path with three motion primitives, which are denoted by {S, L,. R} and applies a constant action over a given section, was published by Lester Dubins in [3].
6th International Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey. 5-7 October 2016 CAMI2016-10
Shortest Path Estimation Considering Kinematical Constraints of Main Haulage Roads in Underground Mines: A Heuristic Algorithm Yardimci, A. G., Karpuz C. Mining Engineering Department, Middle East Technical University, Ankara, TURKEY
Abstract— Main haulage road carries the traffic of men, equipment, extracted rock and ventilation air in underground mines. These roads may contain straight or spiral portions with gradient or a combination. For a few decades, sophisticated underground mine design solutions based on Computer Aided Design (CAD) principles are extensively used by industry professionals. Today, companies still rely on the subjective solutions of skilled mine design specialists for the main haulage road design. However, there is no unique analytical solution for the shortest path that considers the kinematical constraints. Determination of the shortest path with maximum gradient and minimum turning radius constraints for any path is a complex problem that exceeds the limit of human intelligence. Considering the direct effect on the initial capital investment and operating costs, it is obvious that a solution is required. This paper explains the results of a study to create a heuristic algorithm to determine the shortest main haulage road path by considering kinematical constraints. Estimates of the algorithms compared by a real mine case and improvement in terms of length are described.
Objective of this study is to propose a heuristic path planning algorithm to calculate the shortest path for a navigable main haulage way visiting all the crosscut entry points in an underground production scheme. The algorithm is implemented in Matlab 2014 environment. There are two common types of sections in any main haulage way layout: straight sections with ascending or descending gradient (named as incline or decline) or helical sections (similar to a curve travelling around a cylindrical surface and joining the upper elevations to the lower elevations by a fixed gradient). The algorithm searches for an appropriate combination of section types that join each of the crosscut entry points while considering minimum turning radius and maximum gradient constraints. Although an optimization algorithm might be useful to determine the global optimum solution, it requires to search for every possible alternative path. For complex mine layouts with hundreds of crosscut entries, the computation time will increase dramatically. Adding some extra constraints will reduce the search space and increase the computation efficiency. However, the additional constraints depend on the expert knowledge of mine haulage way design specialists and transforms the solution into a heuristic approach. A heuristic solution most probably does not give a global optimum but a near optimum solution. If the near optimum solution does not show remarkable variation from the global optimum, then heuristic approach might be preferred.
Keywords: shortest path, main haulage way, mine planning, underground mining. 1. Introduction In an underground mine, main haulage way establishes a connection between the topographical surface and the underground ore production levels. It is constructed in the early stages of the mine life, before the production starts. Thus, development cost adds to the initial capital investment. Besides the short term effects on the mine economy, operating costs are also influenced by the length of this path. Obviously, determining the shortest path for the main haulage way is an advantageous issue in terms of short term and long term economy.
The proposed algorithm makes use of a mathematical phenomenon called as Dubins path to satisfy the minimum turning radius constraint with the shortest path length. Appropriate Dubins path connecting each crosscut entry to the next level with a variable buffer distance are generated and the shortest one is selected using Dynamic Programming (DP).
In spite of some early attempts on the underground network optimization [1], complex nature of optimal mine planning in underground mines has been noted in [2] to be the main reason of limited number of research in this field compared to the successful attempts of open-pit mine planning optimization.
The mathematical proof of presenting the shortest curved path with three motion primitives, which are denoted by {S, L, R} and applies a constant action over a given section, was published by Lester Dubins in [3]. While the straight sections are symbolized by ‘S’ primitive, the L and R primitives imply
1
6th International Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey. 5-7 October 2016 CAMI2016-10 the turns to the left and right. Each possible kind of shortest path can be denoted by a sequence of three symbols. Such a sequence is called as a “word”. A word corresponds to the sequence of motion primitives applied. Dubins showed that a shortest curved path can be any of the six words of the form: {RSL, LSR, RSR, LSL, RLRL, LRL} If any of the ‘R’ or ’L’ primitives are denoted by ‘C’ then the Dubins paths can be shortly represented by two base words, which are {CCC, CSC}.
A
In Unmanned Air Vehicle (UAV) path planning, Dubins path principle has a long history of use to determine the optimum route between the waypoints. Sharp turns may damage UAVs thus, they have a certain turning radius, which is similar to the case of underground mining vehicles. Dubins curve principle was originally proposed for 2 dimensional curved paths. However, underground mine ramp is a 3 dimensional mine design element. Therefore, the principle requires to be modified.
Branch 1
B
E
C
F
D
G
Branch 2
Branch 3
H
Branch 4
Figure 1 Schematic view of a sample travelling salesman problem The schematic in Figure 1 is very similar to an underground mine main haulage way design problem. City A can be thought as the portal location. Branches are the production levels and nodes in each branch are possible cross-cut entries. The algorithm selects the best node for giving the overall shortest path. The final city H is the end of the main haulage way.
Dynamic Programming is another concept that is used in the algorithm workflow. In mathematics, computer science, economics, and bioinformatics, DP solves a complex problem by breaking it down into groups of simpler sub problems. Solutions of sub problems are evaluated to obtain the least cost solution in the overall.
Path planning problems lack of benchmark problems. In order to validate the proposed algorithm, a simple underground mine layout with an easy to predict shortest path for the main haulage way is generated. The path predicted by the algorithm is compared by the obvious shortest path.
Advantage of using DP lies in the solution mechanism. The most primitive approach is to generate many sub problems and solve them in each of the problem combinations. However, DP seeks to solve each sub problem only once. If the solution to a sub problem has been computed, it is stored: the next time the same solution is needed, it is simply selected from a look up list. By this way, the number of computations are reduced. This approach is especially useful when the number of repeating sub problems grows exponentially as a function of the size of the input. A simple illustration for the solution mechanism of DP can be seen in Figure 1 for a travelling salesman problem. Each of the nodes denoted by a capital letter are cities. There are four branches of cities. A Travelling salesman plans to start from city A and arrive city H by travelling the shortest path between the other cities. Initially, DP generates a distance matrix showing the distance between each paired cities and later determines the shortest route which passes from at least 1 city in branch 2 and branch 3. While doing this, the search can start from city A and go towards city H or in the opposite direction. These methods are named as forward or backward induction. In this study, backward induction is preferred.
Main haulage way design of Bizmisen underground iron mine is used for the purpose of verification. Conventionally designed path is compared with the path predicted by the proposed heuristic algorithm. A remarkable decrease in the path length confirms that the algorithm works. 2. Problem Description The problem is finding the shortest path that visits the desired set of main nodes without violating the kinematical constraints. Main nodes are defined as the portal location and crosscut entry points of each production level. Between each of the main nodes, six alternative Dubins paths are generated and the shortest one is selected. The algorithm is planned to be capable of searching for the appropriate buffer distance. Buffer distance search starts from the closest point on the crosscut entry and moves away from the orebody in the perpendicular direction until a desired range. A simple underground mine layout can be seen in Figure 2. Assumptions, givens and outputs of the algorithm are listed below.
2
6th International Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey. 5-7 October 2016 CAMI2016-10 If two main nodes cannot be connected by neither of a straight section, a CSC or CCC type of path with the maximum allowable gradient because of the high elevation difference, then the final portion is connected by a helical path with a gradient up to the maximum allowable gradient. In case a smaller gradient is possible in this portion, it is preferred. Number of turns of a helical path depends on the level difference between two succeeding nodes and the preferred gradient.
Figure 2 Benchmark problem layout
Givens: Main node coordinates.
Assumptions: The algorithm makes valid shortest path predictions for the case of single orebody problems. The portal location and the crosscut entry points (main nodes) are defined in terms of x, y and z coordinates (in meters). Heading angles for each of the main nodes are defined
Main node heading angles
Buffer distance away from the crosscut entry
Kinematical constraints: -
Minimum turning radius (m)
-
Maximum gradient (%)
Outputs: x, y, z coordinates of the main and dummy nodes for
in degrees. Upper limit for the buffer distance search for each of the
the valid shortest path.
crosscut entries is defined in terms of meters. Visiting sequence of main nodes is known.
3. Path Planning Using a Heuristic Algorithm 3.1. Kinematical Model of an Underground Vehicle In a path planning problem, location of the moving object is required to be predicted throughout the simulation. Unmanned Air Vehicle (UAV) path planning researchers claims two types of kinematical models to represent the motion of a vehicle, which are the accurate and simple models. True motion of a vehicle can be accurately modelled using nonlinear fully coupled ordinary differential equations of motion for a vehicle moving along three axes with six degrees of freedom [4]. This approach is also capable to include the forces and moments acting on the vehicle body, which are driven by gravity, propulsion and aerodynamic forces. However, no closed form solutions exist for these complex to handle equations. Therefore, numerical solutions are required for steady state solutions. Dubins vehicle approach is a more simplistic and appropriate kinematical model to represent the motion of an underground mining vehicle. A Dubins vehicle is a bounded speed and no reversing planar vehicle with constriction to move along paths of bounded curvature [5]. The equations for an underground mining equipment modelled as a Dubins vehicle are given in (1), (2), (3) and (4).
Elevation of the main nodes decrease gradually. The valid path between two main nodes can be in one of the forms stated below: -
A straight section.
-
A Curve-Straight-Curve (CSC) type section
-
A Curve-Curve-Curve (CCC) type section
If two main nodes can be connected by a straight path within the allowable limits of gradient, then it is preferred. If it is not possible to connect two main nodes by a straight path, shortest one of the CSC or CCC type section is used. If two main nodes can be connected by a CSC or CCC type of path with a smaller gradient than the maximum allowable gradient, than it is preferred.
3
6th International Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey. 5-7 October 2016 CAMI2016-10 𝑥𝑖̇ = 𝑣𝑖 cos(𝜃𝑖 ) 𝑖 = 1…𝑛 𝑦𝑖̇ = 𝑣𝑖 sin(𝜃𝑖 ) 𝑖 = 1…𝑛 𝑧𝑖̇ = −𝑣𝑖 sin(𝜓𝑖 ) 𝑖 = 1…𝑛 𝑢𝑖 𝑣𝑖 𝜃𝑖̇ = 𝑖 = 1…𝑛 𝑟𝑚𝑖𝑛 Where; 𝑖 = 𝑛𝑜𝑑𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑛 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑑𝑒𝑠 𝑥𝑖̇ = 𝑥 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖 𝑡ℎ 𝑛𝑜𝑑𝑒 𝑦𝑖̇ = 𝑦 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖 𝑡ℎ 𝑛𝑜𝑑𝑒 𝑧𝑖̇ = 𝑧 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖 𝑡ℎ 𝑛𝑜𝑑𝑒 𝜃𝑖̇ = ℎ𝑒𝑎𝑑𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖 𝑡ℎ 𝑛𝑜𝑑𝑒 𝜓̇ 𝑖 = 𝑚𝑎𝑥. 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑖 𝑡ℎ 𝑛𝑜𝑑𝑒 𝑢𝑖 = 𝑡𝑢𝑟𝑛 𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑖 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑟𝑚𝑖𝑛 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑡𝑢𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑑𝑖𝑢𝑠
Curves of minimal length with a constrained turning radius and prescribed initial and terminal points have been proven by Lester Dubins [3] to be any one of the six alternative forms. In Figure 3, results of a sample run of the proposed heuristic algorithm between two nodes in Matlab environment provides a visual description for the six alternative Dubins paths. The initial node and final node coordinates are (0,0,100) and (100,100,60), Heading angle is 135° for the initial node and 215° for the final node. Reference plane showing the heading angle increase from 0° to 360° can be seen in the same figure on the upper right part. Minimum turning radius is set to 25m and the maximum gradient is 12%. As shown in the figure, the shortest path is of RSR type Dubins path. Changing the node coordinates and heading angles, one can receive any other form of curved path as the shortest alternative. Following the above procedure, the shortest curved path is calculated between each main node.
(1) (2) (3) (4)
Figure 3 A sample constrained path calculation between two nodes by the heuristic algorithm in Matlab (plan view). number of possible paths and the calculation time. Next, according to the Dubins path method, six valid constrained paths with curvature are calculated in X-Y plane. For this purpose, dummy nodes are generated between the main nodes and x, y coordinates are calculated for each of them to present a valid path. In each computation time, the algorithm works between two main nodes with certain heading angles. After computing a valid path, another main node in the sequence is included into the calculation procedure. Elevation for each dummy node is assigned by a different module. If two main nodes can be joined by dummy nodes within the limits of constraints, then no helical ramp is created. Otherwise, the path
3.2. Algorithm Workflow Flowchart of the algorithm is presented in Figure 4. To briefly describe, main node coordinates (in terms of x, y, z) of the portal location and crosscut entry points are defined inside an Excel file as the problem inputs. Level entry points are presented by numerous alternatives perpendicularly sequenced to the orebody in the outward direction. Later, kinematical constraints are defined. Minimum turning radius (m) and maximum gradient (%) must be selected to conform the equipment capabilities. Later, path direction, which is defined as the heading angle of each node, is defined as input. As can be predicted increasing number of main nodes increases the
4
6th International Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey. 5-7 October 2016 CAMI2016-10 starts by a maximum gradient portion and connects to the final main node by a helical ramp with minimum number of turns and maximum possible gradient. In the second part, distance for each possible 3D path is calculated and Dynamic Programming is used to select the shortest valid path.
depth. Obviously, the shortest path is a helical pattern turning around a cylinder. Illustration of the benchmark problem geometry can be seen in Figure 2. Compared to the easy to predict shortest path for the simple mine layout in Figure 2, the proposed heuristic algorithm predicted exactly the same path. Therefore, the algorithm can be claimed to be capable of calculating near-optimal solution even in the complex problems.
Inputs •Main node coordinates •Minimum turning radius •Maximum gradient
4. Case Study 4.1. Research Area Performance of the heuristic algorithm is checked on the conventionally designed main haulage way of an underground iron mine located in the North East of Bizmisen district of Erzincan / Turkey. Satellite view of the Donentas iron mine, plan and cross-section view of the 3D orebody model and orebody dimensions can be seen in Figure 5.
Repeat between each main nodes
Compute six Dubins Paths •RSL, LSR, RSR, LSL, RLR, LRL
Donentas outcropping orebody dips with 23° in the opposite direction to the overlying hillside. orebody depth is estimated to be 345 m, by referencing to the outcrop. Due to the opposite dip direction to the topographical surface, open pit depth for the case of exploiting the complete orebody would extent more than this depth. Based on the economic and legal issues, upper part of the orebody was planned to be exploited by an open pit mine with an overall slope angle of 36°.
Select the shortest curved path
End Repeat
Select the shortest route visiting all of the main nodes by Dynamic Programming
In order to sustain safety in the underground operation, while producing the remaining ore in the lower parts, a crown pillar was planned to be left below the open pit. Empirical studies supported by numerical analyses have shown that the optimum thickness of the crown pillar is 30m.
Figure 4 Flowchart of the heuristic algorithm Although the orebody can be completely extracted by open pit mining, high depth is a potential for slope instability. In this case, a safe pit requires lower overall slope angle; however, cost is the larger open pit boundaries on the topographical surface. Due to the insufficient extent of license boundary, mine administration decided to extract the ore by an open pit as deep as possible. Remaining part was planned to be operated by underground mining method. Cut and fill method was selected. The portal is on the east side of the open pit base. Crosscuts were planned to be situated on the zones of high grade concentration.
3.3. Verification of the Algorithm Path planning suffers from the lack of benchmark problems. It is beyond the limits of human cognitive capacity to present an analytical solution for a complex geometry shortest path validation problem without violating kinematical constraints. However, validity of the predicted shortest paths by the algorithm requires to be proven. In this study, a simplistic approach has been followed. A simple mine layout with a flat topography and orebody in the shape of a rectangular prism was generated. Crosscut entry coordinates in the x-y plane are the same and elevation difference between the successive crosscut entries are equal
5
6th International Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey. 5-7 October 2016 CAMI2016-10
Donentas
Plan view A
A’
Section view
Front view Figure 5 Mine location, orebody geometry and Donentas open pit layout (Google Earth, 2016) 4.2. Design Procedure Initially, the main haulage way is designed by conventional CAD technique with a minimum turning radius constraint of 25 m and maximum gradient constraint of 12%. Length of the navigable path is 1192 m. Layout of the open pit and underground mine with the conventionally designed main haulage way can be seen in Figure 5 from the perspective view. In the next stage, using the heuristic algorithm, a new main haulage way path is computed. One initial node, one final node and three crosscut entry nodes with 100 alternative entries in each level with 0.5m buffer distance from each other can be seen in Figure 6. Totally 1x100x100x100x1=1,000,000 node combinations and considering there are six valid Dubins paths, 6,000,000 paths are calculated in 190 minutes by a low level workstation (i7 processor, 32GB RAM). Length of the valid shortest path is 963 m.
Figure 6 The shortest path predicted by the heuristic algorithm (plan view) 4.3. Results and Discussion The heuristic algorithm computes a main haulage way path, which is 229 m shorter than the conventionally designed path. Considering the average cost of tunnel construction is around 2000$ per meter, net profit is 458,000 $. While the initial capital investment is reduced, operating cost is also decreased by avoiding the fuel cost of underground mine vehicles for
6
6th International Conference on Computer Applications in the Minerals Industries, Istanbul, Turkey. 5-7 October 2016 CAMI2016-10 229m x 2 in each sortie. Considering numerous sorties are required throughout the mine of life, it is obvious that there will be a considerable amount of saving from the fuel costs.
The future studies are planned to focus on adapting evolutionary algorithms to find the global optimum solution for the shortest underground mine main haulage way problem. Another issue is avoiding the forbidden zones, which can be low rock quality zones, faults, joints or aquifers.
5. Conclusions Initial capital investment and operating cost are functions of underground mine design. Contrary to its importance optimization of underground mine morphology has been an underestimated subject in mining. Main haulage way is the most critical structure of the design because vast of the traffic is carried via this long term opening. Therefore, length of the main haulage way directly effects the operating cost.
Acknowledgement The authors would like to thank the staff of Bilfer Mining Co. for their valuable supports. The company is also acknowledged for the permission to use and publish the data. References [1] D. H. Lee, "Industrial case studies of Steiner trees," in NATO, Denmark, 1989. [2] M. Brazil, P. A. Grossman, J. H. Rubinstein and D. A. Thomas, "Improving Underground Mine Access Layouts Using Software Tools," Interfaces, vol. 44, no. 2, pp. 195203, March-April 2014. [3] L. E. Dubins, "On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents," American Journal of Mathematics, pp. 497-516, 1957. [4] H. Ergezer and K. Leblebicioglu, "Path Planning for UAVs for Maximum Information Collection," IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no. 1, pp. 502-520, January 2013. [5] K. Savla, E. Frazzoli and F. Bullo, "Traveling Salesperson Problems for the Dubins Vehicle," IEEE Transactions on Automatic Control, vol. 53, no. 6, pp. 1378-1391, July 2008.
Mine design has long been carried out by skilled experts. Although it may be easier to predict the shortest path for a simpler geometry, complex underground mine geometries may not allow to predict even the near-optimum paths. This study presents a heuristic algorithm to predict the shortest path for underground mine main haulage ways. A simple underground mine geometry was prepared as a benchmark problem to verify the algorithm. Later, a real main haulage way design for an underground mine in Turkey is compared with the predicted path by the algorithm. In addition to the economical parameters like initial capital investment and operating cost, improvement in the path length is noted. In other words, although the solution is improved, there exists a better solution. In spite of its success in the verification problem, the algorithm does not provide the global-optimum solution mainly due to the extra constraints. The computation time is improved by decreasing the search space for the cost of risk to trap inside a local optimum.
7
