Bias estimating method for a target tracking system

[0002] The present invention relates in general to tracking of targets by means of measurements from various sensors and in particular to automatic sensor alignment.

[0003] Target tracking is the process by which the location and motion of objects like aircraft, ships, missiles, etc., are established and presented to users, for example, situated at a command and control centre. The detection of the targets is accomplished by one or more sensors. The most widely used sensor for targets in the air is the radar. However, radars are not the universal solution since they are impaired by serious inherent drawbacks. At the state of the art, their accuracy is limited. Moreover, they can't be used under water. Additionally, by radiating energy, radars reveal their locations and form easy targets for enemy forces. Depending on the particular application, one therefore uses other types of sensors instead of or in complement of radars. Examples of such sensors, which, however, should not be considered as a complete set of possibilities, are Global Navigation Satellite System (GNSS) and its predecessors GPS and GLONASS, Electronic Support Measures (ESM) equipment, Electro-Optical (EO) sensors and their subclass Infrared Search and Track (IRST), Jam strobe detectors (ECM), air pressure based altitude measuring equipment downlinked via ‘Secondary Radars’ (Mode C), as well as active and passive sonars. Out of these sensors, only certain radars (3D) and the satellite based navigation data provide full three-dimensional determination of the target location. ESM, EO, IRST, ECM, and passive sonars only give the direction to the target. Such measurements are referred to as “strobes”.

[0004] The sensors produce measurements at regular or irregular time intervals. A measurement is a piece of information including e.g. range and azimuth of the target. A tracker is a device, which will create a track file for each detected target. A track is a set of usually filtered data which are associated with a certain target. The tracker updates the track data with incoming measurements in such a way that measurement noise is reduced by filtering, and speed and heading is computed. The track is given a label and is presented to the operators. The general art of tracking is well known and can be found at many different places, e.g. in S.S. Blackman “Multiple-Target Tracking with Radar Applications”, Artech House, Inc, MA, USA 1986, in particular pages 1-17 and 357-393.

[0005] There are many reasons to use multiple sensors for target tracking. Evidently, the combination of passive sensors at different locations makes it possible to obtain the distance to the target. By using several radars, gaps in coverage of individual radars may be eliminated. Moreover, satellite data will improve accuracy, if available. ECM gives information even if a jammer on the target makes the radar data useless, and so forth. In general, several sensors contribute to more frequent updates, better accuracy, higher signal to noise ratio, and more reliable tracking.

[0006] There is, however, a serious problem in combining data from several sensors: alignment errors, also known as systematic errors, bias errors, or registration errors. The alignment errors may give the impression that the number of targets is larger than the actual number because the measurements are displaced by some, often very significant, amount. Even if there is only one track per target, accuracy will clearly be impaired. Therefore, estimation and compensation of the bias errors are necessary for successful tracking.

[0007] To illustrate these problems, a simple example will be discussed. FIG. 1 shows an example of a tracking situation. The tracking system have measurements available from two 3D sensors, D₁³and D₂³, respectively, providing range, azimuth as well as height information, and one passive sensor D¹, only providing azimuth angle measurements. One true target X is present in the area covered by the system. The sensors exhibit some systematic errors. The local coordinate system of sensor D₁³is misaligned with respect to the common coordinate system with a certain angle α. The sensor D₂³has an error in the range measement, which gives rise to systematic errors in the range values. In the situation in FIG. 1, sensor D₁³detects a target in the direction of a real azimuth angle A_R1³, but due to the misalignment, the apparent azimuth angle A_A1³is shifted an angle α, so the apparent location for the detected target is M₁³. The range measuring error of sensor D₂³causes the apparent target position as measured by sensor D₂³to be M₂³, offset a distance r along the azimuth angle from the true position. Sensor D¹provides a true azimuth angle measurement. In this situation, it is probable that a target tracking system will interpret the situation as if two targets are present, at M₁³and M₂³. The M₁³position is reasonably consistent with measurements from both D₁³and D¹, while the M₂³position is far away. From this example it is obvious that there is a need for some alignment procedure.

[0008] It turns out that the bias estimation or automatic alignment is quite a difficult task, and it is further complicated by drifts in electronic and/or mechanical systems, which call for recurrent recalibration. The methods existing so far have various severe limitations. Some approaches according to prior art are described below.

[0009] A first approach concerns estimating and compensating for the deviations in x, y and z for each single track separately. The patent DE 3,132,009 gives an example of this approach. The drawback is that the estimation process is initiated for each track, and it may take considerable time before the estimates are good enough for accurate tracking.

[0010] Most other methods aim at modelling each sensor with a set of bias parameters. In this document, the term “bias parameters” is used for parameters, each of which represents a characteristic of the sensor like a location error, a north alignment error, or a range offset etc. Another term used in the literature, basically denoting the same measures, is “registration errors”. Most methods according to the state of the art handle a very limited set of sensor types and a very limited set of parameters.

[0011] A second approach of automatic alignment uses the above described principle which is applied by using reference targets with exactly known positions. By comparing a set of measured positions with a set of true positions one can adapt the bias parameters so that the deviations are minimised. There are, however, severe disadvantages. The positions of the reference targets have to be measured very accurately. The position of the reference target should optimally also be situated somewhere in the volume, where normal targets appear, which often may be over sea or even over international or other nations territories. Fixed reference targets with well defined positions are also normally restricted to ground level or close to ground level, which makes calibrations of altitude and inclination measurements difficult to perform. Moreover, reference targets are also easily destroyed or manipulated with during hostile situations.

[0012] In the absence of reference targets, a third approach involves designation of one of the sensors as a reference sensor, and alignment of the others with respect to the reference. One example of such a system is found in the proceedings of the SPIE—The International Society for Optical Engineering, Vol. 2235, 1994, in an article in the name of R. Helmick et al, with the title “One-step fixed-lag IMM smoothing for alignment of asynchronous sensors”, pages 507-518. Here a method for compensating a number of 3D sensors is disclosed. Each sensor has its own tracker and the filtered data is brought to a common time. One sensor is selected as a master and the tracks of the other sensors are compared with the master to calculate correction factors. This method requires 3D sensors since each sensor has its own complete track. Furthermore, the selection of a master also presents problems if it happens to have relatively large systematic errors itself or if it at any occasion is not working properly, problems which are common for all systems working according to this approach. Finally, this particular method does only discuss range, north alignment, inclination and altitude errors of the sensors.

[0013] In the above described example, shown in FIG. 1, it is obvious that neither sensor D₁³nor sensor D₂³are suitable as reference sensors. Furthermore, sensor D¹can not be used as reference, since it is unable to provide a complete target track on its own.

[0014] There are also methods in prior art, which do not to have one sensor as reference or do not use a reference target. These methods may be collected in a fourth approach. Bias parameters will then be estimated for all the participating sensors. In the proceedings of the International Conference on Radar, Paris May 3-6 1994, an article with the title “Least squares fusion of multiple radar data”, page 364-369, in the name of H. Leung et al discloses such a method for bias compensation of data from two 2D radars. Values of the range and north misalignment errors, which gives the lowest feast squares discrepancy for the radar measurements, are calculated. Poorly conditioned systems of equations are handled by singular value factorisation. Only one single pair of 2D radars is discussed, and only range and north misalignment errors are compensated for.

[0015] In “Maximum Likelihood Registration of Dissimilar Sensors” by Daniel W. McMichael and Nickens N. Okello, Proceedings of First Australian Data Fusion Symposium (ADFS-96), Adelaide November 21-22, 1996, page 31-34, another method for automatic alignment of this approach is disclosed. The method comprises a procedure for maximising a probability function using a maximum likelihood approach. The procedure is performed in an iterative manner to obtain a requested accuracy. The disadvantages with this particular method is that it does not give any clue about how to use measurements performed at different times. The method is furthermore an off-line method, since it does not handle any time propagation features.

[0016] In general, systems according to the fourth approach, that do not have a reference sensor tend to give ill-conditioned estimation problems as will be explained in the sequel, and one has to be very restrictive in the selection of bias parameters.

[0017] In prior art, there are also techniques which are somewhat similar, but not directly related to the present scope of concerns. In the U.S. Pat. No. 5,208,418, the relative alignment of sensors for different weapons are discussed. It discloses a method for compensation of solely misalignments between pairs of sensors and guns. The method does not concern multi-sensor tracking. The U.S. Pat. Nos. 5,072,389 and 5,245,909, disclose methods for compensation of dynamic, rapidly changing errors using accelerometers, which are not of interest within the scope of the present invention.

[0018] An object of the present invention is to provide a method for sensor alignment, which is not limited by the use of a fixed reference sensor or reference target, and which is applicable to all combinations of sensors, passive or active.

[0019] Another object of the present invention is to provide a method for sensor alignment, which is able to estimate a large set of bias parameters.

[0020] Yet another object of the present invention is to provide a method for sensor alignment, which provides updated estimates whenever requested, regularly or occasionally.

[0021] Still another object of the present invention is to provide a method for sensor alignment, which enables handling of each bias parameter with individual time characteristics.

[0022] The objects of the present invention is achieved by a process exhibiting the features set forth in the claims. The process of the invention repeatedly generates estimates for sensor bias errors by minimising a function, given on one hand by the magnitude of the discrepancy between measurements and a measuring model, where the measuring model is a function of the unknown target location and unknown bias parameters, and on the other by the bias parameters and their predetermined statistical distributions. In a preferred embodiment of the present invention, the minimising step is performed by linearising the function around an approximate target position (normally obtained from the tracker) and around nominal (typically zero) bias errors, and the function is subsequently minimised with respect to target positions as well as to the bias parameters. In addition, possible time dependence of the bias parameters are modelled by the incorporation of process noise.

[0023] Embodiments according to the present invention are presented in detail in the following, in connection with the associated drawing, in which:

[0024] FIG. 1 illustrates a tracking situation where automatic alignment is not applied;

[0025] FIG. 2 shows a block diagram illustrating the flow of information in a tracking system comprising an automatic alignment function according to the present invention;

[0026] FIG. 3 illustrates a tracking situation presenting a physical singularity;

[0027] FIG. 4 shows the information flow in an embodiment of the automatic alignment function according to the present invention;

[0028] FIG. 5 shows the information flow in an alternative and preferred embodiment of the automatic alignment function according to the present invention;

[0029] FIG. 6 shows an example of a tracking situation, where automatic alignment according to the present invention is applied, with one target present in the area of detection; and

[0030] FIG. 7 shows an example of a tracking situation, where automatic alignment according to the present invention is applied, with several targets present in the area of detection.

[0031] In a general tracking system, a target track is normally generated in some stage of the process. The target track is determined by a number of parameters, such as position, velocity, acceleration etc., which normally are results of some kind of filtering process. Certain measurements from the individual sensors are possible to associate with a certain target track. The target tracks are intended to correspond to true targets.

[0032] The method according to the present invention, which will be described in detail below, provides an automatic alignment procedure, featuring several attractive properties. In FIG. 2, a target tracking system comprising an embodiment of the present invention is illustrated. A target tracking section 10 performs the actual target tracking, which among other things generates target tracks T and measurements M associated to these target tracks T. This information is transferred 11 to an automatic alignment section 12, which generates estimated values of bias parameters b and preferably also their associated covariances P_b. This information is retransferred 13 back to the target tracking section 10. Although bias compensated measurements are used for the tracking, the corresponding measurements M entering the alignment procedure have uncompensated values. Except for this exchange of information 11 and 13, the alignment procedures are performed separately from the target tracking section 10, which means that the process according to the invention may be used together with any tracking system which provides target tracks T and associated measurements M. Besides the contribution from the target tracking section, the input to the automatic alignment section 12 comprises various triggering information 14 and a priori system data 15, which will be. described more in detail below.

[0033] The method according to the present invention is based on comparing the measurements with the true positions in measurement space. The measurement space is given by the coordinates used by the sensor in question—range and azimuth for a 2D radar; height above earth for mode C data; latitude, longitude, and height for satellite navigation data; etc. The true positions are transformed to the measurement space for a comparison. Of course, the true positions are in general unknown, and according to the present procedure, these are regarded as parameters to be eliminated in the minimisation process.

[0034] The various parameters in an error model are referred to as the bias parameters. The most important parameters tend to be range offset (for active sensors), north alignment, sensor location (north, east) error, timing error, and range gain errors. Range offset and range gain errors will make data to appear farther away or closer than the true position, and the location errors may shift the data in north-south and east-west directions, respectively. An error in the north alignment will shift the azimuth data and a timing error will cause the data to be delayed or time tagged at a wrong time. There are also various errors in axis or sensor tilting that can be important for sensors with low measurement noise, such as axis tilt (north, east), antenna roll as well as antenna pitch. The axis tilt errors may cause data to appear to high or low in north-south and east-west directions, respectively. An antenna pitch error will make data appear at a too high or too low elevation, and an antenna roll error will shift data clockwise or counterclockwise at high elevations. Mode C data will give incorrect altitudes as they relate to a standard air pressure. The selection of the bias parameters may of course be different, e.g. the sensor position bias parameters may be expressed in a coordinate system other than east/north.

[0035] For each sensor s, a vector of bias parameters b_sis introduced, whose components are the parameters (e.g. range offset, etc.) to be estimated. The vectors corresponding to the different sensors are not necessarily of the same type and may subsequently have different sizes and different components.

[0036] It is convenient to stack all the b_svectors into one large bias vector $\begin{array}{cc}b=\left(\begin{array}{c}{b}_1\\ b_2\\ ⋮\end{array}\right)& \left(1\right)\end{array}$

[0037] A set of measurements of one target stemming from different sensors is denoted a bias observation, and given an index i. The measurements should be made at nearly identical times (time differences typically less than the measurement period of the sensors). The true target location when sensor s obtained the measurement belonging to bias observation i is written x_is. This location is mapped onto the measurement space of the sensor by the projection function h_s(x_is,b). This function thus expresses the target location in measurement space, when bias errors are included. (If the bias vector includes a timing error, the projection function should also include target velocity. This will not be written explicitly in the sequel.) The measurement will also be subject to a random displacement, the measurement noise ε_is, so the actual measured position y_isis possible to express as

[0038] y_is=h_s(x_is, b)+ε_is  (2)

[0039] Here y_isand the function h_sare knowns, while x_is, b and ε_isare unknowns. The measured position will now be compared to the model (2), and b (and implicitly x_is) is found through a minimisation procedure. In this comparison it is useful to employ the known variances of the measurement noise:

[0040] R_is=E(ε_isε_is^T)  (3)

[0041] where E is the expectation operator and the noise is assumed to have zero mean. For purposes of the present invention it is, however, more convenient to use its inverse, the information matrix u_is=R_is⁻¹. The advantage of this is that a sensor giving, e.g., azimuth only can be assigned zero information for the other components, and subsequently the same algorithm can be used for 2D and 3D radars, and 1D and 2D passive sensors.

[0042] The goal for the optimisation procedure is to find the b (and x_is) which minimises the discrepancy between measurements y_isand the projected target location. A candidate for an expression to be minimised with respect to b and x_isis thus $\begin{array}{cc}V=\sum _{i,s}\left|\right|y_{\mathrm{is}}-h_s\left(x_{\mathrm{is}},b\right){\left|\right|}^2& \left(4\right)\end{array}$

[0043] The norm is defined such that for any vector q, ∥q∥²=q^Tuq. However, the different x_ispertaining to the same bias observation are not independent. They can be expressed as

[0044] x_is=x_i+r_i(t_is−t_i)  (5)

[0045] where t_isis the time of the measurement, t_iis a nominal time in the vicinity of the t_is, x_iis the true position at time t_i, and r_i(•) is a function considered to be known. For example, during uniform motion one could write (5) as x_is=x_i+{dot over (x)}_i.(t_is−t_i). Combining expressions (4) and (5) yields $\begin{array}{cc}V=\sum _{i,s}\left|\right|y_{\mathrm{is}}-h_s\left(x_i+r_i\left(t_{\mathrm{is}}-t_i\right),b\right){\left|\right|}^2& \left(6\right)\end{array}$

[0046] which thus will be minimised with respect to x_irather than all the x_is. There is, however, a serious problem. The solutions of the minimising procedure will not in general be unique, and this may later show up as singular matrices and severe instability. This singularity is physical, and singular value decomposition will not help to solve this problem. To illustrate this difficulty, a short example will be described with reference to FIG. 3. Suppose that radar locations are among the bias parameters. Now, if Radar A puts the measurement (M_A) 1 km more to the east than Radar B (M_B), this could be explained by Radar A being located 1 km east of its nominal position, while Radar B is correct. However, Radar A may instead be correct, while Radar B is located 1 km to the west of its nominal position. Moreover, it could also be that Radar A is located 50 km to the east and Radar B 49 km to the east, and so on. All such combinations would minimise the candidate expression (6). Obviously, the 50/49 km combination is unreasonable, and it is desirable that the algorithm should take that into account.

[0047] The means for including such information in the method of the present invention is by a priori uncertainties. The user has the information about something that standard algorithms do not take into account, namely expected variations of the different bias parameters. The inclusion of a priori uncertainties is thus an important part in the present invention. The expression to be minimised is according to the present invention: $\begin{array}{cc}V=\sum _{i,s}\left|\right|y_{\mathrm{is}}-h_s\left(x_i+r_i\left(t_{\mathrm{is}}-t_i\right),b\right){\left|\right|}^2+{\left(b-\hat{b}\right)}^{T}v\left(b-\hat{b}\right)& \left(7\right)\end{array}$

[0048] where v is a matrix, typically diagonal, whose components are the inverses of the squared a priori standard deviations of the corresponding parameters. Here {circumflex over (b)} stands for a priori values of the bias parameters. In most applications {circumflex over (b)} can be chosen to be zero. Note that with the inclusion of the a priori term, it is possible to minimise the expression (7) even at the absence of measurements.

[0049] To illustrate the dramatic improvement in stability that has been obtained, the above discussed example of FIG. 3 will be reexamined. The algorithm according to the present invention will put Radar A 0.5 km to the east and Radar B 0.5 km to the west (provided that the a priori information gives equal uncertainties to both radars). The true values can, of course, not be found without additional data, but this is a reasonable solution that eliminates the relative bias. The effect of the introduction of a priori uncertainties of bias parameters is now clear. The a priori uncertainties introduce “soft boundary conditions” on the system. In this way, the otherwise expected singularities are avoided at the same time as all important parameters are free to vary.

[0050] One way of finding the minimum of the expression (7) is to apply some numerical method such as the Levenberg-Marquardt technique, see e.g. William H. Press “Numerical Recipes in Pascal”, Cambridge University Press (1989), pages 574-580). However, for real-time applications it is preferable to use a non-iterative method based on linearisation of h_s. Thus, in a preferred embodiment, the error function is expanded about a nominal target position {circumflex over (x)}_iand the a priori bias vector {circumflex over (b)}. The efficiency of the algorithm will depend on the validity of the Taylor expansion around these values.

[0051] The present invention is intended to work in conjunction with a tracking system. The tracker is suitable for providing the nominal position to perform the Taylor expansion about. However, other means can also be considered, for example, a value computed directly from the measurements by averaging or triangulation. The only requirement is that {circumflex over (x)}_iis close enough to the true target position x_iso that the expansion is valid. If the measurements do not coincide in time, an approximation {circumflex over (r)}_tof the function r_iis needed. This can be obtained from the track state using its velocity ({circumflex over (r)}_t(Δt)={dot over ({circumflex over (x)})}₁·Δt) or higher order terms.

[0052] The first order expansion of h_sruns

[0053] h_s(x_i+r_i(t_is−t_i),b)≈ŷ_is+H_is·(x_t−{circumflex over (x)}_t)+A_isΔb  (8)

[0054] where

[0055] ŷ_is=h_s({circumflex over (x)}_t+{circumflex over (r)}_t(t_is−t_i),{circumflex over (b)}),  (9)

[0056] i.e., the nominal position projected into the measurement space, $\begin{array}{cc}{H_{\mathrm{is}}=\frac{\partial h_s\left(x_i,\hat{b}\right)}{\partial x_i}|}_{x_i={\hat{x}}_i+\hat{r}\left(t_i-t_{\mathrm{is}}\right)},& \left(10\right)\end{array}$

[0057] the Jacobian giving the standard projection matrix, $\begin{array}{cc}{A_{\mathrm{is}}=\frac{\partial h_s\left({\hat{x}}_i+{\hat{r}}_i\left(t_{\mathrm{is}}-t_i\right),b\right)}{\partial b}|}_{b=\hat{b}},& \left(11\right)\end{array}$

[0058] which is called the “design matrix”, and

[0059] Δb=b−{circumflex over (b)}.  (12)

[0060] It is now possible to write the expression to be minimised as: $\begin{array}{cc}V=\sum _{i,s}\left|\right|K_{\mathrm{is}}-H_{\mathrm{is}}{X}_i{\left|\right|}^2+\Delta b^{T}v\Delta b& \left(13\right)\end{array}$

[0061] where

[0062] K_is=d_is+H_is{circumflex over (x)}_i−A_isΔb  (13)

[0063] and

[0064] d_is=y_is−ŷ_is.  (15)

[0065] Now minimising the expression (13) with respect to all x_i, i.e. solving

[0066] ∇_XV=0  (16)

[0067] yields

[0068] H_isx_i={overscore (K)}_is  (17)

[0069] where the “bar” notation for any vector q is defined as $\begin{array}{cc}{\stackrel{\_}{q}}_{\mathrm{is}}={H_{\mathrm{is}}\left(\sum _t{H}_{\mathrm{it}}^T{u}_{\mathrm{it}}{H}_{\mathrm{it}}\right)}^{-1}\sum _t{H}_{\mathrm{it}}^T{u}_{\mathrm{it}}{q}_{\mathrm{it}}& \left(18\right)\end{array}$

[0070] (There may be occasions when geometry is such that the inverse in the expression (18) cannot be calculated. Such cases should be excluded from the summing in the sequel). By inserting expression (17) into the expression (13) the error function expression runs $\begin{array}{cc}V=\sum _{i,s}\left|\right|K_{\mathrm{is}}-{\stackrel{\_}{K}}_{\mathrm{is}}{\left|\right|}^2+\Delta b^{T}v\Delta b& \left(19\right)\end{array}$

[0071] and is now to be minimised with respect to Δb. Writing (19) more explicitly gives $\begin{array}{cc}V=\sum _{i,s}\left|\right|d_{\mathrm{is}}-{\stackrel{\_}{d}}_{\mathrm{is}}+H_{\mathrm{is}}{\hat{x}}_i-\stackrel{\_}{H_{\mathrm{is}}{\hat{x}}_i}-A_{\mathrm{is}}\Delta b+\stackrel{\_}{A_{\mathrm{is}}\Delta b}{\left|\right|}^2+\Delta b^{T}v\Delta b& \left(20\right)\end{array}$

[0072] Now, after verifying that {overscore (H_is+L {circumflex over (x)})}_t=H_is{circumflex over (x)}_i, all that remains is $\begin{array}{cc}V=\sum _{i,s}\left|\right|d_{\mathrm{is}}-{\stackrel{\_}{d}}_{\mathrm{is}}-\left(A_{\mathrm{is}}-{\stackrel{\_}{A}}_{\mathrm{is}}\right)\Delta b{\left|\right|}^2+\Delta b^{T}v\Delta b& \left(21\right)\end{array}$

[0073] Minimising this expression with respect to Δb finally gives Δb as the solution of

[0074] (U+v)Δb=m  (22)

[0075] where $\begin{array}{cc}U=\sum _{i,s}{\left(A_{\mathrm{is}}-{\stackrel{\_}{A}}_{\mathrm{is}}\right)}^T{u}_{\mathrm{is}}\left(A_{\mathrm{is}}-{\stackrel{\_}{A}}_{\mathrm{is}}\right)& \left(23\right)\\ \mathrm{and}& \\ m=\sum _{i,s}{\left(A_{\mathrm{is}}-{\stackrel{\_}{A}}_{\mathrm{is}}\right)}^T{u}_{\mathrm{is}}(d_{\mathrm{is}}-{\stackrel{\_}{d}}_{\mathrm{is}}& \left(24\right)\end{array}$

[0076] and finally b={circumflex over (b)}+Δb. The structure of the expressions (23) and (24) facilitates continuous on-line processing: When a bias observation i is available, the corresponding data is added to the matrix on the left hand side and to the array on the right hand side of expression (22). The system of equations can now be solved for Δb when needed since all requested information is already updated. As b is assumed to vary slowly, there is normally no need to perform the solving at every bias observation i. One can decide to solve the equations at regular time intervals, or to do it by an external trigger. It is thus not necessary that the equations are solved periodically. An operator may therefore intervene with an automatic trigger mechanism and request additional updates of bias parameters. The solution is subsequently sent to the tracking system, and one can proceed updating the left and right hand sides of the expression (22).

[0077] Just as the inverse of v is the a priori covariance matrix, the covariances P_bof the estimated bias parameters are given by (U+i,)⁻¹and these quantities are thus easily provided to the tracking system. For most practical purposes it suffices to take the diagonal elements of (U+v)⁻¹as the squared uncertainties of the corresponding parameter.

[0078] The flow of the calculations is illustrated in FIG. 4. The process starts in step 100 with a matrix allocation. If the total number of bias parameters is n, an n×n matrix for U and an n×1 matrix for m are allocated. In step 100, these matrices are also filled with zeros.

[0079] As a track and its associated measurements enter the alignment estimator, the design matrix A_is, the projection matrix H_isand the other quantities required for the expressions (23) and (24) are computed in 104. These are added to U and m. This is repeated for each corresponding sensor. The matrices U and m are thus progressively filled with more and more data.

[0080] If a trigger signal 106 arrives, manually or automatically, indicating the need for an updated bias vector in step 108, then the system of equations (U+v)Δb=m is solved for Δb in step 110. If required, also the inverse of U+v is calculated to get the variances of b. In step 112, b and its uncertainties are provided to the tracking system.

[0081] In step 120, the next bias observation is awaited, and when it arrives, the process returns to step 104 for a new repetition.

[0082] It should be noted, in connection with the issue of selecting the nominal true positions {circumflex over (x)}_i, then in some cases it may happen that the true target locations are indeed known. This may be the case if calibration points are available, or if satellite navigation data are so accurate that they may be regarded as truth. In this case one should of course use the true positions as x_i. There is then no need to minimise with respect to x_t, and the subsequent calculations are thus simplified considerably.

[0083] The procedure described so far has the limitation that only completely static bias parameters can be handled properly. The system bases its estimates on the complete history of data, so recent changes will have little impact on the estimates. One way of handling drifting bias parameters would be to restart the processing at certain time intervals, e.g. one hour. The drawback of such an approach is that some sensors may not provide data continually, and so any estimates for these would be forgotten. Moreover, the batches should be short for rapidly varying parameters, while they should be long for obtaining reliable estimates where data are sparse. These are contradictory requirements.

[0084] A preferred embodiment of the present invention thus involves a process which gradually supresses old data. To this end the preferred embodiment of the present invention includes “process noise” to model time dependence of the parameters. This results in that the system gradually forgets past data. The time constant is selectable individually for each parameter. Some parameters are likely to vary slowly, like the location of a fixed sensor, and some parameters may vary more rapidly, like e.g. time delays due to congestion of data communication. At regular time intervals, the left and right hand sides of expression (22) are modified to incorporate process noise. It is then important to add v only just before solving the expression (22), otherwise the a priori data would fade away.

[0085] The inclusion of process noise amounts in a preferred embodiment to adding a noise covaricance matrix Q to the covariance matrix U⁻¹. The resulting U is expressed as

[0086] Ũ=(U⁻¹+Q)⁻¹=[U⁻¹(I+UQ)]⁻¹=(I+UQ)⁻¹U  (25)

[0087] where the tilde denotes a quantity after adding noise.

[0088] The solution of Ub=m should not be changed when adding the noise, so the right hand side must be multiplied with the same factor as U:

[0089] {tilde over (m)}=(I+UQ)⁻¹m  (26)

[0090] In general U is not invertible. Although the motivation of (25) and (26) did require invertibility, the end results do not, and it is an assumption that they are valid in all cases.

[0091] The question how to define Q still remains. A convenient way to perform this is to define a time constant τ for each bias parameter. If the a priori uncertainty for that parameter is

[0092] δ_i=(v_ii)^−½,  (27)

[0093] then an assumption that the added variance during the time interval Δt is $\begin{array}{cc}{Q}_{\mathrm{it}}=\frac{\Delta t}{{\tau }_i}\frac{I}{v_{\mathrm{it}}}& \left(28\right)\end{array}$

[0094] is made, that is, the variance grows linearly with time.

[0095] As the inclusion of process noise implies the inversion of a large matrix, this should not be performed more often than necessary. The order of magnitude of the time interval for the process noise calculations should preferably be the same as the shortest time constant of the involved bias parameters. The triggering mechanisms for the process noise inclusion may be automatic or manual, or a combination thereof, but preferably substantially periodical.

[0096] In a practical implementation of the present invention, one should generally not use all the incoming measurements. Therefore, in a preferred embodiment the incoming measurements should be sorted by a set of filters. The filtering in the preferred embodiment will consider the following issues.

[0097] One must make sure that the measurements are correctly associated to the track. If there is a case of ambiguous association (as detected by the association algorithms in the tracker) the measurement should be discarded. The tracker often gives information about the quality of the association, so a simple comparison of such a measure with a threshold value in such a case is enough.

[0098] Furthermore, the track should be in a state of reliable tracking and in reasonably uniform motion. This request is in order to ascertain that the given nominal position and velocity are close to the true values.

[0099] One must make sure that no measurement is used twice, that is, in two different sets i. Furthermore, one should discard measurements that are too old. The measurements included in a set i should be at sufficiently close times. This ensures that the procedure of bringing the set to a common nominal time, described above, is valid.

[0100] In reality it may happen that there is an abundance of measurements in one area (e.g. around an airport) while other areas give very few measurements. In such a case, the algorithm tends to optimise the bias estimation results over the set of measurements, rather than over the complete coverage area. It may, however, be desirable that the system provides good estimates whether the traffic is dense or sparse. If this is the case, it may be preferable to give extra weight to the measurements in the sparse areas, for example by not using all incoming data from the dense areas.

[0101] As a final assurance that the associated measurements belong to the same target, a certain fraction (e.g. 10%) of the bias observations should be discarded based on the size of, for example, the quantity d_is²=∥d_is−{overscore (d)}_is−(A_is−{overscore (A)}_is)(b−{circumflex over (b)})∥², (cf. equation (21), where now b stands for the currently valid bias vector rather than for an unknown).

[0102] After a filtering process according to the above suggestions, the remaining filtered set of measurements and tracks are behaving well, and this in turn ensures that the used approximations are valid.

[0103] In a preferred embodiment, the above steps of time dependence introduction and measurement filtering are included in the alignment process. FIG. 5 illustrates the flow of information in such a case. This flow of information is very similar to that shown in FIG. 4, so therefore only the differences will be described.

[0104] When the measurements becomes available to the automatic alignment section, the process enters into step 102. In this step, the above described filtering of the measurements takes place and the information that is added into the matrices in step 104 is the filtered, well behaving measurements.

[0105] When the process has passed the point, where a bias estimate could be requested, i.e. steps 108-112, the time dependence may be entered. Since the calculations in this respect are quite time consuming, this part should not be performed every time the process comes to this point. Thus, in step 116, the process determines if there is a request for including the time dependence in the measurements data. This is determined by a trigger signal 114, which may be supplied automatically or manually. If that is the case, the process enters into step 118, which suppresses the older data according to the above described routines. More specifically, U and m are recomputed using (25) and (26). The flow then continues at step 120.

[0106] A simple example, with references to FIG. 6, will clarify the algorithms and also illustrate some properties of the invention. Consider a situation where three passive sensors are located on a flat earth described by a Cartesian coordinate system (ξ,η). Each of the sensors measures the direction θ to the target. As is common practice, θ is measured from the η axis in the clockwise direction. Three bias parameters are considered for each sensor s: Sensor location errors b_sξ and b_sη, and a north alignment error (azimuth error), b_sθ. All bias parameters are taken together in the bias vector

[0107] b=(b_1ξ b_1η b_1θ b_2ξ b_2η b_2θ b_3ξ b_3η b_3θ )^T

[0108] The nominal footpoint of the sensor s is denoted x_s0=(ξ_s0,η_s0). The transformation h_s(x,b), which takes target locations x=(ξ,η) down to the measurement space of sensor s, runs $\begin{array}{cc}{h}_s\left(x,b\right)=\arctan \left(\frac{\xi -\left({\xi }_0+b_{s\xi }\right)}{\eta -\left({\eta }_{\mathrm{s0}}+b_{s\eta }\right)}\right)-b_{s\theta }& \left(27\right)\end{array}$

[0109] where it is understood that ±π may have to be added to put θ in the right quadrant.

[0110] The following conditions are assumed. The sensor nominal locations are: x₁₀=(−10,5), x₂₀=(20,0) and x₃₀=(0,−10) kilometres. The measurement noise is 0.005 radians for all sensors, that is, the measurement information matrices μ_isare all 4.10⁴rad⁻². The a priori bias uncertainties are 0.1 km for all locations, and 0.02 radians for all north alignment errors, so the a priori information matrix is v=diag(100 100 2500 100 100 2500 100 100 2500). The actual bias errors are all zero except for the north alignment of sensor 1, b₃, which is 0.02 rad. (This is exaggerated in FIG. 6). For simplicity, all measurements pertaining to the same occasion are at the same time, and the bias vector that one expands about is zero.

[0111] The first observation stems from a target at location x₁=(30,20). To keep the notation simple, henceforth observation index i=1 is dropped. The measurement is given by equation (27) applied to the true location and the true bias vector, and then distorted by measurement noise. In this example, however, no measurement noise is added. (The results will be easier to interpret this way. By adding a long sequence of measurements, the effect of the noise will anyway be smoothed out.) The obtained measurements can then be computed as y₁=1.192,y₂=0.464,y₃=0.785 radians. The true position is unknown to the system, but an approximate nominal position is assumed to be available: {circumflex over (x)}=(30.1,19.8) km.

[0112] Now, the projection matrices are computed by differentiating equation (27) at the nominal target position and assumed zero bias vector:

[0113] H₁=(0.008−0.022)

[0114] H₂=(0.04−0.02)

[0115] H₃=(0.017−0.017)

[0116] Likewise, the design matrices are computed by differentiating (27) with respect to the bias parameters:

[0117] A₁=(−0.008 0.022 −1 0 0 0 0 0 0)

[0118] A₂=(0 0 0 −0.04 0.02 −1 0 0 0)

[0119] A₃=(0 0 0 0 0 0 −0.017 0.017 −1)

[0120] Also the nominal positions in measurement space are calculated by applying (27) to the nominal target position in target space x and zero bias vector:

[0121] ŷ₁=1.217

[0122] ŷ₂=0.472

[0123] ŷ₃=0.79

[0124] The differences between actual and nominal measurements are:

[0125] d₁=−0.025

[0126] d₂=−0.008

[0127] d₃=−0.005

[0128] Next one needs (cf. (18)) ${\left(\sum _s{H}_s^T{\mathrm{uH}}_s\right)}^{-1}=\left(\begin{array}{cc}0.044& 0.047\\ 0.047& 0.072\end{array}\right)$

[0129] Hereby the barred versions of d_sand A_scan be computed:

[0130] {overscore (d)}₁=−0.022

[0131] {overscore (d)}₂=−0.006

[0132] {overscore (d)}₃=−0.012

[0133] {overscore (A)}₁=(−0.007 0.018 −0.835 0.005 −0.002 0.113 −0.006 0.006 −0.353)

[0134] {overscore (A)}₂=(0.0009 −0.002 0.113 −0.037 0.019 −0.922 −0.004 0.004 −0.243)

[0135] {overscore (A)}₃=(−0.003 0.008 −0.353 −0.01 0.005 −0.243 −0.004 0.004 −0.242)

[0136] This is all that is needed to calculate U from (23) and m from (24). Equation (22) can now be solved with the result

[0137] b=(0.0006 −0.002 0.003 0.002 −0.001 0.002 −0.003 0.003 −0.007)^T

[0138] The positional parameters are small, as they should be, but the azimuth components (b₃, b₆, and b₉) may at first sight seem like a failure. However, referring to FIG. 6, rotating the strobes from sensor 1 and 2 clockwise, and rotating the strobe from sensor 3 counterclockwise would indeed compensate for the bias errors visible in this situation. This sole set of measurements cannot uniquely determine the parameters, which also follows from the fact that there are nine unknown parameters, and only three measured quantities, The algorithm finds values for the parameters that make the strobes meet in the least expensive way in terms of consistency with the a priori data.

[0139] To get better estimates one should add more data. In FIG. 7 there are two more bias observations, and it now appears more clearly that it is the strobes from sensor 1 that need correction. The true target locations are x₂=(20,−50) and x₃=(−25,−5) km, and the nominal locations are {circumflex over (x)}₂=(20.2,−50.1) and {circumflex over (x)}₃=(−24.9,−5.2). These are the data that are needed to repeat the calculations shown above. After adding the contributions to U and m and solving together with v, the result runs

[0140] b=(−0.008 −0.001 0.015 −0.004 0.001 −0.002 0.012 −0.0004 −0.004)^T

[0141] which clearly singles out b₃with 15 mrad (true value 20 mrad). In this way, by adding more and more data one can continually improve the estimates.

[0142] The uncertainties can be found by inverting U+v, and taking the square roots of the diagonal elements. These are

[0143] δb=(0.099 0.100 0.010 0.099 0.099 0.008 0.097 0.099 0.010)^T

[0144] The measurements thus have so far done very little to improve the location estimates beyond the a priori data, while the north alignment uncertainties have been cut down by a factor 2. So, e.g., b₃has been narrowed in from 0±20 mrad to 15±10 mrad.

[0145] The present invention is a method with several attractive properties. It handles a multitude of sensors of different characteristics. It even works if all sensors are of a passive type. The method uses targets of opportunity for the estimation and there is no need for special calibration targets. Furthermore, the method does not require that any of the sensors is used as a reference. The present invention provides updated estimates as often as requested by the tracking system. The first estimates can be obtained based on very few measurements. The estimates are improved as more measurements enter the system. Moreover, the method allows for defining a time constant for each bias parameter, handling the measurement history, so that some parameters are regarded as almost stationary while others may be varying more rapidly. Finally, measurements alone do not always contain enough information for unambiguous bias estimates, which is often reflected in singularities and numerical instability in traditional methods. The present invention utilises a priori data that removes this problem.

[0146] Although a preferred embodiment of the present invention have been described in detail, the scope of the invention should not be limited thereto, but should instead solely be limited by characterising part of the following claims.