Patent Publication Number: US-2013231860-A1

Title: Systems and methods to incorporate master navigation system resets during transfer alignment

Description:
BACKGROUND 
     Transfer alignment is the process of transmitting navigation information from an onboard navigation system of an aircraft to the onboard navigation system of a payload that is deployed from the aircraft while in flight. The navigation information typically takes the form of periodic messages that convey position, velocity and attitude as measured by the aircraft navigation system. The navigation system on the payload receives these periodic messages from the aircraft as measurement updates used to derive error statistics for calibrating its own navigation solution. A problem occurs from the fact that the onboard navigation system of the aircraft periodically needs to adjust its own navigation solution to correct for accumulated drift error. This adjustment will appear as a step change in the measurement updates received by the navigation system on the payload. If the step change is too large, the navigation system on the payload may reject the measurement update. Even if the navigation system on the payload accepts the step change measurement update, it faces the conundrum of how to account for the step change and allocate what it must interpret as a measurement error generated by one or more of its own sensors. 
     For the reasons stated above and for other reasons stated below which will become apparent to those skilled in the art upon reading and understanding the specification, there is a need in the art for systems and methods to incorporate master navigation system resets during transfer alignment. 
     SUMMARY 
     The Embodiments of the present invention provide methods and systems that address the master navigation system resets during transfer alignment and will be understood by reading and studying the following specification. 
     In one embodiment, a navigation system for performing a transfer alignment comprises: a set of local inertial sensors; a local strap-down navigation processor coupled to the set of local inertial sensors, the local strap-down navigation processor receiving inertial navigation data from the set of local inertial sensors and converting the inertial navigation data into a navigation solution; a local Kalman filter coupled to the local strap-down navigation processor, the local Kalman filter receiving the navigation solution produced by the local strap-down navigation processor, the local Kalman filter further coupled to a master Kalman filter; wherein the local Kalman filter receives a first message from the master Kalman filter comprising a Precision Transfer Alignment Message (PTAM) that includes at least one navigation aid measurement; wherein the local Kalman filter inputs the navigation aid measurement into a measurement formation algorithm and calculates a measurement residual based on the navigation aid measurement; wherein the local Kalman filter receives a second message from the master Kalman filter comprising a Reset Transfer Alignment Message (RTAM) that includes a bias correction applied by the master Kalman filter as a navigation reset; wherein the local Kalman filter inputs the bias correction into a state propagation algorithm where the bias correction provided by the bias correction is added to a navigation state of the local Kalman filter and propagated. 
    
    
     
       DRAWINGS 
       Embodiments of the present invention can be more easily understood and further advantages and uses thereof more readily apparent, when considered in view of the description of the preferred embodiments and the following figures in which: 
         FIG. 1  is a diagram illustrating a navigation system of one embodiment of the present invention; 
         FIG. 2  is a diagram illustrating a local Kalman filter of one embodiment of the present invention; and 
         FIG. 3  is a flow chart illustrating a method of one embodiment of the present invention. 
     
    
    
     In accordance with common practice, the various described features are not drawn to scale but are drawn to emphasize features relevant to the present invention. Reference characters denote like elements throughout figures and text. 
     DETAILED DESCRIPTION 
     In the following detailed description, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of specific illustrative embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to be understood that other embodiments may be utilized and that logical, mechanical and electrical changes may be made without departing from the scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense. 
     Embodiments of the present description address the needs to address bias corrections in transfer alignment configurations through a modified Kalman filter algorithm implemented in the local navigation system of the payload. In addition to having utilizing navigation information from a master Kalman filter as a navigation aid, with embodiments of the present invention, the local Kalman filter also receives a bias correction via a specific messages generated by the master Kalman filter when the master Kalman filter performs a navigation reset: The local Kalman filter utilizes the bias correction to adjust the values of its own navigation states so that when the local Kalman filter calculates a measurement residual, deviations in the navigation aid measurements due to the reset at the master Kalman filter are offset by the adjustments made to the local navigation states. As a result, no additional residual at the local Kalman filter is produced as a result of the bias correction reset performed at the master Kalman filter. 
       FIG. 1  is a block diagram of a transfer alignment system  100  of one embodiment of the present invention. System  100  comprises a host aircraft navigation system  102  located onboard an aircraft (such as an airplane or helicopter, for example) and a payload navigation system  104  located onboard a payload mounted to the aircraft which is subsequently launched from the aircraft. 
     Aircraft navigation system  102  comprises aircraft inertial sensors  110 , an Aircraft Strap-Down Navigation Algorithm  112 , a navigation aid  114  and a master Kalman filter. The aircraft inertial sensors  110  provide 3-axis gyroscope and 3-axis accelerometer measurements of the aircraft frame to Strap-Down Navigation Algorithm  112 . From these measurements, Strap-Down Navigation Algorithm  112  produces a navigation solution for the aircraft, including the aircrafts position, velocity and attitude in 3 dimensions. Navigation aid  114  provides a secondary source for one or more parameters of a navigation solution for the aircrafts position, velocity and attitude. In one embodiment, Navigation aid  114  comprises a navigation receiver for a Global Navigation Satellite System such as, for example, a Global Positioning System (GPS) receiver. 
     The Strap-Down Navigation Algorithm  112  is a basic navigation algorithm that takes the output from accelerometers and gyros in the aircraft inertial sensors  110  and integrates them up to produce position, velocity and attitude data. This data is referred to as a navigation solution. The values provided in the navigation solution will drift over time. For example, a small error in acceleration or rotation will compound over time if no correction is performed. Therefore, Aircraft navigation system  102  needs to periodically correct for these drift error using a measurement from navigation aid  114 . Master Kalman filter  120  receives the navigation data from Strap-Down Navigation Algorithm  112  and navigation aid  114  and using this information estimates the error in the strapdown navigation solution given a particular measurement from the navigation aid  114 . Once an error is estimated, Master Kalman filter  120  computes and applies a reset (which comprises a step change correction to the strapdown navigation solution) to adjust the Strap-Down Navigation Algorithm  112  output to accommodate for the drift. Resets are generally performed on a periodic basis, but in some embodiments may be held until an accumulated error to reaches a threshold. In one embodiment, update to the navigation solution from the Strap-Down Navigation Algorithm  112  are provided at relatively high rate as compared to updates from Master Kalman filter  120 . For example, in one embodiment, Strap-Down Navigation Algorithm  112  provides updates to its navigation solution output at a rate of 100 Hz (100 times/sec) while the Master Kalman filter  120  generates reset at a rate of 1 Hz (once/sec). In one embodiment, each time the Master Kalman filter  120  estimates an error, in applies a reset to the Strap-Down Navigation Algorithm  112 . 
     Payload navigation system  104  operates in a very similar way to aircraft navigation system  102 , which the notable exception that Master Kalman filter  120  functions as the source of Navigation aid data for Payload navigation system  104 . As shown in  FIG. 1 , Payload navigation system  104  comprises inertial sensors  130 , a Local Strap-Down Navigation Algorithm  132 , and a Kalman filter (referred to herein as the local Kalman filter  140 ). The local inertial sensors  130  provide 3-axis gyroscope and 3-axis accelerometer measurements of the payload body to Strap-Down Navigation Algorithm  132 . From these measurements, Strap-Down Navigation Algorithm  132  produces a navigation solution for the payload, including the payload&#39;s position, velocity and attitude in 3 dimensions. Local Kalman filter  140  uses measurements from the master Kalman filter  120  as the navigation aid for calculating error in the navigation solution provided by Strap-Down Navigation Algorithm  132 . That is, it uses messages from master Kalman filter  120  form a measurement from which it will estimate its own error in its own strap-down navigation system and performing a reset based on that estimate of error. 
     In one embodiment, local Kalman filter  140  comprises an extended Kalman filter algorithm executed by a processing element  142 . In one embodiment, in order to implement Kalman filter  140 , processing element  142  executes a series of algorithms components, shown at  150 - 160  in  FIG. 1  and using the corresponding respective reference numbers in the following table: 
                                     Ref.       Equation Implemented        No.   Algorithm Component   by Algorithm Component                  150   State propagation (including   Δx k   −  = φ k Δx k−1   +  − r k  + T k             addition of RTAM)       152   Measurement Formation   y k  = PTAM n  − h(x k   + )       154   Covariance Propagation   P k   −  = φ k P k−1   + φ k   T  + Q k         156   Gain computation   K k  = P k   − H k   T (H k P k   − H k   T  + R) −1         158   Measurement update   Δx k   +  = Δx k   −  + K k (y k  − H k  Δx k   − )       160   Compute Resets to Plant and   r k  = f(Δx k   + )           states                    
Where the variable notation in the equations follow standard Kalman filter nomenclature except as indicated herein. As would be appreciated by one of ordinary skill in the art upon reading this disclosure, the values held the states of the navigation state vector, x k , can be either in absolute navigation terms (e.g., a position or velocity value), or alternate values such as values based on a linear function of a navigation measurement value.
 
     Local Kalman filter  140  receives two different types of messages from master Kalman filter  120 , which communicate two different measurements. The first message type is a Precision Transfer Alignment Message (PTAM). A PTAM conveys conventional position, velocity and attitude navigation solution measurements (or a subset thereof) that the local Kalman filter  140  uses as navigation aid measurements. In one embodiment, a PTAM provides absolute measurements. However, in other embodiments, the PTAM may provide the navigation solution measurements in other ways. In Local Kalman filter  140 , the values provided by PTAM are entered into the Measurement Formation algorithm component  152  as shown in the table above and in  FIG. 1 . 
     The second type of message from master Kalman filter  120  is generated each time the master Kalman filter  120  generates a reset to its own Strap-Down Navigation Algorithm  112 . For this reason, the second type message is referred to as a Reset Transfer Alignment Message (RTAM). In essence, one purpose of the RTAM is to provide notice to the local Kalman filter  140  that the measurements provided by the next PTAM message may include a significant step change in one or more measurements from those provided in the previously received PTAM. More than that, however, with embodiments of the present invention, the local Kalman filter  140  is constructed to utilize a received RTAM within its Kalman algorithms so that its calculated residual will not increase due to a unexpected measurement error caused by a reset in the master Kalman filter  120 . In local Kalman filter  140 , the values provided by RTAM are entered into the State propagation algorithm component  150  as shown and  FIG. 1  and as represented by the variable Tk as shown in the table above. For iterations of local Kalman filter  140  where no RTAM has been received, the value of Tk used to calculate State propagation algorithm component  150  is simply zero for that iteration. 
     In local Kalman filter  140 , the Measurement Formation algorithm component  152  determines y k , which represents the difference between the most recent measurement (e.g. PTAM n ) and its prediction (represented by h(x k   + )) of what the next measurement (e.g. PTAM n+1 ) will be. Measurement Formation algorithm component  152  does not directly include the value of incoming RTAM messages, Tk, in its computation, but does include the values from PTAM messages. When an RTAM message is generated by master Kalman Filter  120 , the Measurement Formation algorithm component  152 &#39;s prediction h(x k   + ) of the next PTAM measurement will be off by Tk because of the step change of Tk that will appear in the next PTAM message. For example, assume that the master Kalman Filter  120  outputs a PTAM message with the position value p 1 , and subsequently outputs an RTAM message with a position step change correction of Δp. The PTAM message next received from the master Kalman Filter  120  will have a position value p 2  that reflects both any actual change in position since p 1  plus the position step change correction of Δp. Without properly addressing the effects of the master Kalman Filter  120  reset that caused the RTAM message, the Measurement Formation algorithm component  152 &#39;s prediction h(x k   + ) of p 2  will reflect the change in position, but will be off by the Δp position step change correction Δp. This error in the prediction would manifest itself in the residual calculation performed by local Kalman filter  140  as an unexpected occurrence of error. That is, the value of the residual, v, (shown in  FIG. 1  at  162 ) calculated as part of the Measurement Update algorithm component  158  and represented by the expression: 
     
       
      
       v=y 
       k 
       −H 
       k 
       Δx 
       k 
       − 
      
     
     will be off by a factor that is a function of the step change value Tk, which can be illustrated by the expression: 
     
       
      
       v=y 
       k 
       +H 
       k 
       T 
       k 
       −H 
       k 
       Δx 
       k 
       − 
      
     
     To offset the effect of the master Kalman Filter  120  reset and subsequent step change in the next PTAM message, when an RTAM message is received, the value of Tk, is added with the navigation solution, φ k Δx k−1   + , calculated from payload&#39;s the strap-down sensors and propagated forward in time as part of the navigation state vector Δx k   −  by the State propagation algorithm component  150 . When the Measurement update algorithm component  158  is subsequently executed, the residual calculation will end up with two instances of H k T k  that cancel each other out as shown be the expression: 
     
       
      
       v=y 
       k 
       +H 
       k 
       T 
       k 
       −H 
       k 
       Δx 
       k 
       − 
       −H 
       k 
       T 
       k  
      
     
     As shown in this expression, any step change introduced through the value of PTAM caused by a reset of the master Kalman filter  120  will cause an error in the prediction of that PTAM resulting in a change to the residual, v, by +H k T k . This error will be canceled out by the effect of adding the RTAM value Tk to the navigation state vector, resulting in a change to the residual, v, by −H k T k . These values of +H k T k  and −H k T k  sum to have a net zero effect on the value of the residual. In other words, the residual will not indicate the presence of any non-anticipated error caused by the master Kalman filter  120  reset. Further, in this embodiment, because receiving an RTAM message has no net effect on the calculation of the residual, the effects of receiving such an RTAM will not trigger local Kalman filter  140  to reject subsequent incoming PTAM messages. 
       FIG. 2  is a diagram providing an alternate illustration of local Kalman filter  140 . The filter measurement y(n) is the difference between the data from the payload&#39;s strap-down navigation algorithm  132  the measurement values from PTAM messages provided by the master Kalman filter  120 . The output of the H(h) block is H(n)x(n) which represents the local Kalman filter  140 &#39;s prediction of what the next value of y(n) will be. The difference between the two is the measurement residual v(n), which as mentioned above, represents an amount of error in the strapdown navigation data not anticipated by the local Kalman filter  140 . 
     As explained above, when the master Kalman filter  120  generates a bias correction reset and RTAM message, the effect of that reset also appear in subsequent PTAM messages. Thus, the value of y(n) will increase by a corresponding value of H k T k . To accommodate this increase in PTAM message values without affecting the measurement residual v(n), the value Tk of the corresponding RTAM message is added to the navigation state x(n) and then propagated. This increase the value of H(n)x(n) by H k T k . When the difference between y(n) and H(n)x(n) is determined, the two instances of H k T k  are canceled out. Kalman Gain matrix K(n) then produces a weighted vector that determines how much to compensate due to the errors represented by the measurement residual v(n). The output of K(n) is added to x(n) (which includes an uncanceled H k T k  to implement the necessary reset) to produce navigation error states {circumflex over (x)}(n). The navigation solution output of the local Kalman filter  140  is thus a function of the strapdown navigation data as adjusted by the navigation error states {circumflex over (x)}(n). As indicated by the diagram, the navigation error states x {circumflex over (x)} (n) provide the necessary drift compensation determined by the master Kalman filter  120 , without causing any corresponding net increase in the measurement residual v(n). 
       FIG. 3  is a flow chart illustrating a method for performing a transfer alignment from a host navigation system to a payload navigation system. In one embodiment, the method is implemented using the host aircraft navigation system  102  located onboard an aircraft and payload navigation system  104  located onboard a payload mounted to the aircraft as described above with respect to  FIG. 1 . The method begins at  310  with receiving a navigation solution from a local strap down navigation processor. In one embodiment, the local strap down navigation processor is coupled to a set of local inertial sensors aboard the payload that comprises at least a triad of orthogonal accelerometers and a triad of orthogonal gyroscopes. 
     The method proceeds to  320  with receiving a Precision Transfer Alignment Message (PTAM) at a local Kalman filter, the PTAM comprising at least one navigation aid measurement generated by a master Kalman filter. As explained above, the bias correction provided by the RTAM may include a position correction, a velocity correction, an attitude correction, or a combination thereof. In one embodiment, the bias correction comprises a nine-dimensional vector comprising a 3-dimentional position bias correction, a 3-dimentional velocity correction, and a 3-dimentional attitude correction. 
     The method proceeds to  330  with receiving a Reset Transfer Alignment Message (RTAM) at a local Kalman filter, the RTAM comprising a bias correction applied by the master Kalman filter as a navigation reset. In one embodiment, the master Kalman filter receives navigation data from the aircraft&#39;s Strap-Down Navigation Algorithm and navigation aid and using this information estimates the error in its strapdown navigation solution given a particular measurement from the navigation aid. When a drift error is estimated, the master Kalman filter computes and applies a reset to adjust the aircraft&#39;s Strap-Down Navigation Algorithm output to accommodate for the drift. Resets are generally performed on a periodic basis, but in some embodiments may be held until an accumulated error to reaches a threshold. 
     In one embodiment, the PTAM communicates the at least one navigation aid measurement as absolute navigation solution values while the RTAM communications the bias correction as a change in navigation solution values. 
     The method proceeds to  340  with propagating a navigation state at the local Kalman filter using the navigation solution from a local strap down navigation processor, wherein the local Kalman filter adds the bias correction to the navigation state. The method proceeds to  350  with computing a measurement update at the local Kalman filter as a function of the at least one navigation aid measurement and the bias correction, wherein the navigation solution from a local strap down navigation processor is compensated for the bias correction by the measurement update. 
     In one embodiment, the local Kalman filter implements the algorithm components  150 - 160  illustrated in the table above and in  FIG. 1 . As such, in one embodiment, propagating the navigation state at the local Kalman filter is calculated using an equation equivalent to: Δx k   − =φ k Δx k−1   + −r k +T k , where T k  is the bias correction provided by the RTAM. Computing a measurement update at the local Kalman filter may be calculated using an equation equivalent to: Δx k   + =Δx k   − +K k (y k −H l Δx k   − ) where y k =PTAM n −h(x k   + ). As explained above, the local Kalman filter utilizes the bias correction to adjust the values of its own navigation states so that when the local Kalman filter calculates a measurement residual, deviations in the navigation aid measurements due to the reset at the master Kalman filter are offset by the adjustments made to the local navigation states. As a result, no additional residual at the local Kalman filter is produced as a result of the bias correction reset performed at the master Kalman filter. 
     Example Embodiments 
     Example 1 includes a navigation system for performing a transfer alignment, the navigation system comprising: a set of local inertial sensors; a local strap-down navigation processor coupled to the set of local inertial sensors, the local strap-down navigation processor receiving inertial navigation data from the set of local inertial sensors and converting the inertial navigation data into a navigation solution; a local Kalman filter coupled to the local strap-down navigation processor, the local Kalman filter receiving the navigation solution produced by the local strap-down navigation processor, the local Kalman filter further coupled to a master Kalman filter; wherein the local Kalman filter receives a first message from the master Kalman filter comprising a Precision Transfer Alignment Message (PTAM) that includes at least one navigation aid measurement; wherein the local Kalman filter inputs the navigation aid measurement into a measurement formation algorithm and calculates a measurement residual based on the navigation aid measurement; wherein the local Kalman filter receives a second message from the master Kalman filter comprising a Reset Transfer Alignment Message (RTAM) that includes a bias correction applied by the master Kalman filter as a navigation reset; wherein the local Kalman filter inputs the bias correction into a state propagation algorithm where the bias correction provided by the bias correction is added to a navigation state of the local Kalman filter and propagated. 
     Example 2 includes the navigation system of Example 1, wherein the bias correction provided by the RTAM includes one of a position correction, a velocity correction, an attitude correction, or a combination thereof. 
     Example 3 includes navigation system of Example 1 or 2, wherein the bias correction comprises a nine-dimensional vector comprising a 3-dimentional position bias correction, a 3-dimentional velocity correction, and a 3-dimentional attitude correction. 
     Example 4 includes the navigation system of Example 1, 2 or 3, wherein the PTAM communicates the at least one navigation aid measurement as absolute navigation solution values; and wherein the RTAM communications the bias correction as a change in navigation solution values. 
     Example 5 includes the navigation system of any of Examples 1-4, wherein the set of local inertial sensors comprises at least a triad of orthogonal accelerometers and a triad of orthogonal gyroscopes. 
     Example 6 includes the navigation system of any of Examples 1-5, wherein the local Kalman filter is implemented using an Extended Kalman Filter algorithm. 
     Example 7 includes the navigation system of any of Examples 1-6, further comprising a processing element; wherein the local Kalman filter is implemented as an algorithm executed by the processing element. 
     Example 8 includes the navigation system of any of Examples 1-7, wherein the local strap-down navigation processor and the local Kalman filter are both implemented using the same processor. 
     Example 9 includes the navigation system of any of Examples 1-8, wherein the measurement formation algorithm is calculated using an equation equivalent to: 
         y   k =PTAM n   −h ( x   k   + ). 
     Example 10 includes the navigation system of any of Examples 1-9, wherein the state propagation algorithm is calculated using an equation equivalent to: Δx k   − =φ k−1   + −r k +T k , where T k  is the bias correction provided by the RTAM. 
     Example 11 includes a method for perforthing a transfer alignment from a host navigation system to a payload navigation system, the method comprising: receiving a navigation solution from a local strap down navigation processor; receiving a Precision Transfer Alignment Message (PTAM) at a local Kalman filter, the PTAM comprising at least one navigation aid measurement generated by a master Kalman filter; receiving a Reset Transfer Alignment Message (RTAM) at a local Kalman filter, the RTAM comprising a bias correction applied by the master Kalman filter as a navigation reset; propagating a navigation state at the local Kalman filter using the navigation solution from a local strap down navigation processor, wherein the local Kalman filter adds the bias correction to the navigation state; computing a measurement update at the local Kalman filter as a function of the at least one navigation aid measurement and the bias correction, wherein the navigation solution from a local strap down navigation processor is compensated for the bias correction by the measurement update. 
     Example 12 includes the method of Example 11, wherein the bias correction provided by the RTAM includes one of a position correction, a velocity correction, an attitude correction, or a combination thereof. 
     Example 13 includes the method of Examples 11 or 12, wherein the bias correction comprises a nine-dimensional vector comprising a 3-dimentional position bias correction, a 3-dimentional velocity correction, and a 3-dimentional attitude correction. 
     Example 14 includes the method of any of Examples 11-13, wherein the PTAM communicates the at least one navigation aid measurement as absolute navigation solution values; and wherein the RTAM communications the bias correction as a change in navigation solution values. 
     Example 15 includes the method of any of Examples 11-14, wherein the local strap down navigation processor is coupled to a set of local inertial sensors that comprises at least a triad of orthogonal accelerometers and a triad of orthogonal gyroscopes. 
     Example 16 includes the method of any of Examples 11-15, wherein the local Kalman filter is implemented using an Extended Kalman Filter algorithm. 
     Example 17 includes the method of any of Examples 11-16, wherein propagating the navigation state at the local Kalman filter is calculated using an equation equivalent to: Δx k   − =φ k Δx k−1   + −n k +T k , where T k  is the bias correction provided by the RTAM. 
     Example 18 includes the method of any of Examples 11-17, wherein computing a measurement update at the local Kalman filter is calculated using an equation equivalent to: Δx k   + =Δx k   − +K k (y k −H k Δx k   − ) where y k =PTAM n −h(x k   + ). 
     Example 19 includes a system comprising: an aircraft having a aircraft navigation system comprising a master Kalman filter; a payload loaded onto the aircraft, the payload having a payload navigation system comprising: a set of local inertial sensors; a local strap-down navigation processor coupled to the set of local inertial sensors, the strap-down navigation processor, the local strap-down navigation processor receiving inertial navigation data from the set of local inertial sensors and converting the inertial navigation data into a navigation solution; a local Kalman filter coupled to the local strap-down navigation processor, the local Kalman filter receiving the navigation solution produced by the local strap-down navigation processor, the local Kalman filter further coupled to the master Kalman filter; wherein the local Kalman filter receives a first message from the master Kalman filter comprising a Precision Transfer Alignment Message (PTAM) that includes at least one navigation aid measurement; wherein the local Kalman filter inputs the navigation aid measurement into a measurement formation algorithm and calculates a measurement residual based on the navigation aid measurement; wherein the local Kalman filter receives a second message from the master Kalman filter comprising a Reset Transfer Alignment Message (RTAM) that includes a bias correction applied by the master Kalman filter as a navigation reset; wherein the local Kalman filter inputs the bias correction into a state propagation algorithm where the bias correction provided by the bias correction is added to a navigation state of the local Kalman filter and propagated. 
     Example 20 includes the system of Example 19 wherein the measurement formation algorithm is calculated using an equation equivalent to: y k =PTAM n −h(x k   + ) and wherein the state propagation algorithm is calculated using an equation equivalent to: Δx k   − =φ k Δx k−1   + −n k +T k , where T k  is the bias correction provided by the RTAM. 
     Several means of hardware are available to implement the systems and methods of the current invention as discussed in this specification. These means of hardware include, but are not limited to, digital computer systems, microprocessors, general purpose computers, programmable controllers and field programmable gate arrays. Therefore additional embodiments of the present invention include program instructions resident on computer readable storage media which when implemented by such devices, enable them to implement embodiments of the present invention. Computer readable storage media include any form of physical computer data storage hardware, including but not limited to punch cards, magnetic disk or tape, any optical data storage system, flash read only memory (ROM), non-volatile ROM, programmable ROM (PROM), erasable-programmable ROM (E-PROM), random access memory (RAM), or any other form of permanent, semi-permanent, or temporary memory storage system or device. Program instructions and code include, but are not limited to computer-executable instructions executed by computer system processors and hardware description languages such as Very High Speed Integrated Circuit (VHSIC) Hardware Description Language (VHDL). 
     Although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that any arrangement, which is calculated to achieve the same purpose, may be substituted for the specific embodiment shown. This application is intended to cover any adaptations or variations of the present invention. Therefore, it is manifestly intended that this invention be limited only by the claims and the equivalents thereof.