Decentralized Kalman filtering theory & its applications
(one technologically logical approach to sensor fusion; another is to use merely one Kalman filter, depending on the situation)
Consulting for engineering design, analysis, and performance evaluation of alternative estimation algorithms (including decentralized versions for sensor fusion, if decentralized KF are really needed). Sometimes use of a centralized KF suffices.
An Historical Account of our experience therein (more detail below):
Sharing some of our Insights:
DECENTRALIZED OR DISTRIBUTED EXTENDED KALMAN-LIKE FILTERING AS A BASIS FOR SENSOR FUSION IN NEXT GENERATION EARLY WARNING RADAR
Historical Precedents and Benefits to be Accrued from using Decentralized Kalman Filters
perspective conveyed here on the utility of Decentralized Kalman filtering is
the direct result of our (TeK Associatesา)
research and navigation applications experience in this area for over three
decades -, ,
, as cited below. We
later became familiar with the models and constraints of radar target tracking
for strategic Reentry Vehicle (RV) targets ,
We cite the important building block contributions of others (as done in each of
our extensive bibliographies appearing in our references, especially in ,
We do not overlook the use of Kalman filters in image restoration ,
,  nor the use
of more advanced random process statistical techniques in the image processing
arena (e.g., ).
A variation on the above approach uses decentralized Kalman filters -, one dedicated to its own particular navaid sensor, then the results are incorporated into a final single result. The motivation for this particular architecture is three-fold:
Decentralized EKF or Kalman Filters
To avoid single point vulnerabilities, it is prudently recommended (in -) that more than just one collating filter be used (each having the capability of putting together the results of all of the participating sub-system filters). This is not an Interactive Multiple Model (IMM) formulation. However, IMM could constitute any of the subsystem filters.
The motivation for decentralized Kalman filtering
in radar or optical target tracking is different from that arising in
Navigation. Two (or more) synchronized decentralized estimates of the same
target state vector, as viewed from different sensors with, perhaps, different
perspective views, different segments of the electromagnetic spectrum utilized
(e.g., as with conventional radar and laser radar) to exploit inherent target
characteristics, different independent noise contamination intensities (that can
be averaged out) is the primary motivation for such a pursuit (along with the
fact that the result of several separate Multi-Target Tracking (MTT) decisions
no longer need to be reconciled later if the MTT algorithm only resides with the
higher level collating filter). However, the initially unaltered covariances of
estimation error associated with each separate radar or optical sensors
view would not necessarily exhibit a partial-ordering in a completely general
sensor fusion application but, instead, likely be skewed off from each other in
tilt and overall size. However, use of collocated conventional radar along with
laser radar may yield one target ellipsoid contained entirely within another due
to the greater resolution and smaller azimuth error incurred for laser optics.
Sometimes the term sensor
fusion is used in the
Target Tracking context.
It is suspected that certain U.S. strategic radars currently operate autonomously in a stand-alone manner but take direction and report to a Master Center. At some time in the future, it may be desirable to provide a joint solution to target tracking and at that time decentralized filtering will likely provide the theoretical basis for accomplishing this. Such an endeavor should be especially helpful for situations involving multi-target tracking by helping to eliminate some of the ambiguities associated with crossing targets (as seen from just one radar sensor) by simultaneously providing more than one geographic perspective and a natural solution since targets would then be seen from more than one aspect angle (without needing to incur the delay of waiting for more time to elapse to see the same targets later in their trajectory, as was previously the case before the individual targets could be resolved).
An example of a radar for tracking Strategic Targets (e.g., Reentry Vehicles)
The use of decentralized filters would be a rational basis for combining both optical and radar target tracking (or for combining any other sensors). [Missile depicted is a U.S. Peacekeeper.]
Ref.  is an
excellent overview, in depth numerical ranking, and clear interpretation of all
of the various approaches to sensor fusion that have occurred in the target
tracking literature over the last two decades and culminates in an algorithmic
improvement to Covariance Intersection (CI) that 
attributes to the pioneering work of the late Fred C. Schweppes unknown but
bounded approach , where, instead of embracing
the assumption of Gaussian noises being present, Schweppe uses an entirely
deterministic approach of circumscribing the set of potential outcomes arising
at each discrete-time step of a linear systems output within
an ellipsoid. Schweppes approach, although creative, was
notoriously conservative and was never used in applications nor was it admired
as frequently as what is reported on in , as
more representative of Schweppes genius. (Although
overlooked in , parallel developments were
occurring in navigation, as similar techniques were being investigated, e.g.,
some Federated Kalman Filter approaches were
criticized by Larry Levy (JHU/APL) ; but he was
refuted in  regarding Jason Speyers exact 1979 version of decentralized
estimation , where just the estimation portion
was extricated by Chang  (also see
In , it was also pointed out that the crux of
several more recent approaches to decentralized filtering harkens back to the
structural equations presented in 
they neglected to reference , as a published
precedent). Ref.  demonstrates that the
approximate algorithms of the CI approach should only be used with extreme
caution since uncertainty increases as more measurements are used (and fused) as
a counterintuitive, extremely unsettling result stated and proved simply and
convincingly in [10, following Eq. 22]. In a
standard Kalman filter, the covariance of estimation error, P(k), usually
decreases (or at least should not increase) as measurements from more
independent sensors are used at that discrete time step k
(where multiplication by the step size delta is implied by convention). The same is also the
case with an Extended Kalman Filter (EKF) or even with an Iterated Extended
Kalman Filter (IEKF), as used in radar target tracking for nonlinear system
models and/or measurement models, as arise in reentry vehicle (RV) target
Certain attempts to handle non-zero process noise
in decentralized filtering have their own drawbacks .
The four different decentralized filtering architectures for possible sensor
fusion developed by the SDI Panel 10+ years ago (by Dr. Oliver Drummond and also
presented at his 1997 short course at SPIE conference in Orlando, FL) were all predicated (as he
rigorously acknowledged) on there being zero process noise present, otherwise
these structures are useless or detrimental in situations where process noise is
present (so they are inappropriate to use for indoatmospheric tracking wherethe
process noise intensity matrix Q is definitely not zero). Ref. 
discusses a decentralized version of filter fusion where the presence of
non-zero process noise covariance was not well accommodated because only a small
fraction of Q was apportioned to each participating filter (thus causing each
participating filter to perceive the system as being more benign than actual and
as a consequence was not tuned to the true underlying situation) and therefore
likely to fail in its objective of adequately tracking for each, which then
adversely affects the collated whole (since each of N subset filters fails to
track a single target system that is much noisier than it expects) the aggregate
cannot be much better and the computer burden is N times larger than would be
the case if just one correctly modeled single filter were used [thus this
particular approach actually defies the fundamental reason for seeking a
A Recent Approach that Almost Scores a Goal
appears to be much further along in usefulness and creativity in this area but
evidently still misses the mark regarding practical realism, as explained next.
Interactive Multiple Model (IMM) Filtering approach is implemented in
(and elsewhere) only for targets described by linear systems, even under
maneuvers. Reentry Vehicles are known to be nonlinear in both the dynamics and
in the algebraic measurement model. Ref. 
considers Particle Filtering (PF) and Probabilistic Data Association (PDA)
approaches as well. As with almost all PF examples to date, the state space
target dynamics model used within the calculations is planar and consists of
only 4 states. The complexity of PF implementation as a CPU computational burden
increases drastically as the state dimension increases. Practical Early Warning
Radar target models must consist of at least six states (3 position, 3
velocity), and must be nonlinear to realistically reflect the action of an
inverse square gravity and the oblateness of the earth (effects that cannot be
ignored in a realistic EWR application). Ref. 
not reflect Bit Error Rates (BER) nor possible requests for retransmission within
the communications channels as a consequence of BER (an activity that can cause
measurements to the filter to get out of sequence but can be routinely handled
appropriately as long as
each measurement is properly time-tagged or time-stamped). Ref. 
does not consider having adequate processor capability at each sensor site, as
would be the case for EWR with two way communication links present to Colorado
Cheyenne Mountain, and which is more compatible with methodology of
decentralized filtering, reported in References ,
Tricks that Result in Greater Estimation Accuracy in Simulations but fail in
Several open literature simulation evaluations of alternative estimation approaches for strategic target tracking have in fact been previously reported (e.g., ) but all with accuracy results obtained under a somewhat artificial scenario of use (viz., invoking only 4 tracking filter states and assuming only planar motion despite target object not being under the influence of central forces exclusively but projectile treated as if it were and not only that but the observing radar is only at the launch point and in the same plane as the target trajectory) thus leaving prior accuracy quantifications somewhat questionable at best as they relate to actual missile defense since this prior overly benign scenario lacks realism. We appreciate novelty and encourage creativity but we also desire that all major test conditions for algorithm evaluation be realistic and representative of the actual application scenario, as enforced and explained further within our study.
Further elucidating our perception regarding lack of realism by several recent evaluations (e.g., ) using reentry model being completely planar, central forces in 3-D give rise to trajectories that are confined entirely to be within a specific plane (known, historically, as the osculating plane). From a celestial mechanics course, one learns about central force motions and associated properties. Such studies reveal that for a central force field (like inverse squared gravity) the following cross-product , where r is the position vector from the origin of a coordinate system erected at the center of gravity of the earth as focus to the location of the projectile, defines the normal to the plane in which the motion of the projectile is confined. In actual radar applications, the ideal behavior is not precisely obtainable because of the range-Doppler ambiguity encountered (as associated with use of practical radar measurements), so there are slight errors present between measured range and its associated range-rate to some degree which, further, corrupts the accuracy of the effective estimates and that degrades the estimate of the normal to the osculating plane to which the projectile is confined, to the degree of departure from the ideal as indicated from the calculation of , where the individual contributing errors are and . The effect of regression of nodes and rotation of apsides (defined above) aggravates the problem of instantaneously estimating the correct plane of projectile confinement even more since, instead of being merely constant and fixed, the osculating plane now moves due to the oblateness of the earth. While several recent open literature simulations of RV target tracking treat it as a problem that is entirely planar, the real world problem in Missile Defense is to actually figure out what plane the trajectory is situated in and how it is posed. By ignoring these real world effects, such simulations, as a consequence reap greater accuracy than is likely in practice, where four other out-of-plane errors arise (that are ignored). Results were better in these simulations than can be obtained in practice since they assumed the plane of motion was already known precisely (and treat the component of out-of-plane errors as being nonexistent and zero in the tally of total error incurred). Part of the real problem should be to deduce in what plane the target is traveling. In some cases, such as in benign pre-planned test shots from Vandenberg AFB in CA toward the Kwajalein Atoll [where the Tradex radar used for tracking these RV test shots is located there at KREMS (along with Altair, Alcor, and MMW). The L-band Tradex radar has MTT for up to 63 simultaneous targets appearing within the same mechanically scanned 0.61o, 6 dB beamwidth pencil-beam of the 25.6 meter diameter antenna, but has a 600 meter blind zone behind the primary target cluster grouping] in the Marshall Islands using our own cooperative target Reentry Vehicles, one may know the launch point precisely as well as the aim point and our own missiles may have the reentry angle completely known to us for tracking purposes (but for actual missile defense, the defenders radar usually only has a view of the target during a portion of midcourse through reentry and we know that anticipated launch points can be varied via use of wheeled or rail vehicles or via submerged submarines and that a sophisticated enemy can also suddenly vary several parameters relating to reentry drag during end-game). Some recent simulations assume that the observing stationary ground-based radar is at the launch point (and in the plane of the target trajectory, as a considerable advantage of having almost perfect initial conditions for the target tracking algorithms almost from the start, which is an unrealistic assumption for Missile Defense [where, typically, initial conditions for each target must be deduced from detection and confirmation waveforms, available pulse patterns, or from other supplementary information and EWR uses phased arrays and electronic scanning]). More realistic approaches usually have more of a handicap in this aspect and must approximate the initial conditions used in the Extended Kalman filter for each candidate target.
Moreover, when atmospheric drag enters into the picture, the problem is no longer an exclusively central force problem (otherwise guaranteed to be planar). Any tilt of the reentry body forces the trajectory out of the expected osculating plane, which recent simulation studies treat as being perfectly known but did not list this among the assumptions, as were missed in .
awarded a contract, TeK
Associates can Illustrate the Efficacy of Decentralized EKFs for UEWR using
We will implement the existing RVCC EKF  (with parameters as in ) and view target RVs simultaneously from two UEWR sites and combine the EKF estimates from these two sites to demonstrate the benefit of this approach for resolving two crossing targets early on in their trajectory.
Viewing RV Target Simultaneously from Two Radar and Combining the Tracking Results
General Principles Associated with Sensor Fusion and Kalman Filters for Networked Sensor Arrays and Virtual Sensors: straight talk and common sense
Despite the evolution and abstraction
terminology, nomenclature, or current parlance in vogue in sensor fusion and
net-centric considerations, as identified in ,
the situations that must be properly handled and the underlying principles
involved are fairly simple and straight forward, as discussed below.
Terrain map-matching (TERCOM) historically involved using 3 or 4 Janus configured simultaneous active radar views aboard a single platform in order to correlate actual current terrain measurements with an onboard master map (quantized to a certain practical resolution) to obtain a navaid position fix in order to update and correct the primary onboard Inertial Navigation System present within many terrestrial airborne applications (e.g., in cruise missiles, Unmanned Air Vehicles, aircraft).
In the 1980s, and 1990s
for navigation applications, the use of decentralized Kalman filters were
proposed (and used) to handle situations where some navaid sensors were failed
or unavailable because of environmental concerns (sunspot activity, enemy
jamming), onboard electronic failures, or during owncraft maneuvers that incur
wing shading of antennas, or from battle damage. Such approaches were pursued to
obtain margins of safety via redundancy. There was only a single state variable
model of primary interest throughout: position and velocity of owncraft,
presented as aided
INS readouts. All that needs to be conveyed to a central collator (Master Filter
onboard the parent platform) from each participating sensor sub-filter is the
following 2n+1 entity: (state estimate, computed inverse covariance times the
state estimate, time of fix), where n is the state size of the filter model in
use (usually 6 states). Since the databuses stretch for relatively short
distances aboard aircraft from where the measurement data was received, the
delay incurred is small and for a single master filter, the out of sequence
participant reports can still be handled to give the best current state estimate
without any reservations (other than those caused by certain approximations
perhaps being invoked merely to lessen the processing burden). The point is that
no uncalibrated simplifying approximations need be invoked and the result is
that the current best estimate of owncraft position and velocity can be
calculated in a straight forward way even if navaid sensor measurements are all
taken at different times (as long as these times are known and conveyed). In
ship-borne applications, the adverse effect of sensor measurement data
senescence is worse because data buses are usually much longer and incur greater
We have an abiding interest in Bayesian Nets and appreciate their potential both in defense applications (click for PowerPoint presentation or click here for a pdf version) and in commercial applications.
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 Yozevitch, R., Ben Moshe, B., A Robust Shadow Matching Algorithm for GNSS Positioning, Navigation: Journal of the Institute of Navigation (ION), Vol. 66, No. 2, pp. 95-109, Summer 2015 [Notice that they did not say that their PF was real-time] and some of their pertinent references: (1) Crow, F. C., Shadow Algorithms for Computer Graphics, ACM SIGGRAPH Computer Graphics, Vol. 11, No. 2, pp. 242-248, 1977; (2) Bourdeau, A., Sahmoudi, M., and Tourneret, J. Y., Constructive Use of GNSS NLOS-MUltipath: Augmenting the Navigation Kalman Filter with a 3D Model of the Environment, 15th International Conference on Information Fusion (FUSION), pp. 2271-2276, IEEE, 2012; (3) Thrun, S., Burgard, W., and Fox, D., Probabilistic Robotics, MIT Press, 2005; (4) Muralidharan, K. Khan, A. J., Misra, A., Balan, R. K., and Agarwal, S., Barametric Phone Sensors: More Hype than Hope!, Proceedings of the 15th Workshop on Mobile Computing Systems and Applications, ACM, 2014, 12; (5) DeBerg, M., Van Kreveld, M., Overmars, M., and Schwarzkopf, O. C., Computational Geometry, Springer, NY, 2000.
 Uwe D.
Hanbeck, Recursive Nonlinear Set-Theoretic Estimation Based on
Pseudo-Ellipsoids, Proceedings of the IEEE Conference on Multisensor Fusion and Integration for Intelligent
Systems, pp. 159164 (MFI
Fuqiang You, Hualu Zhang, Fuli Wang, A new set-membership estimation method based on zonotopes and
Transactions of the Institute of Measurement and Control, Vol. 40, issue 7, pp. 2091-2099, Article first published online:
27 July 2016; Issue published: 1 April 2018