This set of blogs will not be considered complete until at least seventy (or possibly a hundred) are available for visitors to download and/or print.  Each individual blog, with links to references (which in some cases can also be downloaded and printed from this site), summarizes a specific aspect from a chosen set of topics.  A smaller number of these “one-pagers” will address topics from my earlier, more fundamental, book Integrated Aircraft Navigation.  An additional few (very few) will deal with topics not covered in either of those two books.  An example of the latter publicizes some useful facets of the ultra-familiar classical low-pass filter which (believe it or not – after all these years) have remained obscure.

Over time, dozens more will be added from a wide span of topics (all firmly supported by experience as well as theory, ranging from elementary to advanced, in some cases relatively new and therefore largely unknown) that will include

  • Modern estimation in both block (weighted least squares) and sequential (Kalman filtering, with Battin’s derivation – much easier to follow than Kalman’s) form, with their interrelationship developed quite far, enabling “plant noise” levels to be prescribed in closed-form, also providing highly unusual insight into sequentially correlated measurement errors; chi-squared residuals; implications of optimality during transients; need for conservatism in modeling; sensitivity of matrix-vs-vector extrapolation (“do’s and don’ts”); application-dependence of commonality and uniqueness features; quantification of observability and effects of augmentation on it; duality among a wide scope of navigation modes; commonly overlooked duality between tracking and short-term inertial nav error propagation; when “correction-to-the adjustment” terms can and can’t be omitted; suboptimal (equal-eigenvalues) estimation with steady-state performance indistinguishable from optimal; all fully supported by theory and experience
  • Basic building-blocks for attitude expressions: superiority of quaternions and direction cosines over Euler angles, due to singularity (“gimbal lock” at 90-deg for x-y-z sequence) and at 0-deg for z-x-z sequences used for orbits
  • GPS issues related to the top-priority goal of robustness: beyond elementary (4-state and 8-state) formulations; duality of pseudorange and phase ambiguity; exploitation of modern processing capabilities in GPS/GNSS receivers; carrier phase as integrated doppler vs frequency data; 1-sec sequential phase changes (much easier to mix across constellations, negligible sequential changes in IONO/TROPO propagation, ambiguity resolution not needed, instant reacquisition, no-mask angle needed); streaming velocity for dead reckoning with segmentation of position fixes; differential operation – differencing across satellites, receivers, and time; handling correlations from differencing; orthogonalization for simple QR factorization; measurement relocation in time and lever-arm adjustment; E(Extended)RAIM;  D(Differential)RAIM; necessity of weighting in single-measurement RAIM with pseudoranges and carrier phases, concurrently; sample flight test results showing state-of-the-art accuracies in dynamics (e.g., cm/sec RMS velocity error and tenths-mrad leveling) with a low-cost IMU; revisit of the same flight segment, achieving decimeter/sec RMS velocity error without any IMU
  • Tracking (with subdivision into over a dozen topics including a littoral environment operation with hundreds of ships present; orbit determination; usage of Lambert’s laws; surface-to-air (subdivided into ground-to-air and tracking from ships), air-to-surface and surface-to-surface (again with the same subdivision),  air-to-air; reentry vehicles; usage of stable coordinate frames; linearity in both dynamics and measurements; Mode-S squitters for mutual surveillance and collision avoidance in crowded airspace; multiple track output usage (placement of gates, antenna steering, file maintenance); crucial importance of transmitting measurements rather than coordinates (publication #66); extension to noncooperative objects, critical distinction (often blurred) between errors in tracking and stabilization; sucessfully accomplished concurrent track of multiple objects with electronically steered beams; bistatic and multistatic operation; postprocessing to form familiar parameters from estimator outputs; short-range projectiles over “flat-earth” – plus many more)
  • Processing of inertial data – incrementing of position, velocity, attitude; straightforward state-of-the-art algorithms for complete metamorphosis from raw gyro and accelerometer samples into final 3-D position, velocity, and attitude; motion-sensitive inertial instrument errors; coning; sculling; critical distinction between misalignment (imperfect mechanical mounting) vs misorientation; adaptive accommodation of gyro scale factor and misalignment errors; instability of unaided vertical channel; azimuth pseudomeasurement; near-universal misconceptions connected to free-inertial coast
  • Support functions (transfer alignment; SAR motion compensation; stabilization of images; sensor control mechanizations; synchronization; determination of retention probability)
  • Vision-for-the-future with maximum situation awareness for all cooperating participants in a scenario; critical role of interfaces (implications of singularities, RAIM, Differential GPS, etc.), software modularity, reuse, coordination).  Full validation in GNSS Aided Navigation and Tracking.

For steady-state a suboptimal estimator can be designed with near-optimal performance.  A Kalman filter, though, optimizes accuracy during transients too – provided that the model is known and linear.  Immediately we’ll  invoke the “almost/most/if” qualification: an extended Kalman filter (EKF) is almost optimum, throughout most of its operation, if the model is almost linear and modeling errors are held in check via process noise.  Rather than presenting justification here I’ll cite a set of “do’s-and-dont’s” – validated by long experience – from Section 2.9 of GNSS Aided Navigation and Tracking, with GPS/INS flight test data included.  Eqs. (9.9)-(9.19) of that same reference provides simple design equations for alpha-beta and alpha-beta-gamma trackers that have consistently produced success in operation.

First we’ll note that suboptimal is not equated to “constant gain” – if for no other reason, the time between measurements will vary in many systems.  That’s quite easily accommodated by the alpha-beta[-gamma] designs just mentioned.  There are additional reasons, though, that can be illustrated by addressing a taxing situation for initiating a radar track file in close range air-to-air encounters between two fighter jets.  The target’s (i.e., tracked object’s) cross-range velocity at lock-on time is unknown.  It could be 800 ft/sec, for example, in which case the tracker’s initial velocity error has at least that 800-ft/sec component.  With any additional unknown component of along-range velocity the target may have at that instant (doppler, if observed, might not yet be trusted to represent range rate dynamics), the tracker’s initial velocity error will then exceed 800 ft/sec.  The transient at acquisition could easily be further complicated by acceleration.  Anyone familiar with servo pull-in dynamics will immediately see how the transient can reach significantly beyond the initial error – very fast – in multiple directions (e.g., East/West and North/South).  Since we’re not at all comfortable with velocity errors on the order of 1000 ft/sec, the task is to wash that out ASAP.

A Kalman filter having accurate knowledge of the initial [P] matrix would breeze through this challenge.  An excellent example of its role is provided by this transient behavior.  Knowledge of that matrix is tantamount to knowing whether – at initiation time – the tracker’s North velocity error is positively or negatively correlated (i.e., likely to have the same or opposite sign) as its North position error, likewise for East velocity error with same-vs-opposite sign of North acceleration error, and likewise … all combinations – not only the signs but also the RMS amounts.  Of course that’s completely unrealistic.  So now what?

Suboptimal gain sets phased in at the right times can handle this.  For a simple illustration, let a 3-dimensional tracker, divided into three separable 3-state (position/speed/acceleration) single-direction channels, have a 20-Hz update rate.  If the first few updates have gains of 0.5 or more for position only, even a huge position error can be quickly brought down near sensor-error levels before accompanying errors in dynamics have much time to propagate. Then after that many (“K1”) position corrections, a position-&-speed update phase can be initiated, using the alpha-beta tracker gains related as shown in Eq. (9.12) of the reference cited above.  Duration of that phase is devised to last only as long as necessary to reduce speed error to design levels (which will be proportional to measurement error divided by that duration).  After the total number of corrections has reached that intended design value (“K2”), the alpha-beta-gamma phase can start with gains related according to Eqs. (9.18-19) of that same reference.  That phase continues until the total corrections count reaches “K3” at which time acceleration error is reduced to an amount inversely proportional to the square of (K3-K2).  Gains thereafter may conform to Kalman filter weighting.

This example is not intended to advocate substituting suboptimal for optimal designs just anywhere.  Separation of 3-dimensional trackers into 3-state single-direction channels is often permissible (and sometimes even highly advisable), but – as shown in the cited reference – sometimes inappropriate.  Where it is permitted, use it; solving the unknown-P-zero problem is especially important in applications of this type.  A word to the wise: Do not (repeat: do not) make the update counts K1,K2,etc. programmable.  If you do, someone unfamiliar with the reasoning above will experiment, allowing resets to values producing very prolonged back-and-forth transfer of errors among position and dynamics (one gets worse as another improves; then vice-versa).  When that spectacle is seen by nontechnical administrators, your image in their minds will forever be indelibly painted with that long drawn-out transient veering back and forth between plus and minus extreme levels.

Another slice-of-advice: Even if inputs are extremely erratic, your tracker must maintain high responsiveness (for sensor sightline stabilization at short range and for range[/doppler] gate placement at any range) – but – the outside world doesn’t have to witness the results of that “hitchy-hatchy” from wildly erratic inputs.  So: don’t change the tracker but do low-pass filter what goes outside.  If “hiding the system’s warts” thereby produces criticism, the justification is: the filtered output, even with the resulting delay (and possibly an accompanying distortion), is easier to interpret.

As an alternative to TCAS in air and ASDE on ground, all facets of collision avoidance (see 9-minute video) can be supplanted with vast improvement:

  • INTEGRATION – one system for both 2-D (runway incursions) and 3-D (in-air)
  • AUTONOMY – no ground station corrections required
  • COMMUNICATION – interrogation/response replaced by ModeS squitter operation
  • COORDINATION – coordinated squitter scheduling eliminates garble
  • TRACKING – all tracks maintained with GPS pseudoranges in data packets
  • DYNAMICS – tracks provide optimally estimated velocity as well as position
  • TIMELINESS – history of dynamics with position counteracts latency
  • MULTITARGET HANDLING – every participant can track every other participant
  • CONTROL – collisions avoided by deceleration rather than climb/dive

My previous investigations (publication #61 and #66, combined with publication #85 as well as Chapter 9 of GNSS Aided Navigation and Tracking) provided in-depth analyses for all but the last of these items.  The control aspect of the problem is addressed here.  This introductory discussion involves only two participants, initially on a coaltitude collision course.  One (the “intruder”) continues with his path unchanged (so that the method could remain applicable for encounters between a participant and a non-participant tracked by radar or optical sensors).  The other (“evader”) decelerates to change projected miss distance to a chosen design value.  This simplest-of-all scenarios can readily be extended to encounters at different altitudes and, by reapplying the method to all users wherever projected miss distance falls below a designated threshold, to multiple-participant cases.

Considered here are simple scenarios with aircraft initially on a collision course at angles from 30 to 130 degrees between their velocity vectors.  Those limits can of course be changed but, the closer the paths are to collinear the more deceleration is required to prevent a collision (in the limit – direct head-on – no amount of deceleration can suffice; turns are required instead).  Turns can be addressed in the future; here we briefly discuss the 30-to-130 degree span.

In Coordinates Magazine and again as applied to UAVs it was shown that, over a wide combination of intruder speed, evader speed, and angles (within the 30-to-130 degree span just noted), the required amount of evader speed reduction is modest.  A linearized approximation can be derived intuitively from scenario parameter values.  The speeds and the angle determine a closing range rate, while closest approach time is near the initial time-to-go (ratio of initial distance to closing rate) though deceleration produces a difference.  The projection of evader speed reduction along the relative velocity vector direction has approximately that much time to build up 500 to 1000 meters of accumulated horizontal separation.  Initiation of the speed change that far in advance allows the dynamics to be gradual, in marked contrast to the sudden TCAS maneuver.  To avoid a wake problem, the evader’s aim point can be directed to a few hundred feet above the original coaltitude.  Continuous tracking of the intruder allows the evader to perform repetitive trim adjustments.

A program with results illustrating this scheme will not fit on a one-page summary, but it comes as no surprise that, with accurate tracks established well in advance (a minute or two prior to closest approach time), a modest deceleration can successfully avert collisions.

A widespread missed opportunity began many years ago and continues to this day.  It is still widely believed that significant nonlinearity is inescapable in a tracker – whether for dynamics with spherical coordinates or for measurements with Cartesian coordinates.  The first half of that is definitely true for dynamics using the classical (range/elevation/azimuth) frame in air-to-air encounters at close range; we’ll take that issue first here.

One question that could be raised immediately might sound something line this: Since expressing dynamics in a range/elevation/azimuth frame causes so many problems (not only linearity but many more, to be discussed shortly), why even consider that as a candidate approach?  Old-timers will readily recall that analog radars provided no other choice.  They had range trackers, while angle tracking was done separately in outer loops with bandwidths narrow in comparison to inner stabilization loops maintained via antenna-mounted gyros.  Couldn’t that still be done after digitization?  Yes – in fact, it was done.  In the early 1970s I reviewed a paper for IEEE showing just that.  I gave it the green light because it was correct.  For my own design, though, I used a Cartesian frame (publication #24, 26, 28, 30, 32, 36, 60, 61, 66, and 69).  No contest.

The widely recognized nonlinear degradation of a range/elevation/azimuth representation for dynamics only begins to explain why that choice is fraught with problems.  For one issue, that loop-within-a-loop situation imposed a demand on inner stabilization bandwidth; if too low it would compromise the overall operation’s stability.   Alternatively that requirement could be viewed as a constraint on track loop responsiveness.  Another very serious issue was the interfacing requirements; gyro and accelerometer data had to be used in that track configuration (which added another burden of time-tagging plus another degradation from time-tag imperfections).  Still another limitation was inability to represent multiple track files; they aren’t all in the same direction and their Line-of-Sight rates are anything but uniform.  Finally, if handoff of a track file became necessary (e.g., from a forward-looking to a side-looking sensor during fly-by), I wouldn’t want to inherit that task with (range/elevation/azimuth) dynamics at close but wildly changing distances and geometries.

The item just noted brings up a favorite example raised by my coauthor of publication #24 and 32: Two aircraft approach each other on parallel tracks separated by 1000 ft.  With each having constant 1000 ft/sec groundspeed, the centripetal acceleration (see Ex. 8-12c on page 295 of Integrated Aircraft Navigation) at the instant of closest approach (sidelong position) is


or about 125 g.  The example is extreme but, even with more moderate encounters no range track loop can perform adequately in both responsiveness and noise rejection.  The second derivative of scalar distance is very high in close range scenarios.

Now a comparison to the Cartesian frame can begin.  Instantly the “problem” just shown disappears.  There isn’t any acceleration at all; all Cartesian velocity components are constant.  Then handoff is easy too.  We’re off to an excellent start.  Next: interfacing – that’s also easy – never mind the gyros and accelerometers, just adopt the INS nav frame as the Cartesian reference for tracking and use the nav computer output.  Eq. (2.13) of GNSS Aided Navigation and Tracking shows that it’s just as easy to track from a maneuvering fighter jet as it is while at rest.  Stabilization loop?  It isn’t inside any other loop; let it have whatever bandwidth it has.  Separate the stabilization offsets from tracker inputs as illustrated in Figure (9.2) of GNSS Aided Navigation and Tracking.  That same “dot-off-the-crosshairs” figure, with its accompanying analysis (Section 9.2.2), readily reduces to negligible levels any measurement nonlinearities as well.  Multiple targets?  Again, easy – just maintain one track file for each object being tracked.

That was fast, wasn’t it?

Low pass filter

Decisions are made, understandably, on the basis of a decision-maker’s beliefs.  In general, the better the knowledge base, the better the anticipated outcome.  Inevitably there are times when choices must be made from incomplete information.  Even that can still produce success, but the likelihood of a favorable outcome depends on recognition of those uncertainties.  Likelihood of an unfavorable outcome, then, increases when those information gaps go unrecognized.  That is, when we are unaware of the fact that we don’t know (“don’t-know-squared”).  To make that case for this site I’ll use an example from an area outside of navigation and tracking:

One field that has received thorough investigation is the study of a low-pass filter.  Users of those commonly believe that they know all that is needed to make the wisest design selection.  Quite often they know much – but not everything that would be useful to them.  It is not unusual for a maximally-flat (Butterworth) attenuation characteristic to be chosen while assuming that nothing much can be done about the accompanying nonlinear phase; latency often precludes usage of phase equalizers.  It is known – but not widely known – that a trade-off has been available for decades.  A near-linear phase characteristic over the passband can be realized if some of the attenuation requirements can be relaxed.  Full details can be found in

Handbook of Filter Synthesis by Anatol I. Zverev
ISBN 10: 0471986801 / 0-471-98680-1     ISBN 13: 9780471986805                                                           and
Filtering in the Time and Frequency Domains
by Herman J. Blinchikoff and Anatol I. Zverev
ISBN-10: 1884932177     ISBN-13: 978-1884932175

Already I’ve said as much as I intend to say here about low-pass filters.  To go this far without misinterpreting some points I found it necessary to consult a coauthor (Blinchikoff) of the second reference just cited.  The rest of the blogs on this site involve navigation and tracking – where avoidance of don’t-know-squared is still very much an issue.  Examples from those areas won’t all be obvious (e.g., a pilot believing his broken altimeter), but there is much to be gained from “looking under the hood” and uncovering missed opportunities.  If we’re willing to pursue that, let me assure you that vast improvements in performance are available.