-w>~ ^•w/ \;^^V* %/W*>^ \JwV* %°°W>^ ^- G *.**^>o >*\.^&/V G°*.-^il->o >*..iife/V 0°*..iKl->o ,**V *%*> *°«* 3j; S». ^*, \ 4°* '•>°° "V^'V V^V' V^'V 5.^?o "o^^» aV* <^-v t"^ •% <& **■' &* £&£*%, **>*&'-*+ *****£:>+ 4?*^"i-. ^ ^\^>>< BUREAU OF MINES INFORMATION CIRCULAR/1990 Utilizing Mechanical Linear Transducers for the Determination of a Mining Machine's Position and Heading: The Concept By Christopher C Jobes .P.NT O^ ' a 80 V YEARS (ft 3 O & U.S. BUREAU OF MINES 1910-1990 THE MINERALS SOURCE Mission: As the Nation's principal conservation agency, the Department of the Interior has respon- sibility for most of our nationally-owned public lands and natural and cultural resources. This includes fostering wise use of our land and water resources, protecting our fish and wildlife, pre- serving the environmental and cultural values of our national parks and historical places, and pro- viding for the enjoyment of life through outdoor recreation. The Department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. The Department also promotes the goals of the Take Pride in America campaign by encouraging stewardship and citizen responsibil- ity for the public lands and promoting citizen par- ticipation in their care. The Department also has a major responsibility for American Indian reser- vation communities and for people who live in Island Territories under U.S. Administration. / Information Circular 9254 Utilizing Mechanical Linear Transducers for the Determination of a Mining Machine's Position and Heading: The Concept By Christopher C Jobes UNITED STATES DEPARTMENT OF THE INTERIOR Manuel Lujan, Jr., Secretary BUREAU OF MINES T S Ary, Director <> \ <> o' Library of Congress Cataloging in Publication Data: Jobes, Christopher C. Utilizing mechanical linear transducers for the determination of a mining machine's position and heading: the concept / by Christopher C. Jobes. p. cm. - (Bureau of Mines information circular; 9254) Includes bibliographical references. Supt. of Docs, no.: I 28.27:9254. 1. Mining machinery-Automatic control-Data processing. 2. Transducers. I. Title. II. Series: Information circular (United States. Bureau of Mines); 9254. TN295.U4 [TN345] 622 s-dc20 [622\2] 90-1416 CIP CONTENTS Page Abstract 1 Introduction 2 Global navigation 2 Local navigation 2 Face navigation . 3 Background 3 Problem definition 3 Topological requirements 3 Functional requirements 4 Constraints 4 Dimensional constraints 5 Inertial constraints 5 Position and heading system 5 Sensor selection 5 Sensor configuration 6 Redundant sensor configuration 6 Position and heading algorithm 7 Definition of reference frames 7 Coordinate transformation of reference frames 8 Closed-form solution 9 Loop equation development 9 Three-transducer solution 9 Four-transducer solution 11 Testing and backup strategy 12 Error analysis 13 Linear transducers 13 Analog-to-digital converter 13 Sample rate 13 Conclusions 13 Appendix A.-Linear position transducer calibration data 14 Appendix B— Calibration equations and error frequency distribution 15 Appendix C.-Conversion specifications for Intel remote control board 44/20A analog-to-digital converter ... 16 ILLUSTRATIONS 1. British Coal trolley-pole-type articulated boom . 4 2. Instrumented degrees of freedom on trolley-pole-type articulated boom 4 3. Linear position transducer 6 4. Kinematic equivalent of linear position transducer 6 5. Three-transducer configuration 7 6. Directly solvable redundant four-transducer configuration 7 7. Local and machine reference frames and position vectors 8 8. Rectangular and polar coordinate components 8 9. Configuration for position and heading algorithm 9 B-l. General calibration equation error frequency distribution 15 UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT dB decibel M v microvolt °C degree Celsius /iV/°C microvolt per degree Celsius ft foot mV millivolt ft/s foot per second oz ounce Hz hertz pet percent in inch ppm part per million kHz kilohertz V volt kohm kilohra Vdc volt, direct current lb pound V/in volt per inch US microsecond 7s degree per second UTILIZING MECHANICAL LINEAR TRANSDUCERS FOR THE DETERMINATION OF A MINING MACHINE'S POSITION AND HEADING: THE CONCEPT By Christopher C. Jobes 1 ABSTRACT This U.S. Bureau of Mines report describes a system to determine the position and heading of a mining machine during maneuvers in the face area of an operating mine section. The system is the first step in the development of a guidance system for automated mining machines. The position and heading algorithm is described, and a preliminary error analysis is performed. The sensors employed were linear position transducers mounted on a mining machine with their cables attached to points on a stationary reference. Four linear position transducers were used to provide a data redundancy that increased the reliability of the guidance system. The linear position information obtained from the transducers was processed mathematically via the position and heading algorithm to provide the desired position and heading information. The algorithm takes advantage of sensor redundancy to continually test the accuracy of the sensor data. The major sources of error were determined to be the linear position transducers, the analog-to- digital (A/D) converter used to interpret the data, and the sampling frequency of the measurement system. These sources of error were evaluated to determine their inaccuracies for use in calculating the overall system accuracy. 'Mechanical engineer, Pittsburgh Research Center, U.S. Bureau of Mines, Pittsburgh, PA. INTRODUCTION Navigation is one of the key systems that must be de- veloped if a mining machine can be automated and the machine operator can be moved to a safe, protected area. This report describes the approach taken by the U.S. Bureau of Mines to solve navigation requirements for maneuvering a continuous miner at the coal mine face. This work is part of the Bureau's program to enhance mine safety. The function of a position and heading system for a mining machine is to provide guidance information for navigation in the face area. This system must be attained before face operations can be automated under current continuous mining practices. 2 The automation of equip- ment at the face will increase health, safety, and efficiency in this area. To get a better feel for navigational require- ments in mining, it is necessary to briefly look at naviga- tion in mines as it is currently performed by workers. Navigation in a mine can be divided into three cate- gories: global navigation, local navigation, and face nav- igation. 3 Each of these navigation categories requires the performance of various navigational tasks. GLOBAL NAVIGATION Global navigation consists of the navigation of a piece of mobile mining equipment from one place in a mine to another that cannot be seen. This is similar to driving an automobile from one city to another. The travel of a man- trip is a good example of this form of navigation as it may cover several miles. Operators of supply and maintenance vehicles also perform this type of navigation on a regular basis. The operators of all mobile mining equipment, at one time or another, must perform this type of navigation to get from the surface to their assigned workplace. Then- tasks usually include, among other things, mine mapping, path generation, and path following. The primary task in global navigation is to move a min- ing machine from one place in the mine to another. The major resource for accomplishing a navigational task is the mine map. A mine map is continually updated as changes are made to, or occur in, the mine. These changes are usually noted on the mine map as they occur since they may affect navigational decisions. Such changes may include roof falls, stoppings, dangerous conditions, unnavigable portions, etc. Schnakenberg, G. H., Jr. Computer-Assisted Continuous Coal Mining System-Research Program Overview. BuMines IC 9227, 1989, 15 pp. Anderson, D. L. Framework for Autonomous Navigation of a Continuous Mining Machine: Face Navigation. BuMines IC 9214, 1989, 23 pp. Using a mine map, knowledge of the initial position and desired destination, and path generation rules, a guidance system can usually generate a path from one place in a mine to another. The generation of this path may take into consideration several items: the normally used path, mine conditions along that path, traffic, etc. This form of path generation usually determines which entries must be traveled and where turns must be made. When a guidance system follows a generated path, the current position must be continually updated on the mine map as well as the generated path. As the m inin g machine follows the generated path, it must negotiate entries and corners and must also perform collision avoidance, imple- mented by manipulating the configuration, position, and orientation of the machine. LOCAL NAVIGATION Local navigation is performed within the area of the mine that the guidance system sensors can see. Local navigation includes traversing entries and turning crosscuts in the performance of some navigational task (e.g., path following). The local navigational tasks include scheduling as well as the normal mine mapping, path generation, and path following functions. The local map must be of greater detail than the mine map. It should be updated more frequently and should include the location of all pieces of mining equipment and their function. This information will be of use to the scheduling algorithm. The movement of one piece of mining equipment from one place in the local area to another requires some form of scheduling since there is a good amount of activity and equipment present in the average continuous mining sec- tion. Scheduling is required since there is a cycle of tasks to be performed at the face requiring the application of several pieces of equipment (continuous miners, shuttle cars, roof bolters, rock dusters, etc.) among the faces being mined. The performance of these tasks needs to be coor- dinated so that there is a minimal amount of interference among the tasks performed, thus maximizing the efficiency and therefore the productivity of the section. The path generation and path following tasks are much the same as in global navigation, with only a few added complications. If shuttle cars are present in the mining section, it may be necessary to avoid their trailing cables if they are located in the entry being traveled. Also, there is a greater need of care in collision avoidance since there are more items hanging from the roof, attached to the ribs, or parked in the entries. FACE NAVIGATION Face navigation is performed by mobile mining equip- ment preparing to perform a task in the face area of an entry or crosscut development. The type of equipment in the face area may include mining equipment such as con- tinuous mining machines and roof bolters. The face nav- igational tasks include mine mapping, path generation, and path following. The map of the face area should be of the greatest detail and should be updated frequently. The updated information should include the volume created by coal extraction, the location and types of roof bolts applied in the face area, the local geology (which may be determined by monitoring the virtual work during the drilling cycle), etc. The path generation and path following tasks are sim- ilar to those of local and global navigation, but include some additional tasks. Path generation for coal extraction must be in accordance with predetermined mining patterns and must develop the entry according to the mine plan. Path following must be fairly exact and must function in accordance with the type of machine being guided (i.e., the cutting drum diameter would determine the distance of sump for a continuous mining machine). The actual guidance of the mining machine is of great importance to the mining cycle since much depends on face navigation. This guidance task is important since the conditions and obstacles are a hindrance to the implemen- tation of a guidance system. BACKGROUND Many designs of navigational systems for use with autonomous self-guided mining machines have been at- tempted in prior years. These attempts can be categorized into electromagnetic, stress wave, and mechanical methods. While there is much work being done in the electromag- netic (optical, laser, radar, magnetic compass, natural gamma, etc.) and stress wave (ultrasonic, seismic, vibra- tion, etc.) areas, very little attention is being paid to me- chanical guidance systems. The apparent reason for the lack of interest in mechan- ical guidance systems is that this method usually involves either dead reckoning (e.g., sensing wheel motion) or mechanical attachment. In dead reckoning, the errors introduced into the system are cumulative, and therefore a system relying on this form of navigation alone is unre- liable, particularly in a mining environment. Most mining machine designers seem unwilling to restrict the motion of their machines enough for a mechanical attachment system to be a viable option. One known attempt has been made at using an attach- ment method for the guidance of a mining machine. In particular, a trolley-pole-type articulated boom (fig. 1) is used to guide a roadheader. 4 This articulated boom has six degrees of freedom (fig. 2) that are instrumented (one prismatic and five revolute). This method uses standard robot kinematics techniques to determine the position and orientation of the roadheader with respect to the local reference frame. The system performed adequately, but was not considered for use since there was not enough overhead room available to the continuous mining machine. PROBLEM DEFINITION A system for machine guidance was required to deter- mine the position and heading of a mining machine during maneuvers required in the face area of an entry. To ad- equately design the mechanical system, it was necessary to first define the system's requirements and constraints. TOPOLOGICAL REQUIREMENTS Topological requirements are the minimum set of pa- rameters to be defined before the kinematic chain of a mechanism can systematically be enumerated. Usually this includes the "space" (planar or spatial) in which the mechanism moves, the degree of freedom, and either the number of links or independent loops needed to obtain a finite solution set. At this point in the design procedure, any additional specifications should be made to reduce the number of solutions even further. The nature of motion of a mining machine in a coal seam was determined to be spatial rather than planar since it cannot be assumed that the coal seam would be abso- lutely smooth for even a short distance. Thus, the mining machine was determined to have six degrees of freedom (x, y, z, roll, pitch, and yaw). British Coal, Headquarters Technical Department (Burton on Trent, England). Alignment and Profile Guidance of Roadheaders. Final report on European Coal and Steel Community Research Project 7220-AB/810, 1987, 45 pp. Figure 1. -British Coal trolley-pole-type articulated boom. //////// f* KEY P - r° r3 - Prismatic joint - ( ft - Revolute joint Roadheader The degree of freedom of the system was dependent on the number of independent inputs to the system. In this case, the mining machine was the input to the guidance system and therefore the system had six degrees of free- dom. This degree of freedom required that any kinematic chain connecting the mining machine to the local refer- ence frame have at least six degrees of freedom to work properly. FUNCTIONAL REQUIREMENTS Functional requirements are the tasks to be performed by the mechanism. The only task of this mechanism is the tracking of the position and orientation of the mining machine. In order for the tracking to be accomplished without unnecessary redundancy, no more than six degrees of freedom were required to connect the mining machine to the local reference frame. CONSTRAINTS Figure 2.-lnstrumented degrees of freedom on trolley-pole- type articulated boom. When the mechanism was designed, there were two types of constraints: dimensional and inertial. These constraints determined the feasibility of any mechanisms that were found to satisfy the topological and functional requirements. Dimensional Constraints Some of the dimensional constraints on the mechanism connecting the mining machine to the local reference frame were that the mechanism— 1. Should perform the required task within the entry in which the mining machine is to be working while min- imizing contact with the ribs or uncut coal blocks since such contact may or may not result in damage to the mechanism. 2. Should minimize its interference with the mining machine's task since its purpose is to measure, not to be an avoided object. 3. Should allow up to a 40-ft advance or a 20-ft, 90° crosscut to maximize productivity during the automated cutting cycle. 4. Must avoid interference with the mining machine conveyor boom and shuttle car (if continuous haulage is not being used) since such contact would interfere with reliable measurement and may result in damage to the mechanism. 5. Must avoid the ventilation curtain (if present) since contact with the curtain may affect one or more of the transducers. Inertia I Constraints Some of the inertial constraints on the mechanism con- necting the mining machine to the local reference frame were, as follows: 1. The inertia of the mechanism should be minimized to reduce the loading of the mining machine and the mechanism's resolving joints. 2. The mass of the mechanism must be minimized to reduce its interference with the operation of the mining machine. 3. The mechanism must be stiff enough to maintain accurate measurements during the movements of the min- ing machine. POSITION AND HEADING SYSTEM To develop the position and heading system, the trans- ducers to be used were selected first. Next, the config- uration of the sensors was addressed so that the accuracy and reliability of the position and heading system were maximized. Finally, the number of sensors was made redundant so that the reliability of the position and head- ing system could be continuously monitored. SENSOR SELECTION In the average coal seam, the headroom available to the position and heading system is less than that available to the British Coal roadheader guidance system. Therefore, an articulated boom was not considered to be the best type of connection to make. Aside from the fact that it was a cumbersome way to go about measuring the position and heading of a mining machine, it would have been big and heavy. Because of the weight and length of the system, it was possible to foresee a problem with oscillations in the boom that would reduce the life of the measurement sys- tem and reduce the accuracy of the measurement readings. It would be very difficult to make a 40- to 50-ft-long boom stiff enough for accurate measurement without also making it too big for the job (i.e., cumbersome with a correspond- ing lack of mobility). Since not all of the degrees of freedom of the links connecting the mining machine to the local reference frame needed to be instrumented, several groups of links (called chains) or their equivalents were used. Further- more, while the motion of the mining machine was in six degrees of freedom, not all of the motions were considered large enough to require measurement. (Although coal seams are not "flat," the generally accepted change in gradient does not typically exceed 5° over an occasional roll between 20 to 50 ft wide within the northern Appalachian coal seams.) Therefore, for a relatively smooth seam, the z, roll, and pitch measurements could be assumed to have a negligible effect on the x, y, and yaw measurements (position and heading) with respect to other sources of error. If such information was needed, it would be possible to use roll and pitch sensors on the mining machine to adjust for such errors relative to a leveled local reference. Thus, the sensor selection task was one of determining which six-degree-of-freedom chain could best perform the position and heading measurements, but not all of whose degrees of freedom need to be instrumented. Given that most transducers measure only one degree of freedom and few had the necessary linear range, the list of possible mechanical attachment measurement schemes was reduced to a linear position transducer, or wire pull (fig. 3), as it is sometimes called. Although each linear position transducer technically has an infinite number of degrees of freedom, each wire was modeled as a six- degree-of-freedom kinematic chain equivalent (fig. 4). Thus, the position and heading system development task became a method of configuring linear position trans- ducers to perform the position and heading measurement task. ScaSe, in Figure 3.-Linear position transducer. R KEY - ( - Revolute joint -CIZD- Cylindrical joint Mining machine Figure 4. -Kinematic equivalent of linear position transducer. SENSOR CONFIGURATION The task of finding a configuration in which the linear position transducers could be used to determine the posi- tion and heading of a mining machine was relatively sim- ple. While it was possible to perform the required meas- urement with one linear transducer and two revolute transducers to measure the departure angle of the wire relative to the local and machine reference frames, there was no such transducer on the market. This configuration was deemed to be low in accuracy because the linear transducers had a relatively light "pull" (about 3 lb), and this configuration would not be redundant and therefore could be tested for accuracy. Therefore, three linear po- sition transducers were deemed required to determine the three measured degrees of freedom (position and head- ing). The placement of these transducers and the attach- ment of their cables made little difference in the ability to calculate the position and heading of the mining machine, provided the cables were not parallel and the attachment points and transducer locations were separated by a suffi- cient distance to reduce the linear position transducer error to a reasonable percentage. The placement of the transducers did, however, affect the computation required for the calculation. If the linear position transducers were located as shown in the top portion of figure 5, a set of nonlinear transcendental equations occurred and an itera- tive solution method was required. If the linear position transducers were located as shown in the bottom portion of figure 5, the solution to the position and heading equa- tions could be arrived at easily using elementary trigonom- etry. However, because of the nature of the trigonometric equations, some ambiguities arose that resulted in multiple possible machine positions and headings for a given set of sensor readings. REDUNDANT SENSOR CONFIGURATION An easy method to solve a portion of the multiple so- lution problem arising from ambiguities in the trigono- metric derivation was to add a redundant sensor (fig. 6). This sensor served two purposes. The first purpose was to identify which of the multiple solutions was the correct one by first performing a very simple four-transducer solution. The other purpose was to compare the four three-transducer solutions with the four-transducer solu- tion to see if one of the four linear position transducers was furnishing erroneous data. A transducer error would show up as all of the four three-transducer solutions being different from each other, since only one solution did not use the suspect transducer. Of course in this case, the four-transducer solution was also incorrect. If more than one transducer was giving erroneous data, every three-transducer solution could yield a different position and heading. This was, however, pref- erable to the case with a single three-transducer solution where it was never really known if the solution was correct. Using this information, the mining machine could be guided using the four-transducer solution. A correct solu- tion could still be found, however, if the initial configura- tion and the sign of angular rotation of the mining ma- chine was known. If a valid solution was not obtainable, then the mining machine controller could shut down the m inin g machine and request maintenance. Reference "Wlpt, Entry ^-"AA \ LPT 2\\ Reference /\ Mining yk \ LPT u^w - ^ LPTw r V^LPTV^ Scale, ft KEY LPT Linear position transducer 10 Scale, ft KEY LPT Linear position transducer Figure 5.-Three-transducer configuration. Top, requiring iterative solution; bottom, directly solvable. Figure 6.-Directly solvable redundant four-transducer configuration. POSITION AND HEADING ALGORITHM In developing the position and heading algorithm, the first task was to define the reference frames used. The next task was to develop the coordinate transformation of the reference frames. Finally, a closed-form solution of the position and heading algorithm was developed. DEFINITION OF REFERENCE FRAMES In mechanism analysis, the location of an object was of some concern. In order to describe an object's position in space, a coordinate system or frame was attached to the object. The position and orientation of the frame was then described with respect to some reference coordinate frame. For the purpose of this research, a local reference frame was attached to ground at some distance behind the min- ing machine. The machine reference frame was attached to some point on the mining machine (fig. 7). A reference frame was denoted by the prefix superscripts, A, for the local reference frame, and B, for the mining machine's reference frame. It was essential that some notation be defined before coordinate transformations could be addressed. The po- sition vector was represented by a bold-faced character (R). This character represented the position in space of a point relative to the associated reference frame. For a two-dimensional example, the position vector either de- scribed an (x,y) point in rectangular coordinate form or an (r,0) point in polar coordinate form (fig. 8). The no- tation for the components of a vector in the rectangular coordinate system was where and Xj or yj , i = vector number, x = x component of position vector, y = y component of position vector. The notation for the components of a vector in the polar coordinate system was where i = vector number, r = magnitude of position vector, and 6 = direction of the position vector relative to positive x axis. Scale, ft KEY A Local reference B Machine reference P Point on machine R Vector Figure 7.-Local and machine reference frames and position vectors. 1 i P(: \ R,/ Yi \ -«— Xj— - P(r,0) RECTANGULAR COORDINATE SYSTEM POLAR COORDINATE SYSTEM KEY P Point R Vector Figure 8.-Rectangular and polar coordinate components. The relationship between the rectangular and polar coor- dinate systems was x = r cos(0) and y = r sin(0), with the corresponding inverses r = (x 2 + y 2 ) 1 / 2 and 6 = tan'^y/x). Finally, in matrix notation, the rectangular form was shown as or "r cos(0)l _r sin(0)J COORDINATE TRANSFORMATION OF REFERENCE FRAMES Since the mining machine reference frame moved with respect to the local reference frame with three degrees of freedom, coordinate transformations were required to transform position vectors expressed in the mining ma- chine coordinate system into position vectors relative to the local reference frame. This transformation is standard procedure in robot kinematics. The process was almost trivial since there were only two reference frames in this case. Consider again figure 7. If only the position vectors A R t and B R 2 were known, elementary vector algebra would have dictated that the sum of the two would equal A R 3 . That was not true in this case, however, since B R2 was not measured in the local coordinate frame. If a transform %T was available to convert the position vector from the ma- chine coordinate frame into the local coordinate frame by a rotation operation of V> degrees, then the resulting equa- tion would be If and V R ^D _ An , ArpBwj K 3 ~ K l + B 1 K 2- (17,-18) in local reference frame, (1) R 2 = (20,10) in machine reference frame, V> = heading of 20° in local reference frame, then it could be shown that R 3 = (32,-2) in local reference frame. Letting the rotation transform cos(V>) -sin(V>)l sin(V>) cos(V»)J (2) and substituting the known values into equation 1 yielded [-£] sin(20°) cos(20°)_ |~cos(20°) -sin(20°)l [20l 32 -2 m + [3 It should be noted here that i R,. This is the standard method for performing kinematic analysis of mechanisms in a plane, which was used during the development of the closed-form solution of the posi- tion and heading algorithm. CLOSED-FORM SOLUTION The standard solution method for kinematic problems involving more than one reference frame (coordinate transformation analysis of a planar mechanism) was ap- plied to this position and heading system. While the cho- sen configuration of the position and heading system made the problem simple enough to handle by elementary trig- onometric solution methods, the coordinate transformation analysis method was determined to be general enough to be applied to all possible configurations of the linear trans- ducers, whereas of the infinite number of configurations, only a finite number could be addressed utilizing standard trigonometric techniques. Before a closed-form solution for the position and heading algorithm could be performed, the linear position transducers were configured for use and all the position vectors of interest were defined (fig. 9). The first step in developing the position and heading algorithm was to write the loop equations for a three-transducer solution and to determine if the equations could be reduced into a direct- ly solvable form. The next step was to determine the closed-form solution. A four-transducer solution was then found from the closed-form solutions for the four three- transducer solutions. Finally, a strategy was developed in which the three-transducer solutions were used to test the validity of the four-transducer solution and to provide a backup solution should one transducer provide erroneous data. Loop Equation Development Four loop equations were written from the configura- tion shown in figure 9, as follows: eliminated by subtracting equations 5 and 4 from equation 3, yielding A Rj + A R 4 ArrBn An ■ B 1 *9 ." K ll = o, (3) A R X + % ArpBwJ Awj - gl K 1Q - K n = 0, (4) A R 2 + A R, ArpBp An ■ gl Kg - K n = 0, (5) A R 2 + A R 7 ArpBn Awj ■ B 1 K 10" K ll = 0. (6) and These four vector loop equations represented eight scalar equations and seven unknowns. Thus, the four-transducer solution was overconstrained, which was to be expected since the fourth transducer was added to provide redun- dancy. Thus, the three-transducer solutions are derived first using the four combinations of three equations rep- resenting six scalar equations and six unknowns. Three-Transducer Solution The three-transducer solution derived here used vector loop equations 3, 4, and 5 (i.e., ignored data from trans- ducer 4). To reduce the order of the equations, A R U was and A R : - A R 2 + A R 4 - A Rg = (7) X-^-^T^ + ^Xo-O. (8) Entry Figure 9.-Configuration for position and heading algorithm. 10 Then, by substituting K 2 - Kj - K3 (9) and Kq - *^io = ^8> (10) equations 7 and 8 became A R 4 - A K 3 - A K6 =0 (11) and A R 4 - A R5 - ^T B Rg = 0. (12) Vector equation 11 had two scalar equations and only two unknowns (9 4 and 9 6 ) and could, therefore, be solved directly in the following manner: Letting and a = 8 = 3 + n (13) (14) The scalar equations were separated and the trigonometric identities cos(a + n) = -cos(a) and sin(a + n) = -sin(a) are used. (Hereafter, cd and s0, the kinematic shorthand for cos(0) and sin(0), will be used.) H + 7r-0{ Separating the scalar equations rr 4 c(0 8 + v - ol + M* + h)] [r 4 s(9 8 + rf> - e)J Lr 5 s(i/ + 8 )J (22) (23) N-sV-1 kcflgl = [SV> C0J L r 8 stf sJ (24) The x scalar equation component was solved for 1/ "r 8 c(0 8 + +) - T A d s + * - V = ±cos -9* (25) and v was substituted into the y scalar equation compo- nent of equation 24: r 4 s(0 8 + + - + (4 - (r 8 c(0 8 + V) - r 4 c(0 8 + V - 0) 2 ) V2 - r 8 s(0 8 + *) = 0. (26) The square root was isolated on the left side of the equa- tion and both sides were squared yielding r 2 - r 2 c 2 (0 8 + r(>) + 2r g r 4 c(0 8 + i>)c(9 8 + V - -2„2 2.2/ i^(* 8 + v - = n & Xh + v» - 2„2, 2r 4 r 8 s(^ 8 + ^)s(0 8 + V - + W(9 8 + j>). (27) 11 Equation 27 reduced to r 5 = r 4 + r 8 " 2r 4 r 8 ce > (28) which was essentially an application of the law of cosines to a triangle whose sides are known. Thus, since r 2 , 2 2 r 4 + r 8 * r 5 € = ±COS 2r 4 r 8 (29) equation 22 was solved to find the mining machine heading. V» 4 = ±e (The sign to be used for e cannot be determined unless both the initial configuration and the sign of the angular velocity of the mining machine are known.) The position of the mining machine could then be de- termined from equation 3 to have components r^ii + r r 4 c ^i -Mv^i iv^i =r x iii: od therefore, x n = r l c6 l + r 4 c0 4 - r 9 c(0 9 + V> 4 ) (32) and y n = r l s6 1 + r 4 s0 4 - r 9 s(0 9 + V 4 ). (33) In summary, the three-transducer solution, which did not use transducer 4 data (i.e., using loop equations 3, 4, and 5), for the mining machine's heading was given by using equations 20, 21, 29, and 30. The mining machine's position was given by equations 32 and 33. In a similar manner, in a three-transducer solution, which excludes transducer 3 (i.e., using loop equations 3, 4, and 6), the mining machine's heading is given by -l and 7 = ±cos e 5 = e 3 ± 7 , v = ±cos" V>-> = ft - ±u - r 2 , 2 2 1 r 3 + r 5 - r 7 (34) . 2r 3 r 5 . (35) r 2 , 2 2 l r 5 + r 8 * r 4 (36) . 2r 5 r 8 J (37) The mining machine's position was determined by equation 4 to have components x ll = r i c ^i + r 5 c *5 - r io c (*iO + ^3> (38) and y n = t 1 s6 1 + r 5 s0 5 - r lo s(0 lo + V> 3 ). (39) Next, in a three-transducer solution, which excludes transducer 2 (i.e., using loop equations 3, 5, and 6), the mining machine's heading was given by (30) and _! r 2 , _2 2 - r 3 + r 6 " r 4 (40) — ±cos I 2r 3 r 6 - = 6 3 - n T S, (41) j " r 6 + r l - r 7 " (4 2 ) — ± cos - 2r 6 r 8 - = ±r,-0 s + *6- (43) The mining machine's position was determined by equation 5 to have components and x n - r 2 c0 2 + r 6 c0 6 - r 9 c(0 9 + V> 2 ) (44) y n = r 2 s0 2 + r 6 s0 6 - r 9 s(0 9 + V 2 )- (45) Finally, in the last three-transducer solution, which excludes transducer 1 (i.e., using loop equations 4, 5, and 6), the mining machine's heading was given by -1 and = ±cos = 3 - n T yS, f = ±cos" V>! = n- ±r- r 2 ,2 2i r 3 + r 7 r 5 (46) . 2r 3 r 7 . : A (47) " r 7 + r 8 * r 6 ' (48) - 2r 7 r 8 - - e s + e 7 . (49) The mining machine's position was determined by equation 4 to have components x ll = r 2 c ^2 + x -f- B l - r io c (^io + V>i) (50) and y n = r 2 s0 2 + r 7 s0 7 - r lo s(0 lo + ^i)- (51) Four-Transducer Solution The four-transducer solution was derived through equa- tions already derived in the three-transducer solution sec- tion. Many viable combinations of these equations could perform the desired task, but these combinations relied on all four transducers yielding identical results. For the purpose of this derivation, equations 20, 21, 46, and 47 were used. 12 With 6 4 and 7 known from the derivations in the pre- vious section, the vector loop equation that described this four-transducer solution was A R X + A R 4 - £1% + bT^IO " AR 7 " AR 2 = °> ( 52 ) which by substituting equations 9 and 10, became R ArrB 4 " B T°Rg - ^R 7 % = o. (53) The only unknown in this equation was V>, but to avoid the ambiguities that arise from the use of the inverse sine or cosine, information from both scalar equations was taken into consideration. This was done by isolating the un- known quantity on the left side of the equation, dividing the y scalar component equation by the x scalar compo- nent equation, and solving for the unknown variable, re- sulting in V> = tan" 1 TaSVa " T-jSu-j - ToSP-j r 4 c^ 4 -r 7 c^ 7 -r 3 c^ 3 . (54) Of course, to determine the correct heading of the ma- chine, the quadrant information yielded by the x and y scalar equation components was utilized to eliminate am- biguities caused by the inverse tangent function. The mining machine's position was calculated in the standard manner, yielding x n = T i c h + r 4 c ^4 " r 9 c (^9 + V>) (55) and y n = t 1 &0 1 + r 4 s0 4 - r 9 s(0 9 + \j>). (56) Testing and Backup Strategy When measurements of any kind are performed, it is usually more important to know the reliability of the data than the actual data itself. If the data are not known to be reliable, the data are useless, since in mining operations, safety of personnel is of extreme importance. Thus, the redundancy provided by an additional sensor supplied a means of determining the accuracy of the data furnished. The four-transducer solution would always yield the cor- rect position and heading, provided that the constraints of operation were met and the transducers were furnishing reliable data. In order for the guidance system to operate properly, it was assumed that the attachment points of the transducers on the mining machine would always remain on the posi- tive x side of the imaginary line connecting the transducer cable attachment points (fig. 9). If this constraint was not met, the heading and position data developed would be inaccurate. The best strategy, therefore, was to keep the transducer and cable attachment points separated by a suitable distance. To determine if one of the transducers was furnishing unreliable data, a continuous check could be kept by cross comparing the four three-transducer solutions. If only one transducer was furnishing unreliable data, then the error could be determined to exist by the fact that all four so- lutions would disagree. If more than one transducer was furnishing inaccurate data, then it was possible that no error could be identified. While the possibility of more than one transducer failing at the same time in a manner yielding no identifiable error is small, it was theoretically possible in a roof fall or mechanical obstruction of mul- tiple cables. The transducer data could be checked in such cases for unrealistically large rates of change in length, for underrange errors, and for overrange errors. Since each three-transducer solution method yielded two possible positions and headings, in order for the three-transducer solutions to track the four-transducer solution, knowledge of the initial configuration of the mining machine and the subsequent changes in sign of the three-transducer solution equations was required. The initial configuration of the mining machine was assumed to be similar to that shown in figure 9 and was to be cor- roborated by the four-transducer solution. This config- uration allowed the initialization of the signs in the head- ing equations V» 4 = ae '4' and where V\ V> 3 = 7r - ai> - 8 + 5 , V-2 = b»? - e g + e 6 , 0i = * - bf - 8 + e 7 , (57) (58) (59) (60) and = heading determined by a three-transducer solution, which excludes transducer i, = sign of e and v since both change signs at same time, = sign of r\ and £* since both change signs at same time, to be such that both a and b were positive in the initial position and never change signs at the same time. Chang- ing the signs of a and b could be performed by keeping track of both headings for each solution and choosing the appropriate sign and/or by noting the transition of the solution equations and using the sign of the angular veloc- ity of the mining machine (u>). Determining the signs of a and b using the angular velocity of the mining machine was done in the following manner (fig. 9): if [(u^O) and (w<0)] or [(u~ir) and (w>0)], then a = -1, if [(^=0) and (w>0)] or [(y~ir) and (w<0)], then a = 1, if [(»?M)) and (w<0)] or [(»?~7r) and (w>0)], then b = -1, and if [(f?~0) and (w>0)] or [(r?~7r) and (w<0)], then b = 1. 13 Both of these methods relied on information external to the three-transducer solutions to determine the signs of a and b. If a transducer did fail, the error could be determined using equations 57 to 60 instead of equations 30, 37, 43, and 49. ERROR ANALYSIS Once the position and heading algorithm had been designed to reduce the possibility of mathematical errors induced by the configuration of the position and heading system, it was necessary to determine the source of other possible errors. These errors were introduced into the position and heading calculation through the inherent inaccuracies in the linear position transducers, the analog- to-digital (A/D) converter, and the sample rate. The effects of these errors on the position and heading calculation depended to a large extent on the position and heading of the mining machine. While the determination of the maximum error required either an exhaustive trig- onometric proof involving the taking of the partial deriv- atives of the governing equations or a numerical methods analysis, a preliminary numerical analysis utilizing a com- puter model of the position and heading system indicated that the overall position and heading system error was on the order of 0.25 ft and 3°, respectively. While error prop- agation could not be easily derived, the errors introduced, however, could be quantized. LINEAR TRANSDUCERS Linear transducers of the length necessary to perform the necessary tasks in the position and heading system (approximately 750 in) had a stated accuracy of 0.1 pet full scale, but typically had an accuracy of 0.05 pet full scale. Factory calibration information on five Rayelco 5 P-750A linear position transducers selected for use in the position and heading system can be found in appendix A. A pre- liminary analysis was performed on these calibration data to determine the calibration equations for each linear position transducer and for all five combined. The fre- quency distribution of error from the combined calibration equation was determined and appeared to be Gaussian in nature. (The calibration equations and error frequency distribution can be found in appendix B.) The only other accuracy concern for the linear position transducers was the effect of the dust and accumulations of dust on the transducer's wire itself. If this was discovered to be a problem, a submersible type of linear transducer could be used with water to flush the interior of the cable takeup housing. ANALOG-TO-DIGITAL CONVERTER The A/D conversion circuitry on the Intel remote con- trol board (iRCB) 44/20A used to implement the position and heading system had an accuracy of about 0.035 pet full scale. These errors were due in part to nonlinearity, inher- ent quantizing errors, gain error, zero error, noise, and sample and hold dynamic error, and were found in the iRCB 44/20A's hardware user's manual. The specifica- tions for the A/D converter can be found in appendix C. SAMPLE RATE Because of the processing capability available to the position and heading system through the Intel 8051 micro- processor running in a distributed control executive (iDCX 51) real-time multitasking environment, the number of position and heading calculations performed was on the order of 5 Hz. Considering, for example, that a Joy 16CM mining machine can attain speeds of up to 0.54 ft/s and angular velocities of up to 3.23°/s, 6 the sampling errors incurred of 0.1 ft and 0.6° could be considered acceptable in a mining environment. If greater accuracy is desired, a slower speed could be used or the machine could be stopped altogether. CONCLUSIONS A system to determine the position and heading of a mining machine during maneuvers required in the face area of an operating mine section was described. This system utilized four commercially available linear position transducers to obtain the required sensory data. The sys- tem employed sensor redundancy in its sensor fusion scenario to increase the system reliability. The derivation of the position and heading algorithm was presented and a preliminary error analysis was performed, which showed that the position and heading system had sufficient ac- curacy for mining application. Reference to specific products does not imply endorsement by the U.S. Bureau of Mines. Sammarco, J. J. Closed Loop Control for a Continuous Mining Machine. BuMines RI 9209, 1988, 17 pp. 14 APPENDIX A.-LINEAR POSITION TRANSDUCER CALIBRATION DATA (Range, 750 in; cable tension, 24 oz; potential resistance, 500 ohms; excitation, 10 V dc) Calibration step Travel, in Output, V Ideal, V Delta, mV TRANSDUCER 1 1 0.00 2 187.50 3 357.00 4 562.50 5 750.00 1 0.00 2 187.50 3 357.00 4 562.50 5 750.00 1 0.00 2 187.50 3 357.00 4 562.50 5 750.00 1 0.00 2 187.50 3 357.00 4 562.50 5 750.00 1 0.00 2 187.50 3 357.00 4 562.50 5 750.00 '0.05 pet full scale, 1.29 mV/(V/in) position. ^.OS pet full scale, 1.30 mV/(V/in) position. 3 0.07 pet full scale, 1.29 mV/(V/in) position. 4 0.04 pet full scale, 1.30 mV/(V/in) position. 0.035 2.463 4.889 7.314 9.747 0.035 2.463 4.891 7.319 9.747 TRANSDUCER 2 2 0.028 2.462 4.891 7.321 9.758 0.028 2.460 4.893 7.325 9.758 TRANSDUCER 3 3 0.034 2.454 4.887 7.314 9.743 0.034 2.461 4.888 7.316 9.743 TRANSDUCER 4 4 0.030 2.461 4.898 7.339 9.771 0.030 2.465 4.901 7.336 9.771 TRANSDUCER 5 3 0.026 2.448 4.877 7.298 9.731 0.026 2.452 4.878 7.305 9.731 15 APPENDIX B.-CALIBRATION EQUATIONS AND ERROR FREQUENCY DISTRIBUTION Output (Qo), in volts, for the five linear position trans- q . 0.02780 ducers, as described in appendix A, was used to calculate Qi = ± 3.201727 in, the slopes, intercepts, and standard deviations of the cali- 0.012992 bration equations for each of the five linear position trans- ducers and for all five combined. The general calibration with maximum error = 0.246305 in. equation was determined to be The calibration equation for transducer 5 was determined Qo - 0.02944 t0 be Qi = ± 2.050786 in, °- 012959 Qo - 0.02580 Qi = ± 2.772926 in, with maximum error = 1.694514 in. 0.012940 The calibration equation for transducer 1 was determined with maximum error = 0.030911 in. to be Qo - 0.03460 Qi = ± 0.89625 in, 0.012947 with maximum error = 0.239444 in. The error frequency distribution for the general calibration equation is as shown in figure B-l. The calibration equation for transducer 2 was determined to be Qo - 0.02820 Qi = ± 1.014279 in, 0.012970 with maximum error = 0.223591 in. The calibration equation for transducer 3 was determined to be Qo - 0.03080 Qi = ± 1.120895 in, 0.012948 with maximum error = 0.35526 in. The calibration equation for transducer 4 was determined to be 6 5 4h > o z lit => 3 o LlI ft i ■ i ■ i ■ / V\ * \ / / / / /. / / V $ * ', — -T i 1 1 1 1 : I I i 1 -2.0 -1.5 -1.0 -0.5 0.5 ERROR, in 1.0 1 .5 2.0 Figure B-1 .-General calibration equation error frequency distribution. 16 APPENDIX C.-CONVERSION SPECIFICATIONS FOR INTEL REMOTE CONTROL BOARD 44/20A ANALOG-TO-DIGITAL CONVERTER Linearity Differential linearity Inherent quantizing error System accuracy: Gain = 1 Gain = 10 Gain = 100 Gain = 500 Gain error Zero error Channel crosstalk Noise (A/D converter) Instrumentation amplifier settling time: Gain = 1 Gain = 10 Gain = 100 Gain = 500 A/D conversion time: All gains Maximum A/D throughput: Gain = 1 Gain = 10 Gain = 100 Gain = 500 Sample and hold feedthrough attenuation Sample and hold dynamic error Offset drift (gain = 1) Input offset drift multiplied by gain Gain drift: Gain = 1 Gain = 10 Gain = 100 Gain = 500 Monotonocity Common mode rejection ratio: Gain = 1 Gain = 500 Amplifier input noise Channel-to-channel input voltage error . . Resolution FSR Full-scale reading. LSB Least significant bits. RMS Root mean square. To within ±0.75 LSB. To within ±0.75 LSB. ±0.5 LSB. To within ±0.035 pet FSR. To within ±0.05 pet FSR. To within ±0.07 pet FSR. To within ±0.15 pet FSR. Adjustable to zero. Do. -80 dB at 1 kHz. 0.2 LSB RMS. 20 /is. 20 lis. 100 ms. 100 its. 30 lis. 20,000 samples per second. Do. 7,500 samples per second. Do. -80 dB at 1 kHz. ±0.75 LSB. 100mV/°C. 3 mV/°C. 32 ppm of FSR per degree Celsius. 40 ppm of FSR per degree Celsius. 65 ppm of FSR per degree Celsius. 75 ppm of FSR per degree Celsius. Monolithic, 0° to +60° C. 70 dB at 60 Hz, 1 kohm unbalance. 100 dB at 60 Hz, 1 kohm unbalance. 2 ilV RMS. ±40 /iV. 12 bits. 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