# Patent

## Title
Centralizer-based survey and navigation device and method

## Abstract

A Centralizer based Survey and Navigation (CSN) device designed to provide borehole or passageway position information. The CSN device can include one or more displacement sensors, centralizers, an odometry sensor, a borehole initialization system, and navigation algorithm implementing processor(s). Also, methods of using the CSN device for in-hole survey and navigation.

## Background
This application claims priority to U.S. Provisional Application Ser. No. 60/635,477, filed Dec. 14, 2004, the entirety of which is incorporated by reference herein.


FIELD OF THE INVENTION
The present invention relates, but is not limited, to a method and apparatus for accurately determining in three dimensions information on the location of an object in a passageway and/or the path taken by a passageway, e.g., a borehole or tube.
BACKGROUND OF THE INVENTION
The drilling industry has recognized the desirability of having a position determining system that can be used to guide a drilling head to a predestined target location. There is a continuing need for a position determining system that can provide accurate position information on the path of a borehole and/or the location of a drilling head at any given time as the drill pipe advances. Ideally, the position determining system would be small enough to fit into a drill pipe so as to present minimal restriction to the flow of drilling or returning fluids and accuracy should be as high as possible.
Several systems have been devised to provide such position information. Traditional guidance and hole survey tools such as inclinometers, accelerometers, gyroscopes and magnetometers have been used. One problem facing all of these systems is that they tend to be too large to allow for a “measurement while drilling” for small diameter holes. In a “measurement while drilling” system, it is desirable to incorporate a position locator device in the drill pipe, typically near the drilling head, so that measurements may be made without extracting the tool from the hole. The inclusion of such instrumentation within a drill pipe considerably restricts the flow of fluids. With such systems, the drill pipe diameter and the diameter of the hole must often be greater than 4 inches to accommodate the position measuring instrumentation, while still allowing sufficient interior space to provide minimum restriction to fluid flow. Systems based on inclinometers, accelerometers, gyroscopes, and/or magnetometers are also incapable of providing a high degree of accuracy because they are all influenced by signal drift, vibrations, or magnetic or gravitational anomalies. Errors on the order of 1% or greater are often noted.
Some shallow depth position location systems are based on tracking sounds or electromagnetic radiation emitted by a sonde near the drilling head. In addition to being depth limited, such systems are also deficient in that they require a worker to carry a receiver and walk the surface over the drilling head to detect the emissions and track the drilling head location. Such systems cannot be used where there is no worker access to the surface over the drilling head or the ground is not sufficiently transparent to the emissions.
A system and method disclosed in U.S. Pat. No. 5,193,628 (“the '628 patent”) to Hill, III, et al., which is hereby incorporated by reference, was designed to provide a highly accurate position determining system small enough to fit within drill pipes of diameters substantially smaller than 4 inches and configured to allow for smooth passage of fluids. This system and method is termed “POLO,” referring to POsition LOcation technology. The system disclosed in the '628 patent successively and periodically determines the radius of curvature and azimuth of the curve of a portion of a drill pipe from axial strain measurements made on the outer surface of the drill pipe as it passes through a borehole or other passageway. Using successively acquired radius of curvature and azimuth information, the '628 patent system constructs on a segment-by-segment basis, circular arc data representing the path of the borehole and which also represents, at each measurement point, the location of the measuring strain gauge sensors. If the sensors are positioned near the drilling head, the location of the drilling head can be obtained.
The '628 patent system and method has application for directional drilling and can be used with various types of drilling apparatus, for example, rotary drilling, water jet drilling, down hole motor drilling, and pneumatic drilling. The system is useful in directional drilling such as well drilling, reservoir stimulation, gas or fluid storage, routing of original piping and wiring, infrastructure renewal, replacement of existing pipe and wiring, instrumentation placement, core drilling, cone penetrometer insertion, storage tank monitoring, pipe jacking, tunnel boring and in other related fields.
The '628 patent also provides a method for compensating for rotation of the measuring tube during a drilling operation by determining, at each measurement position, information concerning the net amount of rotation relative to a global reference, if any, of the measuring tube as it passes through the passageway and using the rotation information with the strain measurement to determine the azimuth associated with a measured local radius of curvature relative to the global reference.
While the '628 patent provides great advantages, there are some aspects of the system and method that could be improved.
SUMMARY
The Centralizer-based Survey and Navigation (CSN) device is designed to provide borehole or passageway position information. The device is suitable for both closed traverse surveying (referred to as survey) and open traverse surveying or navigation while drilling (referred to as navigation). The CSN device can consist of a sensor string comprised of one or more segments having centralizers, which position the segment(s) within the passageway, and at least one metrology sensor, which measures the relative positions and orientation of the centralizers, even with respect to gravity. The CSN device can also have at least one odometry sensor, an initialization system, and a navigation algorithm implementing processor(s). The number of centralizers in the sensor string should be at least three. Additional sensors, such as inclinometers, accelerometers, and others can be included in the CSN device and system.
There are many possible implementations of the CSN, including an exemplary embodiment relating to an in-the-hole CSN assembly of a sensor string, where each segment can have its own detector to measure relative positions of centralizers, its own detector that measures relative orientation of the sensor string with respect to gravity, and/or where the partial data reduction is performed by a processor placed inside the segment and high value data is communicated to the navigation algorithm processor through a bus.
Another exemplary embodiment relates to a CSN device utilizing a sensor string segment which can utilize capacitance proximity detectors and/or fiber optic proximity detectors and/or strain gauges based proximity detectors that measure relative positions of centralizers with respect to a reference straight metrology body or beam.
Another exemplary embodiment relates to a CSN device utilizing an angular metrology sensor, which has rigid beams as sensor string segments that are attached to one or more centralizers. These beams are connected to each other using a flexure-based joint with strain gauge instrumented flexures and/or a universal joint with an angle detector such as angular encoder. The relative positions of the centralizers are determined based on the readings of the said encoders and/or strain gauges.
Another exemplary embodiment relates to a CSN device utilizing a strain gauge instrumented bending beam as a sensor string segment, which can use the readings of these strain gauges to measure relative positions of the centralizers.
Another exemplary embodiment relates to a CSN device utilizing a bending beam sensor, which can utilize multiple sets of strain gauges to compensate for possible shear forces induced in the said bending beam.
Another exemplary embodiment relates to a compensator for zero drift of detectors measuring orientation of the sensor string and detectors measuring relative displacement of the centralizers by inducing rotation in the sensor string or taking advantage of rotation of a drill string. If the detector measuring orientation of the sensor string is an accelerometer, such a device can calculate the zero drift of the accelerometer detector by enforcing that the average of the detector-measured value of local Earth's gravity to be equal to the known value of g at a given time, and/or where the zero drift of detectors measuring relative displacement of the centralizers is compensated for by enforcing that the readings of the strain gauges follow the same angular dependence on the rotation of the string as the angular dependence measured by inclinometers, accelerometers, and or gyroscopes placed on the drill string or sensor string that measure orientation of the sensor string with respect to the Earth's gravity.
Another exemplary embodiment relates to a device using buoyancy to compensate for the gravity induced sag of the metrology beam of the proximity-detector-based or angular-metrology-based displacement sensor string.
Another exemplary embodiment relates to centralizers that maintain constant separation between their points of contact with the borehole.
These exemplary embodiments and other features of the invention can be better understood based on the following detailed description with reference to the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 shows a system incorporating a CSN device in accordance with the invention.
FIG. 2a through FIG. 2e show various embodiments of a CSN device in accordance with the invention.
FIG. 3 shows a system incorporating a CSN device as shown in FIG. 2a, in accordance with the invention.
FIG. 4 illustrates a CSN device utilizing a displacement or strain metrology as shown in FIGS. 2b, 2c, and 2e, in accordance with the invention.
FIGS. 5a through 5d show a global and local coordinate system utilized by a CSN device, in accordance with the invention. FIG. 5b shows an expanded view of the encircled local coordinate system shown in FIG. 5a. 
FIG. 6 is a block diagram showing how navigation and/or surveying can be performed by a CSN system/device in accordance with the invention.
FIGS. 7a and 7b show a displacement metrology CSN device, in accordance with the invention; FIG. 7b shows the device of FIG. 7a through cross section A-A.
FIG. 8 shows a CSN device utilizing strain gauge metrology sensors in accordance with the invention.
FIG. 9 shows forces acting on a CSN device as shown in FIG. 8, in accordance with the invention.
FIG. 10 is a block diagram of strain gauge data reduction for a CSN device as shown in FIG. 8, in accordance with the invention.
FIG. 11 shows strains exhibited in a rotating bending beam of a CSN device in accordance with the invention.
FIG. 12 is a block diagram illustrating how data reduction can be performed in a rotating strain gauge CSN device, such as illustrated in FIG. 11, in accordance with the invention.
FIG. 13 shows vectors defining sensitivity of an accelerometer used with a CSN device in accordance with the invention.
FIG. 14 is a block diagram showing how data reduction can be performed in an accelerometer used with a CSN device in accordance with the invention.
FIGS. 15 to 17 show a universal joint strain gauge CSN device in accordance with the invention.
FIG. 18 is a block diagram of a CSN assembly in accordance with the invention.
FIGS. 19, 20a, and 20b show embodiments of centralizers in accordance with the invention.
FIGS. 21a and 21b show gravity compensating CSN devices.



DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
The invention relates to a Centralizer-based Survey and Navigation (hereinafter “CSN”) device, system, and methods, designed to provide passageway and down-hole position information. The CSN device can be scaled for use in passageways and holes of almost any size and is suitable for survey of or navigation in drilled holes, piping, plumbing, municipal systems, and virtually any other hole environment. Herein, the terms passageway and borehole are used interchangeably.
FIG. 1 shows the basic elements of a directional drilling system incorporating a CSN device 10, a sensor string 12 including segments 13 and centralizers 14 (14a, 14b, and 14c), a drill string 18, an initializer 20, an odometer 22, a computer 24, and a drill head 26. A metrology sensor 28 is included and can be associated with the middle centralizer 14b, or located on the drill string 18. The odometer 22 and computer 24 hosting a navigation algorithm are, typically, installed on a drill rig 30 and in communication with the CSN device 10. A CSN device 10 may be pre-assembled before insertion into the borehole 16 or may be assembled as the CSN device 10 advances into the borehole 16.
As shown in FIG. 1, the CSN device 10 can be placed onto a drill string 18 and advanced into the borehole 16. The centralizers 14 of the CSN device 10, which are shown and discussed in greater detail below in relation to FIGS. 19-20b, are mechanical or electromechanical devices that position themselves in a repeatable fashion in the center of the borehole 16 cross-section, regardless of hole wall irregularities. A CSN device 10, as shown in FIG. 1, uses at least three centralizers 14: a trailing centralizer 14a, a middle centralizer 14b, and a leading centralizer 14c, so named based on direction of travel within the borehole 16. The centralizers 14 are connected by along a sensor string 12 in one or more segments 13, which connect any two centralizers 14, to maintain a known, constant spacing in the borehole 16 and between the connected centralizers 14. Direction changes of the CSN device 10 evidenced by changes in orientation of the centralizers 14 with respect to each other or with respect to the sensor string 12 segments 13 can be used to determine the geometry of borehole 16.
The initializer 20, shown in FIG. 1, provides information on the borehole 16 and CSN device 10 insertion orientation with respect to the borehole 16 so that future calculations on location can be based on the initial insertion location. The initializer 20 has a length that is longer than the distance between a pair of adjacent centralizers 14 on the sensor string segment 13, providing a known path of travel into the borehole 16 for the CSN device 10 so that it may be initially oriented. Under some circumstances, information about location of as few as two points along the borehole 16 entranceway may be used in lieu of the initializer 20. Navigation in accordance with an exemplary embodiment of the invention provides the position location of the CSN device 10 with respect to its starting position and orientation based on data obtained by using the initializer 20.
As shown in FIGS. 2a-2e, there are various types of centralizer-based metrologies compatible with the CSN device 10; however, all can determine the position of the CSN device 10 based on readings at the CSN device 10. The types of CSN device 10 metrologies include, but are not limited to: (1) straight beam/angle metrology, shown in FIG. 2a; (2) straight beam/displacement metrology, shown in FIG. 2b; (3) bending beam metrology, shown in FIG. 2c; (4) optical beam displacement metrology, shown in FIG. 2d; and (5) combination systems of (1)-(4), shown in FIG. 2e. These various metrology types all measure curvatures of a borehole 16 in the vertical plane and in an orthogonal plane. The vertical plane is defined by the vector perpendicular to the axis of the borehole 16 at a given borehole 16 location and the local vertical. The orthogonal plane is orthogonal to the vertical plane and is parallel to the borehole 16 axis. The CSN device 10 uses this borehole 16 curvature information along with distance traveled along the borehole 16 to determine its location in three dimensions. Distance traveled within the borehole 16 from the entry point to a current CSN device 10 location can be measured with an odometer 22 connected either to the drill string 18 used to advance the CSN device 10 or connected with the CSN device 10 itself. The CSN device 10 can be in communication with a computer 24, which can be used to calculate location based on the CSN device 10 measurements and the odometer 22. Alternatively, the CSN device 10 itself can include all instrumentation and processing capability to determine its location and the connected computer 24 can be used to display this information.
Definitions of starting position location and starting orientation (inclination and azimuth), from a defined local coordinate system (FIGS. 5b) provided by the initializer 20, allows an operator of the CSN device 10 to relate drill navigation to known surface and subsurface features in a Global coordinate system. A navigation algorithm, such as that shown in FIG. 6, can combine the readings of the sensor string segment(s) 12, the odometry sensor(s) 22, and the initializer 20 to calculate the borehole 16 position of the CSN device 10.
A CSN device 10 provides the relative positions of the centralizers 14. More precisely, an ideal three-centralizer CSN device 10 provides vector coordinates of the leading centralizer 14c in a local coordinate system, as shown by FIG. 5b, where the “x” axis is defined by the line connecting the centralizers 14a and 14c and the “z” axis lies in a plane defined by the “x” axis and the global vertical “Z.” Alternately, the position of the middle centralizer would be provided in a coordinate system where the “x” axis is defined by the line connecting the centralizers 14a and 14b and the “y” axis and “z” axis are defined same as above. Coordinate systems where the x axis connects either leading and trailing centralizers, or leading and middle centralizer, or middle and trailing centralizers, while different in minor details, all lead to mathematically equivalent navigation algorithms and will be used interchangeably.
FIG. 3 illustrates a CSN device 10 in accordance with the metrology technique shown in FIG. 2a, where angle of direction change between the leading centralizer 14c and trailing centralizer 14a is measured at the middle centralizer 14b. As shown, the CSN device 10 follows the drill head 26 through the borehole 16 as it changes direction. The magnitude of displacement of the centralizers 14 with respect to each other is reflected by an angle θ between the beam forming segment 13 connecting the centralizers 14c and 14b and the beam forming segment 13 connecting the centralizers 14b and 14a, which is measured by angle-sensing detector(s) 29 (a metrology sensor 28) at or near the middle centralizer 14b. Rotation φ of the sensor string 12 can also be measured.
FIG. 4 shows a CSN device 10 configured for an alternative navigation/survey technique reflecting the metrology techniques shown in FIGS. 2b, 2c, and 2e, i.e., both displacement and bending/strain metrology. Displacement metrology (discussed in greater detail below in relation to FIGS. 7a and 7b) measures relative positions of the centralizers 14 using a straight displacement metrology beam 31 (as a sensor string 12 segment 13) that is mounted on the leading and trailing centralizers, 14c and 14a. Proximity detectors 38 (a metrology sensor 28) measure the position of the middle centralizer 14b with respect to the straight metrology beam 31.
Still referring to FIG. 4, strain detector metrology (discussed further below in relation to FIGS. 8-12) can also be used in the CSN device 10, which is configured to measure the strain induced in a solid metrology beam 32 (another form of sensor string segment 12) that connects between each of the centralizers 14. Any deviation of the centralizer 14 positions from a straight line will introduce strains in the beam 32. The strain detectors or gauges 40 (a type of metrology sensor 28) measure these strains (the terms strain detectors and strain gauges are used interchangeably herein). The strain gages 40 are designed to convert mechanical motion into an electronic signal. The CSN device 10 can have as few as two strain gauge instrumented intervals in the beam 32. Rotation φ of the sensor string 12 can also be measured.
In another implementation, both strain detectors 40 and proximity detectors 38 may be used simultaneously to improve navigation accuracy. In another implementation, indicated in FIG. 2d, the displacement metrology is based on a deviation of the beam of light such as a laser beam. In a three centralizer 14 arrangement, a coherent, linear light source (e.g., laser) can be mounted on the leading centralizer 14c to illuminate the trailing centralizer 14a. A reflecting surface mounted on trailing centralizer 14a reflects the coherent light back to a position sensitive optical detector (PSD, a metrology sensor 28) mounted on middle centralizer 14b, which converts the reflected location of the coherent light into an electronic signal. The point at which the beam intersects the PSD metrology sensor 28 is related to the relative displacement of the three centralizers 14. In a two centralizer 14 optical metrology sensor arrangement, light from a laser mounted on a middle centralizer 14b is reflected from a mirror mounted on an adjacent centralizer 14 and redirected back to a PSD metrology sensor 28 mounted on the middle centralizer 14b. The point at which the beam intersects the PSD metrology sensor 28 is related to the relative angle of the orientation of the centralizers 14.
As mentioned above, a CSN navigation algorithm (FIG. 6) uses a local coordinate system (x, y, z) to determine the location of a CSN device 10 in three dimensions relative to a Global coordinate system (X, Y, Z). FIG. 5a indicates the general relationship between the two coordinate systems where the local coordinates are based at a location of CSN device 10 along borehole 16 beneath the ground surface. A CSN navigation algorithm can be based on the following operation of the CSN device 10: (1) the CSN device 10 is positioned in such a way that the trailing centralizer 14a and the middle centralizer 14b are located in a surveyed portion (the known part) of the borehole 16 and the leading centralizer 14c is within an unknown part of the borehole 16; (2) using displacement metrology, a CSN device 10 comprises a set of detectors, e.g., metrology sensor 28, that calculates the relative displacement of the centralizers 14 with respect to each other in the local coordinate system; (3) a local coordinate system is defined based on the vector connecting centralizers 14 a and 14c (axis “x” in FIG. 5b) and the direction of the force of gravity (vertical or “Z” in FIG. 5b) as measured by, e.g., vertical angle detectors, as a metrology sensor 28; and (4) prior determination of the positions of the middle and trailing centralizers 14b and 14a. With this information in hand, the position of the leading centralizer 14c can be determined.
An algorithm as shown in FIG. 6 applied by, e.g., a processor, and functioning in accordance with the geometry of FIG. 5c can perform as follows: (1) the CSN device 10 is positioned as indicated in the preceding paragraph; (2) the relative angular orientations θy, θz and positions (y, z) of any two adjacent sensor string segments 13 of a CSN device 10 in the local coordinate system are determined using internal CSN device 10 segment 13 detectors; (3) three centralizers 14 are designated to be the leading 14c, trailing 14a, and middle 14b centralizers of the equivalent or ideal three-centralizer CSN device 10; (4) relative positions of the leading, middle, and trailing centralizers 14 forming an ideal CSN device 10 are determined in the local coordinate system of the sensor string 12.
FIG. 7a shows a CSN device 10 according to an alternative exemplary embodiment of the invention that utilizes straight beam displacement (such as shown in FIGS. 2b and 4) and capacitance measurements as metrology sensors 28 to calculate the respective locations of the centralizers 14a, 14b, and 14c. As shown in FIG. 7a, a stiff straight beam 31 is attached to the leading and trailing centralizers 14c and 14a by means of flexures 33 that are stiff in radial direction and flexible about the axial direction (τ). A set of proximity detectors, 38 can be associated with the middle centralizer 14b. The proximity detectors 38 measure the displacement of the middle centralizer 14b with respect to the straight beam 31. An accelerometer 36 can be used to measure the orientation of the middle centralizer 14b with respect to the vertical. Examples of proximity detectors include, capacitance, eddy current, magnetic, strain gauge, and optical proximity detectors. The Global and Local coordinate systems (FIGS. 5a-5d) associated with the CSN device 10 of this embodiment are shown in FIG. 7a. 
The relationship between these proximity detectors 38 and the straight beam 31 is shown in FIG. 7b as a cross-sectional view of the CSN device 10 of FIG. 7a taken through the center of middle centralizer 14b. The proximity detectors 38 measure position of the middle centralizer 14b in the local coordinate system as defined by the vectors connecting leading and trailing centralizers 14a and 14c and the vertical. The CSN device 10 as shown in FIGS. 7a and 7b can have an electronics package, which can include data acquisition circuitry supporting all detectors, including proximity detectors 38, strain gauges 40 (FIG. 8), inclinometers (e.g., the accelerometer 36), etc., and power and communication elements (not shown).
Data reduction can be achieved in a straight beam displacement CSN device 10, as shown in FIG. 7a, as explained below. The explanatory example uses straight beam displacement metrology, capacitance proximity detectors 38, and accelerometer 36 as examples of detectors. The displacements of the middle centralizer 14b in the local coordinate system (x, y, z) defined by the leading and trailing centralizers 14c and 14a are:

dhorizontal=dz cos φ+dy sin φ

dvertical=−dz sin φ+dy cos φ  (Eq. 1)

Where dhorizontal and dvertical are displacements in the vertical and orthogonal planes defined earlier, dz and dy are the displacements measured by the capacitance detectors 38, and as indicated in FIG. 4, φ is the angle of rotation of the capacitance detectors 38 with respect to the vertical as determined by the accelerometer(s) 36. Thus, the centralizer 14 coordinates in the local (x, y, z) coordinate system are:




  
    
      
        
          
            
              u
              ⇀
            
            1
          
          =
          
            [
            
              
                
                  0
                
              
              
                
                  0
                
              
              
                
                  0
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              u
              ⇀
            
            2
          
          =
          
            [
            
              
                
                  
                    
                      
                        L
                        1
                      
                      -
                      
                        d
                        horizontal
                        2
                      
                      -
                      
                        d
                        vertical
                        2
                      
                    
                  
                
              
              
                
                  
                    d
                    horizontal
                  
                
              
              
                
                  
                    d
                    vertical
                  
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              u
              ⇀
            
            3
          
          =
          
            [
            
              
                
                  
                    
                      
                        
                          L
                          1
                          2
                        
                        -
                        
                          d
                          horizontal
                          2
                        
                        -
                        
                          d
                          vertical
                          2
                        
                      
                    
                    +
                    
                      
                        
                          L
                          2
                          2
                        
                        -
                        
                          d
                          horizontal
                          2
                        
                        -
                        
                          d
                          vertical
                          2
                        
                      
                    
                  
                
              
              
                
                  0
                
              
              
                
                  0
                
              
            
            ]
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          2
        
        )
      
    
  




where ui are position of the leading (i=3), trailing(i=1) and middle (i=2) centralizers 14c, 14b, and 14a, respectively, and; L1 and L2 are the distances between the leading and middle 14c and 14b and middle and trailing centralizers 14b and 14a. 

The direction of vector u2 is known in the global coordinate system (X, Y, Z) since the trailing and middle centralizers are located in the known part of the borehole. Therefore, the orientations of axes x, y, and z of the local coordinate system, in the global coordinate system (X, Y, Z) are:




  
    
      
        
          
            x
            ⇀
          
          =
          
            
              
                u
                ⇀
              
              2
            
            
              
              
                
                  u
                  ⇀
                
                2
              
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            z
            ⇀
          
          =
          
            
              
                g
                ⇀
              
              -
              
                
                  g
                  ⇀
                
                ·
                
                  x
                  ⇀
                
              
            
            
              
              
                
                  g
                  ⇀
                
                -
                
                  
                    g
                    ⇀
                  
                  ·
                  
                    x
                    ⇀
                  
                
              
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            y
            ⇀
          
          =
          
            
              z
              ⇀
            
            ⊗
            
              x
              ⇀
            
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          
            g
            ⇀
          
          =
          
            [
            
              
                
                  0
                
              
              
                
                  0
                
              
              
                
                  1
                
              
            
            ]
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          3
        
        )
      
    
  




The displacement of the leading centralizer 14c (FIG. 5b) in the coordinate system as determined by the middle and trailing centralizers 14b and 14a (respectively, FIG. 5b) can be written as:

ūx= x·(ū3−ū2)

ūy= y·(ū3−ū2)

ūs= z·(ū3−ū2)   (Eq. 4)

Calculating u3 in the global coordinate system provides one with the information of the position of the leading centralizer 14c and expands the knowledge of the surveyed borehole 16.

As discussed above, an alternative to the straight beam displacement CSN device 10 is the bending beam CSN device 10, as shown in FIG. 2c and FIG. 4. FIG. 8 shows a CSN device 10 with strain gauge detectors 40 attached to a bending beam 32. The circuit design associated with the resistance strain gauges 40 and accelerometer(s) 36 is shown below the CSN device 10. Any type of strain detector 40 and orientation detector, e.g., accelerometer 36, may be used. Each instrumented sensor string 12 segment 13, here the bending beam 32 (between centralizers 14) of the CSN device 10 can carry up to four, or more, sets of paired strain gauge detectors 40 (on opposite sides of the bending beam 32), each opposing pair forming a half-bridge. These segments 13 may or may not be the same segments 13 that accommodate the capacitance detector 38 if the CSN device 10 utilizes such. In the device 10 shown in FIG. 8, strain gauge detector 40 and accelerometer 36 readings can be recorded simultaneously. A displacement detector supporting odometry correction (Δl) can also be placed on at least one segment 13 (not shown). Several temperature detectors (not shown) can also be placed on each segment 13 to permit compensation for thermal effects.
It is preferred that, in this embodiment, four half-bridges (strain detector 40 pairs) be mounted onto each sensor string segment 13 (between centralizers 14) as the minimum number of strain detectors 40. The circuit diagrams shown below the CSN device 10, with voltage outputs Vz1, Vy1, Vz2, and Vy2, represent an exemplary wiring of these half-bridges. These detectors 40 can provide the relative orientation and relative position of the leading centralizer 14c with respect to the trailing centralizer 14a, or a total of four variables. It is also preferred that at least one of the adjacent sensor string segments 13 between centralizers 14 should contain a detector (not shown) that can detect relative motion of the CSN device 10 with respect to the borehole 16 to determine the actual borehole 16 length when the CSN device 10 and drill string 18 are advanced therein.
Shear forces act on the CSN device 10 consistent with the expected shape shown in FIG. 8 where each subsequent segment 12 can have slightly different curvature (see chart below and corresponding to the CSN device 10). The variation of curvatures of the beam 32 likely cannot be achieved without some shear forces applied to centralizers 14. The preferred strain gauge detector 40 scheme of the CSN device 10 shown in FIG. 8 accounts for these shear forces. The exemplary circuit layout shown below the CSN device 10 and corresponding chart shows how the sensors 40 can be connected.
FIG. 9 illustrates two dimensional resultant shear forces acting on centralizers 14 of a single sensor string segment 13 comprised of a bending bean 32 as shown in FIG. 8. Four unknown variables, namely, two forces and two bending moments, should satisfy two equations of equilibrium: the total force and the total moment acting on the bending beam 32 are equal to zero. FIG. 9 shows the distribution of shear force ( T) and moments ( M) along the length of bending beam 32. The values are related in the following bending equation:




  
    
      
        
          
            
              ∂
              θ
            
            dx
          
          =
          
            M
            
              E
              ·
              I
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            ⇀
          
          =
          
            
              
                M
                ⇀
              
              1
            
            +
            
              
                
                  
                    
                      M
                      ⇀
                    
                    2
                  
                  -
                  
                    
                      M
                      ⇀
                    
                    1
                  
                
                L
              
              ·
              x
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          5
        
        )
      
    
  




Where θ is the angle between the orientation of the beam 32 and the horizontal, E is the Young Modulus of the beam 32 material, I is the moment of inertia, and L is the length of the segment 12 as determined by the locations of centralizers 14.

According to FIG. 9, in a small angle approximation, the orientation of the points along the axis of the segment 12 in each of two directions (y, z) perpendicular to the axis of the beam (x) may be described such that the relative angular orientation of the end points of the segment 12 with respect to each other can be represented by integrating over the length of the segment:




  
    
      
        
          θ
          =
          
            
              
                ∫
                0
                x
              
              ⁢
              
                
                  M
                  
                    E
                    ·
                    I
                  
                
                ⁢
                
                  
                
                ⁢
                
                  d
                  x
                
              
            
            =
            
              
                
                  M
                  1
                
                ·
                
                  
                    ∫
                    0
                    x
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      d
                      x
                    
                    
                      E
                      ·
                      I
                    
                  
                
              
              +
              
                
                  (
                  
                    
                      M
                      2
                    
                    -
                    
                      M
                      1
                    
                  
                  )
                
                ·
                
                  
                    ∫
                    0
                    x
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      x
                      ⁢
                      
                        d
                        x
                      
                    
                    
                      E
                      ·
                      I
                      ·
                      L
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          or
          ,
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          6
        
        )
      
    
  
  
    
      
        θ
        =
        
          
            
              M
              1
            
            ·
            
              
                ∫
                0
                L
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    (
                    
                      L
                      -
                      x
                    
                    )
                  
                  ·
                  
                    d
                    x
                  
                
                
                  E
                  ·
                  I
                  ·
                  L
                
              
            
          
          +
          
            
              M
              2
            
            ·
            
              
                ∫
                0
                L
              
              ⁢
              
                
              
              ⁢
              
                
                  x
                  ·
                  
                    d
                    x
                  
                
                
                  E
                  ·
                  I
                  ·
                  L
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          7
        
        )
      
    
  




The values of the integrals are independent of the values of the applied moments and both integrals are positive numbers. Thus, these equations (Eqs. 6 and 7) can be combined and rewritten as:

θ=M1·Int1θ+M2·Int2θ  (Eq. 8)

where Int1θ and Int2θ are calibration constants for a given sensor string segment 12 such as that shown in FIG. 9).

If two sets of strain gauges 40 (R1, R2 and R3, R4)are placed on the beam 32 (see FIG. 9) at positions x1 and x2 (see charts below drawings in FIG. 9), the readings of these strain gauges 40 are related to the bending moments applied to CSN device 10 segment as follows:




  
    
      
        
          
            ɛ
            1
          
          =
          
            
              
                
                  M
                  ⁡
                  
                    (
                    
                      x
                      1
                    
                    )
                  
                
                ·
                
                  d
                  1
                
              
              
                2
                ·
                E
                ·
                
                  I
                  1
                
              
            
            =
            
              
                
                  d
                  1
                
                
                  2
                  ·
                  E
                  ·
                  
                    I
                    1
                  
                
              
              ·
              
                (
                
                  
                    M
                    1
                  
                  +
                  
                    
                      (
                      
                        
                          M
                          2
                        
                        -
                        
                          M
                          1
                        
                      
                      )
                    
                    ·
                    
                      
                        x
                        1
                      
                      L
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            ɛ
            2
          
          =
          
            
              
                
                  M
                  ⁡
                  
                    (
                    
                      x
                      2
                    
                    )
                  
                
                ·
                
                  d
                  2
                
              
              
                2
                ·
                E
                ·
                
                  I
                  2
                
              
            
            =
            
              
                
                  d
                  2
                
                
                  2
                  ·
                  E
                  ·
                  
                    I
                    2
                  
                
              
              ·
              
                (
                
                  
                    M
                    1
                  
                  +
                  
                    
                      (
                      
                        
                          M
                          2
                        
                        -
                        
                          M
                          1
                        
                      
                      )
                    
                    ·
                    
                      
                        x
                        2
                      
                      L
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          9
        
        )
      
    
  




where I1 and I2 are moments of inertia of corresponding cross-section (of beam 32 at strain gauges 40) where half bridges are installed (FIG. 9), and d1 and d2 are beam diameters at corresponding cross-sections.

If the values of the strain gauge outputs are known, the values of the moments (M) can be determined by solving the preceding Eq. 9. The solution will be:




  
    
      
        
          
            M
            1
          
          =
          
            
              2
              
                
                  d
                  1
                
                ·
                
                  d
                  2
                
              
            
            ·
            
              
                
                  E
                  ·
                  
                    I
                    1
                  
                  ·
                  
                    ɛ
                    1
                  
                  ·
                  
                    x
                    2
                  
                  ·
                  
                    d
                    2
                  
                
                -
                
                  E
                  ·
                  
                    I
                    2
                  
                  ·
                  
                    ɛ
                    2
                  
                  ·
                  
                    x
                    1
                  
                  ·
                  
                    d
                    1
                  
                
              
              
                
                  
                    (
                    
                      L
                      -
                      
                        x
                        1
                      
                    
                    )
                  
                  ·
                  
                    x
                    2
                  
                
                -
                
                  
                    x
                    1
                  
                  ·
                  
                    (
                    
                      L
                      -
                      
                        x
                        2
                      
                    
                    )
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            M
            2
          
          =
          
            
              2
              
                
                  d
                  1
                
                ·
                
                  d
                  2
                
              
            
            ·
            
              
                
                  
                    -
                    E
                  
                  ·
                  
                    I
                    1
                  
                  ·
                  
                    ɛ
                    1
                  
                  ·
                  
                    (
                    
                      L
                      -
                      
                        x
                        1
                      
                    
                    )
                  
                  ·
                  
                    d
                    1
                  
                
                +
                
                  E
                  ·
                  
                    I
                    2
                  
                  ·
                  
                    ɛ
                    2
                  
                  ·
                  
                    (
                    
                      L
                      -
                      
                        x
                        2
                      
                    
                    )
                  
                  ·
                  
                    d
                    1
                  
                
              
              
                
                  
                    (
                    
                      L
                      -
                      
                        x
                        1
                      
                    
                    )
                  
                  ·
                  
                    x
                    2
                  
                
                -
                
                  
                    x
                    1
                  
                  ·
                  
                    (
                    
                      L
                      -
                      
                        x
                        2
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          10
        
        )
      
    
  




which may also be rewritten as:

M1=m1,1·ε1+m1,2·ε2 

M2=m2,1·ε1+m2,2·ε2   (Eq. 11)

where mi,j are calibration constants. Substitution of Eq. 11 into Eq. 8 gives:

θ=ε1·(Int1θ·m1,1+Int2θ·m2,1)+ε2·(Int1θ·m1,2+Int2θ·m2,2)   (Eq. 12)

Similarly, vertical displacement of the leading end of the string segment 12 may be written as:




  
    
      
        
          y
          =
          
            
              
                M
                1
              
              ·
              
                
                  ∫
                  0
                  L
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    x
                  
                  ⁢
                  
                    
                      ∫
                      0
                      x
                    
                    ⁢
                    
                      
                        
                          (
                          
                            L
                            -
                            x
                          
                          )
                        
                        
                          E
                          ·
                          I
                          ·
                          L
                        
                      
                      ·
                      
                        
                      
                      ⁢
                      
                        d
                        x
                      
                    
                  
                
              
            
            +
            
              
                M
                2
              
              ·
              
                
                  ∫
                  0
                  L
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    x
                  
                  ⁢
                  
                    
                      ∫
                      0
                      x
                    
                    ⁢
                    
                      
                        x
                        
                          E
                          ·
                          I
                          ·
                          L
                        
                      
                      ·
                      
                        
                      
                      ⁢
                      
                        d
                        x
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          y
          =
          
            
              
                M
                1
              
              ·
              
                
                  (
                  
                    
                      
                        ∫
                        0
                        L
                      
                      ⁢
                      
                        
                          
                            
                              (
                              
                                L
                                -
                                x
                              
                              )
                            
                            ·
                            L
                          
                          
                            E
                            ·
                            I
                            ·
                            L
                          
                        
                        ·
                        
                          
                        
                        ⁢
                        
                          d
                          x
                        
                      
                    
                    -
                    
                      
                        ∫
                        0
                        L
                      
                      ⁢
                      
                        
                          
                            
                              (
                              
                                L
                                -
                                x
                              
                              )
                            
                            ·
                            x
                          
                          
                            E
                            ·
                            I
                            ·
                            L
                          
                        
                        ·
                        
                          
                        
                        ⁢
                        
                          d
                          x
                        
                      
                    
                  
                  )
                
                ++
              
            
            ⁢
            
              
                M
                2
              
              ·
              
                (
                
                  
                    
                      ∫
                      0
                      L
                    
                    ⁢
                    
                      
                        
                          x
                          ·
                          L
                        
                        
                          E
                          ·
                          I
                          ·
                          L
                        
                      
                      ·
                      
                        
                      
                      ⁢
                      
                        d
                        x
                      
                    
                  
                  -
                  
                    
                      ∫
                      0
                      L
                    
                    ⁢
                    
                      
                        
                          x
                          2
                        
                        
                          E
                          ·
                          I
                          ·
                          L
                        
                      
                      ·
                      
                        
                      
                      ⁢
                      
                        d
                        x
                      
                    
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          y
          =
          
            
              
                M
                1
              
              ·
              
                
                  ∫
                  0
                  L
                
                ⁢
                
                  
                    
                      
                        (
                        
                          L
                          -
                          x
                        
                        )
                      
                      2
                    
                    
                      E
                      ·
                      I
                      ·
                      L
                    
                  
                  ·
                  
                    
                  
                  ⁢
                  
                    d
                    x
                  
                
              
            
            +
            
              
                M
                2
              
              ·
              
                
                  ∫
                  0
                  L
                
                ⁢
                
                  
                    
                      
                        L
                        ·
                        x
                      
                      -
                      
                        x
                        2
                      
                    
                    
                      E
                      ·
                      I
                      ·
                      L
                    
                  
                  ·
                  
                    
                  
                  ⁢
                  
                    d
                    x
                  
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          13
        
        )
      
    
  




As was the case in relation to Eqs. 6 and 7, both integrals of Eq. 13 are positive numbers independent of the value of applied moment. Thus, Eq. 13 may be rewritten as:

y=M1·Int1y+M2·Int2y   (Eq. 14)

and also

y=ε1·(Int1y·m1,1+Int2y·m2,1)+ε2·(Int1y·m1,2+Int2y·m2,2)   (Eq. 15)

Note that the values of mi,j are the same in both Eq. 12 and Eq. 15. In addition, the values of the Int factors satisfy the following relationship:

Int1i +Int2y=L·Int1θ  (Eq. 16)

which may be used to simplify device calibration.

For a bending beam 32 (FIG. 9) with a constant cross-section, the values of the integrals in Eq. 16 are:




  
    
      
        
          
            Int
            1
            θ
          
          =
          
            
              1
              2
            
            ⁢
            
              L
              
                E
                ·
                I
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Int
            2
            θ
          
          =
          
            
              1
              2
            
            ⁢
            
              L
              
                E
                ·
                I
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Int
            1
            y
          
          =
          
            
              1
              3
            
            ⁢
            
              
                L
                2
              
              
                E
                ·
                I
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Int
            2
            y
          
          =
          
            
              1
              6
            
            ⁢
            
              
                L
                2
              
              
                E
                ·
                I
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          17
        
        )
      
    
  




The maximum bending radius that a CSN device 10, as shown in FIG. 9, is expected to see is still large enough to guarantee that the value of the bending angle is less than 3 degrees or 0.02 radian. Since the cos(0.02) ̃0.999, the small angle approximation is valid and Eqs. 6-17 can be used to independently calculate of projections of the displacement of the leading centralizer 14 relative to a trailing centralizer 14 in both “y” and “z” directions of the local coordinate system.
FIG. 10 shows a block diagram for data reduction in a strain gauge CSN device 10, such as that shown in FIG. 9. Calibration of the bending beam 32 of the CSN device 10 should provide coefficients that define angle and deflection of the leading centralizer 14c with respect to the trailing centralizer 14a, as follows:

y=ε1Y·p1Y+ε2Y·p2Y 

z=ε1Z·p1Z+ε2Z·p2Z 

θY=ε1Y·p1Yθ+ε2Y·p2Yθ

θZ=ε1Z·p1Zθ+ε2Z·p2Zθ  (Eq. 18)

where coefficients piα are determined during calibration. These coefficients are referred to as the 4×4 Influence Matrix in FIG. 10. Additional complications can be caused by the fact that the CSN device 10 may be under tension and torsion loads, as well as under thermal loads, during normal usage. Torsion load correction has a general form:





  
    
      
        
          
            [
            
              
                
                  
                    ɛ
                    j
                    Y
                  
                
              
              
                
                  
                    ɛ
                    j
                    Z
                  
                
              
            
            ]
          
          Corrected
        
        =
        
          
            [
            
              
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          p
                          τ
                        
                        ⁢
                        τ
                      
                      )
                    
                  
                
                
                  
                    -
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            p
                            τ
                          
                          ⁢
                          τ
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          p
                          τ
                        
                        ⁢
                        τ
                      
                      )
                    
                  
                
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          p
                          τ
                        
                        ⁢
                        τ
                      
                      )
                    
                  
                
              
            
            ]
          
          ·
          
            [
            
              
                
                  
                    ɛ
                    j
                    Y
                  
                
              
              
                
                  
                    ɛ
                    j
                    Z
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          19
        
        )
      
    
  




where τ is the torsion applied to a CSN device 10 segment 13 as measured by a torsion detector and pτ is a calibration constant. The factors in Eq. 19 are the 2×2 rotation matrix in FIG. 10.

Still referring to FIG. 10, the thermal loads change the values of factors piα. In the first approximation, the values are described by:

pjCorrectdα(1+CTEX·ΔT)·pjα

pjCorrectdαθ(1+CTEθ·ΔT)·pjα  (Eq. 20)

The CTE's are calibration parameters. They include both material and material stiffness thermal dependences. Each value of piα has its own calibrated linear dependence on the axial strain loads, as follows:

pjCorrectdα=(1+Yjα·εX)·pjα

pjCorrectdαθ=(1+Yjαθ·εX)·pjα  (Eq. 21)

The correction factors described in the previous two equations of Eq. 21 are referred to as Correction Factors in FIG. 10.

Now referring to FIG. 11, if the strain gauge detectors 40 can be placed on an axially rotating beam 32 constrained at the centralizers 14 by fixed immovable borehole 16 walls forming a sensor string segment 12. Advantages in greater overall measurement accuracy from CSN device 10 that may be gained by rotating the beam 32 to create a time varying signal related to the amount of bending to which it is subjected may result from, but are not limited to, signal averaging over time to reduce the effects of noise in the signal and improved discrimination bending direction. The signals created by a single bridge of strain gauge detectors 40 will follow an oscillating pattern relative to rotational angle φ and φm, and the value of the strain registered by the strain gauge detectors 40 can be calculated by:

ε(φ)=εmax sin(φ−φm−ψ)=εsin sin(φ)+εcos cos(φ)+εoffset   (Eq. 22)

where φ and φm are defined in FIG. 11 and ψ is the angular location of the strain detector 40.

One can recover the value of the maximum strain and the orientation of the bending plane by measuring the value of the strain over a period of time. Eq. 22 may be rewritten in the following equivalent form:




  
    
      
        
          ɛ
          ⁡
          
            (
            φ
            )
          
        
        =
        
          
            
              [
              
                
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                    
                  
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                    
                  
                
              
              ]
            
            ·
            
              [
              
                
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ψ
                    
                  
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      ψ
                    
                  
                
                
                  
                    
                      
                        -
                        sin
                      
                      ⁢
                      
                        
                      
                      ⁢
                      ψ
                    
                  
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      ψ
                    
                  
                
              
              ]
            
            ·
            
              [
              
                
                  
                    
                      ɛ
                      z
                    
                  
                
                
                  
                    
                      ɛ
                      y
                    
                  
                
              
              ]
            
          
          +
          
            ɛ
            offset
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          23
        
        )
      
    
  




where εz and εy are strain caused by bending correspondingly in the “xz” and “yz” planes indicated in FIG. 11.

Thus, if the value ε(φ) is measured, the values of the εz and εy may be recovered by first performing a least square fit of ε(φ) into sine and cosine. One of the possible procedures is to first determine values of εsin, εcos, and εoffset by solving equations:




  
    
      
        {
        
          
            
              
                
                  ɛ
                  C
                
                =
                
                  
                    
                      ɛ
                      sin
                    
                    ·
                    CC
                  
                  +
                  
                    
                      ɛ
                      cos
                    
                    ·
                    CS
                  
                  +
                  
                    
                      ɛ
                      offset
                    
                    ·
                    C
                  
                
              
            
          
          
            
              
                
                  ɛ
                  S
                
                =
                
                  
                    
                      ɛ
                      sin
                    
                    ·
                    CS
                  
                  +
                  
                    
                      ɛ
                      cos
                    
                    ·
                    SS
                  
                  +
                  
                    
                      ɛ
                      offset
                    
                    ·
                    S
                  
                
              
            
          
          
            
              
                
                  ɛ
                  dc
                
                =
                
                  
                    
                      ɛ
                      sin
                    
                    ·
                    C
                  
                  +
                  
                    
                      ɛ
                      cos
                    
                    ·
                    S
                  
                  +
                  
                    
                      ɛ
                      offset
                    
                    ·
                    T
                  
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          24
        
        )
      
    
  




where:





  
    
      
        
          
            ɛ
            S
          
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              ɛ
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                sin
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                
                  d
                  φ
                
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            ɛ
            C
          
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              ɛ
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                cos
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                
                  d
                  φ
                
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            ɛ
            dc
          
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              ɛ
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                
                  d
                  φ
                
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          CC
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              cos
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                cos
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                
                  d
                  φ
                
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          SC
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              sin
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                cos
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                
                  d
                  φ
                
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          SS
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              sin
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                sin
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                
                  d
                  φ
                
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          C
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              cos
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                φ
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          S
          =
          
            
              ∫
              0
              T
            
            ⁢
            
              sin
              ⁢
              
                
              
              ⁢
              
                
                  (
                  φ
                  )
                
                ·
                φ
              
              ⁢
              
                
              
              ⁢
              
                (
                t
                )
              
            
          
        
      
    
    
      
        (
        
          Eqs
          .
          
            
          
          ⁢
          25
        
        )
      
    
  




The values of εy and εz can be recovered from:





  
    
      
        
          [
          
            
              
                
                  ɛ
                  z
                
              
            
            
              
                
                  ɛ
                  y
                
              
            
          
          ]
        
        =
        
          
            [
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    ψ
                  
                
                
                  
                    
                      -
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    ψ
                  
                
              
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    ψ
                  
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    ψ
                  
                
              
            
            ]
          
          ·
          
            [
            
              
                
                  
                    ɛ
                    sin
                  
                
              
              
                
                  
                    ɛ
                    cos
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          26
        
        )
      
    
  




The matrix in Eq. 26 is an orientation matrix that must be determined by calibrated experiments for each sensor string segment 12.

Now referring to FIG. 12, the block diagram shows a reduction algorithm for the rotating strain gauge 40 data. Since the strain gauge 40 bridges have an unknown offset, Eq. 23 will have a form as follows:

ε(φ)=(εmax+error)·sin(φ−φm−ψ)+offset   (Eq. 27)

Correspondingly, εY and εZ are determined by solving the least square fit into equations Eq. 26, where:





  
    
      
        
          
            ∑
            i
          
          ⁢
          
            
          
          ⁢
          
            error
            i
            2
          
        
        =
        min
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          28
        
        )
      
    
  




In a more general case, where two approximately orthogonal bridges (a and b) are used to measure the same values of εY and εZ, then a more general least square fit procedure may be performed instead of the analytic solution of the least square fit described by Eq. 28 for a single bridge situation. The minimization function is as follows:




  
    
      
        {
        
          
            
              
                
                  
                    
                      
                        
                          ɛ
                          a
                        
                        ⁡
                        
                          (
                          φ
                          )
                        
                      
                      =
                      
                        
                          
                            
                              ɛ
                              max
                            
                            ·
                            sin
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            (
                            
                              φ
                              -
                              
                                φ
                                m
                              
                              -
                              
                                ψ
                                a
                              
                            
                            )
                          
                        
                        +
                        
                          offset
                          
                            
                              
                            
                            ⁢
                            a
                          
                        
                        +
                        
                          error
                          
                            
                              
                            
                            ⁢
                            a
                          
                        
                      
                    
                  
                
                
                  
                    
                      
                        
                          ɛ
                          b
                        
                        ⁡
                        
                          (
                          φ
                          )
                        
                      
                      =
                      
                        
                          
                            
                              ɛ
                              max
                            
                            ·
                            sin
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            (
                            
                              φ
                              -
                              
                                φ
                                m
                              
                              -
                              
                                ψ
                                b
                              
                            
                            )
                          
                        
                        +
                        
                          offset
                          
                            
                              
                            
                            ⁢
                            b
                          
                        
                        +
                        
                          error
                          
                            
                              
                            
                            ⁢
                            b
                          
                        
                      
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
                  ∑
                  i
                
                ⁢
                
                  
                
                ⁢
                
                  
                    (
                    
                      error
                      i
                      a
                    
                    )
                  
                  2
                
              
            
            +
            
              
                (
                
                  error
                  i
                  b
                
                )
              
              2
            
          
          =
          min
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          29
        
        )
      
    
  




where indexes a and b refer to the two bridges (of strain gauge detectors 40, FIG. 9), index i refers to the measurement number, and ψa and ψb are the Gauge Orientation Angles in FIG. 12 and Eq. 29. The Gauge Orientation Angles shown in FIG. 12 are determined by calibrated experiments for each sensor string segment 12.

Now referring to FIG. 13, which relates to the accelerometer 36 described above as incorporated into the CSN device 10 electronics package as discussed in relation to FIGS. 7a and 8. A tri-axial accelerometer 36 can be fully described by the following data where, relative to the Global vertical direction “Z,” each component of the accelerometer has a calibrated electrical output (Gauge factor), a known, fixed spatial direction relative to the other accelerometer 36 components (Orientation), and a measured angle of rotation about its preferred axis of measurement (Angular Location):















Gauge
Angular




factor
Location
Orientation





















Accelerometer X
mV/g
ψyz
NX, NY, NZ



Accelerometer Y
mV/g
ψyz
NX, NY, NZ



Accelerometer Z
mV/g
ψyz
NX, NY, NZ










The coordinate system and the angles are defined in FIG. 13. Based on the definition of the local coordinate system, rotation matrices may be defined as:




  
    
      
        
          
            R
            zy
          
          ⁡
          
            (
            
              φ
              ZY
            
            )
          
        
        =
        
          
          
            
              
                1
              
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      φ
                      ZY
                    
                    )
                  
                
              
              
                
                  
                    -
                    sin
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      φ
                      ZY
                    
                    )
                  
                
              
            
            
              
                0
              
              
                
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      φ
                      ZY
                    
                    )
                  
                
              
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    (
                    
                      φ
                      ZY
                    
                    )
                  
                
              
            
          
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          30
        
        )
      
    
  
  
    
      
        
          
            
              R
              zx
            
            ⁡
            
              (
              
                φ
                ZX
              
              )
            
          
          =
          
            
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        φ
                        ZX
                      
                      )
                    
                  
                
                
                  0
                
                
                  
                    
                      -
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        φ
                        ZX
                      
                      )
                    
                  
                
              
              
                
                  0
                
                
                  1
                
                
                  0
                
              
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        φ
                        ZX
                      
                      )
                    
                  
                
                
                  0
                
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      
                        φ
                        ZX
                      
                      )
                    
                  
                
              
            
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            g
            _
          
          =
          
            
            
              
                
                  0
                
              
              
                
                  0
                
              
              
                
                  
                    -
                    g
                  
                
              
            
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          31
        
        )
      
    
  




Thus, for a CSN device 10 going down a borehole 16 at an angle φYZ=−θ after it has been turned an angle φzy=φ, the readings of the accelerometer 36 located on the circumference of a CSN device 10 can be determined as:




  
    
      
        
          
            a
            =
            
              
                
                
                  
                    N
                    z
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    N
                    y
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    N
                    z
                  
                
                
              
              ·
              
                
                  R
                  zy
                
                ⁡
                
                  (
                  
                    φ
                    +
                    
                      ψ
                      zy
                    
                  
                  )
                
              
              ·
              
                
                  R
                  zx
                
                ⁡
                
                  (
                  
                    -
                    θ
                  
                  )
                
              
              ·
              
                
                
                  
                    
                      0
                    
                  
                  
                    
                      0
                    
                  
                  
                    
                      
                        -
                        g
                      
                    
                  
                
                
              
            
          
          ⁢
          
            

          
          ⁢
          a
          =
          
            
              
              
                
                  N
                  x
                
                ⁢
                
                  
                
                ⁢
                
                  N
                  y
                
                ⁢
                
                  
                
                ⁢
                
                  N
                  z
                
              
              
            
            ·
            
              
              
                
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        θ
                        )
                      
                    
                  
                
                
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          φ
                          +
                          
                            ψ
                            zy
                          
                        
                        )
                      
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        θ
                        )
                      
                    
                  
                
                
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        
                          φ
                          +
                          
                            ψ
                            zy
                          
                        
                        )
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        θ
                        )
                      
                    
                  
                
              
              
            
            ·
            g
          
        
        ⁢
        
          

        
        ⁢
        
          a
          =
          
            
              
                
                  c
                  0
                
                ·
                g
                ·
                sin
              
              ⁢
              
                
              
              ⁢
              
                (
                θ
                )
              
            
            +
            
              
                g
                ·
                cos
              
              ⁢
              
                
              
              ⁢
              
                
                  (
                  θ
                  )
                
                ·
                
                  (
                  
                    
                      
                        
                          c
                          1
                        
                        ·
                        sin
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        φ
                        )
                      
                    
                    +
                    
                      
                        
                          c
                          2
                        
                        ·
                        cos
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        (
                        φ
                        )
                      
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          32
        
        )
      
    
  




where fit parameters c0, c1, and c2 are determined during initial calibration of the tri-axial accelerometer 36 and g is the Earth's gravitational constant. The equations describing all three accelerometer 36 readings will have the following form:





  
    
      
        {
        
          
            
              
                
                  
                    a
                    X
                  
                  g
                
                =
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        (
                        θ
                        )
                      
                      ·
                      
                        (
                        
                          
                            
                              
                                c
                                1
                                X
                              
                              ·
                              sin
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              φ
                              )
                            
                          
                          +
                          
                            
                              
                                c
                                2
                                X
                              
                              ·
                              cos
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              φ
                              )
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      
                        c
                        0
                        X
                      
                      ·
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      θ
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    a
                    Y
                  
                  g
                
                =
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        (
                        θ
                        )
                      
                      ·
                      
                        (
                        
                          
                            
                              
                                c
                                1
                                Y
                              
                              ·
                              sin
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              φ
                              )
                            
                          
                          +
                          
                            
                              
                                c
                                2
                                Y
                              
                              ·
                              cos
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              φ
                              )
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      
                        c
                        0
                        Y
                      
                      ·
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      θ
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    a
                    Z
                  
                  g
                
                =
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        (
                        θ
                        )
                      
                      ·
                      
                        (
                        
                          
                            
                              
                                c
                                1
                                Z
                              
                              ·
                              sin
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              φ
                              )
                            
                          
                          +
                          
                            
                              
                                c
                                2
                                Z
                              
                              ·
                              cos
                            
                            ⁢
                            
                              
                            
                            ⁢
                            
                              (
                              φ
                              )
                            
                          
                        
                        )
                      
                    
                  
                  +
                  
                    
                      
                        c
                        0
                        Z
                      
                      ·
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      θ
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          33
        
        )
      
    
  




For ideal accelerometers 36 with ideal placements ψzy=0, Eq. 33 reduces to:




  
    
      
        
          
            
              a
              X
            
            g
          
          ≈
          
            sin
            ⁢
            
              
            
            ⁢
            
              (
              θ
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              a
              Y
            
            g
          
          ≈
          
            cos
            ⁢
            
              
            
            ⁢
            
              
                (
                θ
                )
              
              ·
              sin
            
            ⁢
            
              
            
            ⁢
            
              (
              φ
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              a
              Z
            
            g
          
          ≈
          
            cos
            ⁢
            
              
            
            ⁢
            
              
                (
                θ
                )
              
              ·
              cos
            
            ⁢
            
              
            
            ⁢
            
              (
              φ
              )
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          34
        
        )
      
    
  




Now referring to FIG. 14, a data reduction algorithm as shown corrects accelerometer 36 readings for zero offset drift and angular velocity. Such an algorithm can be used by a zero drift compensator, including a processor, with a CSN device 10 as shown in FIG. 11, for example. The zero drift compensator works by rotating the CSN device 10. A zero drift compensator can operate by enforcing a rule that the average of the measured value of g be equal to the know value of g at a given time. Alternatively, a zero drift compensator can operate by enforcing a rule that the strain readings of the strain gauges 40 follow the same angular dependence on the rotation of the string 12 as the angular dependence recorded by the accelerometers 36. Alternatively, a zero drift compensator can operate by enforcing a rule that the strain readings of the strain gauges 40 follow a same angular dependence as that measured by angular encoders placed on the drill string 18 (FIG. 1) or sensor string 12.
Because the zero offset of the accelerometers will drift and/or the accelerometers 36 are mounted on a rotating article, a more accurate description of the accelerometer reading would be:

aα=c0α·g·sin(θ)+g·cos(θ)·(c1α·sin(φ)+c2α·cos(φ))+offα+c3α·ω2   (Eq. 35)

where off is the zero offset of the accelerometer, ω is the angular velocity of rotation, and index α refers to the local x, y, and z coordinate system. Equation 35 can be solved for the angles. The solution has a form:





  
    
      
        {
        
          
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        (
                        θ
                        )
                      
                      ·
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      φ
                      )
                    
                  
                  =
                  
                    
                      
                        d
                        1
                        X
                      
                      ·
                      
                        a
                        X
                      
                    
                    +
                    
                      
                        d
                        1
                        Y
                      
                      ·
                      
                        a
                        Y
                      
                    
                    +
                    
                      
                        d
                        1
                        Z
                      
                      ·
                      
                        a
                        Z
                      
                    
                    -
                    
                      
                        d
                        1
                        
                          
                            
                          
                          ⁢
                          ω
                        
                      
                      ·
                      
                        ω
                        2
                      
                    
                  
                
              
            
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        (
                        θ
                        )
                      
                      ·
                      cos
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      φ
                      )
                    
                  
                  =
                  
                    
                      
                        d
                        2
                        X
                      
                      ·
                      
                        a
                        X
                      
                    
                    +
                    
                      
                        d
                        2
                        Y
                      
                      ·
                      
                        a
                        Y
                      
                    
                    +
                    
                      
                        d
                        2
                        Z
                      
                      ·
                      
                        a
                        Z
                      
                    
                    -
                    
                      
                        d
                        2
                        
                          
                            
                          
                          ⁢
                          ω
                        
                      
                      ·
                      
                        ω
                        2
                      
                    
                  
                
              
            
            
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    
                      (
                      θ
                      )
                    
                  
                  =
                  
                    
                      
                        d
                        3
                        X
                      
                      ·
                      
                        a
                        X
                      
                    
                    +
                    
                      
                        d
                        3
                        Y
                      
                      ·
                      
                        a
                        Y
                      
                    
                    +
                    
                      
                        d
                        3
                        Z
                      
                      ·
                      
                        a
                        Z
                      
                    
                    -
                    
                      
                        d
                        2
                        
                          
                            
                          
                          ⁢
                          ω
                        
                      
                      ·
                      
                        ω
                        2
                      
                    
                  
                
              
            
          
           
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          36
        
        )
      
    
  




The values of the twelve constants djα are determined during calibration. Equations 36 are subject to a consistency condition:

cos2(θ)·sin2(φ)+cos2(θ)·cos2(φ)+sin2(θ)=1   (Eq. 37)

The notation may be simplified if one defines variables, as follows:





  
    
      
        {
        
          
            
              
                
                  
                    V
                    i
                    1
                  
                  =
                  
                    
                      
                        d
                        1
                        X
                      
                      ·
                      
                        a
                        i
                        X
                      
                    
                    +
                    
                      
                        d
                        1
                        Y
                      
                      ·
                      
                        a
                        i
                        Y
                      
                    
                    +
                    
                      
                        d
                        1
                        Z
                      
                      ·
                      
                        a
                        i
                        Z
                      
                    
                  
                
              
            
            
              
                
                  
                    V
                    i
                    2
                  
                  =
                  
                    
                      
                        d
                        2
                        X
                      
                      ·
                      
                        a
                        i
                        X
                      
                    
                    +
                    
                      
                        d
                        2
                        Y
                      
                      ·
                      
                        a
                        i
                        Y
                      
                    
                    +
                    
                      
                        d
                        2
                        Z
                      
                      ·
                      
                        a
                        i
                        Z
                      
                    
                  
                
              
            
            
              
                
                  
                    V
                    i
                    3
                  
                  =
                  
                    
                      
                        d
                        3
                        X
                      
                      ·
                      
                        a
                        i
                        X
                      
                    
                    +
                    
                      
                        d
                        3
                        Y
                      
                      ·
                      
                        a
                        i
                        Y
                      
                    
                    +
                    
                      
                        d
                        3
                        Z
                      
                      ·
                      
                        a
                        Z
                      
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            {
            
              
                
                  
                    
                      OF
                      1
                    
                    =
                    
                      
                        
                          d
                          1
                          X
                        
                        ·
                        
                          off
                          X
                        
                      
                      +
                      
                        
                          d
                          1
                          Y
                        
                        ·
                        
                          off
                          Y
                        
                      
                      +
                      
                        
                          d
                          1
                          Z
                        
                        ·
                        
                          off
                          Z
                        
                      
                    
                  
                
              
              
                
                  
                    
                      OF
                      2
                    
                    =
                    
                      
                        
                          d
                          2
                          X
                        
                        ·
                        
                          off
                          X
                        
                      
                      +
                      
                        
                          d
                          2
                          Y
                        
                        ·
                        
                          off
                          Y
                        
                      
                      +
                      
                        
                          d
                          2
                          Z
                        
                        ·
                        
                          off
                          Z
                        
                      
                    
                  
                
              
              
                
                  
                    
                      OF
                      3
                    
                    =
                    
                      
                        
                          d
                          3
                          X
                        
                        ·
                        
                          off
                          X
                        
                      
                      +
                      
                        
                          d
                          3
                          Y
                        
                        ·
                        
                          off
                          Y
                        
                      
                      +
                      
                        
                          d
                          3
                          Z
                        
                        ·
                        
                          off
                          Z
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          38
        
        )
      
    
  




where index i refers to each measurement performed by the accelerometers. Note that offsets OF1, OF2, OF3 are independent of measurements and do not have index i. Consistency condition Eq. 37 can be rewritten as:

(Vi1−OF1−d1ω·ω2)2+(Vi2−OF2−d2ω·ω2)2+(Vi3−OF3−d3ω·ω2)2=1   (Eq. 39)

Since ω is small and the value of cos(θ)≈1, the value of ω is determined using:




  
    
      
        
          
            
              
                ω
                2
              
              =
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          ∂
                          
                            V
                            i
                            l
                          
                        
                        
                          ∂
                          t
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          ∂
                          
                            V
                            i
                            2
                          
                        
                        
                          ∂
                          t
                        
                      
                      )
                    
                    2
                  
                
                
                  
                    cos
                    2
                  
                  ⁡
                  
                    (
                    
                      θ
                      i
                    
                    )
                  
                
              
            
          
        
        
          
            
              ≈
              
              ⁢
              
                
                  
                    
                      (
                      
                        
                          ∂
                          
                            V
                            i
                            1
                          
                        
                        
                          ∂
                          t
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          ∂
                          
                            V
                            i
                            2
                          
                        
                        
                          ∂
                          t
                        
                      
                      )
                    
                    2
                  
                
                
                  1
                  -
                  
                    
                      (
                      
                        V
                        i
                        3
                      
                      )
                    
                    2
                  
                
              
              ≈
              
                
                  
                    (
                    
                      
                        ∂
                        
                          V
                          i
                          1
                        
                      
                      
                        ∂
                        t
                      
                    
                    )
                  
                  2
                
                +
                
                  
                    (
                    
                      
                        ∂
                        
                          V
                          i
                          2
                        
                      
                      
                        ∂
                        t
                      
                    
                    )
                  
                  2
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          40
        
        )
      
    
  




The necessity for any correction for cos(θ)≠1 must be determined experimentally to evaluate when deviation from this approximation becomes significant for this application.
Since the accelerometers 36 have a zero offset that will change with time, equation 40 will not be satisfied for real measurements. The value of offsets OF1, OF2, OF3, are determined by the least square fit, i.e., by minimizing, as follows:




  
    
      
        
          
            
              
                min
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    
                      ∑
                      i
                    
                    ⁢
                    
                      [
                      
                        
                          
                            (
                            
                              
                                V
                                i
                                1
                              
                              -
                              
                                OF
                                1
                              
                              -
                              
                                
                                  d
                                  1
                                  ω
                                
                                ·
                                
                                  ω
                                  2
                                
                              
                            
                            )
                          
                          2
                        
                        +
                      
                    
                  
                
              
            
          
        
        ⁢
        
          
            
              
                
                  
                    
                      
                        
                          (
                          
                            
                              V
                              i
                              2
                            
                            -
                            
                              OF
                              2
                            
                            -
                            
                              
                                d
                                2
                                ω
                              
                              ·
                              
                                ω
                                2
                              
                            
                          
                          )
                        
                        2
                      
                      +
                      
                        
                          (
                          
                            
                              V
                              i
                              3
                            
                            -
                            
                              OF
                              3
                            
                            -
                            
                              
                                d
                                3
                                ω
                              
                              ·
                              
                                ω
                                2
                              
                            
                          
                          )
                        
                        2
                      
                      -
                      1
                    
                    ]
                  
                  2
                
                )
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          41
        
        )
      
    
  




Once the values of the offsets OF1, OF2, OF3 are determined, the rotation angle can be defined as:




  
    
      
        
          
            sin
            ⁡
            
              (
              
                φ
                i
              
              )
            
          
          =
          
            
              
                V
                i
                1
              
              -
              
                OF
                1
              
              -
              
                
                  d
                  1
                  ω
                
                ·
                
                  ω
                  2
                
              
            
            
              
                
                  
                    (
                    
                      
                        V
                        i
                        1
                      
                      -
                      
                        OF
                        1
                      
                      -
                      
                        
                          d
                          1
                          ω
                        
                        ·
                        
                          ω
                          2
                        
                      
                    
                    )
                  
                  2
                
                +
                
                  
                    (
                    
                      
                        V
                        i
                        2
                      
                      -
                      
                        OF
                        2
                      
                      -
                      
                        
                          d
                          2
                          ω
                        
                        ·
                        
                          ω
                          2
                        
                      
                    
                    )
                  
                  2
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            cos
            ⁡
            
              (
              
                φ
                i
              
              )
            
          
          =
          
            
              
                V
                i
                2
              
              -
              
                OF
                2
              
              -
              
                
                  d
                  2
                  ω
                
                ·
                
                  ω
                  2
                
              
            
            
              
                
                  
                    (
                    
                      
                        V
                        i
                        1
                      
                      -
                      
                        OF
                        1
                      
                      -
                      
                        
                          d
                          1
                          ω
                        
                        ·
                        
                          ω
                          2
                        
                      
                    
                    )
                  
                  2
                
                +
                
                  
                    (
                    
                      
                        V
                        i
                        2
                      
                      -
                      
                        OF
                        2
                      
                      -
                      
                        
                          d
                          2
                          ω
                        
                        ·
                        
                          ω
                          2
                        
                      
                    
                    )
                  
                  2
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          42
        
        )
      
    
  




When values of the offsets OF1, OF2, OF3 are known, the values of offsets of individual accelerometers 36 and the values of φi and cos(θi) can be determined.
Now referring to FIGS. 15-17, each of which shows a universal joint angle measurement sensor 50, which is an alternative embodiment to the strain gauge displacement CSN device 10 embodiments discussed above in relation to, e.g., FIGS. 2c and 8. As shown in FIG. 15, the universal joint 50 can be cylindrical in shape to fit in a borehole 16 or tube and is comprised of two members 56 joined at two sets of opposing bendable flexures 54 such that the joint 50 may bend in all directions in any plane orthogonal to its length. The bendable flexures 54 are radially positioned with respect to an imaginary center axis of the universal joint 50. Each one of the two sets of bendable flexures 54 allows for flex in the joint 50 along one plane along the imaginary center axis. Each plane of flex is orthogonal to the other, thus allowing for flex in all directions around the imaginary center axis. The strain forces at the bendable flexures 54 are measured in much the same way as those on the strain gauge detectors 40 of the CSN device 10 of FIG. 8 using detectors 52. Spatial orientation of universal joint 50 relative to the vertical may be measured by a tri-axial accelerometer 57 attached to the interior of universal joint 50.
The universal joint 50 may be connected to a middle centralizer 14b of a CSN device 10 as shown in FIG. 16. A spring 58 can be used to activate the centralizer 14b (this will be explained in further detail below with reference to FIGS. 19-20b). The universal joint 50 and middle centralizer 14b are rigidly attached to each other and connected with arms 44 to leading and trailing centralizers 14a and 14c. 
As shown in FIG. 17, the universal joint 50, when located on a CSN device 10 for use as a downhole tool for survey and/or navigation, is positioned at or near a middle centralizer 14b of three centralizers 14. The two outer centralizers 14a and 14c are connected to the universal joint 50 by arms 44, as shown in FIG. 17, which may house electronics packages if desired. The universal joint 50 includes strain gauges 52 (FIG. 15) to measure the movement of the joint members 56 and arms 44.
As discussed above, the CSN device 10 of the various embodiments of the invention is used for the survey of boreholes 16 or passageways and navigation of downhole devices; the goal of the navigation algorithm (FIG. 6) is to determine relative positions of the centralizers 14 of the CSN device 10 and to determine the borehole 16 location of the CSN device 10 based on that data. Now referring to FIG. 18, which is a block diagram of the assembly of a CSN device 10, the first local coordinate system (#1) has coordinate vectors as follows:




  
    
      
        
          
            X
            ⇀
          
          ⁢
          
            
          
          =
          
            [
            
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
              
              
                
                  0
                
              
              
                
                  
                    
                      -
                      sin
                    
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            Y
            ⇀
          
          =
          
            [
            
              
                
                  0
                
              
              
                
                  1
                
              
              
                
                  0
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            Z
            ⇀
          
          =
          
            [
            
              
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
              
              
                
                  0
                
              
              
                
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            g
            ⇀
          
          =
          
            [
            
              
                
                  0
                
              
              
                
                  0
                
              
              
                
                  
                    -
                    1
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          43
        
        )
      
    
  




where cosθ is determined by the accelerometers 57 and g is the Earth gravity constant. Given a local coordinate system (FIGS. 5a-5d) with point of origin ri and orientation of x-axis Xi ↑↑ āi, and the length L of an arm 44, the orientation of axis would be:





  
    
      
        {
        
          
            
              
                
                  
                    X
                    ⇀
                  
                  i
                
                =
                
                  
                    
                      a
                      ⇀
                    
                    i
                  
                  
                    
                    
                      
                        a
                        ⇀
                      
                      i
                    
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      Z
                      ⇀
                    
                    ⁢
                    
                      
                    
                  
                  i
                
                =
                
                  
                    
                      -
                      
                        g
                        ⇀
                      
                    
                    +
                    
                      
                        
                          X
                          ⇀
                        
                        i
                      
                      ·
                      
                        (
                        
                          
                            
                              X
                              ⇀
                            
                            i
                          
                          ·
                          
                            g
                            ⇀
                          
                        
                        )
                      
                    
                  
                  
                    
                    
                      
                        -
                        
                          g
                          ⇀
                        
                      
                      +
                      
                        
                          
                            X
                            ⇀
                          
                          i
                        
                        ·
                        
                          (
                          
                            
                              
                                X
                                ⇀
                              
                              i
                            
                            ·
                            
                              g
                              ⇀
                            
                          
                          )
                        
                      
                    
                    
                  
                
              
            
          
          
            
              
                
                  
                    
                      Y
                      ⇀
                    
                    ⁢
                    
                      
                    
                  
                  i
                
                =
                
                  
                    
                      Z
                      ⇀
                    
                    i
                  
                  ⨯
                  
                    
                      X
                      ⇀
                    
                    i
                  
                
              
            
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          44
        
        )
      
    
  




Referring again to FIG. 5d, which shows the local coordinate system previously discussed above, the reading of strain gauges, e.g., 52 as shown in FIG. 15, provide the angles θy, θz of the CSN device 10 segment leading centralizer 14c position in the local coordinate system. Correspondingly, the origin of the next coordinate system and the next centralizer 14b would be:




  
    
      
        
          
            r
            ⇀
          
          
            i
            +
            1
          
        
        =
        
          
            
              r
              ⇀
            
            i
          
          +
          
            
              
                X
                ⇀
              
              i
            
            ⁡
            
              (
              
                
                  L
                  i
                
                -
                
                  
                    2
                    3
                  
                  ·
                  
                    
                      
                        y
                        i
                        2
                      
                      +
                      
                        z
                        i
                        2
                      
                    
                    
                      L
                      i
                    
                  
                
              
              )
            
          
          +
          
            
              
                Y
                ⇀
              
              i
            
            ·
            
              y
              i
            
          
          +
          
            
              
                Z
                ⇀
              
              i
            
            ·
            z
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          45
        
        )
      
    
  




The orientation of the next coordinate system will be defined by Eq. 46 where the new vectors are:




  
    
      
        
          
            
              
                a
                ⇀
              
              ⁢
              
                
              
            
            
              i
              +
              1
            
          
          =
          
            
              a
              i
            
            +
            
              
                tan
                ⁡
                
                  (
                  
                    θ
                    i
                    Y
                  
                  )
                
              
              ·
              
                
                  Y
                  ⇀
                
                i
              
            
            +
            
              
                tan
                ⁡
                
                  (
                  
                    θ
                    i
                    Z
                  
                  )
                
              
              ·
              
                
                  Z
                  ⇀
                
                i
              
            
          
        
        ⁢
        
          

        
        ⁢
        and
        ⁢
        
          

        
        ⁢
        
          
            g
            ⇀
          
          =
          
            [
            
              
                
                  0
                
              
              
                
                  0
                
              
              
                
                  
                    -
                    1
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        
          Eq
          .
          
            
          
          ⁢
          46
        
        )
      
    
  




Using Eq. 45 and 46, one can define the origin and the orientation of the CSN device 10 portion in the unknown region of a borehole 16 in the first local coordinate system. After applying equations 45 and 46 to all CSN device 10 segments 13, the location of the CSN device 10 portion in the unknown region of a borehole 16 is determined. The shape of the CSN device 10 is defined up to the accuracy of the strain gauges 40 or 52. The inclination of the CSN device 10 with respect to the vertical is defined within the accuracy of the accelerometers 36 or 57. The azimuth orientation of the CSN device 10 is not known.
Now referring to FIGS. 19, 20a, and 20b, embodiments of centralizers for use with CSN devices 10 are shown. As previously discussed, centralizers 14 are used to accurately and repeatably position the metrology sensors 28 (FIG. 1) discussed above within a borehole 16. Additionally, the centralizer 14 has a known pivot point 60 that will not move axially relative to the metrology article to which it is attached. The centralizer 14 is configured to adapt straight line mechanisms to constrain the centralizer 14 pivot point 60 to axially remain in the same lateral plane. This mechanism, sometimes referred to as a “Scott Russell” or “Evan's” linkage, is composed of two links, 64 as shown in FIG. 19, and 64a and 64b as shown in FIGS. 20a and 20b. The shorter link 64b of FIGS. 20a and 20b has a fixed pivot point 60b, while the longer link 64a has a pivot point 60a free to move axially along the tube housing 34. The links 64a and 64b are joined at a pivot point 66, located half-way along the length of the long link 64a, while the short link 64b is sized so that the distance from the fixed point 60b to the linked pivot 66 is one half the length of the long link 64a. 
This centralizer 14 mechanism is formed by placing a spring 68 behind the sliding pivot point 60a, which provides an outward forcing load on the free end of the long link 64a. This design can use roller bearings at pivot points, but alternatively they could be made by other means, such as with a flexure for tighter tolerances, or with pins in holes if looser tolerances are allowed. A roller 62 is positioned at the end of the long link 64a to contact the borehole 16 wall.
According to this centralizer 14 concept, all pivot points are axially in line with the pivot point 60b of the short link 64b, and thus, at a known location on the CSN device 10. Additionally, this mechanism reduces the volume of the centralizer 14. FIG. 19 shows a centralizer 14 embodiment with a double roller, fixed pivot point 60. This embodiment has two spring-loaded 68 rollers 62 centered around a fixed pivot point 60. FIGS. 20a and 20b have a single roller structure, also with a single fixed pivot point 60, but with one spring-loaded 68 roller 62.
In an alternative embodiment of the invention, a device is utilized for canceling the effects of gravity on a mechanical beam to mitigate sag. As shown in FIGS. 21a and 21b, using buoyancy to compensate for gravity-induced sag of a metrology beam of a CSN device 10 having a proximity-detector-based or angular-metrology-based displacement sensor string, accuracy of the survey or navigation can be improved. As shown in FIG. 21a, an angle measuring metrology sensor CSN device 10 can enclose the sensor string segments 13 within a housing 34 containing a fluid 81. This fluid 81 provides buoyancy for the segments 13, thus mitigating sag. Alternatively, as shown in FIG. 21b, a displacement measuring metrology sensor CSN device 10 can likewise encase its straight beam 31 within a fluid 81 filled housing 34. In this way, sagging of the straight beam 31 is mitigated and with it errors in displacement sensing by the capacitor sensor 38 are prevented.
Various embodiments of the invention have been described above. Although this invention has been described with reference to these specific embodiments, the descriptions are intended to be illustrative of the invention and are not intended to be limiting. Various modifications and applications may occur to those skilled in the art without departing from the true spirit and scope of the invention as defined in the appended claims.

## Claims

1. A metrology device, comprising:
at least one sensor string segment;
at least three centralizers;
at least one metrology sensor; and
at least one odometry device; and
wherein said metrology sensor is a displacement detector; and
wherein said displacement detector is configured to measure the displacement of a straight beam relative to one of said at least three centralizers; and
wherein said straight beam is fixed to a first centralizer and a third centralizer, where said one of said three centralizers is a second centralizer between said first and second centralizers.

2. The metrology device of claim 1, wherein said displacement detector comprises a capacitance proximity detector.

3. The metrology device of claim 2, wherein said capacitance proximity detector comprises a plurality of capacitor plates.

4. A metrology sensing device, comprising:
at least three centralizers;
a beam connecting at least two of said at least three centralizers;
a metrology sensor associated with said beam, said metrology sensor being located between said centralizers;
a housing encasing said beam and said metrology sensor; and
fluid within said housing and at least partially supporting said beam.

5. The metrology sensing device of claim 4, wherein said metrology sensor is an angle measuring sensor.

6. The metrology sensing device of claim 4, wherein said metrology sensor is a displacement sensor.