Patent Publication Number: US-2022228934-A1

Title: Hanger bearing mounted torque sensor

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of and claims priority to International Patent Application No. PCT/US2020/055018, which was filed on Oct. 9, 2020, and which claimed priority to U.S. Provisional Patent Application Ser. No. 62/912,900, filed Oct. 9, 2019, the disclosures of which are incorporated by reference herein in their entireties. 
    
    
     TECHNICAL FIELD 
     The subject matter disclosed herein relates to torque measurement, as well as associated methods of use and manufacture thereof. 
     BACKGROUND 
     Methods for torque measurement using variable reluctance (VR) sensors to measure twist across a shaft segment are well-known. Typically, a reference tube is used in conjunction with ferrous target teeth to assess twist across a length of shaft. Variable reluctance (VR) sensors are employed to measure changes in the timing of pulses produced by the passage of the ferrous targets. Twist in the shaft can be related to the relative change in pulse timing. Then, by knowing the torsional spring rate of the shaft, torque can be derived from twist. 
     Existing torque sensing systems are known to be, in many applications, prohibitively complicated, expensive, and large when there exists a large gap between the target region (e.g., the outer circumferential surface) of the shaft and the sensor(s), for example, a gap above about 0.5 inches. In designing such known torque sensing systems, the maximum radial deflection of the shaft at the location along its length where the sensor is to be located must be determined and the sensor must be positioned at least that distance away from the shaft in the radial direction in order to ensure that the shaft does not strike the sensor during operation. An algorithm may be used in aiding in accuracy of such known torque sensing systems by compensating for all of the various motions of the shaft, relative to the sensor, during normal operation. This radial gap between the sensor(s) and the rotating shaft is the predominant factor associated with the inherent inaccuracy of such known systems. One presently known solution for reducing the gap between the sensors and the shaft is to stiffen the drivetrain and the structure to which the drivetrain is attached (e.g., a fixed structure, such as a frame) so that the gap between the fixed frame components (e.g., the sensors and fixed structure) and the rotating frame components (e.g., the shaft) is reduced. However, this is typically not a priority in aircraft design or may pose integration challenges. 
     Many different Torque technologies are limited by the operating gap between the fix frame sensor and the rotating shaft. VR sensor/target Technology has a large sensor amplitude and phase changes over operating gap variations which must be compensated for in order for such systems to be operable. Magnetoelastic torque sensors mildly magnetize the shaft material and would also see sensor coil amplitude dramatically change over operating gaps, and perhaps not work at all at large gaps. Magnetic ring phase shift technology using fixed frame sensing coils would also see large amplitude and phase changes over operating gap variations, which must be compensated for in order for such systems to be operable. Strain gauge base sensors need to communicate data across the operating gap with technologies such as near field communication, as well as transmit power across to the rotating frame. Near field communication and power transmission can be difficult to manage at gaps above 0.5 inches. 
     There is a need to provide highly accurate twist measurement on a rotating shaft as well as multi-axis shaft motion with a light-weight and minimally invasive solution. Monopole VR sensor-based solutions are light weight and minimally invasive but have limitations in terms of provided twist measurement accuracy. Conventional multi-plane sensing solutions are able to provide high twist accuracy as well as measurement of additional shaft motions, but typically require more than six VR sensors disposed across multiple measurement planes and can present integration challenges. The presently disclosed subject matter is directed towards a torque sensing architecture that addresses such disadvantages known from conventional systems. 
     SUMMARY 
     The presently disclosed subject matter provides devices, systems, and/or methods for accurately measuring safety critical torque and speed (e.g., angular speed) of a flexible drivetrain suspended in a compliant manner from a sufficiently fixed structure in and/or on which the flexible drivetrain is installed and/or associated with. As used here, the term “fixed structure is generally used to refer to a frame element or other structural element that is rigidly attached to and/or integrally formed with, the structure in and/or on which the flexible drivetrain is installed. Non-limiting examples of structures in which such a flexible drivetrain may be installed include aircraft (e.g., helicopters, airplanes, and the like), boats/ships, motor vehicles, and heavy machinery, which can be mobile and/or stationary. In some embodiments, the flexible drivetrain is suspended by hanger bearings compliantly mounted to the fixed structure, such that the hanger bearings, as well as the drivetrain components to which the hanger bearings are attached, are capable of movement relative to the fixed structure. 
     By mounting a sensor to a frame rigidly attached to a bearing supporting the drivetrain, the bearing being substantially incapable of radial movement relative to the drivetrain component to which it is attached, the gap (e.g., as measured in the radial direction of the shaft or other rotary component of the flexible drivetrain) between the sensor (e.g., attached to the bearing frame) and a target region of the drivetrain is minimized (e.g., on the order of about 0.02 inches) compared to gaps between sensors and the rotatable shaft in conventionally known torque sensing systems. For such conventionally known devices, systems, and methods using flexible drivetrains, the gaps between the drivetrain and non-hanger bearing components (e.g., the sensor, or sensors) can be on the order of 0.5 inches or more, which leads to significant error in such conventionally known torque sensing devices, systems, and methods. As such, the significant reduction of the gap between the sensor(s) and the target region allows the presently disclosed devices, systems, and methods to be considerably more accurate than such conventionally known devices, systems, and/or methods. While minimizing the gap between the shaft and the sensor(s) removes the error associated with the presence of the gap, the fact that the gap is so small is also known to potentially introduce some errors due to the fact that minimal shaft motion can be tolerated, resulting in a smaller twisting section of the shaft (e.g., less than 0.2 degrees), which means that the twist measurement must be extremely precise to determine torque with such a small amount of total twist. 
     In an example embodiment, a system for sensing torque in a rotatable shaft is provided. According to this example embodiment, the system comprises: a target region extending along at least a portion of a length of the shaft; at least one sensor configured to measure a torque transmitted through the shaft over the target region; a bearing having an inner race and an outer race, the inner race being supported by, and in contact with, an outer surface of the shaft, such that the inner race and the shaft are rotatably locked together; a frame fixedly mounted to the outer race of the bearing, such that the frame maintains a substantially constant radial distance from the shaft; and a compliant mount configured to attach the frame to a fixed structure, such that the frame is configured to move substantially in unison with the shaft, relative to the fixed structure, in at least two dimensions, the at least two dimensions being in a plane perpendicular to a longitudinal axis of the shaft; wherein the shaft is configured to rotate relative to the frame; and wherein the at least one sensor is rigidly attached to the frame, such that a gap between the at least one sensor and the outer surface of the shaft in the target region is substantially constant. 
     In some embodiments of the system, the target region comprises a first set of target elements and a second set of target elements, wherein the first set of target elements are attached to the outer surface of the shaft at a first position, wherein the second set of target elements are attached to the outer surface of the shaft at a second position, wherein the first and second positions are spaced apart, within the target region, from each other along the longitudinal axis of the shaft, wherein the first and second sets of target elements are interleaved with each other, and wherein the at least one sensor is configured to measure a distance between adjacent target elements of the first and/or second sets of target elements. 
     In some embodiments of the system, the first set of target elements extend towards the second set of target elements, such that at least a portion of each target element of the first set of target elements is positioned within a same plane as the second set of target elements. 
     In some embodiments of the system, the first set of target elements and the second set of target elements extend in a same direction and overlap each other at the at least one sensor. 
     In some embodiments of the system, the plane is substantially perpendicular to the longitudinal axis of the shaft and defines a deflection region, where the at least one sensor is fixedly positioned to measure the distance between adjacent target elements of the first and second sets of target elements. 
     In some embodiments of the system, the first and second sets of target elements extend towards each other along the longitudinal axis of the shaft and overlap each other in a deflection region, which is between the first and second positions, such that at least a portion of each target element of the first and second sets of target elements is positioned within a same plane. 
     In some embodiments of the system, the plane is substantially perpendicular to the longitudinal axis of the shaft, and wherein the at least one sensor is fixedly positioned to measure the distance between adjacent target elements of the first and second sets of target elements. 
     In some embodiments of the system, the at least one sensor comprises a variable reluctance (VR) sensor. 
     In some embodiments of the system, the VR sensor is configured to detect the distance between adjacent target elements of the first and second sets of target elements induced upon torsional deformation of the shaft. 
     In some embodiments of the system, the first and second sets of target elements are interleaved in an alternating pattern, and wherein the distance between adjacent target elements is measured in the circumferential direction of the shaft. 
     In some embodiments of the system, the at least one sensor comprises a plurality of VR sensors spaced apart from each other circumferentially around the shaft. 
     In some embodiments of the system, the compliant mount is configured such that the shaft, the bearing, and the frame are movable in at least three dimensions relative to the fixed structure. 
     In some embodiments of the system, the target region comprises a first set of target elements and a second set of target elements, each of which are arranged about the shaft in a circumferential direction thereof, wherein the first set of target elements are on the outer surface of the shaft at a first position, wherein the second set of target elements are attached to the outer surface of the shaft at a second position. 
     In some embodiments of the system, the at least one sensor comprises at least a first sensor and a second sensor, both of which are variable reluctance (VR) sensors. 
     In some embodiments of the system, the first sensor is attached to the frame over the first position, wherein the second sensor is attached to the frame over the second position, and wherein the system is configured to detect a change in relative position in the circumferential direction between the first and second sets of target elements induced upon torsional deformation of the shaft. 
     In some embodiments of the system, the first position and the second position are spaced apart by a majority of a length of the shaft. 
     In some embodiments of the system, the first sensor is rigidly attached to the frame, such that the first sensor is positioned over the first position; a second bearing is attached to the shaft, adjacent the second position; a second frame is mounted to the second bearing in a fixed manner, such that the second frame maintains a substantially constant radial distance from the shaft; the second sensor is rigidly attached to the second frame, such that the second sensor is positioned over the second position; and the system is configured to detect a change in relative position in the circumferential direction between the first and second sets of target elements induced upon torsional deformation of the shaft. 
     In some embodiments of the system, the bearing comprises an inner race and an outer race, the inner race being supported by, and in contact with, the outer surface of the shaft, such that the inner race of the second bearing and the shaft are rotatably locked together. 
     In some embodiments, the system comprises a second compliant mount that attaches the second frame to the fixed structure, such that the second frame is movable, substantially in unison with the shaft, relative to the fixed structure, in at least two dimensions, the at least two dimensions being in a plane perpendicular to the longitudinal axis of the shaft. 
     In some embodiments of the system, the second frame is attached to the fixed structure via the compliant mount, such that the second frame is movable, substantially in unison with the shaft, relative to the fixed structure, in at least two dimensions, the at least two dimensions being in a plane perpendicular to the longitudinal axis of the shaft. 
     In some embodiments of the system, the target region is a magnetized portion of the outer surface of the shaft configured to generate a magnetic field, and wherein the at least one sensor is configured to detect a change in the magnetic field induced by shear within the outer surface of the shaft, the shear corresponding to torsional deformation of the shaft over at least a portion of the target region due to twisting. 
     In some embodiments of the system, the at least one sensor is configured to detect the change in the magnetic field when the shaft is substantially stationary. 
     In some embodiments of the system, the bearing comprises a redundant bearing. 
     In some embodiments of the system, the target region comprises a first set of target elements and a second set of target elements; the first set of target elements comprise magnets that are attached to the outer surface of the shaft at a first position and are spaced about the shaft in the circumferential direction such that adjacent magnets of the first set of target elements have different polarities from each other; the second set of target elements comprise magnets that are attached to the outer surface of the shaft at a second position and are spaced about the shaft in the circumferential direction such that adjacent magnets of the second set of target elements have different polarities from each other; the first and second positions are spaced apart, within the target region, from each other along the longitudinal axis of the shaft; the at least one sensor comprises at least a first sensor, which is arranged at the first position to detect a magnetic field produced by the magnets of the first set of target elements, and a second sensor, which is arranged at the second position to detect a magnetic field produced by the magnets of the second set of target elements; and the system is configured to determine, based on a relative phase shift of the magnetic fields produced by the magnets of the first and second sets of target elements due to a torsional deformation of the shaft between the first and second sets of target elements, respectively, the torque being transmitted through the rotatable shaft. 
     In some embodiments of the system, the magnets of the first set of target elements are adjacent to each other to form a ring of magnets about the shaft at the first position and/or wherein the magnets of the second set of target elements are adjacent to each other to form a ring magnets about the shaft at the second position. 
     In some embodiments of the system, the magnets of the first set of target elements are in direct contact with each other to form a substantially continuous and uninterrupted ring of magnets about the shaft at the first position and/or wherein the magnets of the second set of target elements are in direct contact with each other to form a substantially continuous and uninterrupted ring of magnets about the shaft at the second position. 
     According to another example embodiment, a method for sensing torque in a rotatable shaft is provided. According to this example embodiment, the method comprises: providing a target region extending along at least a portion of a length of the shaft; attaching a bearing to the shaft, the bearing having an inner race and an outer race, wherein the inner race is supported by, and in contact with, an outer surface of the shaft, such that the inner race and the shaft are rotatably locked together; mounting a frame to the outer race of the bearing in a fixed manner, such that the frame maintains a substantially constant radial distance from the shaft; attaching, via a compliant mount, the frame to a fixed structure, such that the frame is movable, substantially in unison with the shaft, relative to the fixed structure, in at least two dimensions, the at least two dimensions being in a plane perpendicular to a longitudinal axis of the shaft; rigidly attaching at least one sensor to the frame, such that a gap between the at least one sensor and the outer surface of the shaft in the target region is substantially constant; and measuring a torsional deformation of the shaft over the target region. 
     In some embodiments, the method comprises: providing a first set of target elements in and/or on the outer surface of the shaft at a first position within the target region; and providing a second set of target elements in and/or on the outer surface of the shaft at a second position within the target region; wherein the first and second positions are spaced apart, within the target region, from each other along the longitudinal axis of the shaft; and wherein the first and second sets of target elements are interleaved with each other. 
     In some embodiments, the method comprises measuring, using the at least one sensor, a distance between adjacent target elements of the first and/or second sets of target elements. 
     In some embodiments of the method, the first set of target elements extend towards the second set of target elements, such that at least a portion of each target element of the first set of target elements is positioned within a same plane as the second set of target elements. 
     In some embodiments of the method, the plane is substantially perpendicular to the longitudinal axis of the shaft and defines a deflection region, where the at least one sensor is fixedly positioned to measure the distance between adjacent target elements of the first and second sets of target elements. 
     In some embodiments of the method, the first and second sets of target elements extend towards each other along the longitudinal axis of the shaft and overlap each other in a deflection region, which is between the first and second positions, such that at least a portion of each target element of the first and second sets of target elements is positioned within a same plane. 
     In some embodiments of the method, the plane is substantially perpendicular to the longitudinal axis of the shaft, and wherein the at least one sensor is fixedly positioned to measure the distance between adjacent target elements of the first and second sets of target elements. 
     In some embodiments of the method, the first set of target elements and the second set of target elements extend in a same direction and overlap each other at the at least one sensor. 
     In some embodiments of the method, the at least one sensor comprises a variable reluctance (VR) sensor. 
     In some embodiments, the method comprises detecting, using the VR sensor, the distance between adjacent target elements of the first and second sets of target elements induced upon torsional deformation of the shaft. 
     In some embodiments of the method, the first and second sets of target elements are interleaved in an alternating pattern, and wherein the distance between adjacent target elements is measured in the circumferential direction of the shaft. 
     In some embodiments of the method, the at least one sensor comprises a plurality of VR sensors spaced apart from each other circumferentially around the shaft. 
     In some embodiments of the method, the compliant mount allows the shaft, the bearing, and the frame to move in at least three dimensions relative to the fixed structure. 
     In some embodiments, the method comprises: providing a first set of target elements in and/or on the outer surface of the shaft at a first position within the target region, such that target elements of the first set of target elements are arranged circumferentially about the shaft; and providing a second set of target elements in and/or on the outer surface of the shaft at a first position within the target region, such that target elements of the first set of target elements are arranged circumferentially about the shaft. 
     In some embodiments of the method, the at least one sensor comprises at least a first sensor and a second sensor, both of which are variable reluctance (VR) sensors. 
     In some embodiments, the method comprises: attaching the first sensor to the frame over the first position; attaching the second sensor to the frame over the second position; and detecting a change in relative position in the circumferential direction between the first and second sets of target elements induced upon torsional deformation of the shaft. 
     In some embodiments of the method, the first position and the second position are spaced apart by a majority of a length of the shaft 
     In some embodiments, the method comprises: rigidly attaching the first sensor to the frame, such that the first sensor is positioned over the first position; attaching a second bearing to the shaft, adjacent the second position; mounting a second frame to the second bearing in a fixed manner, such that the second frame maintains a substantially constant radial distance from the shaft; rigidly attaching the second sensor to the second frame, such that the second sensor is positioned over the second position; and detecting a change in relative position in the circumferential direction between the first and second sets of target elements induced upon torsional deformation of the shaft. 
     In some embodiments of the method, the bearing comprises an inner race and an outer race, the inner race being supported by, and in contact with, the outer surface of the shaft, such that the inner race of the second bearing and the shaft are rotatably locked together. 
     In some embodiments, the method comprises attaching, via a second compliant mount, the second frame to the fixed structure, such that the second frame is movable, substantially in unison with the shaft, relative to the fixed structure, in at least two dimensions, the at least two dimensions being in a plane perpendicular to the longitudinal axis of the shaft. 
     In some embodiments, the method comprises attaching, via the compliant mount, the second frame to the fixed structure, such that the second frame is movable, substantially in unison with the shaft, relative to the fixed structure, in at least two dimensions, the at least two dimensions being in a plane perpendicular to the longitudinal axis of the shaft. 
     In some embodiments, the method comprises: providing a magnetized portion of the outer surface of the shaft in the target region; generating a magnetic field adjacent to the magnetized portion; transmitting a torque through the shaft to induce shear within the outer surface of the shaft, wherein the magnetic field changes due to the shear, which corresponds to torsional deformation of the shaft over at least a portion of the target region due to twisting; and detecting, using the at least one sensor, a change in the magnetic field. 
     In some embodiments of the method, the shaft is substantially stationary when the at least one sensor is detecting the change in the magnetic field. 
     In some embodiments of the method, the bearing comprises a redundant bearing. 
     In some embodiments of the method, the target region comprises a first set of target elements and a second set of target elements; the first set of target elements comprise magnets that are attached to the outer surface of the shaft at a first position and are spaced about the shaft in the circumferential direction such that adjacent magnets of the first set of target elements have different polarities from each other; the second set of target elements comprise magnets that are attached to the outer surface of the shaft at a second position and are spaced about the shaft in the circumferential direction such that adjacent magnets of the second set of target elements have different polarities from each other; the first and second positions are spaced apart, within the target region, from each other along the longitudinal axis of the shaft; the at least one sensor comprises at least a first sensor, which is arranged at the first position to detect a magnetic field produced by the magnets of the first set of target elements, and a second sensor, which is arranged at the second position to detect a magnetic field produced by the magnets of the second set of target elements; and the system is configured to determine, based on a relative phase shift of the magnetic fields produced by the magnets of the first and second sets of target elements due to a torsional deformation of the shaft between the first and second sets of target elements, respectively, the torque being transmitted through the rotatable shaft. 
     In some embodiments of the method, the magnets of the first set of target elements are adjacent to each other to form a ring of magnets about the shaft at the first position and/or wherein the magnets of the second set of target elements are adjacent to each other to form a ring magnets about the shaft at the second position. 
     In some embodiments of the method, the magnets of the first set of target elements are in direct contact with each other to form a substantially continuous and uninterrupted ring of magnets about the shaft at the first position and/or wherein the magnets of the second set of target elements are in direct contact with each other to form a substantially continuous and uninterrupted ring of magnets about the shaft at the second position. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic side view of multilink drivetrain, including a plurality of longitudinally-extending shafts, which are assembled together at flexible joints (“flexjoints”) and supported by compliantly-mounted bearings attached to a fixed structure, the multilink drivetrain being capable of transmitting torque. 
         FIG. 2  is a cross-sectional view of an example embodiment of a system for sensing torque transmitted through a rotatable shaft via a monopole torque sensor architecture rigidly attached to a bearing that is attached to a fixed structure in a compliant manner and supports the shaft to allow radial movement of the shaft and bearing relative to the fixed structure to which the bearing is attached. 
         FIG. 3  is a cross-sectional view of an example embodiment of a system for sensing torque transmitted through a rotatable shaft via a multiplane torque sensor architecture rigidly attached to a bearing that supports the shaft. 
         FIG. 4  is an isometric view of an example system for sensing torque transmitted through a rotatable shaft via at least one sensor positioned in a radially fixed position relative to the shaft, such that the gap in the radial direction between the shaft and the sensor is substantially constant. 
         FIG. 5  is a schematic illustration of an example embodiment for a redundant communications architecture having a safety critical, redundant measurement of torque on two separate drivetrains, each of which has a system for redundant torque sensing of the respective shaft associated therewith. 
         FIG. 6  is a schematic illustration of another example embodiment for a redundant communications architecture having a safety critical, redundant measurement of torque on two separate drivetrains, each of which has a system for redundant torque sensing of the respective shaft associated therewith. 
         FIG. 7  is a schematic illustration of another example embodiment for a redundant communications architecture having a safety critical, redundant measurement of torque on a single drivetrain, which has a system for redundant torque sensing of the shaft thereof. 
         FIG. 8  is an isometric view of a shaft suspended by a compliantly-mounted (e.g., “floating”) bearing mounted to a fixed structure in and/or on which the shaft is installed, attached to, and/or associated with. 
         FIG. 9A  is a cross-sectional view of a conventional spline shaft adapter. 
         FIG. 9B  is a cross-sectional view of an example embodiment of a spline shaft adapter having interleaved target elements to allow for torque sensing through and/or across the spline shaft adapter. 
         FIG. 10  is an isometric view of an example embodiment of a shaft having a set of interleaved monopole target elements (e.g., “teeth”) supported by a compliantly-mounted bearing mounted, via a bearing frame, to a fixed structure, the system having a sensor mounted on the bearing frame for torque sensing. 
         FIG. 11  shows side and isometric views of an example embodiment of a conical section of a sleeve configured for mounting to a shaft adapter and having a set of interleaved monopole targets attached (e.g., integrally) thereto. 
         FIG. 12  is a graphical representation for conditioning the signal of a variable reluctance (VR) sensor, showing a voltage waveform with arming voltage threshold, a zero-crossing logic level signal with tooth period p k , and a timing clock to capture zero-crossing logic signals. 
         FIG. 13  is a filtering schematic for processing timing values from an example torque sensor. 
         FIG. 14  is a schematic illustration of a processing architecture diagram for the torque sensing. 
         FIG. 15  is a cross-sectional view of an example embodiment of a system for sensing torque transmitted through a rotatable shaft, the system having two sensors attached on physically discrete bearings spaced apart along the length of the shaft to measure relative twist (e.g., torsional deflection) of the shaft across a long span of the shaft. 
         FIG. 16  shows various views of an example embodiment of an integrated wheel where rivets can be installed to store calibration information. 
         FIG. 17  is a cross-sectional view of an example embodiment of a system for sensing torque transmitted through a rotatable shaft, the shaft being supported by redundant bearings to allow the shaft to continue rotating upon failure of one of the bearings. 
         FIG. 18  is a cross-sectional view of an example embodiment of a system for sensing torque transmitted through a rotatable shaft via a sensor configured to detect a change in magnetization due to torsional deformation within a target region of the shaft. 
         FIG. 19  is a signal processing diagram for a system configured to calculate the torque applied to a shaft. 
         FIG. 20  is a diagram showing an unraveled set of targets passing a VR sensor. 
         FIG. 21  is a signal processing diagram for a system augmented to detect axial motion. 
         FIG. 22  is a diagram showing an unraveled set of targets (some of which are slanted) passing a VR sensor. 
         FIG. 23  is a signal processing diagram of a system configured for processing two sensor signals to achieve a more accurate torque measurement. 
         FIG. 24  is a diagram showing an unraveled set of targets passing two VR sensors. 
         FIG. 25  is a signal processing diagram for an example system using dual sensors and axial/slanted teeth to output torque. 
         FIG. 26  is a signal processing diagram for an example system using three sensors. 
         FIG. 27  is a cross-sectional view of an example embodiment of a system for sensing torque transmitted through a rotatable shaft via a sensor configured to detect a magnetic field generated by a set of magnets circumferentially arranged about the shaft to co-rotate with the shaft. 
         FIG. 28  is a cross-sectional view of the example embodiment shown in  FIG. 27 , taken through the plane defined by the first set of magnets. 
         FIG. 29  is an isometric view of an alternative embodiment of the system shown in  FIG. 10 , in which the interleaved target elements are extended towards the bearing to be proximate the sensors. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a side view of an example embodiment of a multilink drivetrain, such as is suitable for use in virtually any machine that transmits torque through a rotatable shaft. For example, such multilink drivetrains are typically used in aircraft, particularly in helicopters. In helicopters, there are generally at least two rotary outputs, or propellers, which are spaced apart from each other along the length of the body of the helicopter. As shown in  FIG. 1 , the multilink drivetrain has a plurality of rotatable shafts  10  that are connected together axially (e.g., in an end-to-end configuration) to transmit torque therebetween, from one end of the drivetrain to the opposite end of the drivetrain. The shafts  10  are supported, either along the lengths of the shafts themselves (e.g., at a point between the ends of a shaft) and/or between adjacent shafts  10 , by bearings  60 , which have a housing, or frame  50 , which is connected to a support structure  1  by a compliant mount  2 . The compliant mount  2  can have, for example, spring and/or damping characteristics and can be made of any suitable material. As such, the bearings  60  can “float” (e.g., move transversely to) relative to the support structure  1  via the compliant mount(s)  2 , thereby allowing each shaft  10 , as well as the bearing(s)  60  and frame(s)  50  attached thereto, to move in unison relative to the support structure in two or more directions (e.g., in the plane perpendicular to the longitudinal axis of the undeflected shaft(s)  10 ). In some embodiments, the shaft(s)  10 , bearing(s)  60  and frame(s)  50  can move in unison relative to the support structure in three or more directions (e.g., including in the direction of the longitudinal axis of the undeflected shaft(s)  10 ). As used hereinabove, the term “direction” can be the same as a “degree of freedom.” Additionally, the term “move in unison” allows for relative movement between the shaft(s)  10 , bearing(s)  60 , and frame(s)  50  due to tolerances introduced during assembly and under structural deflections during operation, which is inherent in virtually any rotatable connection using a bearing  60 . The bearing  60  can be of any suitable type, including, for example, ball bearing, roller bearing, journal bearing, squeeze-film bearing, and the like, depending on the degree of precision desired in allowed relative movement between the shaft  10  and the frame  50  of the bearing  60  and also the expected service life and/or interval of the drivetrain. 
     The presently disclosed torque sensing devices and systems generally comprise at least several components with the purpose of measuring torque on a drivetrain component (such as, for example, a rotatable shaft). Such devices and systems generally include a shaft with a target region extending axially along at least a portion of the length of the shaft, so that, as torque is transmitted through the shaft, there is a detectable shift (e.g., of an electrical, magnetic, physical, and/or mechanical) over some or all of the target region of the shaft.  FIG. 2  shows an example embodiment of a torque sensing system, generally designated  100 . As shown, the system  100  includes a rotatable shaft  10 , which has a plurality of target elements  12 A,  12 B attached to an outer surface thereof and spaced apart from each other in the circumferential direction of the shaft  10  (e.g., in the plane perpendicular to centerline CL). In the example embodiment shown, the target elements  12 A,  12 B can be referred to as “teeth”. The target elements  12 A,  12 B can be any suitable type of fiducial attached to the shaft  10 , so long as the fiducial is detectable by a sensor (e.g.,  120 A,  120 B). The system  100  includes a first set of target elements  12 A that are rigidly attached to the outer circumferential surface of the shaft  10  at a first position  11 A. The system also includes a second set of target elements  12 B that are rigidly attached to the outer circumferential surface of the shaft  10  at a second position  11 B. 
     In the example embodiment shown, the first set of target elements  12 A are offset in the circumferential direction from the second set of target elements  12 B, such that the target elements of the first set of target elements  12 A are interleaved with the target elements of the second set of target elements  12 B. As such, the target elements of the first and second sets of target elements  12 A,  12 B are arranged circumferentially about the shaft in an alternating pattern. While the spacing pattern between adjacent target elements of the first and second sets of target elements  12 A,  12 B may be any suitable pattern, in the example embodiment shown the spacing is uniform, such that a gap between each target element is substantially identical when the shaft is not being twisted (e.g., torsionally deformed) over the target region TR. 
     The target elements of the first set of target elements  12 A extend away from the first position  11 A towards the target elements of the second set of target elements  12 B. Similarly, the target elements of the second set of target elements  12 B extend away from the second position  11 B towards the target elements of the first set of target elements  12 A. The first and second target elements  12 A,  12 B are interleaved with each other so that at least a portion of each target element of the first set of target elements  12 A overlaps in the axial direction of the shaft  10  with at least a portion of each target element of the second set of target elements  12 B to define a deflection region  13 . While the example embodiment disclosed herein has target elements that are substantially aligned with (e.g., parallel to) each other, in some embodiments, the target elements may be inclined relative to each other. In some embodiments, the target elements may be oriented parallel to each other but inclined relative to the centerline CL of the shaft  10 , such that, if an imaginary line having a constant radial distance from the outer surface of the shaft  10  were extended from each target element, in the direction of extension thereof, each such imaginary line would wrap helically about the shaft  10 . 
     The system  100  also has one or more sensors  120 A,  120 B arranged at respective radial positions about the shaft  10  to measure a change in the gap, measured in the circumferential distance of the shaft  10 , between adjacent target elements. In the example embodiment shown, the system  100  has a plurality of (e.g., two or more) sensors  120 A,  120 B that are arranged about the shaft  10  such that the first sensor  120 A is spaced apart from the second sensor  120 B in the circumferential direction. The sensors  120 A,  120 B are variable reluctance (VR) sensors in the example embodiment shown, but any sensor type(s) capable of detecting a change in the gap (e.g., distance, but over an arcuate shape or path) in the radial direction between two adjacent target elements can be used without deviating from the scope of the subject matter disclosed herein. In some embodiments a single sensor (e.g.,  120 A) could be used. Regardless of the number of sensors  120 A,  120 B included in the system  100 , the sensors  120 A,  120 B are provided to measure the phase shifting of adjacent target elements of the first and second sets of target elements  12 A,  12 B to measure the torsional deformation of the shaft over the target region TR. 
     The sensors  120 A,  120 B are attached to a frame  50 , which is radially supported on and/or by a bearing  60 , which is in contact with (e.g., direct or indirect contact) the outer surface of the shaft  10 . The bearing  60  may be of any suitable type, including, for example, ball bearing, roller bearing, journal bearing, and the like. The bearing  60  has an inner race  62  that is in contact with the outer surface of the shaft  10 , such that the inner race  62  is radially locked to the shaft  10 . As such, the inner race  62  co-rotates at a same, or substantially similar, angular velocity or speed as the shaft  10 . The outer race  64  is rotationally decoupled from the shaft  10 , such that the shaft  10  is capable of rotating freely independent of the angular position or velocity of the outer race  64 . The outer race  64  is rigidly attached to a frame  50 , to which one or more of the sensor(s)  120 A,  120 B are rigidly attached. While tolerances of the bearing  60  may allow for minimal radial deflection of the outer race  64  relative to the shaft  10 , the outer race  64 , the frame  50  attached thereto, and the sensor(s)  120 A,  120 B attached to the frame  50  are substantially positionally fixed in the radial direction relative to the shaft  10 . 
     In some example embodiments, such bearings  60  can have small radial clearances which can reduce gaps from about 0.5 inches to about 0.005 inches. It is advantageous to select a bearing  10  with sufficient tolerances to maintain a radial distance between the target elements and the sensors  120 A,  120 B of no more than 0.02 inches. By selecting such a bearing  60  as described herein, the radial distance between the target elements and the sensors  120 A,  120 B can be reduced to a value where relative motion is not really occurring between the shaft  10  and sensor(s)  120 A,  120 B (e.g., except for rotation about centerline CL), and a plurality of sensors  120 A,  120 B can be used to enable greater accuracy of the torque sensing capabilities of the system  100 . The distance between the sensors  120 A,  120 B and the target elements is the primary source of error in conventionally known torque sensors that have large gaps (e.g., greater than about 0.5 inches). However, whenever this radial distance is decreased, the overall observable twist, or torsional deformation, of the shaft  10  necessarily decreases. As such, the primary factor for the accuracy of the torque sensing devices, systems, and methods using VR technology, as disclosed herein in some example embodiments, is the resolution of the change in gap between adjacent target elements. 
     As such, the sensor(s)  120 A,  120 B are fixed at a substantially constant radial distance from the shaft  10  and also from the target elements of the first and second sets of target elements  12 A,  12 B in the deflection region  13 . As such, any radial deflections of the shaft  10  during operation will also occur in substantially the same magnitude and the same direction for the sensor(s)  120 A,  120 B, so that the sensor(s)  120 A,  120 B are essentially static relative to the shaft  10  as the shaft  10  rotates substantially about centerline CL. The sensor(s)  120 A,  120 B move radially in unison with the shaft  10 . Therefore, the sensors  120 A,  120 B, are substantially fixed relative to the shaft and the target elements attached thereto, by virtue of the sensors  120 A,  120 B being rigidly attached to the frame  50 , which is rigidly fixed to the outer race  64  of the bearing  60 , and which is positionally fixed, at least in the radial direction (e.g., in the plane perpendicular to centerline CL), relative to the shaft  10 . The frame  50  may experience vibrations that may result in some perturbation in relative position between the target elements and the sensor(s)  120 A,  120 B, however this flexing movement of the frame  50 , independent of the shaft  10  and/or the target elements, is within the meaning of the terms “positionally fixed” and “in unison” as used herein, just as are any tolerances in the bearing  60  that allows for relative movement between the inner race  62  and the outer race  64 . 
     Still referring to the example embodiment shown in  FIG. 2 , the frame  50  and/or the bearing  60  are attached by a compliant mount, generally designated  2 , to a structural element  1  of the machine (e.g., aircraft, motor vehicle, stationary machine, ship, and the like) in and/or on which the shaft  10  is arranged and/or attached. The compliant mount  2  is configured to have spring and/or damping characteristics and can be made of any suitable material, or combination of materials, that will allow for relative movements between the structural element  1  and the mobile components including, for example, the frame  50 , the bearing  60 , the shaft  10 , and other components associated therewith, which are capable of movement in the radial and/or axial direction of the shaft  10  relative to the structural element  1 . In some embodiments, the compliant mount  2  can comprise an elastomeric material, such as natural rubber. 
     The system  100  also includes at least one temperature sensor  140 A,  140 B. The temperature sensor(s)  140 A,  140 B can be used to detect a temperature at, within, and/or around the system  100 , and/or the components thereof. Detecting the temperature is advantageous because, by knowing the temperature, this parameter to can be used to determine the mechanical parameters that are temperature-dependent in calculating the torque being transmitted through the shaft  10  based on the torsional deformation of the shaft  10  over the target region TR. For example, the shaft may be more ductile at elevated temperatures and undergo additional torsional deformation for a same torque than would occur for the same torque at a lower operating temperature. The temperature sensor(s)  140 A,  140 B shown in  FIG. 2  can be included in any of the various types of torque sensing systems disclosed herein without deviating from the scope of the disclosed subject matter. The temperature sensor(s)  140 A,  140 B can be attached to the frame  50 , such that the shaft  10  and the temperature sensor(s)  140 A,  140 B move in the radial direction of the shaft  10  in unison, or the temperature sensor(s)  140 A,  140 B can be mounted independent from the shaft  10 , such that the shaft  10  moves radially relative to the temperature sensor(s)  140 A,  140 B, meaning that a distance between the temperature sensor(s)  140 A,  140 B and the shaft  10  can change as the shaft  10  moves during normal operation. As such, in any of the example embodiments disclosed herein, one or more (e.g., a plurality of) temperature sensors  140 A,  140 B may be included therein or omitted therefrom. 
     The sensors  120 A,  120 B and the temperature sensors  140 A,  140 B each output a signal corresponding, respectively, to the circumferential distance between adjacent target elements, which corresponds to a torsional deformation of the shaft  10 , and the to the temperature detected in the immediate vicinity of the components of the system  100  (e.g., the shaft  10 ). The signal from each of the sensors  120 A,  120 B and the temperature sensors  140 A,  140 B is transmitted to a signal conditioning unit  210  (SCU), sometimes referred to as a signal conditioner. One or more (e.g., a plurality of, or redundant pair of) SCUs  210  can be provided. In the embodiment shown, the first sensor  120 A and the first temperature sensor  140 A send their respective signals to a first SCU  210  and the second sensor  120 B and the second temperature sensor  140 B send their respective signals to a second SCU  210  to provide for redundant, fail-safe torque sensing capabilities. 
       FIG. 3  shows a different example embodiment of a torque sensing system, generally designated  101 , for sensing (e.g., measuring and/or detecting) torque transmitted through a shaft  10 . Shaft  10  is substantially similar to the shaft  10  shown and described relative to  FIG. 2 , however, unlike in system  100 , in which the target elements  12 A,  12 B were respective interleaved target elements attached at respective first and second positions along the length of the shaft  10 , in the example embodiment shown in  FIG. 3 , the system  101  has a first target element  14 A, or set (e.g., a plurality) thereof, attached at the first position  11 A and a second target element  14 B, or set (e.g., a plurality) thereof, attached at the second position  11 B. The first position  11 A is axially offset (e.g., in the axial direction of the shaft  10 ), from the second position  11 B. The distance between the first and second positions  11 A,  11 B defines the target region TR, which is the axial portion of the shaft along which the torsional deformation is measured. 
     In system  101 , the first and second target elements  14 A,  14 B are not interleaved with each other. The system  101  comprises a plurality of (e.g., two) sensors  120 A,  120 B, which are, for example, variable reluctance (VR) sensors that are rigidly attached to a frame  50  and spaced out along the frame such that a first sensor  120 A is positioned substantially over (e.g., axially aligned with) the first position  11 A and a second sensor  120 B is positioned substantially over (e.g., axially aligned with) the second position  11 B. The first sensor  120 A is used to detect a passage of each first target element  14 A passing adjacent to the first sensor  120 A as the shaft  10  rotates relative to the first sensor  120 A and the frame  50  to which it is rigidly attached. The second sensor  120 B is used to detect a passage of each second target element  14 B passing adjacent to the second sensor  120 B as the shaft  10  rotates relative to the second sensor  120 B and the frame  50  to which it is rigidly attached. As such, upon the shaft  10  receiving and/or transmitting a torque therethrough, the shaft  10  is torsionally deformed (e.g., will twist). As the shaft  10  is twisted, the relative radial positions of the first target elements  14 A relative to the second target elements  14 B will change as the shaft is deformed, or twisted. It is this relative radial offset of the radial positions of the first and second target elements  14 A,  14 B that is detected, using the first and second sensors  120 A,  120 B, which are used to observe the phase shift relative to the output of the first and second sensors  120 A,  120 B when the shaft  10  is undeformed. 
     As shown, the first and second sensors  120 A,  120 B are rigidly attached to a frame  50 , which is radially supported by a bearing  60 . The bearing  60  and the frame  50  are substantially similar to that described in the system  100  of  FIG. 2 . It is advantageous to use a bearing with small radial play to minimize radial motion of the first and second sensors  120 A,  120 B relative to the outer surface of the rotating shaft  10  and, consequently, the first and second target elements  14 A,  14 B affixed thereto. As such, the radial distance between the first target elements  14 A and the first sensor  120 A and the radial distance between the second target elements  14 B and the second sensor  120 B are substantially constant (e.g., allowing for changes due to vibration of the frame  50  and tolerances of the bearing  60  allowing relative radial movement between the inner race  62  and the outer race  64 ). The first and second sensors  120 A,  120 B send signals corresponding to the detection of each of the respective first and second target elements  14 A,  14 B to a Signal Conditioning Unit (SCU)  210 , which processes the timing measurements and calculates the torque being transmitted through the shaft  10  (e.g., substantially in real-time, allowing for the time necessary to perform mathematical calculations by a processor based on the signals sent from the first and second sensors  120 A,  120 B). As shown in  FIG. 2 , the system  101  may include one or more temperature sensors ( 140 A,  140 B,  FIG. 2 ) to compensate for any changes in stiffness (e.g., torsional stiffness) of the shaft  10  as a function of temperature. 
     The systems  100 ,  101  shown in  FIGS. 2 and 3  generate electrical signals (e.g., from the first and second sensors  120 A,  120 B) that are then received by an SCU  210 . The SCU  210  calculates a real-time measurement of the torque on the shaft  10  and can transmit an analog and/or digital signal to a Full Authority Digital Engine Controller (FADEC), Flight Control Computer (FCC), or other critical control system  250 . An isometric view of the torque sensing system  100  is shown in  FIG. 4 .  FIG. 4  shows that the system  100  has a connector, generally designated  125 , for transmitting waveform signals from the first and/or second sensors  120 A,  120 B to an electronics box (e.g., containing an SCU  210 ) for processing (e.g., calculating torque through the shaft  10 ). As shown, the first and second sensors  120 A,  120 B are mounted to a frame  50  (as shown, in an example sensor support cradle) mounted, via a radial bearing  60 , to the shaft  10  to prevent the frame  50  from rotating with the shaft  10 . In the example embodiment shown in  FIG. 4 , the first and second target elements  12 A,  12 B are interleaved teeth to allow a single sensor  120 A,  120 B to measure twist, or torsional deformation, through this section of the shaft  10 , which can be correlated by a temperature corrected torsional stiffness to torque. 
     A system architecture, generally designated  200 , compatible with typical aerospace safety standards that comprises a plurality of torque sensing systems  100  is shown in  FIG. 5 . This system architecture  200  is fully redundant so as to prevent single point failures. As such, this system architecture  200  has four independent torque sensing systems  100  attached at four different locations along the length of the shaft  10  to provide in-line torque measurement. Two of the systems  100  are measuring the same torque value for redundancy. Each torque measurement channel has one or more sensors (e.g.,  120 A,  120 B,  FIGS. 2 and 3 ) that can determine the amount of torsional deformation of the shaft  10 . These systems  100  are connected to two independent SCUs  210 A,  210 B to provide redundancy and process the tachometer outputs received from each system  100 . Each SCU  210  can condition, for example, the output signals from up to seven (7) sensors and multiple temperature sensors, which can be, in some embodiments, a resistance temperature detector (RTD). Because of this, only two signal conditioning units have to be used (instead of the typical 4) in the example system architecture  200  disclosed herein. The RTD&#39;s are used to aid in estimating the shaft temperature stiffness and also account for thermal variations in the sensor mounting. The SCUs  210  are, in some embodiments, capable of communicating with each other for cross checking. Ultimately, redundant digital signals are transmitted to a corresponding critical control computer  250 A,  250 B (e.g. Flight Control Computer—FCC or Full Authority Digital Engine Controller—FADEC). 
     In  FIG. 6 , an alternate embodiment of the system architecture  200  is shown, in which two in-line portions of the drivetrain have torsional loads applied to them. As such, both portions of the drivetrain twist by an amount that the system  100  is measuring. This twist is observed by two independent sets of one or more sensors (e.g.,  120 A,  120 B,  FIG. 2 ), which provide the resulting electrical signals to the SCU  210 . As was shown in  FIG. 5 , each SCU  210  may communicate with each other for redundancy, and ultimately send a digital signal to a corresponding critical control computer  250 A,  250 B. This system architecture  200  minimizes the mechanical hardware that needs to be integrated into a drivetrain. A further example embodiment of a system architecture, generally designated  202 , is shown in  FIG. 7 . The system architecture  202  is generally similar to the system architecture  200 , but includes using only a single system  100  to output a signal corresponding to torsional deformation of the shaft  10  at a single axial position of the shaft  10 . The output of each sensor is transmitted to an independent SCU  210 , which can communicate with the other SCU  210  in some embodiments, and is then transmitted to a corresponding critical control computer  250 A,  250 B for redundancy in signal processing. 
     Additional embodiments of such system architectures can also be implemented by those skilled in the art but are not shown herein for brevity. For example, four SCUs  210  could be used and, furthermore, could be integrated directly into a corresponding one of the critical control computers  250 A,  250 B to advantageously save space. 
     Relative motion between the drivetrain and the support structure (e.g., aircraft fuselage, or frame element) to which it is compliantly attached is accommodated by a series of hanger bearings (e.g., including frame  50  and bearing  60 ) positioned at the ends of each of the sync shafts (see generally  FIG. 1 ). The bearing  60  is typically a precision ball bearing that suspends the shaft  10  and is housed in a frame  50  (e.g., a bracket) that is attached to the support structure  1  through a set of compliant (e.g., vibration isolation) mounts  2 . This allows the bearing  60  and frame  50  to translate in axial direction (e.g., x-direction) and in the radial direction (e.g., y- and z-directions) with the shaft, as shown in  FIG. 8 . The compliant mount  2  has a support rod  3 , along which the frame  50 , along with the bearing  60  and sensor(s) can move in the x-direction as the shaft  10  moves in the x-direction (e.g., in the axial direction of the shaft  10 ). The compliant mount  2  also has a compliant grommet  4 , by which the frame  50  is attached to the support rod  3 , the compliant grommet  4  being made of any suitable material that will allow for compression of the compliant grommet  4  so that the frame  50 , along with the bearing  60  and sensor(s), can move in the radial directions (e.g., the y- and/or z-directions) relative to the support rod  3  as the shaft  10  moves in the radial direction. The compliant mount  2  is rigidly attached to the support structure (e.g.,  1 , see  FIG. 1 ). In some embodiments, sensor cradles may be mounted to the support structure to detect relative motion between the target elements and sensors equal to the relative motion allowed by the compliant mount(s)  2 . Sensors rigidly attached to the frame  50  can move with the shaft  10 , such that relative motion between the sensors and the targets will be predominated by the motion allowed by the precision ball bearing (e.g.,  60 ). 
     An example embodiment of a torque sensing device is shown in  FIG. 9B , in which the target elements are integrated into a standard aerospace spline shaft adapter.  FIG. 9A  is an aerospace spline shaft adapter, generally designated  300 , known from the prior art and is provided for reference to show the difference between it and the example torque sensing device, generally designated  301 , shown in  FIG. 9B . As shown in  FIG. 9A , the adapter  300  comprises a male spline adapter  322  to transmit torque, an adapter coupling  320  that is generally conical, or at least frustoconical, in shape, and a mating coupling  321 , which is attached by fasteners, generally designated  330 , to a shaft  310 . The mating coupling  321  has a balance flange  340  radially attached (e.g., integrally) thereabout. In  FIG. 9B , the adapter  301  has a reference shaft structure  324  that is arranged external to (e.g., concentrically about, in a nested configuration) the adapter coupling  320 . As such, the reference shaft structure  324  generally is conical, or at least frustoconical, and is oriented parallel to the tapered section the adapter coupling  320 . The target elements are positioned near the balance flange  340  (e.g., away from the hanger bearing). The target elements are attached to the reference shaft structure  324  and the balance flange  340  and extend circumferentially about the adapter  301  in an alternating manner. As such, every other target element is attached to the balance flange  340  and the others of the target elements are attached to the reference shaft structure  324 , such that each target element attached to the balance flange  340  is adjacent a target element attached to the reference shaft structure  324  on both sides of the target element attached to the balance flange  340 . Upon torsional deformation of the adapter  301 , the target elements of the reference shaft  324  move towards or away from the adjacent target elements attached to the balance flange  340  (e.g., in the circumferential direction, such that a gap between adjacent target elements changes as the adapter  301  undergoes torsional deformation, or twisting). 
       FIG. 10  shows an isometric view of a system  102  similar to the target arrangement of  FIG. 9B , with a modified hanger bearing frame  50  that has a VR sensor  120  rigidly attached thereto. In the example embodiment shown, the system  102  comprises a first plurality of target elements  72 A attached about a first annular ring  70 A and a second plurality of target elements  72 B attached about a second annular ring  70 B. Each of the first plurality of target elements  72 A is attached to the first annular ring  70 A and extends towards the second annular ring  70 B, such that at least a portion of each target elements of the first plurality of target elements  72 A extends over at least a portion of the second annular ring  70 B. As such, the first and second pluralities of target elements  72 A,  72 B are arranged at least partially in a single plane, such that a single sensor  120  can detect changes in circumferential gaps between adjacent first and second target elements  72 A,  72 B. The first and second annular rings  70 A,  70 B are rigidly attached to the shaft  10  so that relative movement between the first annular ring  70 A and the shaft  10  and/or between the second annular ring  70 B and the shaft  10  is prevented. The first and second annular rings  70 A,  70 B are attached to the shaft  10  so as to be spaced apart from each other in the axial direction. The gap between the first and second annular rings  70 A,  70 B defines the target region, which is the portion of the shaft  10  over which the torsional deformation is to be detected by monitoring relative changes in the gaps between adjacent first and second target elements  72 A,  72 B. 
       FIG. 11  illustrates another example embodiment for a torque sensing device, in which the target elements  82 A,  82 B are shown being positioned closer to the hanger bearing (e.g., adjacent to the male spline  80 B for transmitting torque). Such an arrangement as shown in  FIG. 11  would necessarily require the sensors (e.g.,  120 ) to be positioned closer to the hanger bearing, making the frame (e.g.,  50 ) able to be a smaller and/or lighter component. In this example embodiment, the respective target elements  82 A,  82 B are positioned at a conical section of portions of a sleeve  80 A,  80 B that could be mounted to a shaft adapter like the adapter depicted in  FIG. 9A , thereby allowing the device of  FIG. 11  to be retrofit onto an existing prior art shaft adapter without requiring replacement thereof. 
     In order to process the VR signals, the electrical waveforms are converted to a logic level signal using a zero-crossing detection circuit (or ZCD), as shown in  FIG. 12 . It is advantageous to minimize any signal noise, as it directly affects the accuracy of the twist measurement. These logic level signals are captured at a specific time v k  and assigned to a value in the microcontroller&#39;s memory. The alternating target elements  12 A and  12 B are shown as an overlay corresponding to the electrical waveforms shown in  FIG. 12 . 
     Following the capture of the electrical waveforms (e.g., by a microcontroller), the timing can be filtered according to the diagram shown in  FIG. 13 . The example filtering schematic shown is specifically provided for a single sensor with N interleaved target elements measuring twist (e.g., torsional deformation), but similar processing can be implemented for multi-sensor embodiments and is known to those skilled in the art. There are two main paths for the filtering, the CPR (counts per revolution) path determines the shaft speed, and the AB path determines the twist between the left (or A) and right (or  8 ) side teeth (that are interleaved). 
     The speed can be calculated by applying a digital moving average filter F 00  to the timing measurement v k  (e.g., where the sample rate of this filter is the target element passage frequency). This value can then be subsampled at a lower rate after the application of an Anti-Aliasing (or decimation) filter F AA . 
     
       
         
           
             
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     Where Nis the number of target elements. 
     
       
      
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     The resulting CPR value can be converted to shaft speed using the following equation. 
     
       
         
           
             
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     After the calculation of speed, the twist can be calculated by applying a digital moving average filter F 60  to the timing measurement v k  (where the sample rate of this filter is the target element passage frequency) resulting in the AB k  value. 
     
       
         
           
             
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     The absolute value of AB k  is then determined by applying the Q 2  operator, and the resulting value can be subsampled at a lower rate after the application of an Anti-Aliasing (or decimation) filter F AA . 
         AB=F   AA   Q   2   AB   k   =F   AA   |AB   k | 
     The value AB can be converted to twist θ (in degrees) with the following equation which uses the previously determine CPR value. 
     
       
         
           
             
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     It is also helpful to then convert the twist θ to torque T. 
       τ= K (θ−θ 0 )
 
     Where K is the torsional stiffness of the shaft (usually in units of in-lbs/deg, ft-lbs/deg, or N-m/deg) between the interleaved target elements and θ 0  is a twist offset that is applied based on calibration and can be a function of speed, temperature, operating conditions, or other waveform parameters. The value K for the torsional stiffness can be selected from a table based on a temperature reading from a temperature sensor of the system, the temperature sensor being positioned adjacent the shaft at the target region. 
     The processing architecture for the algorithms described above is divided in software to the task depicted in the schematic shown in  FIG. 14 . 
     Sensor signals from the system  100  are received in a second processor, FPGA, or the control law accelerator  220  (CLA) math accelerator within the SCU  210 . The AB and CPR algorithms shown and described in  FIG. 13  and described hereinabove are executed for each passage of a target element adjacent the sensor and the results are stored in shared random access memory (RAM)  230 . At 500 Hz these sensor values are obtained by the CPU  240  in the tms_task  242  from the shared memory  230  (or the communications bus) between the CLA. 
     The tms_task  242  then applies an algorithm to normalize the data into engineering units, compensate the data for calibration parameters, and provide additional filtering for anti-aliasing before finally storing the output values of speed and torque to the datastore  246 . 
     The comms_task  248  retrieves the data from the datastore  246  at the rate required (e.g., at 100 Hz) for transmission by the consumer  260  of the data. Additional rate limiting and filtering maybe applied prior to transmission over the preferred data bus, usually ARINC-429, RS422, or CAN. 
     If two channels of torque are being measured, the CLA  220  will independently process the AB and CPR calculations for the two channels at the target element passage frequency of both channels. 
     The scope of the subject matter disclosed herein is not limited to a hanger bearing mounted torque sensor with only one sensor at one shaft location. An alternative embodiment is shown in  FIG. 15 , in which VR sensors  120 A,  120 B are mounted on frames  50  respectively mounted to bearings  60  at each end of a long shaft  10 , such that the sensors  120 A,  120 B are separated from each other by a large axial distance. As such, in the example embodiment shown in  FIG. 15 , the target region TR is at least a majority of a length of a shaft  10 . Otherwise, the components of the system  103  of  FIG. 15  are substantially similar to the components of the system  101  of  FIG. 3  and the common elements thereof will not be described further herein in the interest of brevity. Each frame  50  may be connected (e.g., using a compliant mount  2 ) to the support structure  1  of the machine with which the shaft  10  is associated (e.g., installed in and/or attached to). Because the frames  50  are attached to the shaft  10  by a respective bearing  60 , the radial distance between the first sensor  120 A and the first target element(s)  14 A is substantially constant and the radial distance between the second sensor  120 B and the second target element(s)  14 B is substantially constant. In some embodiments, the radial distance between the first sensor  120 A and the first target element(s)  14 A is substantially identical to the radial distance between the second sensor  120 B and the second target element(s)  14 B is substantially constant during operation of the system  103 , even as the shaft  10  may be moving in the radial and/or axial directions relative to the support structure  1 . 
     This longer target region TR, as measured axially along the shaft  10 , will experience a much larger amount of twist than, for example, in system  100  of  FIG. 2  and system  101  of  FIG. 3  under an identical torsional load transmitted through the shaft  10 . This increased magnitude of twist, or torsional deflection, is advantageous in some aspects, in that the sensors  120 A,  120 B can be operated with increased accuracy when the magnitude of the twist of the shaft is commensurately greater due to the increased length of the target region TR. However, other operational parameters and/or motions of the shaft  10  may cause significant errors if the overall twist measured was not so large. For example, large temperature gradients may be present across the axial length of the shaft  10 . It may also be more difficult in such embodiments to align the two frames  50  to yield a consistent phase difference. Furthermore, various deflections of the frame  50  could occur that may erroneously manifest in the measurement as twist of the shaft  10 . 
     Sometimes it is advantageous to include in a torque sensing device, generally designated  500 , system-specific calibration information, such as drivetrain stiffness or twist offset. The HBM torque sensor at a basic level acts as a torque sensor which requires a slope and offset calibration. This calibration may be different for each specific set of hardware (e.g., for each specific drivetrain), thus, it may be advantageous to provide calibration information as part of the torque sensing device  500  that can be read off the shaft without having to program such calibration information into a SCU  210  or by inserting a floppy disk, USB thumb drive, data card, or otherwise transmitting such information to the SCU  210 . An example of how this would be implemented is shown in  FIG. 16 , which shows a torque sensing device  500 , which has holes  512  formed circumferentially about an outer flange member  510 . These holes  512  can be filled (e.g., by inserting a fastener, such as a rivet, therethrough) to store “bits” of information corresponding to an offset and a slope associated with the particular drivetrain component. A sensor (e.g., a VR sensor) could be provided and positioned proximate to the holes  512  to measure bits (e.g., which of the holes  512  is filled) and output a signal to the SCU  210 , which can translate the “bits” into offset and slope calibration information. Verification that the measured data is valid could be performed with a checksum and repeated reads of the data as the shaft rotates. 
       FIG. 17  is an example embodiment of a torque sensing system, generally designated  104 , in which the components of the torque sensing system  104  are substantially identical to, and operate in a substantially similar manner as, the components of the torque sensing system  100  shown in  FIG. 2 . As such, similar components will not be addressed further herein in the interest of brevity. In system  104 , however, the bearing  60 R is a redundant bearing in which the shaft  10  remains capable of rotation, relative to the frame  50 , even upon a failure of a portion of the bearing. The bearing  60 R may be implemented in any of the example torque sensing systems disclosed herein without limitation. 
       FIG. 18  is an example embodiment of a torque sensing system, generally designated  105 . Unlike in other torque sensing systems disclosed herein, in which variable reluctance (VR) sensors are used to detect the relative movements of a set of target elements, system  105  comprises at least one magnetic field sensor  122 A. In the embodiment shown, the system  105  comprises a first magnetic field sensor  122 A and a second magnetic field sensor  122 B. The first and second magnetic field sensors  122 A,  122 B are rigidly attached, as was disclosed already in, for example, system  100 , a frame and a bearing  60 . In some embodiments, the magnetic field sensors  122 A,  122 B can be respective linear Hall sensors. The frame  50  is positionally fixed in the radial direction, relative to the shaft  10 , such that the radial distance between the first and second magnetic field sensors  122 A,  122 B, respectively, and the shaft  10  remains substantially constant. As such, the first and second magnetic field sensors  122 A,  122 B and the shaft  10  move substantially in unison (e.g., allowing only relative movement therebetween caused by vibration of the frame  50  and/or tolerances of the bearing  60 ) upon movement of the shaft  10  relative to a support structure  1  to which it is attached by a compliant mount  2  (see, e.g.,  FIG. 2 ). The shaft has a target region TR, over the surface of which a magnetic polarity MP is provided. When the shaft  10  is torsionally deformed (e.g., twisted, or experiences a shear force), the magnetic polarity MP changes. The magnetic field sensor(s)  122 A,  122 B detect any changes in the magnetic polarity MP over the target region TR. The change in magnetic polarity MP as a function of applied torque is known and the torque transmitted through the shaft  10  can be calculated therefrom in a substantially similar manner to other example embodiments described elsewhere herein. 
       FIG. 19  is a signal processing diagram for a system configured to calculate the torque applied to a shaft. The signal processing is configured for isolating the effect of twist on the timing pattern of the shaft. The signal processing includes a digital filter  1202  configured to isolate a twist measurement from a raw timing measurement. The signal processing includes a low pass filter  1204  configured to output a raw twist measurement. The signal processing includes a combiner  1206  to use a measurement of shaft stiffness with the twist measurement to produce a torque output. 
       FIG. 20  is a diagram showing an unraveled set of targets passing a VR sensor. The timing pattern between the teeth can be written as a series of timing values based on the period of time between two successive tooth passages (or zero crossings). 
     In the example shown in  FIG. 20 , the instant in time that each tooth passes (v k ) can be written as the following: 
     
       
         
           
             
               v 
               k 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             f 
                             clock 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             k 
                           
                           ⁢ 
                           
                             
                               dk 
                               ′ 
                             
                             
                               f 
                               
                                 s 
                                 ⁢ 
                                 haft 
                               
                               
                                 k 
                                 ′ 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             
                               c 
                               ⁢ 
                               l 
                               ⁢ 
                               o 
                               ⁢ 
                               c 
                               ⁢ 
                               k 
                             
                           
                           
                             f 
                             
                               s 
                               ⁢ 
                               haft 
                             
                             k 
                           
                         
                         ⁢ 
                         
                           θ 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             where 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             odd 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           f 
                           
                             c 
                             ⁢ 
                             l 
                             ⁢ 
                             o 
                             ⁢ 
                             c 
                             ⁢ 
                             k 
                           
                         
                         N 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           0 
                           k 
                         
                         ⁢ 
                         
                           
                             
                               dk 
                               ′ 
                             
                             
                               f 
                               
                                 s 
                                 ⁢ 
                                 haft 
                               
                               
                                 k 
                                 ′ 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               where 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               even 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Where f clock  is the clock speed of the timing measurement, N is the total number of teeth, k is the discrete index in time, f shaft  is the shaft speed at time instant k, and θ is the shaft twist. This can be further simplified if the shaft speed, f shaft , is roughly constant. 
     
       
         
           
             
               v 
               k 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             f 
                             
                               c 
                               ⁢ 
                               l 
                               ⁢ 
                               o 
                               ⁢ 
                               c 
                               ⁢ 
                               k 
                             
                           
                           N 
                         
                         ⁢ 
                         
                           k 
                           
                             f 
                             
                               s 
                               ⁢ 
                               haft 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             
                               c 
                               ⁢ 
                               l 
                               ⁢ 
                               o 
                               ⁢ 
                               c 
                               ⁢ 
                               k 
                             
                           
                           
                             f 
                             
                               s 
                               ⁢ 
                               haft 
                             
                           
                         
                         ⁢ 
                         
                           θ 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             where 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             odd 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           f 
                           
                             c 
                             ⁢ 
                             l 
                             ⁢ 
                             o 
                             ⁢ 
                             c 
                             ⁢ 
                             k 
                           
                         
                         N 
                       
                       ⁢ 
                       
                         k 
                         
                           f 
                           shaft 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           where 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           is 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           even 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     The timing value at each discrete index in time, Ts k , can be written as the following (with shaft speed f shaft  assumed to be constant over the small time interval between teeth): 
     
       
         
           
             
               Ts 
               k 
             
             = 
             
               
                 
                   v 
                   k 
                 
                 - 
                 
                   v 
                   
                     k 
                     - 
                     1 
                   
                 
               
               = 
               
                 
                   
                     f 
                     
                       c 
                       ⁢ 
                       l 
                       ⁢ 
                       o 
                       ⁢ 
                       c 
                       ⁢ 
                       k 
                     
                   
                   
                     f 
                     
                       s 
                       ⁢ 
                       haft 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       1 
                       N 
                     
                     + 
                     
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           k 
                         
                         ⁢ 
                         θ 
                       
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Note that the final result of this equation applies to all discrete indices of k. The effect of twist on an interleaved pattern of teeth results in a timing change that adds to one time period and subtracts from the next; this pattern repeats every revolution. A series of digital filtering can therefore isolate the twist. The twist over an entire revolution can be calculated by adding and subtracting all of the timing values. 
     
       
         
           
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 
                   n 
                   = 
                   
                     N 
                     - 
                     1 
                   
                 
               
               ⁢ 
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   n 
                 
                 ⁢ 
                 
                   Ts 
                   
                     k 
                     - 
                     n 
                   
                 
               
             
             = 
             
               
                 
                   Ts 
                   k 
                 
                 - 
                 
                   Ts 
                   
                     k 
                     - 
                     1 
                   
                 
                 + 
                 
                   Ts 
                   
                     k 
                     - 
                     2 
                   
                 
                 - 
                 
                   Ts 
                   
                     k 
                     - 
                     3 
                   
                 
                 + 
                 … 
                 + 
                 
                   Ts 
                   
                     k 
                     - 
                     N 
                     - 
                     2 
                   
                 
                 - 
                 
                   Ts 
                   
                     k 
                     - 
                     N 
                     - 
                     1 
                   
                 
               
               = 
               
                 
                   - 
                   
                     
                       f 
                       clock 
                     
                     
                       f 
                       
                         s 
                         ⁢ 
                         haft 
                       
                     
                   
                 
                 ⁢ 
                 
                   θ 
                   π 
                 
                 ⁢ 
                 
                   N 
                   2 
                 
               
             
           
         
       
     
     Rewriting this equation and solving for θ results in the following: 
     
       
         
           
             
               θ 
               k 
             
             = 
             
               
                 
                   
                     - 
                     2 
                   
                   ⁢ 
                   π 
                 
                 N 
               
               ⁢ 
               
                 
                   f 
                   
                     s 
                     ⁢ 
                     haft 
                   
                 
                 
                   f 
                   clock 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   
                     n 
                     = 
                     
                       N 
                       - 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     n 
                   
                   ⁢ 
                   
                     Ts 
                     
                       k 
                       - 
                       n 
                     
                   
                 
               
             
           
         
       
     
     This can also be rewritten as a digital FIR filter with the following coefficients for a case where there are N=12 teeth. This digital FIR filter is an example of the digital filter  1202  for isolating twist. 
     
       
         
           
             B 
             = 
             
               
                 
                   
                     - 
                     2 
                   
                   ⁢ 
                   π 
                 
                 12 
               
               ⁢ 
               
                 
                   
                     f 
                     shaft 
                   
                   
                     f 
                     
                       c 
                       ⁢ 
                       l 
                       ⁢ 
                       o 
                       ⁢ 
                       c 
                       ⁢ 
                       k 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                       
                         1 
                       
                       
                         
                           - 
                           1 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     In practice, this value of θ should be designed to always be positive, and should also be filtered down to a lower bandwidth with an anti-aliasing filter, F AA ; it is also helpful to apply a calibration offset θ 0  to adjust for any real world imperfections in the amount of twist. 
       θ= F   AA |θ k |−θ 0  
 
     After performing filtering operation, the shaft torsional stiffness, K, can be multiplied in to determine torque, T: 
         T=K (θ−θ 0 )
 
     Similarly, this signal processing can also be augmented to detect axial motion of the shaft. It uses the addition of a specific slant pattern in the teeth, and an additional digital filter used to isolate the effects of the slanted teeth. 
       FIG. 21  is a signal processing diagram for a system augmented to detect axial motion. The signal processing includes a parallel path includes a digital filter  1402  to isolate slanted teeth and a low pass filter  1404  to output an axial measurement. The axial measurement can be used for compensation of the twist measurement and the shaft stiffness to improve the torque output. 
       FIG. 22  is a diagram showing an unraveled set of targets passing a VR sensor. Similar to the case with straight teeth, described above with reference to  FIG. 20 , the timing at each tooth passage can be written in the following form with the addition of a term to account for the effect of the axial motion and the slants of the teeth: 
     
       
         
           
             
               v 
               k 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     f 
                                     clock 
                                   
                                   N 
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       ∫ 
                                       0 
                                     
                                     k 
                                   
                                   ⁢ 
                                   
                                     
                                       dk 
                                       ′ 
                                     
                                     
                                       f 
                                       
                                         k 
                                         shaft 
                                         ′ 
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     f 
                                     clock 
                                   
                                   
                                     f 
                                     shaft 
                                   
                                 
                                 ⁢ 
                                 
                                   θ 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     f 
                                     clock 
                                   
                                   
                                     f 
                                     shaft 
                                   
                                 
                                 ⁢ 
                                 
                                   z 
                                   
                                     2 
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     r 
                                   
                                 
                                 ⁢ 
                                 
                                   tan 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       β 
                                       × 
                                       
                                         
                                           ( 
                                           
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         
                                           
                                             ( 
                                             
                                               k 
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           / 
                                           2 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               where 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               odd 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           f 
                           clock 
                         
                         N 
                       
                       ⁢ 
                       
                         
                           
                             ∫ 
                             0 
                           
                           k 
                         
                         ⁢ 
                         
                           
                             
                               dk 
                               ′ 
                             
                             
                               f 
                               
                                 k 
                                 shaft 
                                 ′ 
                               
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               where 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               k 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               is 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               even 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Where f clock  is the clock speed of the timing measurement, N is the total number of teeth, k is the discrete index in time, and f shaft  is the shaft speed at time instant k, and θ is the shaft twist. Additional parameters introduced to represent axial motion include z, the axial displacement, r the radius of the targets that are on the shaft, and β which is the angle of the tooth slants. While it is possible to make these slants non-uniform, the signal processing complexity is reduced if the slant is equal and opposite in the pattern shown above and the slant is a small angle. This can be further simplified if the shaft speed, f shaft , is roughly constant over the small time interval between teeth. 
     
       
         
           
             
               v 
               k 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             f 
                             clock 
                           
                           N 
                         
                         ⁢ 
                         
                           k 
                           
                             f 
                             shaft 
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             clock 
                           
                           
                             f 
                             shaft 
                           
                         
                         ⁢ 
                         
                           θ 
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             clock 
                           
                           
                             f 
                             shaft 
                           
                         
                         ⁢ 
                         
                           z 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             r 
                           
                         
                         ⁢ 
                         
                           tan 
                           ⁡ 
                           
                             ( 
                             
                               β 
                               × 
                               
                                 
                                   ( 
                                   
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   
                                     ( 
                                     
                                       k 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   / 
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             where 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             odd 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           f 
                           clock 
                         
                         N 
                       
                       ⁢ 
                       
                         k 
                         
                           f 
                           shaft 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           where 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           is 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           even 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     The timing value at each discrete index in time, Ts k , can be written as the following (with shaft speed f shaft  assumed to be constant) pattern that repeats where m is an integer (1, 2, 3, . . . ). 
     
       
         
           
             
               T 
               ⁢ 
               
                 s 
                 
                   k 
                   - 
                   0 
                 
               
             
             = 
             
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     0 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               = 
               
                 
                   
                     v 
                     
                       k 
                       - 
                       0 
                     
                   
                   - 
                   
                     v 
                     
                       k 
                       - 
                       1 
                     
                   
                 
                 = 
                 
                   
                     
                       f 
                       
                         c 
                         ⁢ 
                         l 
                         ⁢ 
                         o 
                         ⁢ 
                         c 
                         ⁢ 
                         k 
                       
                     
                     
                       f 
                       shaft 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         1 
                         N 
                       
                       - 
                       
                         θ 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       + 
                       
                         
                           Z 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             r 
                           
                         
                         ⁢ 
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         β 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               T 
               ⁢ 
               
                 s 
                 
                   k 
                   - 
                   1 
                 
               
             
             = 
             
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     1 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               = 
               
                 
                   
                     v 
                     
                       k 
                       - 
                       1 
                     
                   
                   - 
                   
                     v 
                     
                       k 
                       - 
                       2 
                     
                   
                 
                 = 
                 
                   
                     
                       f 
                       
                         c 
                         ⁢ 
                         l 
                         ⁢ 
                         o 
                         ⁢ 
                         c 
                         ⁢ 
                         k 
                       
                     
                     
                       f 
                       shaft 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         1 
                         N 
                       
                       + 
                       
                         θ 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       - 
                       
                         
                           Z 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             r 
                           
                         
                         ⁢ 
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         β 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               T 
               ⁢ 
               
                 s 
                 
                   k 
                   - 
                   2 
                 
               
             
             = 
             
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     2 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               = 
               
                 
                   
                     v 
                     
                       k 
                       - 
                       2 
                     
                   
                   - 
                   
                     v 
                     
                       k 
                       - 
                       3 
                     
                   
                 
                 = 
                 
                   
                     
                       f 
                       
                         c 
                         ⁢ 
                         l 
                         ⁢ 
                         o 
                         ⁢ 
                         c 
                         ⁢ 
                         k 
                       
                     
                     
                       f 
                       shaft 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         1 
                         N 
                       
                       - 
                       
                         θ 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       - 
                       
                         
                           Z 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             r 
                           
                         
                         ⁢ 
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         β 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               T 
               ⁢ 
               
                 s 
                 
                   k 
                   - 
                   3 
                 
               
             
             = 
             
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     3 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               = 
               
                 
                   
                     v 
                     
                       k 
                       - 
                       3 
                     
                   
                   - 
                   
                     v 
                     
                       k 
                       - 
                       4 
                     
                   
                 
                 = 
                 
                   
                     
                       f 
                       
                         c 
                         ⁢ 
                         l 
                         ⁢ 
                         o 
                         ⁢ 
                         c 
                         ⁢ 
                         k 
                       
                     
                     
                       f 
                       shaft 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         1 
                         N 
                       
                       + 
                       
                         θ 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       + 
                       
                         
                           Z 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             r 
                           
                         
                         ⁢ 
                         tan 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         β 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Or more simply, 
     
       
         
           
             
               T 
               ⁢ 
               
                 s 
                 k 
               
             
             = 
             
               
                 
                   f 
                   
                     c 
                     ⁢ 
                     l 
                     ⁢ 
                     o 
                     ⁢ 
                     c 
                     ⁢ 
                     k 
                   
                 
                 
                   f 
                   shaft 
                 
               
               ⁢ 
               
                 ( 
                 
                   
                     1 
                     N 
                   
                   - 
                   
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         k 
                       
                       ⁢ 
                       θ 
                     
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                   + 
                   
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           
                             k 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           
                             2 
                             Z 
                           
                         
                       
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         r 
                       
                     
                     ⁢ 
                     tan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     β 
                   
                 
                 ) 
               
             
           
         
       
     
     Note that the calculation for twist remains the same, and axial motion does not affect nominally affect this measurement of twist: 
     
       
         
           
             
               
                 θ 
                 k 
               
               = 
               
                 
                   
                     
                       - 
                       2 
                     
                     ⁢ 
                     π 
                   
                   N 
                 
                 ⁢ 
                 
                   
                     f 
                     shaft 
                   
                   
                     f 
                     
                       c 
                       ⁢ 
                       l 
                       ⁢ 
                       o 
                       ⁢ 
                       c 
                       ⁢ 
                       k 
                     
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       n 
                       = 
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       n 
                     
                     ⁢ 
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         n 
                       
                     
                   
                 
               
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               θ 
               = 
               
                 
                   F 
                   
                     A 
                     ⁢ 
                     A 
                   
                 
                 | 
                 
                   θ 
                   k 
                 
                 | 
                 
                   - 
                   
                     θ 
                     0 
                   
                 
               
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               T 
               = 
               
                 K 
                 ⁡ 
                 
                   ( 
                   
                     θ 
                     - 
                     
                       θ 
                       0 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The axial displacement over an entire revolution can be calculated by adding and subtracting all of the timing values. 
     
       
         
           
             
               
                 
                   ∑ 
                   
                     m 
                     = 
                     0 
                   
                   
                     m 
                     = 
                     
                       
                         N 
                         ⁢ 
                         
                           / 
                         
                         ⁢ 
                         4 
                       
                       - 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   T 
                   ⁢ 
                   
                     s 
                     
                       k 
                       - 
                       
                         4 
                         ⁢ 
                         m 
                       
                     
                   
                 
               
               - 
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     1 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               - 
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     2 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               + 
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     3 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               = 
               
                 
                   
                     T 
                     ⁢ 
                     
                       s 
                       k 
                     
                   
                   - 
                   
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         1 
                       
                     
                   
                   - 
                   
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         2 
                       
                     
                   
                   + 
                   
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         3 
                       
                     
                   
                   + 
                   … 
                   ⁢ 
                   
                       
                   
                   + 
                   
                     Ts 
                     
                       k 
                       - 
                       N 
                       - 
                       4 
                     
                   
                   - 
                   
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         N 
                         - 
                         3 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   - 
                   
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         N 
                         - 
                         2 
                       
                     
                   
                   + 
                   
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         N 
                         - 
                         1 
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 = 
                 
                   N 
                   ⁢ 
                   
                     
                       f 
                       clock 
                     
                     
                       f 
                       shaft 
                     
                   
                   ⁢ 
                   
                     Z 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       r 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   tan 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   β 
                 
               
             
           
         
       
     
     Rewriting this equation and solving for z results in the following: 
     
       
         
           
             
               z 
               k 
             
             = 
             
               
                 
                   
                     2 
                     ⁢ 
                     π 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     r 
                   
                   
                     N 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     tan 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       β 
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   
                     f 
                     shaft 
                   
                   
                     f 
                     clock 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       m 
                       = 
                       0 
                     
                     
                       m 
                       = 
                       
                         
                           N 
                           ⁢ 
                           
                             / 
                           
                           ⁢ 
                           4 
                         
                         - 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     T 
                     ⁢ 
                     
                       s 
                       
                         k 
                         - 
                         
                           4 
                           ⁢ 
                           m 
                         
                       
                     
                   
                 
               
               - 
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     1 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               - 
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     2 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
               + 
               
                 T 
                 ⁢ 
                 
                   s 
                   
                     k 
                     - 
                     3 
                     - 
                     
                       4 
                       ⁢ 
                       m 
                     
                   
                 
               
             
           
         
       
     
     This can also be rewritten as a digital FIR filter with the following coefficients for a case where there are N=12 teeth. This digital FIR filter is an example of the digital filter  1404  for isolating axial motion. 
     
       
         
           
             B 
             = 
             
               
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   R 
                 
                 
                   12 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   tan 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     β 
                     ) 
                   
                 
               
               ⁢ 
               
                 
                   
                     f 
                     shaft 
                   
                   
                     f 
                     clock 
                   
                 
                 [ 
                 
                   
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       
                         - 
                         1 
                       
                     
                     
                       
                         1 
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     In practice, this value of z should be designed to always be positive, and should also be filtered down to a lower bandwidth with an anti-aliasing filter, F AA ; it is also helpful to apply a calibration offset z 0  to adjust for any real world imperfections in the axial location. 
     
       
      
       Z=F 
       AA 
       |z 
       k 
       |−z 
       0  
      
     
     Due to real-world machining tolerances, the twist value measured may change as the axial measurement changes. This would adjust the twist offset to be a function of the axial measurement (denoted θ 0 {z}). 
         T=K (θ−θ 0   {z })
 
     In addition, depending on the mechanical construction of the shaft, temperature variation may increase proportionally with the axial measurement. In order to remove a temperature sensor, the axial measurement can be used to adjust the stiffness as a function of the axial measurement, denoted K{z} (instead of being a function of temperature). This would adjust the Torque calculation as follows: 
         T=K{z }(θ−θ 0   {z })
 
     Similar to the single sensor torque calculation, a dual sensor configuration can be used to achieve additional accuracy. This involves placing one of the two sensors over opposite sets of the interleaved teeth, for example, as shown in  FIG. 6 . 
       FIG. 23  is a signal processing diagram of a system configured for processing two sensor signals to achieve a more accurate torque measurement. The signal processing includes a digital filter  1602  to isolate a twist measurement from a raw timing measurement, a digital filter  1604  to isolate radial effects, and a combiner  1606 . The output of the combiner  1606  is input to a low pass filter  1608  that outputs a compensated twist measurement. The signal processing includes another combiner  1610  to use a measurement of shaft stiffness to generate a torque output. 
     In general, these effects become more important as overall twist on the shaft becomes small, such as 0.5 degrees. At large gaps, e.g., &gt;0.2″ there is a noise improvement utilizing two sensors for measurement. Some magnetic effects from multiple sensors cause phase shifts in the twist measurement with radial motion. Multiple sensors can be used such that this effect (observed on the order of 0.030 degrees) to be reduced to negligible levels (e.g., 0.004 degrees). 
       FIG. 24  is a diagram showing an unraveled set of targets passing two VR sensors. In the example shown in  FIG. 24 , the instant in time that each tooth passes (v k ) can be written as the following (note that this is now a vector quantity representing two sensors): 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       v 
                       1 
                       k 
                     
                   
                 
                 
                   
                     
                       v 
                       2 
                       k 
                     
                   
                 
               
               ] 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             f 
                             clock 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             
                               ∫ 
                               0 
                             
                             k 
                           
                           ⁢ 
                           
                             
                               dk 
                               ′ 
                             
                             
                               f 
                               shaft 
                               
                                 k 
                                 ′ 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             clock 
                           
                           
                             f 
                             shaft 
                             k 
                           
                         
                         ⁢ 
                         
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   θ 
                                 
                               
                               
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             where 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             odd 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             f 
                             clock 
                           
                           N 
                         
                         ⁢ 
                         
                           
                             
                               ∫ 
                               0 
                             
                             k 
                           
                           ⁢ 
                           
                             
                               dk 
                               ′ 
                             
                             
                               f 
                               shaft 
                               
                                 k 
                                 ′ 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             clock 
                           
                           
                             f 
                             shaft 
                             k 
                           
                         
                         ⁢ 
                         
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   0 
                                 
                               
                               
                                 
                                   θ 
                                 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             where 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             even 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     Where f clock  is the clock speed of the timing measurement, N is the total number of teeth, k is the discrete index in time, and f shaft  is the shaft speed at time instant k, and θ is the shaft twist. This can be further simplified if the shaft speed, is f shaft , roughly constant over the small time interval between teeth. 
     
       
         
           
             
               [ 
               
                 
                   
                     
                       v 
                       1 
                       k 
                     
                   
                 
                 
                   
                     
                       v 
                       2 
                       k 
                     
                   
                 
               
               ] 
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             f 
                             clock 
                           
                           N 
                         
                         ⁢ 
                         
                           k 
                           
                             f 
                             shaft 
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             clock 
                           
                           
                             f 
                             shaft 
                           
                         
                         ⁢ 
                         
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   θ 
                                 
                               
                               
                                 
                                   0 
                                 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             where 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             odd 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 1 
                               
                             
                             
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             f 
                             clock 
                           
                           N 
                         
                         ⁢ 
                         
                           k 
                           
                             f 
                             shaft 
                           
                         
                       
                       + 
                       
                         
                           
                             f 
                             clock 
                           
                           
                             f 
                             shaft 
                           
                         
                         ⁢ 
                         
                           
                             1 
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   0 
                                 
                               
                               
                                 
                                   θ 
                                 
                               
                             
                             ] 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             where 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             k 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             even 
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The timing value between the two sensors, denoted dab k , can be written as the following (with shaft speed f shaft  assumed to be constant) and is a measurement of twist: 
     
       
         
           
             
               d 
               ⁢ 
               a 
               ⁢ 
               
                 b 
                 k 
               
             
             = 
             
               
                 
                   v 
                   1 
                   k 
                 
                 - 
                 
                   v 
                   2 
                   k 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   k 
                 
                 ⁢ 
                 
                   
                     f 
                     
                       c 
                       ⁢ 
                       l 
                       ⁢ 
                       o 
                       ⁢ 
                       c 
                       ⁢ 
                       k 
                     
                   
                   
                     f 
                     shaft 
                   
                 
                 ⁢ 
                 
                   
                     - 
                     θ 
                   
                   
                     2 
                     ⁢ 
                     π 
                   
                 
               
             
           
         
       
     
     Note that the final result of this equation applies to all discrete indices of k. The effect of twist on an interleaved pattern of teeth results in a timing change that is an alternating positive and negative value of twist; this pattern repeats every revolution. A series of digital filtering can therefore isolate the twist. The twist over an entire revolution can be calculated by adding and subtracting all of the timing values. This equation forms the basis of the filtering coefficients for the digital filter  1602  for isolating twist with two sensors. 
     
       
         
           
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 
                   n 
                   = 
                   
                     N 
                     - 
                     1 
                   
                 
               
               ⁢ 
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   n 
                 
                 ⁢ 
                 d 
                 ⁢ 
                 a 
                 ⁢ 
                 
                   b 
                   
                     k 
                     - 
                     n 
                   
                 
               
             
             ⁢ 
             
               
 
             
             ⁢ 
             
               = 
               
                 
                   
                     d 
                     ⁢ 
                     a 
                     ⁢ 
                     
                       b 
                       k 
                     
                   
                   - 
                   
                     d 
                     ⁢ 
                     a 
                     ⁢ 
                     
                       b 
                       
                         k 
                         - 
                         1 
                       
                     
                   
                   + 
                   
                     d 
                     ⁢ 
                     a 
                     ⁢ 
                     
                       b 
                       
                         k 
                         - 
                         2 
                       
                     
                   
                   - 
                   
                     d 
                     ⁢ 
                     a 
                     ⁢ 
                     
                       b 
                       
                         k 
                         - 
                         3 
                       
                     
                   
                   + 
                   … 
                   ⁢ 
                   
                       
                   
                   + 
                   
                     dab 
                     
                       k 
                       - 
                       N 
                       - 
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   - 
                   
                     d 
                     ⁢ 
                     a 
                     ⁢ 
                     
                       b 
                       
                         k 
                         - 
                         N 
                         - 
                         1 
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 = 
                 
                   
                     - 
                     
                       
                         f 
                         
                           c 
                           ⁢ 
                           l 
                           ⁢ 
                           o 
                           ⁢ 
                           c 
                           ⁢ 
                           k 
                         
                       
                       
                         f 
                         shaft 
                       
                     
                   
                   ⁢ 
                   
                     θ 
                     π 
                   
                   ⁢ 
                   
                     N 
                     2 
                   
                 
               
             
           
         
       
     
     However, in experimental testing, radial motion effects did cause slight phase shifts in the VR sensor Zero-Crossing measurement. The above calculation is a raw twist measurement that requires some adjustment as the target wheel moves radially, this allows a correction of the twist accuracy to levels that are sub 0.004 degrees accurate. This radial correction factor can be isolated by looking at an individual target passing both sensors. 
     The timing value between the two sensors looking at one side of targets, denoted dabz1 k , can be written as the following (with shaft speed f shaft  assumed to be constant): 
     
       
         
           
             
               d 
               ⁢ 
               a 
               ⁢ 
               b 
               ⁢ 
               z 
               ⁢ 
               
                 1 
                 k 
               
             
             = 
             
               
                 
                   v 
                   1 
                   k 
                 
                 - 
                 
                   v 
                   2 
                   
                     k 
                     - 
                     1 
                   
                 
               
               = 
               
                 
                   
                     v 
                     1 
                     k 
                   
                   - 
                   
                     
                       v 
                       2 
                       k 
                     
                     ⁢ 
                     
                       z 
                       
                         - 
                         1 
                       
                     
                   
                 
                 = 
                 
                   
                     f 
                     
                       c 
                       ⁢ 
                       l 
                       ⁢ 
                       o 
                       ⁢ 
                       c 
                       ⁢ 
                       k 
                     
                   
                   
                     N 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       f 
                       shaft 
                     
                   
                 
               
             
           
         
       
     
     Note that this value should remain constant, however, in practice the value changes as the radial position of the shaft or sensor changes, because of this observed fact, this value can be used to compensate the twist measurement and provide a more accurate torque value. This equation forms the basis of the filtering coefficients for the digital filter  1604  for isolating radial motion with two sensors. Filtering over a revolution gives the following relationship: 
     
       
         
           
             
               DABZ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 1 
                 k 
               
             
             = 
             
               | 
               
                 
                   1 
                   N 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       n 
                       = 
                       
                         N 
                         - 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     d 
                     ⁢ 
                     a 
                     ⁢ 
                     b 
                     ⁢ 
                     z 
                     ⁢ 
                     
                       1 
                       
                         k 
                         - 
                         n 
                       
                     
                   
                 
               
               | 
             
           
         
       
     
     In practice, a more accurate twist measurement can be calculated with the following relationship: 
     
       
         
           
             
               θ 
               comp 
               k 
             
             = 
             
               
                 θ 
                 raw 
                 k 
               
               - 
               
                 G 
                 × 
                 
                   
                     2 
                     ⁢ 
                     π 
                     ⁢ 
                     
                       f 
                       shaft 
                     
                   
                   
                     f 
                     
                       c 
                       ⁢ 
                       l 
                       ⁢ 
                       o 
                       ⁢ 
                       c 
                       ⁢ 
                       k 
                     
                   
                 
                 ⁢ 
                 DABZ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   1 
                   k 
                 
               
             
           
         
       
     
     Where G is a scalar value or lookup table that depends on any of the following values: shaft speed, temperature, or the value of DABZ1 k  (if it ends up being a non-linear relationship). In practice, this compensated value of θ should be filtered down to a lower bandwidth with an anti-aliasing filter, F AA ; it is also helpful to apply a calibration offset θ 0  to adjust for any real world imperfections in the amount of twist. 
       θ= F   AA |θ comp   k |−θ 0  
 
     Exactly as before, the shaft torsional stiffness, K, can be multiplied in to determine torque, T: 
         T=K (θ−θ 0 )
 
     Similar to previous concepts, Axial (or other) motions can be measured by incorporated slanted teeth with a single sensor. This process can also be followed with a two sensor setup where the axial measurement can be used to further compensate the dual sensor twist measurement by providing an additional calibration offset for the twist measurement, θ, and/or providing an alternate measurement to temperature for compensating the stiffness, K.  FIG. 25  is a signal processing diagram for an example system using dual sensors and axial/slanted teeth to output torque. The system includes a digital filter  1802  to isolate a twist measurement, a digital filter  1804  to isolate radial effects, and a digital filter  1806  to isolate axial effects. 
     Similar to the dual sensor torque concept with straight teeth, three sensors can be used to determine a more accurate torque. With three sensors, the exact x/y position of the shaft or cradle can be ascertained. This also allows a slightly more accurate compensation of the twist measurement, θ. For example, U.S. Pat. No. 7,093,504 describes methods for determining x/y motion from three sensors. U.S. Pat. No. 7,093,504 is hereby incorporated by reference in its entirety.  FIG. 26  is a signal processing diagram for an example system using three sensors. 
       FIGS. 27 and 28  show various aspects of another example embodiment of a system, generally designated  106 , for sensing torque transmitted through a rotatable shaft  10 . As shown, the system  106  is generally similar to the system  105  shown in  FIG. 18 , in that magnetic field sensors  122 A,  122 B are used to detect phase shifts in a magnetic field while the shaft  10  is rotating. However, unlike in system  105 , in system  106 , the magnetic field is generated by a plurality of magnets arranged circumferentially about the outer surface of the shaft  10 . The magnets are arranged as a first set of magnets  16 A, which are attached to the shaft  10  at a first position  11 A and are positioned adjacent to (e.g., in a position so that the magnetic field produced is detectable by) the first magnetic field sensor  122 A, and as a second set of magnets  16 B, which are attached to the shaft at a second position  11 B and are positioned adjacent to (e.g., in a position so that the magnetic field produced is detectable by) the second magnetic field sensor  122 B. As the shaft  10  transmits a torque therethrough, the shaft  10  will torsionally deform (e.g., twist) in the target region TR, defined as between the first position  11 A and the second position  11 B. As such, during torsional deformation of the shaft  10 , the magnetic fields produced by the first and second sets of magnets  16 A,  16 B will be offset from each other, relative to the magnetic fields present when the shaft is not twisted. This offset in magnetic fields produced by the first and second sets of magnets  16 A,  16 B, as detected by the first and second magnetic field sensors  122 A,  122 B, can be referred to as a phase shift, the phase shift being proportional to the magnitude of torsional deformation (e.g., twist) of the shaft  10 . 
     As is shown in  FIG. 28 , the magnets  17  are arranged circumferentially about the outer surface of the shaft  10 , with the polarity of the magnets  17  alternating in the circumferential direction of the shaft  10  (e.g., such that the polarity of each magnet  17  is different, or opposite, from the polarity of each adjacent magnet  17 ). In some embodiments, the magnets  17  can be integrated into the shaft  10  and/or attached to an inner surface thereof. In the example embodiment shown in  FIG. 28 , the magnets  17  are spaced equally about the perimeter of the shaft  10 , however the magnets  17  may be spaced apart from each other in any pattern. In some embodiments, the spacing pattern of the magnets of the first set of magnets  16 A is substantially identical to the spacing pattern of the magnets of the second set of magnets  16 B. In some embodiments, the spacing pattern of the magnets of the first set of magnets  16 A is substantially similar to, but circumferentially offset from, the spacing pattern of the magnets of the second set of magnets  16 B. In some embodiments, the spacing pattern of the magnets of the first set of magnets  16 A is different from the spacing pattern of the magnets of the second set of magnets  16 B. In another example embodiment, the magnets  17  may be formed as all or a portion of a ring of magnets around the circumference of the shaft  10 . For example, the magnets of the first set of magnets  16 A can be interleaved (e.g., in an alternating pattern, or any other suitable pattern) with the magnets of the second set of magnets  16 B. In some such embodiments, each of the magnets of the first set of magnets  16 A can be directly adjacent to (e.g., in direct contact with, such as in the circumferential direction) correspondingly adjacent magnets of the second set of magnets  16 B, such that the magnets  17  form a substantially continuous and uninterrupted ring of magnets about the circumference of the shaft  10 . In some such embodiments, each magnet  17  has a different (e.g., reverse, or opposite) polarity from magnets  17  immediately adjacent thereto. 
       FIG. 29  is an alternative example embodiment of the system  105  shown in  FIG. 18 , but in which the first and second annular rings  70 A,  70 B are axially spaced apart from the position of the bearing  60 , with the target elements  72 A,  72 B extending in the longitudinal direction of the shaft  10  to be interleaved between each other (e.g., in an alternating pattern) in a plane defined by the sensor  120 . As such, the first targets  72 A have a longer length than the second targets  72 B, such that the first and second targets  72 A,  72 B terminate in a same plane, at or beyond the axial position of the sensor  120  in the longitudinal direction of the shaft  10 . As such, this example embodiment allows for measuring torque across a target region of a shaft  10  spaced apart from the bearing  10  in the longitudinal direction of the shaft  10 . Otherwise, the system  107  is substantially identical to the system  105 . 
     The embodiments described herein are examples only and are not limiting. Many variations and modifications of the systems, apparatus, and processes described herein are possible and are within the scope of the disclosure. Accordingly, the scope of protection is not limited to the embodiments described herein, but is only limited by the claims that follow, the scope of which shall include all equivalents of the subject matter of the claims.