Federated Kalman Filter Simulation Results NEAL A. CARLSON
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`integrity Systems,
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`Inc..
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`Belmont. Massachusetts MICHAEL I? BERARDUCCI
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`Air Force Wright Laboratory, WPAFB. Ohio
`Recerued
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`June 1993 Reused March 1994 ABSTRACT This paper describes federated filter applications to integrated, fault-tolerant navigation systems, with emphasis on real-time implementation issues and numer- ical simulation results. The federated filter is a near-optimal estimator for decen- tralized. multisensor data fusion. Its partitioned estimation architecture is based on theoretically sound information-sharing principles. It consists of one or more sensor-dedicated local filters, generally operating in parallel, plus a master combin- ing filter. The master filter periodically combines Muses) the local filter solutions to form the best total solution. Fusion generally occum at a reduced rate, relative to the local measurement rates. The method can provide significant improvements in fault tolerance, data throughput, and system modularity. Numerical simulation results are presented for an example multisensor navigation system. These results demonstrate federated filter performance characteristics in terms of estimation accuracy, fault tolerance, and computation speed. INTRODUCTION Integrated multisensor navigation systems have the potential
`provide high levels of accuracy and fault tolerance. The presence of multiple data sources provides functional redundancy, as well as greater observability of the desired navigation states. However, that potential has not always been fully realizable through the application of standard (centralized) KaIman filtering techniques for multisensor data fusion. Centralized filters can result in severe computation loads when processing data from multiple sensors. Worse, when used in two-stage (cascaded) filter architectures, standard Kalman filters can exhibit poor accuracy and unpredictable behavior under some conditions. During the past 15 years, the development of decentralized (or distributed) Kalman filtering methods has received increasing attention. Parallel process- ing technology, emphasis on fault-tolerant system design, and availability of multiple specialized sensors strongly motivate the development of such meth- oda. Potential applications include multisensor navigation systems, tracking systems, and other data fusion systems. Early contributions to optimal decentralized filter theory [l-51 provided useful insights about different approaches to partitioned estimation. Building 297
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`298 Navigatron Fall 1994 on this theoretical base. reference 161 proposed a decentralized filtering struc- ture with attractive fault-tolerant features. Reference (71 developed an optimal and practical decentralized filtering method for orbit estimation purposes. Ref- erences IS] and [91 made subsequent extensions of that approach for other applications. More recently, the federated filter method based on rigorous information- sharing principles was developed [ 10-121. This method provides globally opti- mal or near-optimal estimation accuracy with a high degree of fault tolerance. and is practical for real-time distributed navigation systems applications [131. The federated filter structure employs sensor-dedicated local filters ( LFs). and a master filter (MF) to combine or fuse the
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`outputs. The method allows several different information-sharing strategies, or modes, for different applica- tions. Some modes require simultaneous updates from the LFs to the MF, while others permit independent updates from each LF. Recently, the net information approach to decentralized estimation was developed i 14-151. This globally optimal/near-optimal method is based on infor- mation-sharing principles similar to those of the federated method. Each local element employs two LFs (one is locally optimal) and a differencing algorithm to determine the new information gained over each period. Local elements pass their new information to the MF independently, for data fusion. This paper focuses on the performance of the federated filtering technique applied to integrated, fault-tolerant navigation systems. The remaining sec- tions describe the distributed estimation problem, limitations of standard Kal- man filters, federated filter basics, federated configuration options and features, numerical simulation results, and conclusions. DISTRIBUTED ESTIMATION I’ROBLEM The federated filter is a partitioned estimation method. It employs a two- stage (cascaded) data processing architecture, in which the outputs of sensor- related LFs are subsequently combined by a larger MF, as illustrated in Figure 1. As indicated, each LF is dedicated to a separate sensor subsystem, and also uses data from a common reference system, generally an inertial navigation system ( INS). The federated filter technique comprises a weighted least-squares solution to the following linear for linearized) estimation problem. First,
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`consider a
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`system state vector x that propagates from time t’ to t = t’ + At according to the following dynamic model: x=cDx’+Gu (1) Here, @ is the state transition matrix between time points t’ and t, G is the process noise distribution matrix, and u is the additive uncertainty vector due to white process noise acting over the timestep At. The errors
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`in the initial state estimates % have covariance P,,. The sequential process noise values \4 at times t, have covariances Qj, and are uncorrelated with e, and with each other.
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`-Geneml Fedcmted Filter Structure Our system also has access to external measurements ii from i = l...n separate local sensor subsystems. The discrete measurements from sensor #i at time t, are related to the true state 4 by a linear (or linearized) relationship: $= HI; x, + vij (2, Here. HI,T is the sensor Xi measurement observation matrix (often defined without the transpose), and vi, is the additive, random measurement error. The sequential error values vij have covariances R1[1, and are uncorrelated at successive times t,. In addition, measurement errors from different sensors i and m are statistically independent. Disjoint sensor data sets permit the total estimation problem to be divided into sensor-related partitions with indepen- dent measurement processes. LIMITATIONS OF STANDARD KALMAN FILTERS The primary limitations of standard (centralized) Kalman filtering methods when applied to multisensor navigation systems an&or systems with embedded local filters are (1) heavy computation loads, (2) poor fault-tolerance, and I
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`LOCAL L,,,,_,-l L LOCAL PN.xn SENSOR N 1” - FILTER N L
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`inability to correctly process prefiltered data in a cascaded (two-stage) filter structure. The first limitation of a centralized Kalman filter (CFJ is fairly obvious. In Figure
`the presence of several sensors generally implies a relatively large number of filter states n, since each sensor typically introduces one to five measurement bias states. For a single CF, the per-cycle computation load grows roughly in proportion to n3 + Im - n2, where Im is the total number of
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`300 Nawgatlon Fall 1994 measurements across all the sensors. The problem is especially severe when one or more sensors require high-rate measurement processing. (However. computation load is a less critical concern with modem, high-speed processors and algorithms that take advantage of sparseness in the a, Q, and H matrices.) The second limitation of a CF relates to fault tolerance. Like any optimal filter. the CF attempts to make its data inputs agree in a weighted least- squares sense. thereby suppressing their differences. An undetected failure in one sensor gets distributed into all of the navigation state and sensor bias estimates. so that they all tend toward agreement. Thus, while measurement residual tests can readily detect sudden hard failures, they may completely miss gradual sop failures. If the CF does incorporate faulty data from any sensor, its full solution becomes corrupted and must be reinitialized. The third limitation of a CF relates to cascaded filter processing, and can best be illustrated by means of an example. Figure 2 shows the major components of a cascaded filter designed for an integrated navigation system composed of an INS. a GPS Phase IIIA receiver/navigator, and a radar subsystem. GPS receiver measurements and INS outputs are processed locally by an embedded GPS/ inertial Kalman filter. Periodically, position and velocity outputs from the GPS LF are incorporated as discrete “measurements” by a Kalman MF in the central computer. One particular aspect of this ad hoc cascaded filter design can lead to accuracy and/or stability problems, given a Kalman MF: errors in the position and velocity outputs from the.GPS LF are sequentially correlated, not uncorrelated as a Kalman filter requires. Sequential correlations in the GPS filter outputs imply that each output contains some new and some old information. Treating each GPS filter output as entirely new information causes the MF to become overoptimistic regarding its own accuracy (i.e, its covariance gets too small). The resulting problem is especially obvious if we suppose that GPS receiver
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`Fig. 2-Ad Hoc GPSIR#ar Filter
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`Vol 41 No 3 Car/son. ef al.: Federated K&man Filter 301 measurements become unavailable for several mmutes. The .MF will continue to incorporate successive GPS filter outputs as fresh “measurements” and to reduce its covariance accordingly, even though those outputs contain absolutely no new information. A common, ad hoc fix for these correlation problems is to limit the GPS incorporation rate in the Kalman MF to once every 10 or 20 s. Over these intervals. the GPS Phase IIIA filter output errors typically become decorrelated, 30 that they appear sequentially random to the
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`F. (Rapid decorrelation is caused by large process-noise terms in the GPS filter.) This ad hoc cascaded filter design can yield satisfactory estimation performance for many applications. However, questions remain as to whether the same approach will work with different GPS filter tuning parameters or with other sensor subsystems. FEDERATED FILTER BASICS The new federated filter method avoids the theoretical and practical difficult- ies described in the previous section by means of a simple yet effective informa- tion-sharing methodology. We use the term information to denote the filter solution, i.e., the filter state vector and covariance matrix (or equivalently its inverse, commonly called the information matrix,. The basic concept of the information-sharing approach implemented by the federated filter is this: ( 1) divide the total system information among several component (local) filters; ( 2) perform local time propagation and measurement processing, adding local sensor information; and (3) recombine the updated local information into a new total sum. The remainder of this section briefly reviews how the federated filter applies information-sharing principles in its use of the n LFs and one MF in Fig ure 1. The presentation is heuristic, to emphasize the general concept. Refer- ences 110, 11, 161 provide a rigorous theoretical derivation of the federated filter. First, let the full, CF solution be represented by the covariance matrix PF and state vector ft the ith LF solution by PI and ii; and the MF solution by PM and tm. We will use index i = l...n for the EFs alone, and k = l...n.m for the MF plus LFs, where k = m is the MF. Now, if the LF and MF solutions are statistically independent, they can be optimally combined by the following additive information algorithm, where the inverse covariance matrix P- ’ is known as the information matrix: PF-’ = PM-’ + Pl-’ + . . . + PN-’ 13) PF-‘kf = PM-‘im + Pl-lx1 + . . . + PN-‘h (4) The key to the new federated filtering method is to construct individual LF and MF solutions so they can be combined or recombined at any time by the above simple algorithm. In particular, the construction avoids the need to maintain local/local or 1ocaYmast.w crowcovariances. The procedure for doing so is the essence of the federated filter method.
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`Suppose we start with a full solution PF,
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`divide that solution so
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`k = 1 . . . n,m
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`It is clear that
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`these LF and MF solutions can be recombined per equa- tions (3) and
`ft. provided we maintain constant total information across the terms in equations (51 by requiring that
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`share-fraction values PI, sum to unity: n.m s
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`through independent, parallel operations of the LFs and MF. pro- vided the common process noise information is divided in the same fashion as was the common state information:
`QF-’ = QM-’ + Ql-’ + . . . + QN-’
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`(8ai QK-’ = QF-’ 3k or QK = QF p;’ (8b) These scaled QK values are used in the standard covariance propagation equa- tions from time t’ to t for each of the k = I . . . n,m filters:
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`PK = *K PKWKT + GK QK GKT
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`fi=@K~&’
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`PK’, 3dr’ values
`(For the moment, let us assume that the filterok matrices @K, GK equal the full-filter values @F, CF.1 Thus, if the
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`have been initially obtained by equations (5), and the QK values are obtained by equations (8). then the post-propagation values
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`ik can again be summed by the simple fusion equations (3) to (4) to yield the correct full solution PF, %f. Third, consider the measurement update process. Each LF Xi incorporates discrete measurements ii from i& own unique sensor/subsystem. Measurement information is added to LF #i as follows, where RI-’ is the sensor-i measure- ment information matrix:
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`PI:’ = PI-’ + HIRI-‘Hf
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`PI:’ &iii, = PI-‘&i + HI RI-%
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`(11)
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`(12)
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`where subscript + refers to postmeasurement values. Again, combining the above results by the fusion equations (3) to (41 yields the correct full solution, i.e., the solution achieved by a single CF processing all of the i = 1 . . . n sensor measurement sets. Now, equations (5) to (12) demonstrate that the federated filter solution is the same as that of a single, centralized Kalman filter, and hence is itself globally optimal, when certain assumptions are satisfied: ( 1) each filter employs a single @k value for all of the full-system states and process noises, and (2) the
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`Vol 41 No 3 Car/son. et al.; Federared Kalman Filter 303 mformation fusion and reset (division 1 operations are performed
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`measurement cycle. However. much less restrictive conditions can be accommo- dated. with only a modest loss of optimality. First. the federated filter can be implemented such that the LFs are of minimum size. each LF #i containing only the common INS states and its own unique sensor-i bias states. Then the matrices PK, OK, GK, and QK contain only the common and bias-k partitions of the full matrices. The p4 fraction dues apply only to the common INS states, since only those states are shared among the LFs and MF. It can be shown 110, 11. 161 that this mmimum-LF structure still produces a globally optimal solution in some cases. For other cases. the minimum-LF structure introduces a slight loss of information: how- ever, in practice. the resultant estimation accuracy is almost equal to that of the globally optimal filter. Second. the federated filter can perform fusion updates at a reduced rate relative to the LF measurement rates, implying data compression in some or all of the LFs (multiple LF measurement sets are
`into the latest LF state estimates). LF data compression does introduce a small loss of global information, equivalent to neglecting a vector measurement of common process noise dimension at each interior step. This information loss is negligible when Q C-C P over the fusion interval, as is the usual situation. FEDERATED FILTER CONFIGURATION OPTIONS The new federated filtering technique can be implemented in a variety of design configurations, each reflecting a different information-sharing strategy, or mode. This section describes four such configurations. All four are suitable for
`applications, in which the MF may be freely designed, but the LFs are assumed to have been developed elsewhere for stand- alone filter operations. (Only minor modifications to these LFs are assumed possible.) We will consider as an example a hypothetical multisensor navigation system containing sensor subsystems typical of an advanced tactical aircraft. These sensor subsystems and their data output rates (Hz) or intervals (s) are t 1) strap- down INS, bare-aided (50 Hz); (2) GPS receiver (l-2 s); (3) terrain-aided navi- gation (TAN) equipment (l/4-1/2 s) and (4) synthetic aperture radar SAR) ( 100-300 s). The strapdown INS is the common reference system: it puts out indicated position, velocity, body attitude, and angular velocity. The GPS receiver puts out pseudorange and pseudorange-rate (or delta-range) measurement pairs fromfour or more satellites each cycle. (Note that some GPS receiver/navigators output only a filtered GPS/inertial navigation solution, and not raw measure- ments: these GPS sets represent the constrained case mentioned above.) The TAN radar altimeter puts out one height measurement per cycle, used in conjunction with stored terrain elevation data. The
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`landmark range and range-rate measurements; some SAR systems may also put out landmark azimuth and elevation measurements, and up to three high- precision Doppler velocity measurements.
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`SAR typically puts Out
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`shows the federated filter Fusion-Reset #‘RI mode. Again, the LFs collectively maintain the system long-term memory, while the, MF provides temporary short-term memory. The FR mode is unique in that it involves
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`For this example system, all of the federated filter configurations described here consist of one MF and three LFs, as in Figure 3. Each LF processes measurements from one external sensor subsystem (GPS, TAN, or SAR), while all of the filters use common INS data. Each LF provides navigation information derived from its unique sensor(s) and the INS. The MF performs generic fusion operations, and generates the best total navigation solution. The fundamental differences among the four modes relate to information storage. In the first two modes, the LFs collectively store the long-term system information, while the MF acts as a short-term information combiner. Con- versely, in the second two modes, the MF stores the long-term system informa- tion, while the LFs act as short-term data collectors, or data compression filters. The following paragraphs provide more specific details. Figure 3 shows the federated No-Reset t
`mode. Here, the LFs collectively maintain the system long-term memory, while the MF provides short-term memory through propagation of the fused solution after combining (fusing) the LF outputs. There is no information feedback fmm the MF to the LFs; each LF retains its own, unique portion of the total system information. This mode permits the LFs to operate independently, aa stand-alone filters, with estimation accuracies nearly at their normal levels. The LFs ail send solutions to the MF for fusion at the same time. The MF propagates the fused solution to intermediate time points, but does not use it in the next fusion update, since the next set of LF solutions contains all the system information. Figure
`5 / 4 / GPS ---_) LOCAL Pl,Xl Receiver ,T1 - ALTER1 * py--pqx , A 4 TAN 23- LOCAL P3,x3 - FILTER3
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`feedback of fusion-reset solutions (fused state and scaled-up covariance parti- tions) from the MF to the LFs. Thus, the less accurate LFs operate at improved accuracy levels, since they gain information from the more accurate sensors via the MF resets. (The more accurate LFs may operate at somewhat reduced accuracy levels if they gain back less information from the MF than they provide.) The FR mode requires all the LFs to send solutions to the MF for fusion at the same time. Each LF then waits for a reset solution from the MF before proceeding. Figure 5 shows the Zero-Reset (ZR) mode. Here, the MF retains all of the fused information or system long-term memory. The three LFs retain none of the fused information, acting as data compression filters with short-term mem- ory only. Each LF provides the MF with the new information it has gained since the last update. The MF adds each new LF solution input to its total information, in much the same way that a standard Kalman filter adds new sensor measurements. After each fusion update, the LF resets itself to zero information (large covariance), keeping the same state estimates. An attractive feature of the ZR mode is that the MF can incorporate LF solutions at different times, e.g., baaed on how rapidly each LF gains information. Last, the federated filter Rescale (RS) mode comprises a variation of the ZR mode shown in Figure 5. Here, each LF passes some but not all of its information to the MF at each fusion update, and retains the remainder. The LF then rescales its covariance by l/a,, where oi is the fraction of information that LF Xi retains, and keeps the same state estimates. The MF accumulates and retains most of the total system information, while the LFs collectively retain the remainder. The process is similar to that of the ZR mode, except that there
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`is some lag in passage of information from the LFs to the MF, so that the LFs always retain some useful information (as may be needed to support subsidiary LF functions). With regard to constrained applications, the NR mode is best suited for use with existing LFs, since it requires no additional software functions beyond those needed for stand-alone operations. (Even the common process-noise multi- pliers l/p, are often unnecessary, given conservative process noise models.) The RS mode is usable with existing LFs only if they can be modified to accommodate periodic resealing of the covariance matrix. The ZR mode is not directly usable with existing LFs, but can be approximated if each LF can be restarted I at a large, initial covariance value) whenever a zero-reset is required. The FR mode is not usable with existing LFs, since LF state and covariance resets to MF-supplied values are not feasible. FEDERATED PERFORMANCE FEATURES This section compares the four federated filter configurations in terms of data processing rates, real-time control issues, and processor and sensor fault tolerance. With regard to data processing rates, the four federated filter modes all have the capability to adjust the MF fusion rate relative to the LF measurement update rates. When maximum accuracy is required, the MF can perform a fusion update after every measurement cycle of the most accurate LF (e.g., the GPS LF). When maximum MF accuracy is not required, and a reduction in computation burden is desired, the MF fusion rate can be decreased to well below the key measurement rate. Thus, the relatively large fusion computation
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`modes are most flexible in this regard. since they permit fusion to occur at different times for different LFs. whereas the NR and FR modes require all LF solutions to be fused at the same time. With regard to real-time control, NR is the simplest of the four modes, since there are no LF resets. The ZR and RS modes are less simple, since each LF must reset itself after sending its solution to the MF. but can do so immediately, wIthout waiting for a reply from the
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`The FR mode is the most complex, jlnce each LF has to wait for a reset solution from the AMF before it can reset itself and continue its normal operations. Processor fault tolerance denotes the ability to detect and recover from filter dropouts caused by processor or bus failures. All of the federated filter modes provide enhanced processor fault tolerance in a way not possible with a single CF. Each filter sends out periodic status messages to the other filters, and monitors their messages in return. If an LF disappears, the federated controller 1 FC) detects its absence almost immediately via the missing message. The FC then modifies the information shares for the remaining LFs, and tells the MF to use only those LFs in the fusion process. If the missing LF later reappears, the FC can reconfigure the federation once more, to again include that LF. Finally, sensor fault tolerance denotes the ability to detect, isolate, and recover from sensor:subsystem failures. The federated filter modes support sensor fault detection. isolation, and recovery (FDIR) at both the LF and MF levels. ‘For purposes of comparison, we will first consider the sensor FDIR capabilities of a CF. The CF maintains a single, globally optimal solution, and incorporates mea- surements from all the sensors. It has maximum capability to detect single- measurement faults through measurement residual tests, since each measure- ment is checked against all of the accumulated prior information. However, if a gradual sensor failure goes undetected, faulty data will be distributed into all of the navigation state and sensor bias estimates, so that they all tend toward agreement. Thus, while CF measurement residual tests can readily detect sudden hard failures, they may completeIy miss gradual soft failures. If the CF does use bad data from any sensor, its full solution becomes corrupted. When the faulty sensor is eventually detected, the only safe means of recovery is to reinitialize the CF solution and wait for it to reconverge. In contrast, the federated filter NR mode has a high degree of sensor fault tolerance. First, the effects of a (non-INS) sensor fault are localized to one LF. Since there is no feedback of the MF fused solution to the LFs, there is no possibility of LF-to-LF cross-contamination. Each LF haa good sensor FDI capability (except for soft failures) through measurement residual tests against all of the prior local information. Most important, the MF has excellent LF:LF FDI capability, since the effects of a sensor soft failure will be increasingly visible in one LF solution only, greatly increasing its probability of detection. When the faulty LF is identified (by abnormally large fusion residuals). a new MF solution can be generated immediately by combining the remaining good LF solutions (per the normal fusion process). The other federated filter modes are less attractive with respect to sensor fault tolerance. The FR mode provides weak LF:LF FDI capability in the MF, since the fusion-reset process causes all LF solutions to agree except for their
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`308 Navrgar/on Fall 1994 most recent sensor measurements. The ZR and RS modes are also weak in LF:LF FDI. since the effects of a sensor fault can accumulate in one LF only during the relatively short interval between fusion updates, before being checked against prior information in the MF solution (much like the CF case). Finally, none of these other modes support rapid fault recovery, since the accumulated good data has been mixed with the accumuiated bad data; full reinitialization (as for the CF) is required in each case. In summary, the NR mode stands out from the others as having superior sensor FDIR capabilities. It also has the simplest real-time control require- ments, since it requires no LF resets. Its only disadvantages are a slight degra- dation in estimation accuracy, and the need for all LF solutions to be combined by the MF at the same time (generally not a problem). The ZR and RS modes permit LF solutions to be processed independently by the MF, but are not attractive from a sensor FDIR viewpoint. The FR mode provides the best estima- tion accuracy, but exhibits disadvantages with regard to sensor FDIR, computa- tion and databus loads, and real-time control requirements. FEDERATED FILTER SIMULATION RESULTS The federated filter and its several information-sharing modes have been implemented in computer software to support performance simulation testing. A general-purpose federated filter package has been built in FORTRAN-77 for non-real-time simulation testing on VAX and PC (80x861 computers 1161. In addition, a general-purpose federated filter has been built in Ada, with asyn- chronous multitasking capabilities, for real-time testing in an avionics simula- tion testbed (171. The primary goals of the simulation teats reported in this section have been to ( 1) demonstrate the global optimality or near-optimality of federated filter configurations; (21 examine characteristics of MF and LF’ solutions relative to one another and to the equivalent
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`solution; (3) demonstrate improved capability of the federated filter to detect, isolate, and recover from sensor faults; and (41 demonstrate improved processing speed (reduced computation load per processor) of the federated filter as compared with the CF. To ensure fair comparisons, all filters employed the same sensor state models for their associated sensors. Performance Simulation Scenario The federated filter haa been tested in both non-real-time and near-real- time simulation environments. The non-real-time DKF
`contained FORTRAN filters and FORTRAN truth models, including a high-dynamic aircraft trajectory generator, a strapdown INS model, a bare-altimeter model, a GPS satellite/receiver model, a SAR model (with both landmark imaging and precision velocity updates), and a TAN model (radar altimeter plus synthetic terrain-map generator). The near-real-time
`contained Ada filters and FORTRAN truth models, including all those above except for the SAR model. The sensor truth models were generally of medium-high fidelity, containing significantly more error sources than do the corresponding filter models. Both environments permitted the filter navigation state estimates to be compared with the true states, to determine the estimation errors.
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`Vol. 31. No 3 Car/son. et al.. Federared Kalman Filter 309 In the slmulatlon runs. the tactical alrcraft trajectory consisted of a climb to altitude. a straight high-altitude cruise. a descent. and an extended low- altitude segment including several jinkmg maneuvers (zig-zags, and a large heading change. GPS pseudorange and pseudorange-rate 1 or delta-range) mea- surements were processed by the filter at a rate of 0.2 Hz. Some runs included two lengthy GPS data outages (to simulate jamming), separated by a brief period of tracking. SAR position and velocity measurements were processed at a comparatively slow rate of once every 300 s; the SAR was turned off near the end of the low-altitude segment, approaching the target. TAN radar aitimeter height measurements were processed during the entire low-altitude segment, at a relatively rapid rate of 1 Hz. Federated Filter Optimality, Results The results in this subsection were obtained with the non-real-time DKF Simulator [16). Figures 6 to 8 demonstrate the global optimality of the federated filter. Each figure shows the east position estimation error for t