematical pose model of roadheader is established, and the constraint conditions of path planning based on pose analysis are determined. According to the constraints and deviation correction strategy, a path optimization model with goal of the shortest total distance is established.(2)
Because traditional particle swarm optimization algorithm has problems of premature convergence of particles and poor quality in solving the optimal solution, an improved particle swarm optimization algorithm is proposed. The inertia weight is dynamically adjusted by using particle fitness difference as feedback signal to further adjust search demand of the group for the whole or part of search space. In order to accelerate convergence and optimization of particles, the learning factor is also adjusted. In addition, a random factor is introduced to update the position of particles to avoid local traps as much as possible. The I-PSO algorithm is compared with existing improved particle swarm optimization algorithm in terms of performance. The results show that the I-PSO algorithm has higher convergence accuracy in terms of optimization accuracy and evolution speed than existing improved particle swarm optimization algorithm.(3)
Taking EJM340/4-2 full-width horizontal axis roadheader as simulation object, I-PSO algorithm, WPSO algorithm, and CPSO algorithm are used to simulate and verify three actual working conditions of position deviating from roadway center line, direction deviating from roadway center line, and position and direction deviating from roadway center line. The results indicate that I-PSO algorithm proposed can quickly and accurately plan a safe and shorter path than WPSO algorithm and CPSO algorithm in roadheader deviation correction path planning and has better convergence and stability. The total distance of roadheader’s deviation correction path planning will increase with the increase of initial position offset and decrease with the increase of initial deflection angle.



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*Source: 3373873-2023-01-09.xml*