Patent Application: US-201414477191-A

Abstract:
this invention relates to methods and devices for bias estimation and correction , particularly for time - of - arrival based wireless geolocation systems . multipath and non - line - of - sight biases can cause distance estimation errors in the range of tens - hundreds of meters and is particularly problematic in urban and indoor environments . the behaviour of the biases dynamically changes depending on the clutter and / or obstructions between the base station and the mobile device . aspects of the present invention provide practical real - time bias estimation and correction techniques for toa - based systems and are based on inferring and estimating the biases from dynamic time differential measurements . the techniques can operate in real - time and involve simple calculations .

Description:
at their broadest , methods of the present invention provide for methods of estimating the bias in estimated distances between a mobile device and a base station by recursively estimating the bias and , optionally , determining the minimum such biases . a first aspect of the present invention provides a method of estimating the bias in an estimated distance between a mobile device and a base station , the method including the steps of : receiving , in the mobile device , wireless signals from the base station ; estimating , from said wireless signals , the distance of the mobile device from the base station ; and recursively estimating , from said estimated distances and the changes between differences of said estimated distances , the bias experienced in each of said estimates . the method of this aspect is particularly useful in harsh multipath or nlos environments . it is particularly suited to indoor and / or urban environments , in which more accurate localization approaches ( such as gps ) are not available . preferably the estimation of the distance of the mobile device from the base station uses time - of - arrival or time - difference - of - arrival ranging . there may be more than one base station and the bias estimation may be carried out for each base station . the biases can be dynamically estimated from the differential bias changes that occur while the mobile device is moving . once bias transitions occur then it is possible to lock onto the lowest experienced bias . thus the method of this aspect can have learning and convergence properties that result in accurate and robust localization and tracking . the method of this aspect can be computationally efficient as it only requires simple algebraic operations , it does not assume zero bias in los and does not require los / nlos identification . further , no assumption about time invariance ( stability of bias fluctuations ) need be made . furthermore , each range measurement / bias sample can be a completely different realization . this is highly desirable in dynamic and motion intensive environments ( e . g . shopping malls ) where the bias is changing / fluctuating with high frequency . the method can be integrated with a kalman filter framework for position estimation and does not require ( but can work with ) inertial navigation systems ( ins ). in some embodiments , the bias estimate { circumflex over ( b )} i for the bias at time i is estimated using the recursive equation : { circumflex over ( b )} i = δδ { circumflex over ( d )} i , i − 1 + 2 { circumflex over ( b )} i − 1 −{ circumflex over ( b )} i − 2 wherein δδ { circumflex over ( d )} i , i − 1 is the second difference in the estimate of the distance between the mobile device and the base station between the times i and i − 1 . preferably the method further includes the steps of : determining the minimum of said estimated biases within a selected time period ; and updating said estimated biases using said determined minimum bias . preferably the method further includes the steps of determining the absolute minimum bias of all said minimum biases across all time periods ; and updating said estimated biases using the absolute minimum bias . the minimum bias experienced across all time periods is likely to represent the best channel conditions ( e . g . if los exists between the mobile device and the base station ), and so can be used as an accurate base against which the other bias estimates can be calibrated . preferably the step of determining the minimum of said estimated biases includes the steps of : identifying decreases in the estimated bias between successive estimations ; and filtering said decreases to identify decreases which are more than a predetermined threshold . in identifying decreases in the estimated bias , drops in the bias to lower levels of bias can be identified and locked onto for the purposes of correcting earlier and subsequent estimates of the bias . filtering assists by removing those decreases which are caused by noise in the bias estimation . preferably the step of determining the absolute minimum bias further includes the steps of : calculating a running sum of the differences between minimum bias values in subsequent time periods and identifying the absolute minimum bias as the minimum of said running sum . by calculating the running sum of differences between minimum bias values , each bias drop can be tracked and preserved as it occurs , even if the distance or bias estimation that gave rise to that drop is no longer present in the selected time period . in certain embodiments , the step of estimating the biases using the minimum and absolute minimum bias calculates the corrected bias estimate { circumflex over ( b )} i c at time i as : { circumflex over ( b )} i c ={ circumflex over ( b )} i − κ i + σ k = 1 i δ k p −{ circumflex over ( θ )} i wherein { circumflex over ( b )} i is the bias estimate for the bias at time i , κ i is the minimum value of { circumflex over ( b )} i within the selected time period , δ i is the first difference of the minimum values κ i ( δ i = κ i − κ i − 1 ), δ i p is δ i to which a threshold filter has been applied and { circumflex over ( θ )} i is a running estimate of θ , the initial offset in the recursive bias estimation and is calculated as preferably the recursive estimation of bias and the determination of the minimum bias are performed substantially in real time . since the estimations are performed recursively and only on a selected time period , the computational requirements of the method of this aspect can be kept sufficiently low to allow the estimation to be carried out substantially in real time . this can allow the biases to be updated as further distance measurements are received and a real time estimation of the bias in those measurement made . the method of the present aspect may include any combination of some , all or none of the above described preferred and optional features . a second aspect of the present invention provides a method of estimating a position of a mobile device in relation to a base station , the method including the steps of : receiving , in the mobile device , wireless signals from the base station ; estimating , from said wireless signals , the uncorrected distance of the mobile device from the base station ; recursively estimating , from said estimated distances and the changes between differences of said estimated distances , the bias experienced in each of said estimates ; correcting uncorrected distance estimations to take account of the estimated bias in each of said uncorrected estimations ; and estimating the position of the mobile device from said corrected distance estimations . the method of this aspect is particularly useful in harsh multipath or nlos environments . it is particularly suited to indoor and / or urban environments , in which more accurate localization approaches ( such as gps ) are not available . preferably the estimation of the distance of the mobile device from the base station uses time - of - arrival or time - difference - of - arrival ranging . there may be more than one base station and the bias estimation may be carried out for each base station . when more than one base station is present , the position estimation is preferably carried out using multi - lateration . the biases can be dynamically estimated from the differential bias changes that occur while the mobile device is moving . once bias transitions occur then it is possible to lock onto the lowest experienced bias . thus the method of this aspect can have learning and convergence properties that result in accurate and robust localization and tracking . the method of this aspect can be computationally efficient as it only requires simple algebraic operations , it does not assume zero bias in los and does not require los / nlos identification . further , no assumption about time invariance ( stability of bias fluctuations ) need be made . furthermore , each range measurement / bias sample can be a completely different realization . this is highly desirable in dynamic and motion intensive environments ( e . g . shopping malls ) where the bias is changing / fluctuating with high frequency . preferably the step of estimating the position uses a kalman filter applied to the corrected distance estimations . the method does not require ( but can work with ) inertial navigation systems ( ins ). in some embodiments , the bias estimate { circumflex over ( b )} i for the bias at time i is estimated using the recursive equation : { circumflex over ( b )}= δδ { circumflex over ( d )} i , i − 1 + 2 { circumflex over ( b )} i − 1 −{ circumflex over ( b )} i − 2 wherein δ { circumflex over ( d )} i , i − 1 is the second difference in the estimate of the distance between the mobile device and the base station between the times i and i − 1 . preferably the method further includes the steps of , prior to correcting said uncorrected distance estimations , determining the minimum of said estimated biases within a selected time period ; and updating said estimated biases using said determined minimum bias . preferably the method further includes the steps of determining the absolute minimum bias of all said minimum biases across all time periods ; and updating said estimated biases using the absolute minimum bias . the minimum bias experienced across all time periods is likely to represent the best channel conditions ( e . g . if los exists between the mobile device and the base station ), and so can be used as an accurate base against which the other bias estimates can be calibrated . preferably the step of determining the minimum of said estimated biases includes the steps of : identifying decreases in the estimated bias between successive estimations ; and filtering said decreases to identify decreases which are more than a predetermined threshold . in identifying decreases in the estimated bias , drops in the bias to lower levels of bias can be identified and locked onto for the purposes of correcting earlier and subsequent estimates of the bias . filtering assists by removing those decreases which are caused by noise in the bias estimation . preferably the step of determining the absolute minimum bias further includes the steps of : calculating a running sum of the differences between minimum bias values in subsequent time periods ; and identifying the absolute minimum bias as the minimum of said running sum . by calculating the running sum of differences between minimum bias values , each bias drop can be tracked and preserved as it occurs , even if the distance or bias estimation that gave rise to that drop is no longer present in the selected time period . in some embodiments , the step of estimating the biases using the minimum and absolute minimum bias calculates the corrected bias estimate { circumflex over ( b )} i c at time i as : { circumflex over ( b )} i c ={ circumflex over ( b )} i − κ i + σ k = 1 i δ k p −{ circumflex over ( θ )} i wherein { circumflex over ( b )} i is the bias estimate for the bias at time i , κ i is the minimum value of { circumflex over ( b )} i within the selected time period , δ i is the first difference of the minimum values κ i ( δ i = κ i − κ i − 1 ), δ i p is δ i to which a threshold filter has been applied and { circumflex over ( θ )} i is a running estimate of θ , the initial offset in the recursive bias estimation and is calculated as preferably the recursive estimation of bias and the determination of the minimum bias are performed substantially in real time . since the estimations are performed recursively and generally only on a selected time period or using a running sum , the computational requirements of the method of this aspect can be kept sufficiently low to allow the estimation to be carried out substantially in real time . this can allow the biases to be updated as further distance measurements are received and a real time estimation of the bias in those measurement made . in certain embodiments , the step of correcting also includes correcting historic distance estimations to improve the accuracy of previous estimates of the position of the mobile device . this backward correction can be very useful where the overall track of the mobile device is of interest and the tracking starts in severe nlos conditions and so its position in the earlier stages of tracking is very uncertain . backward correction can allow an accurate position estimate for those earlier stages to be made , for example after the mobile device has encountered los communication with a base station . the method of embodiments of this aspect will be demonstrated in the examples below to correct range measurements providing biased measurements commensurate with the lowest experienced bias . for example if a zero bias to the base stations is experienced at least once during the navigation track then unbiased location estimation can be achieved . in particular , these embodiments can dynamically and recursively reduce the bias error and pass the corrected measurements to the position estimation , for example an extended kalman filter . the method of the present aspect may include any combination of some , all or none of the above described preferred and optional features . the methods of the above aspects is preferably implemented by a mobile device or a system according to the third or fourth aspects of this invention , as described below , but need not be . further aspects of the present invention include computer programs for running on computer systems which carry out the methods of the above aspects , including some , all or none of the preferred and optional features of that aspects . at their broadest , systems of the present invention provide wireless mobile devices which can estimate their position compared to a plurality of base stations , having corrected initial uncorrected estimates of the position to take account of the estimated bias in those uncorrected measurements . a third aspect of the present invention provides a wireless mobile device having a memory and a processor which is arranged to estimate the position of the mobile device , wherein : the mobile device is capable of receiving wireless signals from a plurality of base stations and estimating the distance of the mobile device from a base station based on the wireless signals received , and the processor is arranged to : repeatedly estimate , from said wireless signals , the uncorrected distance of the mobile device from the base station and store said uncorrected distance in said memory ; recursively estimate , from said estimated distances and the changes between differences of said estimated distances , the bias experienced in each of said estimates ; correct uncorrected distance estimations to take account of the estimated bias in each of said uncorrected estimations ; and estimate the position of the mobile device from said corrected distance estimations . the mobile device of this aspect is particularly suited to use in harsh multipath or nlos environments . it is particularly suited to indoor and / or urban environments , in which more accurate localization approaches ( such as gps ) are not available . preferably the estimation of the distance of the mobile device from the base station uses time - of - arrival or time - difference - of - arrival ranging . the position estimation is preferably carried out using multi - lateration . the biases can be dynamically estimated from the differential bias changes that occur while the mobile device is moving . once bias transitions occur then it is possible to lock onto the lowest experienced bias . thus the position estimation carried out by the mobile device of this aspect can have learning and convergence properties that result in accurate and robust localization and tracking . the position estimation carried out by the mobile device of this aspect can be computationally efficient as it only requires a sliding window and simple algebraic operations , it does not assume zero bias in los and does not require los / nlos identification . further , no assumption about time invariance ( stability of bias fluctuations ) need be made . this means that the processor of the mobile device does not need to be particularly powerful and / or that the processor is able to carry out the position estimation whilst still carrying out other tasks that may be required of it . furthermore , each range measurement / bias sample can be a completely different realization . this is highly desirable in dynamic and motion intensive environments ( e . g . shopping malls ) where the bias is changing / fluctuating with high frequency . preferably the processor estimates the position uses a kalman filter applied to the corrected distance estimations . the mobile device does not require ( but can work with ) an inertial navigation systems ( ins ). in some embodiments , the processor estimates the bias { circumflex over ( b )} i at time i using the recursive equation : { circumflex over ( b )} i = δδ { circumflex over ( d )} i , i − 1 + 2 { circumflex over ( b )} i − 1 −{ circumflex over ( b )} i − 2 wherein δδ { circumflex over ( d )} i , i − 1 is the second difference in the estimate of the distance between the mobile device and the base station between the times i and i − 1 . preferably the processor is further arranged to , prior to correcting said uncorrected distance estimations , determine the minimum of said estimated biases within a selected time period ; and update said estimated biases using said determined minimum bias . preferably the processor is further arranged to , prior to correcting said uncorrected distance estimations , determine the absolute minimum bias of all said minimum biases across all time periods ; and update said estimated biases by said absolute minimum bias . preferably the processor is further arranged to , when determining the absolute minimum bias , identify decreases in the estimated bias between successive estimations ; and filter said decreases to identify decreases which are more than a predetermined threshold . in identifying decreases in the estimated bias , drops in the bias to lower levels of bias can be identified and locked onto for the purposes of correcting earlier and subsequent estimates of the bias . filtering assists by removing those decreases which are caused by noise in the bias estimation . preferably the processor is further arranged to : calculate a running sum of the differences between minimum bias values in subsequent time periods ; and identify the absolute minimum bias as the minimum of said running sum . by calculating the running sum of differences between minimum bias values , each bias drop can be tracked and preserved as it occurs , even if the distance or bias estimation that gave rise to that drop is no longer present in the selected time period . in some embodiments , the processor estimates the biases using the minimum and absolute minimum bias by calculating the corrected bias estimate { circumflex over ( b )} i c at time i as : { circumflex over ( b )} i c ={ circumflex over ( b )} i − κ i + σ k = 1 i δ k p −{ circumflex over ( θ )} i wherein { circumflex over ( b )} i is the bias estimate for the bias at time i , κ i is the minimum value of { circumflex over ( b )} i within the selected time period , δ i is the first difference of the minimum values κ i ( δ i = κ i − κ i − 1 ), δ i p is δ i to which a threshold filter has been applied and { circumflex over ( θ )} i is a running estimate of θ , the initial offset in the recursive bias estimation and is calculated as preferably the recursive estimation of bias and the determination of the minimum bias are performed substantially in real time . since the estimations are performed recursively and generally only on a selected time period or by using a running sum , the computational requirements on the processor can be kept sufficiently low to allow the estimation to be carried out substantially in real time . this can allow the biases to be updated as further distance measurements are received and a real time estimation of the bias in those measurement made . a fourth aspect of the present invention provides a positioning system for estimating the position of a mobile device , the system including the mobile device and a plurality of base stations , wherein the mobile device is capable of receiving wireless signals from said plurality of base stations , the system further including a processor which is arranged to estimate the position of the mobile device by : repeatedly estimating , from said wireless signals , the uncorrected distance of the mobile device from the base station ; recursively estimating , from said estimated distances and the changes between differences of said estimated distances , the bias experienced in each of said estimates ; correcting uncorrected distance estimations to take account of the estimated bias in each of said uncorrected estimations ; and estimating the position of the mobile device from said corrected distance estimations . the system of this aspect is particularly suited to use in harsh multipath or nlos environments . it is particularly suited to indoor and / or urban environments , in which more accurate localization approaches ( such as gps ) are not available . preferably the estimation of the distance of the mobile device from the base station uses time - of - arrival or time - difference - of - arrival ranging . the position estimation is preferably carried out using multi - lateration . the biases can be dynamically estimated from the differential bias changes that occur while the mobile device is moving . once bias transitions occur then it is possible to lock onto the lowest experienced bias . thus the position estimation carried out by the system of this aspect can have learning and convergence properties that result in accurate and robust localization and tracking . the position estimation carried out by the system of this aspect can be computationally efficient as it only requires a sliding window and simple algebraic operations , it does not assume zero bias in los and does not require los / nlos identification . further , no assumption about time invariance ( stability of bias fluctuations ) need be made . this means that the processor of the mobile device does not need to be particularly powerful and / or that the processor is able to carry out the position estimation whilst still carrying out other tasks that may be required of it . furthermore , each range measurement / bias sample can be a completely different realization . this is highly desirable in dynamic and motion intensive environments ( e . g . shopping malls ) where the bias is changing / fluctuating with high frequency . preferably the processor estimates the position uses a kalman filter applied to the corrected distance estimations . the system does not require ( but can work with ) an inertial navigation systems ( ins ). in some embodiments , the processor estimates the bias { circumflex over ( b )} i at time i using the recursive equation : { circumflex over ( b )} i = δδ { circumflex over ( d )} i , i − 1 + 2 { circumflex over ( b )} i − 1 −{ circumflex over ( b )} i − 2 wherein δδ { circumflex over ( d )} i , i − 1 is the second difference in the estimate of the distance between the mobile device and the base station between the times i and i − 1 . preferably the processor is further arranged to , prior to correcting said uncorrected distance estimations , determine the minimum of said estimated biases within a selected time period and update said estimated biases using said determined minimum bias . preferably the processor is further arranged to , prior to correcting said uncorrected distance estimations , determine the absolute minimum bias of all said minimum biases across all time periods ; and update said estimated biases by said absolute minimum bias . preferably the processor is further arranged to : identify decreases in the estimated bias between successive estimations ; and filter said decreases to identify decreases which are more than a predetermined threshold . in identifying decreases in the estimated bias , drops in the bias to lower levels of bias can be identified and locked onto for the purposes of correcting earlier and subsequent estimates of the bias . filtering assists by removing those decreases which are caused by noise in the bias estimation . preferably the processor is further arranged to , when determining the absolute minimum bias : calculate a running sum of the differences between minimum bias values in subsequent time periods ; and identify the absolute minimum bias as the minimum of said running sum . by calculating the running sum of differences between minimum bias values , each bias drop can be tracked and preserved as it occurs , even if the distance or bias estimation that gave rise to that drop is no longer present in the selected time period . in some embodiments , the processor estimates the biases using the minimum and absolute minimum bias by calculating the corrected bias estimate { circumflex over ( b )} i c at time i as : { circumflex over ( b )} i c ={ circumflex over ( b )} i − κ i + σ k = i δ k p −{ circumflex over ( θ )} i wherein { circumflex over ( b )} i is the bias estimate for the bias at time i , κ i is the minimum value of { circumflex over ( b )} i within the selected time period , δ i is the first difference of the minimum values κ i ( δ i = κ i − κ i − 1 ), δ i p is δ i to which a threshold filter has been applied and { circumflex over ( θ )} i is a running estimate of θ , the initial offset in the recursive bias estimation and is calculated as preferably the recursive estimation of bias and the determination of the minimum bias are performed substantially in real time . since the estimations are performed recursively and generally only on a selected time period or using a running sum , the computational requirements on the processor can be kept sufficiently low to allow the estimation to be carried out substantially in real time . this can allow the biases to be updated as further distance measurements are received and a real time estimation of the bias in those measurement made . in certain embodiments , the step of correcting also includes correcting historic distance estimations to improve the accuracy of previous estimates of the position of the mobile device . this backward correction can be very useful where the overall track of the mobile device is of interest and the tracking starts in severe nlos conditions and so its position in the earlier stages of tracking is very uncertain . backward correction can allow an accurate position estimate for those earlier stages to be made , for example after the mobile device has encountered los communication with a base station . the system of the present aspect may include any combination of some , all or none of the above described preferred and optional features . in order to introduce and explain the embodiments of the present invention a simple scenario set out in fig8 is considered . a md 1 is ranging to a bs 2 and starts out in los with small bias errors . as the device moves along the path as illustrated in fig8 , it is shadowed by an elevator 5 and the channel condition switches to nlos . if we examine the evolution of the actual distance between the bs 2 and the md 1 , we notice that in this case it gradually increases . the evolution of the range measurements should follow that of the actual distance but with positive biases . fig9 illustrates the evolution of the distance and the range measurements . as the md 1 moves behind the obstruction ( an elevator 5 in this example ) the range measurements experience a sudden bias jump due to the blocking of the direct path signal by the obstruction . once the md 1 moves out of the shadow of the obstruction , the range measurements drop back to the original baseline . as the md 1 moves from a los to a nlos channel condition , as this example shows , there is a positive jump between the range measurement { circumflex over ( d )} i and { circumflex over ( d )} ii and the difference can be given by where δd ii , i = d ii − d i , δb ii , i = b ii − b i and δn ii , i = n ii − n i . thus for a small sample time , t s , the difference between actual distances is small , that is δd ii , i ≈ 0 and it is the geometrical change in the distance between the two subsequent locations . thus in this example any significant change in δ { circumflex over ( d )} ii , i will be due to δb ii , i . if we further assume that b i = b min ≈ 0 , the minimum bias experienced , then it is possible to estimate b ii as { circumflex over ( b )} ii = δb ii , i and correct the range measurement { circumflex over ( d )} ii by subtracting { circumflex over ( b )} ii or in a more general scenario the md 1 might move from a severe nlos to a light nlos and then to los . as a result it would be possible to estimate the bias drops and essentially “ lock ” on the best minimum bias , b min . fig1 illustrates an example in which the md 1 is moving and experiencing different nlos obstacles , including “ light ” nlos where the direct path signal can reach the md 1 ( e . g . through a window 6 ). the resulting evolution of the distance and estimated ( biased ) range measurements is illustrated in fig1 . thus if the minimum bias experienced ( the largest differential range measurement drop ) can be tracked , then a baseline with which to compare all subsequent and previous range measurements can be established . it is then possible to continuously estimate subsequent jumps and subtract them from the range measurements . fig1 shows an example of a md ranging scenario in a typical indoor office environment and illustrates how , after establishing that the current measurement sample is the minimum bias , correction can be achieved for subsequent steps . in this example scenario , the md 1 is moving along the path from point a to point f . at point a , a set of cabinets 7 and a wall 8 obstructs the dp from the bs 2 and the nlos condition induces large biases . as the md moves , the condition somewhat improves and at point c the channel condition changes to los resulting in small bias errors . fig1 ( a ) shows the resulting evolution of the distance and respective range measurement for the md in the scenario of fig1 . fig1 ( b ) shows how forward and backward bias correction can be made from this data . the lower smooth curve is the evolution of actual distance in each case . the upper , jagged curve is the evolution of the biased range measurements . the range estimates experience several drops indicating a reduction in the experienced bias as the nlos channel improves . eventually , at point c , the range measurements become los and the bias is reduced significantly ( compared to a and b ). once a minimum bias is established then any subsequent bias errors relative to the minimum can be subtracted from the range measurements ; thus achieving forward bias correction , as shown by the middle line in fig1 ( b ). similarly , backward bias correction can be achieved to correct previous biased measurements . forward bias correction is necessary to improve localization and tracking in real - time . backward bias correction , however , can be of great value as well ; since it can be used to correct the md track / path took prior to reaching the minimum bias location . an example of where backward bias correction might be useful is tracking shoppers in a shopping mall . once the user enters the mall , depending on the location of the bss relative to the md , the range measurements might be in nlos ( heavily biased ). thus the estimated position or track will not be accurate . however as the user moves around the mall , then the range measurements will be recursively corrected and the real - time track and previous track will be corrected . the forward / backward correction will not only improve the real - time tracking , but also provide a more accurate track history and this information can be very useful for many applications , from a commercial point of view . for security applications backward bias correction can be of great importance . if a user enters into a building un - authorized , then obtaining a reliable estimate of the track history can provide an indication of where the user entered and where he / she spent time ; thereby identifying security breach points and activities . bias correction can be easily integrated within a kf framework . the range measurements obtained between the bss and the md can be corrected in real - time and fed to a kf as illustrated in fig1 . the estimated range ( i . e . the distance estimate ) can converge to unbiased range measurements if at least one range measurement experiences zero bias in los . alternatively it can converge to the minimum bias that is experienced while the md moves in an indoor / urban environment . a more formal analysis of the above embodiments will now be set out . the starting point is the range measurements between a md 1 and a bs 2 which can be modeled as where d i is the actual distance between the md and the bs at the ith sample time iε { 1 , n } and b i is the bias induced due to either los ( multipath ) or nlos ( multipath and dp obstruction ) and n i is the zero - mean gaussian measurement noise with variance σ i 2 . given a sequence of range measurements between a md and a bs {{ circumflex over ( d )} 1 , { circumflex over ( d )} 2 , . . . , { circumflex over ( b )} n } obtained while moving through the multipath propagation environment the aim is to estimate the corresponding sequence of biases { b 1 , b 2 , . . . , b n }. if the biases can be accurately estimated then the range measurements in ( 23 ) can be corrected to provide unbiased ( clean ) range measurements to the kf . the method according to an embodiment of the invention provides an approach that examines the differential information available between the range measurements to estimate the biases . the general first difference equation between two subsequent range measurements obtained at time steps t i and t i − 1 can be established as δ { circumflex over ( d )} i , i − 1 ={ circumflex over ( d )} i −{ circumflex over ( d )} i − 1 = δd i , i − 1 + δb i , i − 1 + δn i , i − 1 , ( 24 ) where the sample time is t s = t i − t i − 1 , δb i , i − 1 = b i − b i − 1 and δn i , i − 1 = n i − n i − 1 . using ( 24 ) the second difference equation can be defined by note that the second difference equation is essentially the sum of the second differences between the actual distance , the bias and the measurement noise . after some algebra ( 25 ) simplifies to δ { circumflex over ( d )} i , i − 1 = δδd i , i − 1 + b i − 2 b i − 1 + b i − 2 + δδn i , i − 1 ( 26 ) b i = δδ { circumflex over ( d )} i , i − 1 − δδd i , i − 1 + 2 b i − 1 − b i − 2 − δδn i , i − 1 . ( 27 ) since in practice only δδ { circumflex over ( d )} i , i − 1 and an estimate of the previous two biases { circumflex over ( b )} i − 1 and { circumflex over ( b )} i − 2 are available then instead a recursive estimate of the bias at i can be obtained as , { circumflex over ( b )} i = δδ { circumflex over ( d )} i , i − 1 + 2 { circumflex over ( b )} i − 1 −{ circumflex over ( b )} i − 2 . ( 28 ) fig1 illustrates a block diagram of the recursive estimator . the recursive bias estimation starts with the assumption that the first estimated bias is the minimum bias and it is therefore zero or since in general this assumption is not always true ( except in unique cases as will be discussed later ), this introduces an initial offset in the recursive bias estimation or next in order to compute { circumflex over ( b )} 2 using ( 28 ) both { circumflex over ( b )} 1 and { circumflex over ( b )} 0 are required . since { circumflex over ( b )} 0 is not available then an estimate of { circumflex over ( b )} 2 can be obtained from the first difference equation which is a biased estimate of b 2 with mean and variance given by e [{ circumflex over ( b )} 2 ]= b 2 + δd 2 , 1 − b 1 − n 1 ( 32 ) var [ { circumflex over ( b )} 2 ]= var [ n 2 ]= σ 2 2 = σ i 2 ( 33 ) where it was assumed in ( 32 ) that once n , occurs it is a constant offset to { circumflex over ( b )} 2 and thus can be viewed as an unknown deterministic constant ( since it doesn &# 39 ; t change throughout the iterations ). it is clear from ( 31 ) that if a good estimate of the initial offset θ = b 1 and δd 2 , 1 can be obtained then the bias can be estimated accurately . the bias estimate for the next time step can be similarly obtained using the recursive bias estimation equation given in ( 28 ) { circumflex over ( b )} 3 = δδ { circumflex over ( d )} 3 , 2 + 2 { circumflex over ( b )} 2 −{ circumflex over ( b )} 1 . ( 34 ) using ( 25 ) and the expressions for the bias estimates { circumflex over ( b )} 2 and { circumflex over ( b )} 1 in ( 31 ) and ( 29 ), respectively , gives { circumflex over ( b )} 3 = b 3 + δd 3 , 1 − b 1 − n 1 + n 3 ( 36 ) which is a similar expression we obtained for { circumflex over ( b )} 2 and we can easily see that it is biased by δd 3 , 1 − b 1 − n 1 and its variance is σ 3 2 = σ i 2 . in fact if the recursive exercise is continued for the rest of the bias estimates a generic expression for the estimate can be obtained as { circumflex over ( b )} i = b i + δd i , 1 − b 1 − n 1 + n i ( 37 ) e [{ circumflex over ( b )} i ]= b i + δd i , 1 − b 1 − n 1 ( 38 ) and the variance of the bias estimate is var [{ circumflex over ( b )} i ]= var [ n i ]= σ i 2 . it can be noted that the bias estimates obtained from the recursive relationship ( 37 ) are biased by a slow varying component and a constant offset due to the incorrect assumption that b 1 is zero . the example shown in fig1 illustrates how these components skew and offset the bias estimates . in the example of fig1 , the solid curve represents the true biases experienced ; while the dashed line are the estimated biases obtained from ( 28 ). it is clear that the estimated biases are offset by the initial θ and skewed by δd i , 1 . thus in order to obtain accurate bias estimates then δd i , 1 and θ need to be estimated in order to recover the actual biases . one simple approach to estimate δd i , 1 and θ is to pass the estimated biases through a minimum sliding window ; where essentially the minimum of the window is retained at each sample index . the output of the minimum sliding window is then used to correct for the slowly varying δd i , 1 and track bias transitions ( drops ) relative to the initial assumption of b 1 = 0 . the maximum drop relative to the first bias assumption is then used to correct the offset θ and establish the new baseline . in the example of fig1 , the single los sample that causes the actual bias to drop to zero ( or near zero ) will be used once it falls within the sliding window . to illustrate the approach an example can be considered . in this example it is assumed that the user is ranging in an all nlos scenario with the following bias statistics : exponential distributed , λ = ⅛ and b min = 5 . at three sample times , the user experiences los conditions with the following biases b 100 = 3 , b 300 = 1 and b 500 = 0 . the estimated biases obtained using ( 28 ) are illustrated in fig1 . fig1 ( a ) shows the estimated bias obtained from ( 28 ). fig1 ( b ) is a close - up of the estimated biases focusing on the bias drops occurring at sample times 100 , 300 and 500 . this scenario deals with the most general case , where each bias sample can be a realization from a different channel condition . in simpler cases , a similar range of bias values can persist for several samples as the user is moving . the system diagram in fig1 illustrates the steps involved in correcting the bias estimates . from ( 28 ) a sequence of bias estimates { circumflex over ( b )} 1 . . . { circumflex over ( b )} i are obtained that are skewed with time and offset ( relative to b 1 ), see ( 37 ) and fig1 . the objective is to obtain the corrected bias estimates ( without the skew and the offset ), { circumflex over ( b )} 1 c . . . { circumflex over ( b )} i c . the bias estimates are first passed through a sliding window of length m , w [ i ]=[{ circumflex over ( b )} i − m , { circumflex over ( b )} i − m + 1 , . . . , { circumflex over ( b )} i ] t ( s 100 ) and the minimum is selected from each window or κ i = min { w [ i ]} ( s 101 ). the output at point a in fig1 essentially tracks the minimum baseline and any shift in the baseline can be detected . fig1 shows the tracking of the minimum baseline . fig1 ( a ) shows the output at point a in fig1 . fig1 ( b ) is a close - up view docusing on the bias drops at points 100 , 300 and 500 . if the baseline shift ( the drops ) occur once within a window , then the baseline persists for the duration of the window - length ( in this example , m = 100 ). if multiple bias drops occur then this results in a “ staircase ” of drops . the last bias drop in the window then persists until it is dropped out of the window . it will be noted that the minimum baseline essentially is an estimate of δd i , 1 , except at locations where the bias drops occur . the impact of the bias drops then persists through the length of the window , until the sample is pushed out of the window . next , in order to estimate the bias transitions ( drops ), the first difference of the minimum values κ i is obtained ( see fig1 at point b ) which is the first derivative of κ i and , as expected , it will result in impulses at the transition locations , as illustrated in fig2 ( a ). since the aim is to track baseline shifts due to changes in biases ( bias drops ) then δ i is filtered by applying a simple threshold to obtain a waveform of impulses , δ i p where γ & gt ; 0 is a threshold that should be related to the intensity of the measurement noise , var [ n i ]. the thresholding is used to avoid detecting noise changes as bias transitions . the filtered pulses are illustrated in fig2 ( b ). in order to remove the impact of the windowed response to the transitions , the waveform created as a result of the windowing has to be recreated . if the train of impulses ( fig2 ( b )) is passed through a running sum , then a waveform is obtained that is the same as that obtained from the transitions : compare fig1 ( b ) and fig2 ( a ), where the latter is the output of the running sum ( point d in fig1 ) . the final step is to track the absolute minimum bias drop . this can be achieved by passing the sum of the train of impulses , δ i p , through a running minimum function or note that { circumflex over ( θ )} i is a running estimate of θ ; that is if the channel condition improves with time and better los conditions occur , then the { circumflex over ( θ )} i estimate will converge to θ , or as b i → 0 , then { circumflex over ( θ )} i → 0 . this is illustrated in fig2 ( b ) where the minimum baseline is tracked throughout the measurements and drops to a minimum when a zero bias is experienced . thus if the initial assumption of 0 bias was not true , then the offset can be recursively corrected . finally it is clear from the figures and the illustrative examples that simple addition and subtraction of the estimated sequences can provide a very accurate corrected bias estimate in real - time which can be summarized as in the example , once the minimum baseline at i = 500 occurs then the bias estimates converge to the true value and thus range measurements can be corrected providing very accurate measurements to the ekf block . fig2 illustrates the final corrected biases ( fig2 ( a )) and the bias estimation error ε i = b i −{ circumflex over ( b )} i c ( fig2 ( b )). it is clear from fig2 ( b ) that the error in bias estimation converges to zero after the 500 th sample . fig2 , illustrates the overall system diagram of bias estimation and range measurement corrections according to an embodiment of the present invention . equipped with the final bias estimates , it is then possible to correct the corrupted range measurements as essentially the process to unbias the range measurements involves first estimating an accurate estimate of the biases and then subtracting them from the range measurements . the measurements will be biased by the lowest experienced bias . the corrected measurements will still be corrupted by noise and some minor bias ( due to imperfect estimation and noise in the system ) but this can be easily handled by a suitably designed ekf . the bias estimation and correction approach of the above embodiments can be viewed from an optimization view point as well . since the object is to obtain the minimum of all the range measurements ( correct the biases ) then it is possible to employ a convex hull ( linear programming ) approach to provide corrected range measurements [ 16 ]. in the next section some numerical examples will be provided to illustrate the ability of embodiments of the present invention to estimate the biases and correct the range measurements in an ekf framework . embodiments of the present invention have been set out above and have illustrated how the estimates of the biases can be obtained with accuracy related to the best experienced channel condition while moving in a multipath environment . to demonstrate the functioning of these embodiments , localization and tracking performance is now presented for two examples . in the first example , the md is moving indoors in an all nlos ranging condition with exponential bias statistics with λ nlos and b min = 5 , that is the lowest bias experienced is 5 meters . then at three specific sample times , the user experiences range measurements with smaller biases ( indicating channel improvements ). the three bias values are 3 , 1 and 0 meters and they occur for the bss at the following sample times the example is similar to that introduced in the previous section , but it is more general since we are examining the bias drops experienced across all the bss and how the proposed invention can be implemented in a kf framework to improve the accuracy of the tracking fig2 illustrates the actual ( lower line ) and measured range ( upper line — sampled ) from the md to the bs . note that the lower curve in the four figures is the evolution of the actual distance between the md and the respective bs ; while the highly variable curve is the estimated distance ( distance plus nlos bias ). the three bias drops appear as spikes ( in the highlighted region of each graph ) with the last one touching the actual distance curve since the bias is zero . the tracking performance without any bias correction is illustrated in fig2 which uses a similar actual track 3 to fig6 and 7 described above . fig2 ( a ) shows the actual track 3 and the measured track 4 as well as the positions of the bs 1 . fig2 ( b ) shows the position error as the mobile device moves along the track 3 . since the statistics of the nlos bias are severe the tracking performance is very poor . the range measurements provide almost useless information to the ekf and thus the performance is degraded significantly . an average positioning error of 18 meters can be seen in fig2 ( b ). in contrast , fig2 illustrates the tracking performance for the same track 3 of the mobile device using the proposed bias correction technique of the above embodiments prior to the ekf . fig2 ( a ) demonstrates that the estimated track 4 quickly converges to the actual track 3 and the localization estimate converges to the true location as soon as the bias drops occur . fig2 ( b ) shows the position error as before and after the final bias drop occurring at 5500 , the localization estimate converges due to very accurate bias estimates that are fed to the ekf . from a comparison of fig2 and 26 it will be seen that the performance is significantly better than localization and tracking with no bias correction . fig2 illustrates the impact of the bias correction algorithm on the estimated range ( fig2 ( a )) and the bias estimation error ( fig2 ( b )) to each bs . specifically , fig2 ( a ) illustrates the actual distance evolution and the estimated distance . note that even before the bias drops , the technique locks on the best bias ( which is around 5 meters — since it is the minimum bias that can be experienced statistically ). then once the bias drops occur , the corrected range measurements converge to the true distance and thus improve the localization significantly . this is also illustrated in fig2 ( b ) where the bias estimation error converges to zero . although there are variations in the measurements due to noise , these can be easily handled by the ekf with proper design of the process and noise covariances . the second example illustrates the potential of the bias correction algorithm when the initial measurements are actually zero bias ; which ensures accurate bias estimation from the beginning . this is particularly useful for firefighter localization and tracking applications , in which the firefighters with mobile radios enter a burning building and it is desirable to track them with high accuracy in order to ensure their safety and coordinate rescue operations . since the firefighters ( and their mobile devices ) are going from a los into a nlos then it is possible to calibrate the mobile radios prior to entering the building . by calibrate we mean lock the mobile radios to zero bias conditions , so that any subsequent increase in bias can be estimated and corrected accurately . an example layout is shown in fig2 . the firefighter localization application usually involves a set of mobile bss 1 that are deployed on the scene instantaneously . so prior to final deployment the mobile bss 1 are first congregated close to the firefighters to calibrate the initial biases between the bss 1 and the mds 23 carried by the firefighters . this can be achieved by knowing the exact distance between the firefighter &# 39 ; s mobile device 2 and the bss 1 ; through laser tags , gps , etc . once a zero bias initial condition is established then the mobile bss 1 are deployed to the desired location which is usually required to ensure proper coverage and system optimization , for example as shown in fig2 ( b ). as the mobile bss 1 are moving to their location , the proposed bias estimation and correction technique can maintain the initial baseline . finally , once the base stations are in place , the firefighter starts moving and enters the building , as illustrated in fig2 . with the proposed bias estimation and correction technique it is then possible to continuously correct the corrupted range measurements and enable very accurate and precise localization and tracking performance . fig3 and 31 illustrate the localization and tracking performance with an initial baseline of zero ; obtained from the zero bias calibration . fig3 ( a ) shows the near perfect match between the actual track 3 and the predicted track 4 with a zero initial bias . fig3 ( b ) shows the positioning error with a zero initial bias . fig3 ( a ) shows the actual and estimated distances to each base station 1 after applying the bias correction technique with zero bias calibration and fig3 ( b ) shows the bias estimation error to each base station 1 . the numerical examples in this section highlight the robustness and accuracy of the embodiments of the present invention under different applications and scenarios . in particular , these embodiments can dynamically and recursively reduce the bias error and pass the corrected measurements to the ekf . the simulation results tested highlight the more rigorous scenarios where the biases are significant and fluctuate with high frequency at each measurement sample . the systems and methods of the above embodiments may be implemented in a computer system ( in particular in computer hardware or in computer software ) in addition to the structural components and user interactions described . the term “ computer system ” includes the hardware , software and data storage devices for embodying a system or carrying out a method according to the above described embodiments . for example , a computer system may comprise a central processing unit ( cpu ), input means , output means and data storage . preferably the computer system has a monitor to provide a visual output display ( for example in the design of the business process ). the data storage may comprise ram , disk drives or other computer readable media . the computer system may include a plurality of computing devices connected by a network and able to communicate with each other over that network . the methods of the above embodiments may be provided as computer programs or as computer program products or computer readable media carrying a computer program which is arranged , when run on a computer , to perform the method ( s ) described above . the term “ computer readable media ” includes , without limitation , any non - transitory medium or media which can be read and accessed directly by a computer or computer system . the media can include , but are not limited to , magnetic storage media such as floppy discs , hard disc storage media and magnetic tape ; 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