Abstract:
A system and methods are disclosed for providing the location of a tunnel boring machine (TBM) by establishing of a plurality of known locations or “monuments”; from these monuments located at least on, over or within the TBM&#39;s start point, known in the art as a “pit”. The present invention provides among other things an integrated navigation system that provides real-time parametric guidance information to the TBM, relative to the tunnel origin, past course and current trajectory, while simultaneously employing a non-contact measuring system in concert with said origin and course information for the final provision of an as-built map of tunnel dimensions and centerline.

Description:
COPYRIGHT NOTICE 
       [0001]      
         [0000]    
       
         
               
               
               
               
             
           
               
                   
               
             
             
               
                 Dan Alan Preston 
                 US Citizen 
                 US Resident 
                 Bainbridge Island, 
               
               
                   
                   
                   
                 WA 
               
               
                 Joseph David Preston 
                 US Citizen 
                 US Resident 
                 Bainbridge Island, 
               
               
                   
                   
                   
                 WA 
               
               
                 Marc A. Derenburger 
                 US Citizen 
                 US Resident 
                 Bremerton, WA 
               
               
                 Carin L. Douglass 
                 US Citizen 
                 US Resident 
                 Silverdale, WA 
               
               
                 Paul M. Peterson 
                 US Citizen 
                 US Resident 
                 Bremerton, WA 
               
               
                 Kyle A. Yeats 
                 US Citizen 
                 US Resident 
                 Port Orchard, WA 
               
               
                   
               
             
          
         
       
     
         [0002]    Contained herein is material that is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the United States Patent and Trademark Office patent file or records, but otherwise reserves all rights to the copyright whatsoever. The following notice applies to the software, screenshots and data as described below and in the drawings hereto and All Rights Reserved. 
       FIELD OF THE INVENTION 
       [0003]    This invention generally relates to the establishment of a plurality of known locations known in the art as “monuments”; from these monuments located at least on, over, or within the tunnel boring machine&#39;s (TBM) start point, known in the art as a “launch start”. The present invention provides among other things an integrated navigation system that provides real-time parametric guidance information to the TBM, relative to the tunnel origin, past course and current trajectory, while simultaneously employing a non-contact measuring system in concert with said origin and course information for the final provision of an as-built map of tunnel dimensions and centerline. 
       BACKGROUND OF THE INVENTION 
       [0004]    Tunnel boring machines (TBM) are used to excavate circular cross section tunnels through a variety of soil and rock strata. As tunnels are bored regardless of geology, it is imperative the TBM and resulting excavating tunnel stay on the design alignment within the mandated tolerances. It may be very costly if 1. The tunnel veers off alignment wandering outside of the client&#39;s purchased Right-of-Way (ROW), 2. The TBM encounters unanticipated geological features or utilities in urban settings, or 3. The tunnel alignment and correction curves exceed the tight tolerances required for sustaining the dynamic envelope of train tunnels and highway tunnels. In order to avoid negative impacts on the TBM, the tunnel surroundings, or underground utilities, it is imperative that TBM be precisely locatable and guided when boring through the earth. 
         [0005]    In addition to the need for precise navigation of the TBM, the tunnel itself must be mapped. The need for mapping in tunnels is twofold. Firstly, an as-built map of the tunnel is needed to compare finished tunnel dimensions to plan requirements. Secondly, the as-built map can be maintained after the tunnel is completed and used as a baseline measurement for reference during subsequent surveys to observe changes in tunnel geometry over time. 
         [0006]    The present methods of TBM guidance primarily use lasers and conventional surveying techniques. Lasers and transit theodolites, originating from the tunnel entrance, are relayed through a network of fixed monuments on the tunnel walls and used to identify the position and attitude of the TBM relative to the desired design location. The precision in identifying the exact location (Northing, Easting, Elevation) of this progressive series of monuments and their growing error as the tunnel extends can lead to improper alignment of the tunnel or missing the end target within the stipulated tolerance. This conventional system using sighted theodolites to advance the monuments used by the laser guidance systems is often adversely affected by error inherent to accuracy of the measuring instruments, light refraction, angle of incidence, and reception. From the final measured monument near the TBM, a servo theodolite with distance measuring capability, along with inclinometers on the TBM, are used to identify the axis of the TBM as well as monitor TBM pitch (up and down), yaw, and rotation depending on their installation orientation. The theodolite locates and reports to the underlying guidance computer prisms attached to the TBM with a known orientation and location relative to the reference frame of the TBM. The motorized station can measure their location as the TBM bores the tunnel. The output from the inclinometers and updated target locations is relayed to a central processing unit which outlines the path for the TBM. Monitoring of TBM vertical alignment is derived from the same methods of angle and distance measurement. The series of monuments affixed to the tunnel wall as the TBM advances is measured for elevation using wire line water level instruments to minimize the accumulation of error relative to elevation. Gyroscopes may also be used to monitor the yaw of the TBM, verified by a surveyor. 
         [0007]    The present state of tunnel mapping utilizes a two-step method. Firstly, the mapping positions are precisely located in reference to a known point outside of the tunnel. This is accomplished using a theodolite measurement device. If the tunnel curves, mirrors are used to reflect the beam, and the mirrors&#39; locations are measured by the laser measurement device. Each of these mirrors induces additional error in the final measurement of the mapping positions. With the location and orientation of the mapping stations known, the tunnel walls are then measured at several locations with respect to this position. These measurements are typically done using reflector-less laser measurement system; however, other touch-less measurement systems, such as Electronic Distance Measurements (EDM), may be used to measure the distance to the tunnel walls. 
         [0008]    The process of establishing the mapping locations and obtaining measurement is repeated until the entire tunnel has been measured. The distance measurements are then associated with their respective locations to generate a three dimensional map of the tunnel. This process is costly, time-consuming, and labor-intensive, requiring cessation of any work and traffic in the tunnel until survey completion. 
         [0009]    What is needed is an integrated navigation system that provides real-time parametric guidance information to the TBM, relative to the tunnel origin (hereinafter “the pit”), past course, and current trajectory, while simultaneously employing a non-contact measuring system in concert with said origin and course information for the final provision of an as-built map of tunnel dimensions and centerline. The pit is a known point within the earth-centered-earth fixed global positioning system (GPS), and at least one of GPS retransmission and time modulated wireless triangulation architectures provide availability of positioning signals in the otherwise unavailable underground environment of a newly excavated tunnel. As the TBM proceeds along its excavation heading, a vehicle such as a rubber wheeled vehicle or a locomotive delivers ring assemblies, fabricated on-site in the pit to support the recently excavated portion of the tunnel. The constrained curvilinear path, also known as the design centerline, from pit to TBM is regularly traversed by the locomotive which is, in current systems, employed for transport of ring assemblies and muckout. 
       DESCRIPTION OF RELATED ART 
       [0010]    In a discussion of prior art, European patent application Ser. No. EP20010304645 filed May 25, 2001, titled SELF-CONTAINED MAPPING AND POSITIONING SYSTEM UTILIZING POINT CLOUD DATA generally describes a self-contained mapping and positioning system for underground mining that is capable of mapping the topography of a region, such as a mine tunnel, and further being able to use the mapped data to determine the position of an object, such as a mining vehicle, within the mine tunnel. 
         [0011]    The method described in European patent application Ser. No. EP20010304645 provides only the position of the object, whereas the present invention incorporates positioning as well as automatic course correction as determined by a pre-established path. Furthermore, the present invention includes permanent monuments to be used in post-boring surveys to evaluate changes in tunnel geometry. 
         [0012]    In a discussion of prior art, U.S. patent application Ser. No. 08/304,858 filed Sep. 13, 1994, titled GUIDANCE SYSTEM AND METHOD FOR KEEPING A TUNNEL BORING MACHINE CONTINUOUSLY ON A PLAN LINE generally describes a guidance system and method for keeping a TBM continuously on a plan line. The guidance system requires no machine operator calculations and provides the boring machine operator with a graphic display of past, present, and projected positions of the boring machine from a horizontal and vertical perspective. The system uses a laser beam transmitter placed to the rear of the TBM along with a front opaque target with a horizontal and vertical cross-hair and a rear transparent target with a horizontal and vertical cross-hair. The front and rear targets are disposed on the front and the rear of the boring machine. Also, an on-board programmable computer is installed on the boring machine for imputing data as to horizontal offset and vertical offset readings from the front and rear targets as the boring machine advances forward. Typically the boring machine moves forward in increments of four feet with offset readings taken by the operator after each increment. The offsets are measured in feet up to two decimal places with the readings based on measured positions being wither right or left of the vertical cross-hair and above or below the horizontal cross-hair of the front and rear targets. Further, the on-board computer is programmed to store and provide a laser alignment check for verifying laser setup information and to graphically display alignment errors during a change in the setup of the laser beam transmitter by a survey crew. 
         [0013]    The device described in U.S. patent application Ser. No. 08/304,858 employs a series of lasers to project the path which are prone to error inherent to light refraction, angle of incidence, and reception. The present invention provides the TBM with data via the locomotive, combined with an on-board INS to provide the TBM with the current orientation, direction, and position to compile the projected path and compare with the desired path. 
         [0014]    In a discussion of prior art, U.S. Pat. No. 3,498,673 filed Feb. 19, 1968, titled MACHINE GUIDANCE SYSTEM AND METHOD generally describes a TBM disposed within a tunnel and provided with a guidance system comprising a laser projection unit fixedly supported by a wall of the tunnel and directing its beam onto a mirror-like reflector mounted on the machine, whereby the reflector provides a reflection of the beam on a target also mounted on the machine. The tunnel boring apparatus is steered to maintain the reflection at a predetermined location on the target. 
         [0015]    The method described in U.S. Pat. No. 3,498,673 aligns external lasers with the desired path and is sent through the TBM which is steered such that the TBM keeps the laser within a designated area. Laser guidance systems are prone to error inherent to light refraction, angle of incidence, and reception. The present invention utilizes a locomotive with an INS to position the TBM continuously; this position is then compared to the desired path as programmed into the TBM to provide the TBM with a path that needs to be followed to match the desired path. 
         [0016]    In a discussion of prior art, European patent application Ser. No. EP20030250157 filed Jan. 10, 2003, titled METHOD AND APPARATUS FOR SURVEYING THE GEOMETRY OF TUNNELS generally describes a method and apparatus for surveying the geometry of tunnels comprising measuring the position of a tunnel surface relative to an absolute three-dimensional coordinate system, using at least one reflector-less distance sensor mounted for orientation in three dimensions and calculating a deviation from a predefined geometry for the surface and displaying said deviation in real time. 
         [0017]    The method and apparatus described in European patent application Ser. No. EP20030250157 is well suited to the mapping of tunnels as well as post-boring surveys for maintenance of the tunnel but is not well suited for as-built mapping during the tunneling process. The present invention utilizes permanent monuments for long-term tunnel mapping, utilizes an INS system that is integrated with the self-contained mapping system, and communicates the combined parametric information to the TBM to provide navigational guidance for the TBM. 
         [0018]    So as to reduce the complexity and length of the Detailed Specification, and to fully establish the state of the art in certain areas of technology, Applicant(s) herein expressly incorporate(s) by reference all of the following materials identified in each numbered paragraph below. The incorporated materials are not necessarily “prior art” and Applicant(s) expressly reserve(s) the right to swear behind any of the incorporated materials. 
         [0019]    U.S. Pat. No. 6,707,424 Integrated Positioning System and Method 
         [0020]    U.S. Pat. No. 8,417,490 System and Method for the Configuration of an Automotive Vehicle with Modeled Sensors 
         [0021]    Inertial Navigation, by Kevin J. Walchko, University of Florida, Gainesville, Fla., and Dr. Paul A. C. Mason, NASA Goddard Space Flight Center, Greenbelt Md. Published 2002. 
         [0022]    Real-Time Tunnel Boring Machine Monitoring: A State of the Art Review, by Michael A. Mooney, Bryan Walter, Christian Frenzel. Colorado School of Mines, Golden Co. Published 2012 
         [0023]    Design and Field Testing of an Autonomous Underground Training System, by Joshua A. Marshall and Timothy D. Barfoot. Published Dec. 13, 2007 
         [0024]    Applicant(s) believe(s) that the material incorporated above is “non-essential” in accordance with 37 CFR 1.57, because it is referred to for purposes of indicating the background of the invention or illustrating the state of the art. However, if the Examiner believes that any of the above-incorporated material constitutes “essential material” within the meaning of 37 CFR 1.57(c)(1)-(3), applicant(s) will amend the specification to expressly recite the essential material that is incorporated by reference as allowed by the applicable rules. 
       SUMMARY OF THE INVENTION 
       [0025]    Although the best understanding of the present invention will be had from a thorough reading of the specification and claims presented below, this summary is provided in order to acquaint the reader with some of the new and useful features of the present invention. Of course, this summary is not intended to be a complete litany of all of the features of the present invention, nor is it intended in any way to limit the breadth of the claims, which are presented at the end of the detailed description of this application. 
         [0026]    The present invention employs the regular traverse of the locomotive between the pit and the TBM as the method for accumulation of parametric guidance information to the TBM, relative to the tunnel origin, past course, and current trajectory; the means of transmission of said information to the TBM for navigation; and the means by which a non-contact measuring system is deployed in concert with said origin and course information for the final provision of an as-built map of tunnel dimensions and centerline. 
         [0027]    To implement the present invention, an integrated system of devices is installed in the pit, on the locomotive, on the TBM, and on the tunnel ring assemblies. Installed in the pit are GPS receivers and GPS re-transmitters. Installed on the locomotive are GPS receivers for the retransmitted signals within the pit, a fault-tolerant inertial navigation system (FTINS) that obtains course information, a self-contained mapping system, a central processing unit (CPU), and a wireless transmitter for information transfer to and from the TBM. Installed on the TBM are a transceiver to receive transmitted origin and course information, a microprocessor and attitude heading reference system (AHRS) for calculation of heading, and various input/output devices. Included in the present invention are permanent monuments affixed to tunnel ring assemblies which are utilized in concert with the aforementioned self-contained mapping system installed on the locomotive, as well as being available for post-boring surveys of tunnel geometry. 
         [0028]    Aspects and applications of the invention presented here are described below in the drawings and detailed description of the invention. Unless specifically noted, it is intended that the words and phrases in the specification and the claims be given their plain, ordinary, and accustomed meaning to those of ordinary skill in the applicable arts. The inventors are fully aware that they can be their own lexicographers if desired. The inventors expressly elect, as their own lexicographers, to use only the plain and ordinary meaning of terms in the specification and claims unless they clearly state otherwise and then further, expressly set forth the “special” definition of that term and explain how it differs from the plain and ordinary meaning. Absent such clear statements of intent to apply a “special” definition, it is the inventors&#39; intent and desire that the simple, plain and ordinary meaning to the terms be applied to the interpretation of the specification and claims. 
         [0029]    The inventors are also aware of the normal precepts of English grammar. Thus, if a noun, term, or phrase is intended to be further characterized, specified, or narrowed in some way, then such noun, term, or phrase will expressly include additional adjectives, descriptive terms, or other modifiers in accordance with the normal precepts of English grammar. Absent the use of such adjectives, descriptive terms, or modifiers, it is the intent that such nouns, terms, or phrases be given their plain, and ordinary English meaning to those skilled in the applicable arts as set forth above. 
         [0030]    Further, the inventors are fully informed of the standards and application of the special provisions of 35 U.S.C. §112, ¶6. Thus, the use of the words “function,” “means” or “step” in the Detailed Description or Description of the Drawings or claims is not intended to somehow indicate a desire to invoke the special provisions of 35 U.S.C. §112, ¶6, to define the invention. To the contrary, if the provisions of 35 U.S.C. §112, ¶6 are sought to be invoked to define the inventions, the claims will specifically and expressly state the exact phrases “means for” or “step for, and will also recite the word “function” (i.e., will state “means for performing the function of [insert function]”), without also reciting in such phrases any structure, material or act in support of the function. Thus, even when the claims recite a “means for performing the function of . . . ” or “step for performing the function of . . . ”, if the claims also recite any structure, material or acts in support of that means or step, or that perform the recited function, then it is the clear intention of the inventors not to invoke the provisions of 35 U.S.C. §112, ¶6. Moreover, even if the provisions of 35 U.S.C. §112, ¶6 are invoked to define the claimed inventions, it is intended that the inventions not be limited only to the specific structure, material or acts that are described in the preferred embodiments, but in addition, include any and all structures, materials or acts that perform the claimed function as described in alternative embodiments or forms of the invention, or that are well known present or later-developed, equivalent structures, material or acts for performing the claimed function. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0031]    A more complete understanding of the present invention may be derived by referring to the detailed description when considered in connection with the following illustrative figures. In the figures, like-reference numbers refer to like-elements or acts throughout the figures. The presently preferred embodiments of the invention are illustrated in the accompanying drawings, in which: 
           [0032]      FIG. 1  is a perspective view of the locomotive in the launch pit according to the preferred embodiment of the present invention, depicting the physical method for the accumulation of locomotive initial geo-location data at the launch pit and the retransmission of said data to the locomotive. 
           [0033]      FIG. 2  is a block diagram of the components contained within  FIG. 1 . 
           [0034]      FIG. 3  depicts an alternate embodiment using a robotic total survey station mounted in the launch pit and a system of survey reflectors. 
           [0035]      FIG. 4  is a block diagram depicting the components within  FIG. 3  which provide for the accumulation and transmission of initial geo-location data. 
           [0036]      FIG. 5  depicts an alternate embodiment using a robotic total survey station mounted on the locomotive and a system of survey reflectors in the launch pit. 
           [0037]      FIG. 6  is a block diagram depicting the components contained within  FIG. 5  which provide for the accumulation and transmission of initial geo-location data. 
           [0038]      FIG. 7  depicts an alternate embodiment using a docking station and alignment marks at surveyed locations within the launch pit. 
           [0039]      FIG. 8  is a block diagram depicting the components contained within  FIG. 7  which provide for the accumulation and transmission of initial geo-location data. 
           [0040]      FIG. 9  is a block diagram of the components contained within the locomotive. 
           [0041]      FIG. 10  is a block diagram illustrating the components contained within the TBM. 
           [0042]      FIG. 11  depicts the process elements associated with the utilization of geo-location input data. 
           [0043]      FIG. 12  depicts the process elements associated with the utilization of updated geo-location information. 
           [0044]      FIG. 13  is a block diagram illustrating the components contained within the fault tolerant inertial navigation system. 
           [0045]      FIG. 14  depicts the process elements accomplished within the fault tolerant inertial navigation system componentry. 
           [0046]      FIG. 15  is a block diagram illustrating the components utilized for the self-contained mapping system. 
           [0047]      FIG. 16  depicts the process elements accomplished by the self-contained mapping system. 
           [0048]      FIG. 17  is a block diagram of the SACore IMM for Automatic Guidance of a TBM. 
       
    
    
     DETAILED DESCRIPTION 
       [0049]    In the following description, and for the purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the various aspects of the invention. It will be understood, however, by those skilled in the relevant arts, that the present invention may be practiced without these specific details. In other instances, known structures and devices are shown or discussed more generally in order to avoid obscuring the invention. In many cases, a description of the operation is sufficient to enable one to implement the various forms of the invention, particularly when the operation is to be implemented in software. It should be noted that there are many different and alternative configurations, devices and technologies to which the disclosed inventions may be applied. The full scope of the inventions is not limited to the examples that are described below. 
         [0050]    In the following examples of the illustrated embodiments, references are made to the accompanying drawings which form a part hereof, and in which is shown by way of illustration various embodiments in which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural and functional changes may be made without departing from the scope of the invention. 
         [0051]      FIG. 1  illustrates the preferred embodiment of the physical method for the accumulation of locomotive initial geo-location data within the launch pit  105  and the retransmission of said data to the locomotive  100 . The locomotive  100  starts and ends each travel cycle through the tunnel  110  in the launch pit  105 . Surrounding the launch pit  105  are three or more geo-location and retransmission devices  200 . A locomotive mounted transceiver  115  receives and transmits the position data  225  ( FIG. 2 ). 
         [0052]      FIG. 2  describes the components illustrated in  FIG. 1 . The geo-location and retransmission devices  200  each contain at least one global positioning system (GPS) receiver  205 , microprocessor  210 , and transmitter  215 . These devices  200 , well known in the art of position determination and land survey techniques, are designed to receive a GPS signal  220  by which position data  225  is determined and relayed to a locomotive mounted transceiver  115  ( FIG. 1 ). 
         [0053]    These commercially available systems utilize two positioning and navigation systems in a single unit, the first is used within sight of earth-orbiting Global Navigation Satellite System (GNSS) satellites and the second in less than optimal GNSS locations. The locomotive  100  ( FIG. 1 ) in the launch pit  105  ( FIG. 1 ) will generally not be in line-of-sight of the earth-orbiting satellites. The locomotive mounted transceiver  115  ( FIG. 1 ) receives transmissions from a group of at least three ground-based beacons stationed at precise known locations, each of which transmits a distinct signal, including location information. The means of data transmission may include wireless protocols such as 802.11g, Bluetooth, time modulated ultra-wide band (TM-UWB) or other such wireless methods as may arise with developing technologies. 
         [0054]    Another embodiment of the present invention, illustrated in  FIG. 3  and described by  FIG. 4 , uses a pit mounted robotic total survey station  400  comprising a robotic mount  405 , a controller  410 , and a total station  415  (similar in method and construction to the Trimble Series 6), and a survey station transceiver  420 . The pit mounted robotic total survey station  400  is mounted along the perimeter of the launch pit  105  where it utilizes reflective monuments  310  outside of the launch pit  105  to determine its geo-location. A locomotive mounted reflective monument  305  is used to determine the location of the locomotive  100  relative to the known reflective monuments  310 . The survey station transceiver  420  contained within the pit mounted robotic total survey station  400  transmits position data  225  to the survey station transceiver  420  on the locomotive  100 . 
         [0055]    According to yet another embodiment of the present invention, illustrated in  FIG. 5  and described by  FIG. 6 , a locomotive mounted robotic total survey station  600 , comprising a robotic mount  405 , a controller  410 , and total station  415 . Within the method described by this embodiment, the locomotive mounted robotic total survey station  600  obtains the position of the locomotive  100  by triangulation with at least three reflective monuments  310  at surveyed locations within the launch pit  105 . As depicted in  FIG. 9 , the locomotive mounted transceiver  115  is bypassed with this embodiment and position data  225  is direct fed to self-contained mapping system (SCMS)  900  ( FIG. 9 ) onboard the locomotive  100 . 
         [0056]    A less technical yet viable alternative embodiment of the present invention is illustrated in  FIG. 7  and described by  FIG. 8 , wherein the locomotive  100  stops at a fixed docking station  700  installed at the terminal point of the track in the pit  105 . Alignment marks  705  may also be utilized in this embodiment either independently or in concert with the fixed docking station  700  to establish position data  225 . The surveyed origin point established by the docking station  700  is a direct input to central processing unit one (CPU 1 )  1520  ( FIG. 15 ) onboard the SCMS  900  ( FIG. 9 ), offsets to the inertial navigation system (INS) are calculated and the INS is updated to the current position and time of position. 
         [0057]    Referring now to  FIG. 9 , there is illustrated an embodiment of the present invention describing the componentry mounted on and within the body of the locomotive  100  for the gathering and storage of information related to the guidance of the TBM  1000  ( FIG. 10 ) and the mapping of the tunnel  110  ( FIGS. 1, 3, 5, 7 ). Within the preferred embodiment of  FIG. 1  and the alternate embodiment of  FIG. 3 , a locomotive mounted transceiver  115  establishes a link by which position data  225  ( FIGS. 2, 4, 6, and 8 ) is transferred to and received by CPU 1   1520  ( FIG. 15 ). In the case of the alternate embodiments of  FIGS. 5 and 7 , initial position data is directly provided to CPU 1   1520  ( FIG. 15 ). In each of the described embodiments, the position data  225  ( FIGS. 2, 4, 6, and 8 ) from the SCMS  900  is utilized to either initialize or recalibrate fault tolerant inertial navigation system one (FTINS 1 )  1515  ( FIG. 15 ) such that a point of origin, also known as initial position, within the launch pit (Position  1 ) is calculated. Data is transmitted from CPU 1   920  via SCMS transmitter  1505  ( FIG. 15 ) to the host PC  910  and to the TBM  1000  ( FIG. 10 ). The data transmitted to the host PC  910  may be stored on internet based storage. 
         [0058]    Referring now to  FIG. 10 , the docking process  1135  ( FIG. 11 ) within an embodiment of the present invention establishes the actual position  1020  relative to the locomotive  100  ( FIGS. 1, 2, 3, and 7 ), which is both necessary and sufficient to characterize the actual underground location of the TBM  1000 . Central processing unit  2  (CPU 2 )  1010  receives updated position information (Position  2 )  1015  from CPU 1   1520  ( FIG. 15 ) through TBM transceiver  1005 . The TBM  1000  obtains the presumed position (Position  3 )  1025  from FTINS 2   1030  based on movement of the TBM  1000  (element  1200 ). CPU 2   1010  calculates the difference between the updated position information (Position  2 )  1015  and the presumed position (Position  3 )  1025  (element  1205 ), and the difference is applied as a correction to the Position  3   1025  data in FTINS 2   1030  (element  1210 ). 
         [0059]      FIG. 11  depicts the process elements associated with the utilization of position input data provided by the componentry depicted in  FIG. 1 , the interaction of componentry depicted in  FIG. 9  as the locomotive  100  transits the curvilinear tunnel path  1125 , and the transmission of accumulated data to the TBM  1000  componentry of  FIG. 10 . Initial Position  0  is established at the geo-location and retransmission devices  200  (element  1100 ). The time modulated signal with Position  0  data is then transmitted to CPU 1   1520  ( FIG. 15 ) onboard the locomotive  100  (element  1105 ). CPU 1   1520  ( FIG. 15 ) calculates initial position of the locomotive  100  (Position  1   1400 ) (element  1110 ). The Position  1   1400  data is then provided to FTINS 1   1515  ( FIG. 15 ) (element  1115 ). FTINS 1   1515  ( FIG. 15 ) is then updated to Position  1   1400  (element  1120 ). 
         [0060]    Two processes occur as the locomotive  100  ( FIGS. 1, 3, 5, and 7 ) transits a curvilinear path  1120  through the tunnel  110  ( FIGS. 1, 3, 5, and 7 ) from the launch pit  105  ( FIGS. 1, 3, 5, and 7 ) to the TBM  1000  ( FIG. 10 ). The first process is the establishment of guidance information for the TBM  1000  ( FIG. 10 ) by the delivery of a position update to the FTINS 2   1030  ( FIG. 10 ). The second process is the active mapping of the tunnel  110  ( FIGS. 1, 3, 5, and 7 ) as-built, accomplished by the measurement of distance to the tunnel walls  1130  by a SCMS  900  ( FIGS. 9 and 15 ). These processes may be accomplished simultaneously during the transit from launch pit  105  ( FIGS. 1, 3, 5, and 7 ) to docking with the TBM at Position  2   1135  or separately so as to focus on delivery of position data to the TBM  1000  ( FIG. 10 ) on the incoming trip and to focus on tunnel mapping on the outgoing trip. 
         [0061]    Within these two processes, whether simultaneous or separate, information from FTINS 1   1515  ( FIG. 15 ) and the SCMS  900  ( FIGS. 9 and 14 ) is provided to CPU 1   1520  ( FIG. 15 ), which compiles the aforementioned data with the initial position data  1120 . Upon docking  1135  with the TBM  1000  ( FIG. 10 ), the FTINS microprocessor  1320  ( FIG. 13 ) calculates the actual position and provides this data to the onboard CPU 1   1520  ( FIG. 15 ) (element  1140 ). The docking process  1135  establishes actual position relative to the locomotive  100  ( FIGS. 1, 3, 5 , and  7 ) and the TBM  1000  ( FIG. 10 ). All information relative to travel and tunnel measurement from the SCMS  900  ( FIG. 9 ) is retained onboard the locomotive  100  ( FIGS. 1, 3, 5, and 7 ) until its return to the launch pit  105  ( FIGS. 1, 3, 5, and 7 ), where data collected in terms of TBM  1000  ( FIG. 10 ) position and tunnel measurements are uploaded to the host PC  910  ( FIG. 9 ) and may be backed up to internet-based data storage. 
         [0062]    Referring now to  FIG. 12 , the TBM  1000  ( FIG. 10 ) obtains the presumed position (Position  3 )  1025  ( FIG. 10 ) from FTINS 2   1030  ( FIG. 10 ) based on movement of the TBM  1000  ( FIG. 10 ) (element  1200 ). CPU 2   1010  ( FIG. 10 ) calculates the difference between the updated position information (Position  2 )  1015  ( FIG. 10 ) and the presumed position (Position  3 )  1025  ( FIG. 10 ) (element  1205 ), and the difference is applied as a correction to the Position  3   1025  ( FIG. 10 ) data in FTINS 2   1030  ( FIG. 10 ) (element  1210 ). There is programmed within CPU 2   1010  ( FIG. 10 ) a known offset distance  1215  from FTINS 2   1030  ( FIG. 10 ) to the steering control point of the TBM  1000  ( FIG. 10 ), and CPU 2   1010  ( FIG. 10 ) applies the aforementioned known offset to the Position  3   1025  ( FIG. 10 ) data (element  1220 ) contained within FTINS 2   1030  ( FIG. 10 ). The designed tunnel route  1225  programmed within the read-only memory of CPU 2   1010  ( FIG. 10 ), and the as-designed tunnel route  1225  is now compared to the current position  1230 . CPU 2   1010  ( FIG. 10 ) on the TBM  1000  ( FIG. 10 ) now establishes a new immediate heading  1235  for the TBM  1000  ( FIG. 10 ) to follow. This new heading is applied to a steering correction  1240  to the TBM  1000  ( FIG. 10 ) for either a manual or automatic pilot to follow. 
         [0063]      FIG. 13  depicts the components in the fault tolerant inertial navigation system (FTINS)  1300 . Both the locomotive  100  ( FIGS. 1, 3, 5, and 7 ) and the TBM  1000  ( FIG. 10 ) are equipped with two inertial measurement units (IMUs)  1305  which include one or more angular rate sensors (gyroscopes)  1310  and one or more accelerometers  1315  which provide information to the FTINS microprocessor  1320 . The output of the FTINS microprocessor  1320  describes the physical location of the locomotive  100  ( FIGS. 1, 3, 5, and 7 ) relative to the known initial position data  225  ( FIGS. 2, 4, 6, and 8 ), and the physical location of the TBM  1000  ( FIG. 10 ) relative to its last position update. 
         [0064]      FIG. 14  depicts the process elements accomplished within the FTINS  1300  ( FIG. 13 ) componentry. The gyroscopes provide (ω x , ω y , ω z )  1405  and the accelerometers provide ({umlaut over (x)}, ÿ, {umlaut over (z)})  1410  data. The FTINS microprocessor  1320  ( FIG. 13 ) calculates change in position by integrating the acceleration data  1415 . The FTINS microprocessor  1320  ( FIG. 13 ) then receives the initialized position (Position  1 )  1400  and compares it to the change in position  1420 . The FTINS microprocessor  1320  ( FIG. 13 ) finally calculates the current position  1425 . 
         [0065]    Referring now to  FIG. 15 , there is illustrated an embodiment of the present invention describing the self-contained mapping system (SCMS)  900  mounted within the locomotive  100  ( FIGS. 1, 3, 5, and 7 ). The SCMS  900  uses a vibration isolation device  1510  and reflective monuments  310  installed on, or cast into, the tunnel walls to map the tunnels. The vibration isolation device  1510  comprises: FTINS 1   1515 ; a contact-free 3D scanner  1525 , such as a Light Detection and Ranging or Laser Imaging Detection and Ranging (LiDAR); CPU 1   1520 ; a data storage unit  1530 , such as a hard disk drive or a flash memory device; and a wired or wireless transmitter  1505 . The SCMS  900  is capable of generating a 3D map of the tunnel  110  ( FIGS. 1, 3, 5, and 7 ) during and after the tunnel boring process, generating an accurate measurement of the tunnel centerline, and, through the use of the reflective monuments  310  ( FIGS. 3, 4, 5, and 6 ), the SCMS  900  may be used to observe the movement of fixed points on the tunnel wall during and after the tunnel boring process to determine change in tunnel geometry over time. 
         [0066]      FIG. 16  depicts the process elements accomplished by the SCMS  900  ( FIGS. 9 and 15 ). The SCMS  900  ( FIGS. 9 and 15 ) applies a position offset to the FTINS 1   1515  ( FIG. 15 ) measurements in order to match the FTINS 1   1515  ( FIG. 15 ) reference frame with that of the 3D scanner  935  ( FIGS. 9 and 15 ). The SCMS  900  ( FIGS. 9 and 15 ) then measure the distance to the tunnel wall relative to the locomotive  100  ( FIGS. 1, 3, 5, and 7 ) and geometric tunnel center  1600 . The FTINS 1   1515  ( FIG. 15 ) estimates the absolute position of the locomotive  100  ( FIGS. 1, 3, 5, and 7 ) (element  1605 ). Timestamps  1610 ,  1615  are applied to the two measurements and the data is stored. A position offset to the FTINS 1   1515  ( FIG. 15 ) is applied to the laser reference frame  1625 . The data resulting from  1610  and  1625  is then correlated  1620  and used to: generate the geometric centerline of the tunnel  1630 , extract reflective monument measurements  1635 , and generate a 3D mesh of the tunnel wall  1640 . The position of the reflective monuments  310  ( FIGS. 3, 4, 5, and 6 ) can be extracted  1635  by isolating measurements from the 3D scanner  1525  ( FIG. 15 ) which indicate a higher reflectivity, and individual reflective monuments can be identified by their specific reflectivity. Upon completing the trip through the tunnel, the SCMS  900  ( FIGS. 9 and 15 ) uploads the measurements from the FTINS 1   1515  ( FIG. 15 ) and 3D scanner  1525  ( FIG. 15 ) to an external computer via wired or wireless link  1645  for analysis of the as-built tunnel geometry. 
       Multi-Sensor Data Fusion: 
       [0067]    Those skilled in the art of state estimation, robotics, and advanced defense avionics understand academically that sensor-fusion is the art of combining sensory data or data derived from disparate sources such that the resulting information is in some sense “better” than would be possible when these sources were used individually. This process is predicated on the covariance (or the measure of how much two variables vary together) of non-independent sources. The term “better” in the case above can mean more accurate, more complete, more dependable, or refer to the result of an emerging view or state estimation. 
         [0068]    The data sources for a fusion process are not specified to originate from identical sources or sensors which may or may not be spatially and temporally aligned. Further one can distinguish direct fusion, indirect fusion, and fusion of the outputs of the former two. Direct fusion is the fusion of sensor data from a set of heterogeneous or homogeneous sensors, soft sensors, and history values of sensor data, while indirect fusion uses information sources like a prior knowledge about the environment and human input. Sensor fusion is also known as “multi-sensor data fusion” and is a subset of information fusion through an implementation of the probability theory. 
         [0069]    Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty. More precisely, probability is used for modeling situations when the result of a measurement, realized under the same circumstances, produces different results. Mathematicians and actuaries think of probabilities as numbers in the closed interval from  0  to  1  assigned to “events” whose occurrence or failure to occur is random. Two crucial concepts in the theory of probability are those of a random variable and of the probability distribution of a random variable. 
         [0070]    Implementing the features described above with affordable instruments requires reliable real-time estimates of system state. Unfortunately, the complete state is not always observable. State Estimation takes all the data obtained and uses it to determine the underlying behavior of the system at any point in time. It includes fault detection, isolation and continuous system state estimation. 
         [0071]    There are two parts to state estimation: modeling and algorithms. The overall approach is to use a model to predict the behavior of the system in a particular state, and then compare that behavior with the actual measurements from the instruments to determine which state or states is the most likely to produce the observed system behavior. 
         [0072]    This is not well understood or currently implemented in the construction industry; the approach understood and practiced is logical decisions in linear and deterministic systems. If use cases require higher confidences in measurements, instrument specifications are tightened resulting in the undesired effect of cost and schedule increases. The environment we live and operate in is neither linear nor deterministic; use cases are infinite; and the perverse variability of the systems and potential errors cannot be modeled. The variability of the problem identified above includes aspects other than just spatial (i.e. precise location of the tunnel boring machine); temporal relationships are part of the fundamental intellectual structure (together with space and number) within which events must be sequenced, quantify the duration of events, quantify the intervals between them, and compare the kinematics of objects. 
         [0073]    In any of the embodiments listed above; the use of Fusion Engine (FE) and Kalman filters in the guidance system of the TBM, will greatly improve position accuracy and reduce instrument costs. The FE continuously receives measurements from multiple sources and generates a state estimate and covariance (confidence) of the current position of the TBM; all updated position data measurements received are used to ensure the measurement data is within the FE estimates. 
         [0074]    In order to continuously and accurately estimate the position of the TBM the Kalman filters in the preferred embodiment are implemented as an asynchronous n-scalable Interacting Multiple Model (IMM) estimation Filter. The IMM comprises multiple models of drift from position in order to accurately match the maneuvering and drift expectations. 
         [0075]    Since the drift or progression of the gyros in either FTINS is not known ahead of time, an accurate model cannot be designed, so errors in the position estimation will occur. Adding process noise to model the TBM maneuvers or using a maneuver detector to adapt the filter has been used in the art, but detection delays and large estimation errors during maneuvers are still a problem. It is generally accepted that the Interacting Multiple Model (IMM) estimator provides superior tracking performance compared to a single Kalman Filter. 
         [0076]    The IMM is based on using several models in parallel to estimate the maneuvering TBM&#39;s states. Each Kalman Filter, uses a different model for each maneuver, one models a constant state of the TBM, another models a position change in the longitudinal axis while another models a position change in the lateral axis and vertical axis. Switching between these models during each sample period is determined probabilistically. Unlike maneuver detection systems where only one filter model is used at a time, the IMM uses all filters. The overall state estimate output is a weighted combination of the estimates from the individual filters. The weighting is based on the likelihood that a filter model is the correct maneuvering TBM model. 
         [0077]    The IMM estimator is a state estimation algorithm that uses Markovian switching coefficients. A system with these coefficients is described by r models, M 1 , M 2 , . . . , M r , and given probabilities of switching between these models. M j (k) denotes that model j (M j ) is in effect during the sampling period ending at time t k , [t k-1 , t k ]. The dynamics and measurement for a linear system are given by 
         [0000]        x ( k )=Φ j ( k,k− 1) x ( k− 1)+ G   j ( k,k− 1) w   j ( k− 1),  (1)
 
         [0000]    and 
         [0000]        z ( k )= H   j ( k ) x ( k )+ v   j ( k ),  (2)
 
         [0078]    where x(k) is the system state at time t k , z(k) is the measurement vector at time t k , Φ j (k,k−1) is the state-transition matrix from time t k-1  to time t k  for M j (k), G j (k,k−1) is the noise input matrix, and H j (k) is the observation matrix for M j (k). The process noise vector w j (k−1) and the measurement noise vector v j (k) are mutually uncorrelated zero-mean white Gaussian processes with covariance matrices Q j (k−1) and R j (k) respectively. 
         [0079]    The initial conditions for the system state under each model j are Gaussian random variables with mean  x   j (0) and covariance P j (0). These prior statistics are assumed known, as also is μ j (0)=Pr{M j  (0)}, which is the initial probability of model j at t 0 . 
         [0080]    The model switching is governed by a finite-state Markov chain according to the probability π ij =Pr{M j (k)|M i (k−1)} of switching from M i (k−1) to M j (k). The model switching probabilities, π ij , are assumed known and an example is 
         [0000]    
       
         
           
             
               
                 
                   
                     π 
                     ij 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             .95 
                           
                           
                             .05 
                           
                         
                         
                           
                             .05 
                           
                           
                             .95 
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0081]    A block diagram of the IMM estimator with only two models, for simplicity, is shown in  FIG. 17 . 
         [0082]    The inputs to the IMM estimator are {circumflex over (x)} 1 (k−1|k−1), {circumflex over (x)} 2 (k−1|k−1), P 1 (k−1|k−1), P 2 (k−1|k−1), and μ i|j (k−1|k−1), all from the sampling period ending at t k-1 . Where {circumflex over (x)} 1 (k−1|k−1) is the state estimate from filter  1  at time t k-1  using measurements from time t k-1  and P 1 (k−1|k−1) is the corresponding state covariance matrix. Each of the filters use a different mixture of {circumflex over (x)} 1 (k−1|k−1) and {circumflex over (x)} 2 (k−1|k−1) for their input, For r models, this mixing allows the model-conditioned estimates in the current cycle to be computed using r filters rather than r 2  filters, which greatly decreases the computational burden. The inputs to the filters, {circumflex over (x)} 01 (k−1|k−1), {circumflex over (x)} 02 (k−1|k−1), and the corresponding covariance matrices are computed in the Interaction (Mixing) block. 
         [0083]    For the filter matched to M j (k), the inputs are 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           x 
                           ^ 
                         
                         
                           0 
                            
                           j 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             - 
                             1 
                           
                            
                           
                             k 
                             - 
                             1 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         r 
                       
                        
                       
                         
                           
                             μ 
                             
                               i 
                               | 
                               j 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 - 
                                 1 
                               
                                
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             
                               x 
                               ^ 
                             
                             i 
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 - 
                                 1 
                               
                                
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         P 
                         
                           0 
                            
                           j 
                         
                       
                        
                       
                         ( 
                         
                           
                             k 
                             - 
                             1 
                           
                            
                           
                             k 
                             - 
                             1 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         r 
                       
                        
                       
                         
                           
                             μ 
                             
                               i 
                                
                               j 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 - 
                                 1 
                               
                                
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 P 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                    
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         x 
                                         ^ 
                                       
                                       i 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           - 
                                           1 
                                         
                                          
                                         
                                           k 
                                           - 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         x 
                                         ^ 
                                       
                                       
                                         0 
                                          
                                         j 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           k 
                                           - 
                                           1 
                                         
                                          
                                         
                                           k 
                                           - 
                                           1 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                               ⋆ 
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           x 
                                           ^ 
                                         
                                         i 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             - 
                                             1 
                                           
                                            
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           x 
                                           ^ 
                                         
                                         
                                           0 
                                            
                                           j 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             k 
                                             - 
                                             1 
                                           
                                            
                                           
                                             k 
                                             - 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                 T 
                               
                             
                           
                           } 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the conditional model probability is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             μ 
                             
                               i 
                                
                               j 
                             
                           
                            
                           
                             ( 
                             
                               
                                 k 
                                 - 
                                 1 
                               
                                
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           Pr 
                            
                           
                             { 
                             
                               
                                 
                                   
                                     M 
                                     i 
                                   
                                    
                                   
                                     ( 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     M 
                                     j 
                                   
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 Z 
                                 1 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                             
                            
                           
                             
                               1 
                               
                                 
                                   μ 
                                   j 
                                 
                                  
                                 
                                   ( 
                                   
                                     k 
                                      
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               π 
                               ij 
                             
                              
                             
                               
                                 μ 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     - 
                                     1 
                                   
                                    
                                   
                                     k 
                                     - 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         , 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the predicted model probability is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             μ 
                             j 
                           
                            
                           
                             ( 
                             
                               k 
                                
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           Pr 
                            
                           
                             { 
                             
                               
                                 
                                   M 
                                   j 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                               
                                 Z 
                                 1 
                                 
                                   k 
                                   - 
                                   1 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             r 
                           
                            
                           
                             
                               π 
                               ij 
                             
                              
                             
                               
                                 
                                   μ 
                                   i 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       - 
                                       1 
                                     
                                      
                                     
                                       k 
                                       - 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0084]    Using the measurements, z(k), for the filter matched to M j (k), the updates are computed using the familiar Kalman Filter equations 
         [0000]        {circumflex over (x)}   j ( k|k− 1)=Φ j ( k,k− 1) {circumflex over (x)}   01 ( k|k− 1),  (8)
 
         [0000]        P   j ( k|k− 1)=Φ( k,k− 1) P   0j ( k|k− 1)[Φ j ( k,k− 1)] T   +G   j ( k,k− 1) Q   j ( k− 1)[ G   j ( k,k− 1)] T   (9)
 
         [0000]        v   j ( k )= z ( k )− H ( k ) {circumflex over (x)}   j ( k|k− 1),  (10)
 
         [0000]        S   j ( k )= H   j ( k ) P   j ( k|k− 1)[ H   j ( k )] T   +R   j ( k ),  (11)
 
         [0000]        K   j ( k )= P   j ( k|k− 1)[ H   j ( k )] T   [S   j ( k )] −1 ,  (12)
 
         [0000]        {circumflex over (x)}   j ( k|k )= {circumflex over (x)}   j ( k|k− 1)+ K   j ( k ) v   j ( k ),  (13)
 
         [0000]        P   j ( k|k )=[ I−K   j ( k ) H   j ( k )] P   j ( k|k− 1),  (14)
 
         [0000]    where {circumflex over (x)} j (k|k−1) is the predicted state estimate under M j (k), P j (k|k−1) is the corresponding prediction covariance, v j (k) is the residual, S j (k) is the residual covariance matrix, K j (k) is the Kalman gain matrix, {circumflex over (x)} j (k|k) is the updated state estimate under M j (k), and P j (k|k) is the updated covariance matrix. 
         [0085]    The likelihood of the filter matched to M j (k) is defined by Λ j (k)=f[z(k)|M j (k), Z 1   k-1 ], where f[|] denotes a conditional density. Using the assumption of Gaussian statistics, the filter residual and the residual covariance, the likelihood is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Λ 
                       j 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           det 
                            
                           
                               
                           
                           [ 
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               
                                 S 
                                 j 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                      
                     exp 
                      
                     
                       
                         { 
                         
                           
                             - 
                             
                               
                                 
                                   
                                     
                                       1 
                                       2 
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           v 
                                           j 
                                         
                                          
                                         
                                           ( 
                                           k 
                                           ) 
                                         
                                       
                                       ] 
                                     
                                   
                                   T 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       S 
                                       j 
                                     
                                      
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                   ] 
                                 
                               
                               
                                 - 
                                 1 
                               
                             
                           
                            
                           
                             
                               v 
                               j 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The probability for M j (k) is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         μ 
                         j 
                       
                        
                       
                         ( 
                         
                           k 
                            
                           k 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         Pr 
                          
                         
                           { 
                           
                             
                               
                                 M 
                                 j 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                             
                               Z 
                               1 
                               k 
                             
                           
                           } 
                         
                       
                       = 
                       
                         
                           1 
                           c 
                         
                          
                         
                           
                             μ 
                             j 
                           
                            
                           
                             ( 
                             
                               k 
                                
                               
                                 k 
                                 - 
                                 1 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             Λ 
                             j 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the normalization factor c is 
         [0000]    
       
         
           
             
               
                 
                   c 
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       r 
                     
                      
                     
                       
                         
                           μ 
                           i 
                         
                          
                         
                           ( 
                           
                             k 
                              
                             
                               k 
                               - 
                               1 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           
                             Λ 
                             i 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0086]    These computations are performed in the Model Probability Update block. Finally the combined state estimate {circumflex over (x)}(k|k) and the corresponding state error covariance for the IMM are given by 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           x 
                           ^ 
                         
                          
                         
                           ( 
                           
                             k 
                              
                             k 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           r 
                         
                          
                         
                           
                             
                               μ 
                               j 
                             
                              
                             
                               ( 
                               
                                 k 
                                  
                                 k 
                               
                               ) 
                             
                           
                            
                           
                             
                               
                                 x 
                                 ^ 
                               
                               j 
                             
                              
                             
                               ( 
                               
                                 k 
                                  
                                 k 
                               
                               ) 
                             
                           
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   
                     P 
                      
                     
                       ( 
                       
                         k 
                          
                         k 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       r 
                     
                      
                     
                       
                         
                           μ 
                           j 
                         
                          
                         
                           ( 
                           
                             k 
                              
                             k 
                           
                           ) 
                         
                       
                        
                       
                         
                           { 
                           
                             
                               
                                 P 
                                 j 
                               
                                
                               
                                 ( 
                                 
                                   k 
                                    
                                   k 
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           x 
                                           ^ 
                                         
                                         j 
                                       
                                        
                                       
                                         ( 
                                         
                                           k 
                                            
                                           k 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         x 
                                         ^ 
                                       
                                        
                                       
                                         ( 
                                         
                                           k 
                                            
                                           k 
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           x 
                                           ^ 
                                         
                                         j 
                                       
                                        
                                       
                                         ( 
                                         
                                           k 
                                            
                                           k 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       
                                         x 
                                         ^ 
                                       
                                        
                                       
                                         ( 
                                         
                                           k 
                                            
                                           k 
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                               
                               T 
                             
                           
                           } 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0087]    The final state estimate, {circumflex over (x)}(k|k), is the best estimate of the TBM state and P(k|k) is the error covariance matrix for this optimal state estimate. 
         [0088]    For the sake of convenience, the operations are described as various interconnected functional blocks or distinct software modules. This is not necessary, however, and there may be cases where these functional blocks or modules are equivalently aggregated into a single logic device, program or operation with unclear boundaries. In any event, the functional blocks and software modules or described features can be implemented by themselves, or in combination with other operations in either hardware or software. 
         [0089]    Having described and illustrated the principles of the invention in a preferred embodiment thereof, it should be apparent that the invention may be modified in arrangement and detail without departing from such principles. Claim is made to all modifications and variation coming within the spirit and scope of the following claims.