Once again I am privileged to work with Ohio University Prof. Frank vanGraas, in presenting tutorial sessions at the Institute of Navigation’s GNSS-17 conference. For 2017, as in several consecutive previous years, two sessions will cover integrated navigation with Kalman filtering.  Descriptions of the part 1 session and part 2 session are now available online.

By way of background: The first session is introductory.  Each attendee will be given a book with a development aimed at those learning inertial navigation and/or Kalman filtering for the first time.  Prior to the course, my free-to-members online tutorial is recommended.  Also my three-part matrix tutorial video will be made freely available to attendees.

Prof. vanGraas sponsored, and provided the flight data that enabled, the successful validation of 1-cm/sec RMS velocity vector accuracy obtained from 1-second sequential changes in carrier phase.  Those results for almost an hour in air are provided, with the algorithms used to obtain them, in a more recent book that is given to those attending the second session. attending the second session.



A video completed recently provides just enough matrix theory needed for Kalman filtering. It’s available for  (1) purchase or 72-hour rent at low cost or (2) free to those attending courses I teach in 2014 or after (because the short durations don’t allow time to cover it). The one-hour presentation is divided into three sections. Each section has a preview, freely viewable.



The first section, with almost NO math, begins by explaining why matrices are needed — and then immediately emphasizes that MATH ALONE IS NOT ENOUGH, To drive home that point, a dramatic illustration was chosen. Complex motions of a satellite, though represented in a MATHEMATICALLY correct way, were not fully understood by its designers nor by the first team of analysts contracted to characterize it. From those motions, shown with amplitudes enlarged (e.g., doubled or possibly more) for easy visualization, it becomes clear why insight is every bit as important as the math.


EmailMarketingPicFor some viewers the importance of insight alone will be of sufficient interest with no need for the latter two sections. Others, particularly novices aspiring to be designers, will find the math presentation extremely helpful. Straight to the point for each step where matrices are applied, it is just the type of information I was earnestly seeking years ago, whole “pulling teeth” to extract clarification of ONLY NECESSARY theory without OVER simplification.


The presentation supplies matrix theory prerequisites that will assist aspiring designers in formulating linear(ized) estimation algorithms in block (weighted least squares) or sequential (recursive Kalman/EKF) form. Familiar matrix types (e.g., orthogonal, symmetric), their properties, how they are used — and why they are useful — with interpretation of physical examples, enable important operations both powerful and versatile. An enormous variety of applications involving systems of any order can be solved in terms of familiar expressions we saw as teenagers in college.


Useful for either introduction or review, there is no better way to summarize this material than to repeat one word that matters beyond all else — INSIGHT.

 A comment challenged my video .  I’m glad it included an acknowledgment that some points might have been missed. To be frank that happened a bunch; bear with me while I explain. First, there’s the accuracy issue; doppler &/or deltarange info provided from many receivers is far less accurate than carrier phase (sometimes due to cutting corners in implementation — recall that carrier phase, as the integral of doppler, will be smoother if processing is done carefully). Next, preference for 20-msec intervals will backfire badly. If phase noise at L-band gives a respectable 7mm = 0.7cm, doppler velocity error [(current phase) – (previous phase)] / 1 sec is (1.414) (0.7) = 1 cm/sec RMS for a 1-sec sequential differencing interval.  Now use 20 msec: FIFTY times as much doppler error! Alternatively if division is implicit instead of overt, degradation is more complicated: sequential phase differences are highly correlated (with a correlation coefficient of -1/2, to be precise). That’s because the difference (current phase) – (previous phase) and the difference (next phase) – (current phase) both contain the common value of current phase. In a modern estimation algorithm, observations with sequentially correlated errors are far more difficult to process optimally.  That topic is a very deep one; Section 5.6 and Addendum 5.B of my 2007 book address it thoroughly. I’m not expecting everyone to go through all that but, to offer fortification for its credibility, let me cite a few items:

* agreement from other designers who abandoned efforts to use short intervals
* table near the bottom of a page on this site.

* phase residual plots from Chapter 8 of my 2007 book.

The latter two, it is recalled, came from flight test for an extended duration (until flight recorder was full), under severe test aircraft (DC-3) vibration.

For doppler updating from sources other than satnav, my point is stronger still. Doppler from radar (which lacks the advantage of passive operation) won’t get velocity error much below a meter/sec — and even that is an improvement over unaided inertial nav (we won’t see INS velocity specs expressed in cm/sec within our lifetime).

Additional advantages of what the video offers include (1) no requirement for a mask angle (2) GNSS interoperability, and (3) robustness. A brief explanation:

(1) Virtually the whole world discards all measurements from low-elevation satellites because of propagation errors. But ionospheric and tropospheric effects change very little over a second; 1-sec phase differences are great for velocity information. Furthermore they offer a major geometry advantage while occurrence of multipath would stick out like a sore thumb, easily edited out.
(2) 1-sec differences from various constellations are much easier to mix than the phases themselves. 
(3) For receivers exploiting FFT capability  even short fragments of data, not sufficiently continuous for conventional mechanizations (track loops), are made available for discrete updates.
The whole “big picture” is a major improvement is robust operation 

The challenger isn’t the only one who missed these points; much of our industry, in fact, is missing the boat in crucial areas. Again I understand skepticism, but consider the “conventional wisdom” regarding ADSB: Velocity errors expressed in meters per second — you can hear speculative values as high as ten. GRADE SCHOOL ARITHMETIC shows how scary that is; collision avoidance extrapolates ahead. Consider the vast error volume resulting from doing that 90 seconds ahead of closest approach time with several meters per second of velocity error. So — rely on see-and-avoid? There are beaucoup videos that show how futile that is (and many more videos that show how often near misses occur — in addition there are about a thousand runway incursions each year). That justifies the effort for dramatic reduction of errors in tracking dynamics — to cm/sec relative velocity accuracy.

It’s perfectly logical for people to question my claims if they seem too good to be true. All I ask is follow through, with visits to URLs cited here.

Avionics Commonality

A LinkedIn discussion centered on the Future Airborne Capability Environment (FACE) standard contained an important observation concerning certification.  Granted — requirements for validation, with acceptance by governing agencies, definitely are essential for safety. What follows here is advocacy for a proposed way to realize the common avionics benefits offered by FACE while retaining (and in fact, improving) the process of certification. Reasoning is based on three major items:

* CHANGE. In many respects this has necessitated improved standards. 

* HISTORY. Spectacular failures in what we have now are widely documented.

* COST. The status quo is (and, for a long time, has been) unaffordable.

In regard to the first item: the pace of change in so many areas (hardware, software, operating systems, data communication, etc., etc., etc.) — and the effects on procurement cycles — are well known. How can certification remain unchanged when nothing else does? That argument would be undercut if the process had a rock solid track record — but that theme would not be supported by the second item — history:

Myriad shortcomings of existing operational systems are so pervasive that no one is considered a “loose cannon” for openly discussing them. Any of my horror stories — too strange and too numerous to be revisited here — would be trumped anyway by a document from the government itself. GAO-08-467SP, in 2008, described outlandish cost overruns, schedule delays, and deficient technical performance in the defense industry. That 3-way combination speaks for itself. Now a significant addition: the certification process has not been at all immune to serious flaws. The first-ever certified GPS receiver is now well known to have failed spectacularly in multiple facets of integrity testing by another manufacturer. It is readily acknowledged that correction of those early problems is quite credible, but one issue is inescapable: Historical proof of flightworthiness improperly bestowed — with proprietary rights accepted for algorithms and tests –- happened,  and that was not widely known until much later.

There is still more, including integrity failure probability limits missed by orders-of-magnitude in certified GPS receivers, severe limitations of GO/NO-GO testing, and failed attempts to gain approval to set requirements for correcting those plus other deficiencies. For brevity here, those issues are covered by citing the fifth page from another related reference.

The final item is, after years of fruitless talk about cost reduction, being acknowledged — we can’t do what we’ve been doing any more.  With dollars being the ultimate driver of so many decisions, we might finally see the necessary break from ingrained habits. FACE already addresses the issues and the requisite justifications. To make it all happen, two essential ingredients are

* raw-data-across-the-board, and

* nonproprietary software, with standardization under government control.
Flight-validated algorithms already in existence can be converted (e.g., from proof-of-concept to in-flight real-time form) according to government specification, by small groups more interested in engineering than in dollars (yes, that does exist). The payoff in cost savings can be huge.

Significant momentum is evolving toward a role for Open System Architecture (OSA) applied to radar. My observations in connection with that, voiced in a short LinkedIn discussion, seem worth repeating here.

One step could add major impact to this development: Rather than position (or relative position) outputs, provide measured range, azimuth, elevation (doppler could optionally be added if applicable) on the output interface. That simple step would vastly improve effectiveness of track file maintenance. Before attempting to describe all reasons for improved performance, two obvious benefits can be identified first:
* ability to use partial information (e.g., range-only or, for passive operation, angle-only)
* proper weighting of data for updating track state estimates.
The first item is self-evident. The second arises from common-sense attachment of greater value to the most accurate information. An explanation:

One-sigma error ellipsoids for individual radar fixes are not spherical (not a beachball shape but more like a flattened beachball), even at close range. At longer distances the shape progresses from a frisbee to a pancake to a DVD. Kalman filtering has enabled us to capitalize on that feature for over a half-century. Without exploiting it, we effectively treat separate radar-derived “coordinates” by intersecting volumes in space that are common to overlapping spheres. Resulting uncertainty volume is enormously larger than it should be.

The feature just noted shows up dramatically when mixing data among multiple platforms. Consider cooperative engagement whereby participants, all tracking each other via network-transmitted GPS observations, share radar observations from an unknown non-participant. Share measurements or coordinates? No contest; multiple lines crossing from different directions can offer best (i.e., along-range) accuracies applicable in 3-D.

That fact (i.e., combining data from different sensors and different platforms further dramatizes available improvements) doesn’t diminish the basic issue; even with a time history of data from only one radar, a designer with direct measurements available — instead of, not in addition to, coordinates — can provide incomparably superior performance.

“Send Measurements not Coordinates” (1999; #66 from the “Published Articles” panel, opening with eight rock-solid reasons) was aimed at GPS rather than radar. Many of the principles are the same when mixing data with information from other platforms — and from other sensors such as GPS. There is no reason, in fact, why data from satellite navigation and radar can’t be combined in the same estimation algorithm. That practice hasn’t evolved but the historical separation of operations (e.g., navigation and surveillance), arising from old equipment limitations, should no longer be a constraint. Moreover, with focus shifted from tracking to navigation, integration with additional (e.g., inertial) data offers still more reasons for using direct measurements. Rather than loose integration, superior benefits are widely known to result as the sophistication progresses forward (tight. ultratight, and deep integration).

Further elaborations on “casting off our old habits” appear from different perspectives in various blogs, one-pagers, and a few manuscripts available at this site. If your library has a copy of GNSS Aided Navigation & Tracking  pages 203-4 show a way to implement the cooperative sharing of radar data obtained from a non-participant, among participants tracking each other via the mutual surveillance and tracking approach defined earlier in that same chapter.

Because so many operational systems (in fact, a vast majority) use reports in the form of coordinates, reiteration is warranted. The central issue is the content, not the amount, of data. Rather than coordinates, provide accurately time-stamped direct measurements with links connected to whichever platform observed the data (e.g., for satnav — pseudoranges; for radar — range, azimuth, elevation). Those links are automatically attached when Mode-S extended squiter (e.g., chosen for ADSB) is the means for conveying the data.  For message content, strictly disallow “massaging the data beyond the light of day” (e.g., by unknown processes, with uncertain timing, … ) which invariably results in enormous loss of performance in common occurrence today.


Now that a few years have passed since the LORAN-C budget was killed, it might be a good time to revisit that decision. Unlike other decisions, this one might conceivably be undone; there hasn’t been the widespread demolition of resources (e.g., towers, transmitters) followed by restoration of sites. Something else, though, did occur: recent success achieved by cooperative effort between the Coast Guard and UrsaNav Inc.

For brevity here it suffices to make a few surface-scratching notes. The vast majority of us in the navigation community recognized the potential benefit of LORAN (and an extended form eLORAN) as a crucial backup — at extremely low cost — to be used when GPS is unavailable.  Many of us, furthermore, anxiously pressed for sanity (e.g., my “2-cents worth” written, to no avail, in 2009).

What’s different now, conceivably, is a combined effect of multiple factors:
* The USCG/UrsaNav success surpassed goals that had been stated earlier.
* Awareness of GPS vulnerability (therefore need for backup) has increased.
* Delay in follow-through (site restoration) offers the chance for a remedy.

An utterance appearing in Coordinates Magazine’s March 2012 cover story was reached from a different context, but its importance prompted me to cite it in the April 2012 cover story — and to repeat it here: “Do we really need to wait for a catastrophe before taking action against GNSS vulnerabilities?”

Once again I’m adding my voice to the chorus of those speaking out before it’s too late.


Questions submitted by members of various forums, understandably, frequently involve one or more of the following topics:
 * some or all facets of inertial navigation
 * means of updating and reinitializing the drifting inertial solution
 * satellite navigation (GPS/GNSS) for providilng the updates
 * other means of updating (radar, laser, optics, VOR, DME, hyperbolic, … )
 * best ways to use what’s available for various applications.
The pool of literature that might be offered can be vast, partly due to a vast array of operations – each with application-dependent requirements.  Finding just the relevant information from a mountain of available references can be a daunting task, especially for young designers.  I’ll try to make their search easier, by offering a list they can ask themselves early in the design process:
 * do you need a lat/lon/altitude Earth reference or just a designated point?
 * is the path determined from provisions onboard (nav) or remote (track)?
 * what’s your required accuracy for “absolute” (geolocation) position?
 * what’s your required accuracy for relative position (e.g., from a runway)?
 * do you need precise incremental position history (SAR motion compensation)?
 * do you need precise angular orientation (e.g., laser pointing)?
 * do you need precise angular rates (for image or antenna stabilization)?
 * for direction do you use a North reference or just along-track/cross-track?
 * will you have dependable access to updating information (GPS, radar, …)?
 * if not, how irregular will dynamics be over active parts of your mission?
 * if so, how irregular will the dynamics be during inter-update periods?
 * also if so, what data rate? Longest expected “blind period” between updates?
 * also if so, will measurements need averaging to meet your required accuracy?
 * also if so, how accurate are your measurements AND their time stamps?
 * also if so, can you use postprocessing or do you need everything real-time?
 * are you willing to accept partial updates (some but not all directions)?
 * do you need just position or derivatives too (velocity, acceleration)?
 * if so, how long can your dynamics be trusted to conform to model fidelity?
 * are you doing INS update (e.g., replacing acceleration with tilt states)?
 * if so, will you need to deduce drift rates – and how long will those hold?
 * do your sensors measure distances, angles, doppler, differences of those?
 * for how long does your sensor information content provide observability?
 * how’s your sensor integrity (bad readings at least detectable if present)?
 * for safety-critical operations — what are your backup provisions?
 * are you accommodating multiple modes with time-shared sensing resources?
 * do you need to perform image registration with different imaging sensors?
etc.etc. — the list goes on.  I won’t even try to claim thoroughness; you get the idea.  Designers with new tasks dumped in their lap can understandably feel overwhelmed.  Searching for references can become a trip through a maze of half-relevant sources.
A first step, then, is to separate the relevant (what you need) from the irrelevant (what you don’t need), instantly dismiss any thought of the latter, and do the opposite with the former (nail it).
Brief examples — the first two items from the above list —
 * If you just need to know your location relative to a designated point, irrespective of its latitude and lingitude — this might help.
 * If you’re tracking instead of navigating — check these out —
and one from the last item from that list —
Again, you get the idea — volumes have been written on all facets.  Many won’t apply to your immediate task; disregard those.
The good news is — paths to logical solutions are known and documented.  To avoid abandoning you to an enormous maze of references I’ll point out some fundamental and advanced (state-of-the-art) tracts that address all issues just cited and more.  Several blogs and short “1-pagers” will help individual designers to choose, based on their specific tasks, passages from available references.
Before GPS we struggled hard for accurate measurements in enough places.  That actually produced a benefit — we had to be resourceful.  My biggest challenge was to understand subjects (Kalman filtering, strapdown inertial navigation) then considered exotic.  Again a benefit; pulling information from 1950s books and papers forced me to understand, focus, and reduce concepts to whatever level became necessary.  The experience prompted me to write the first of my two books on navigation.
That first book has been used in myriad courses, including one currently taught by Prof. Hablani who wrote the most recent testimonial shown on that URL .
Some topics that earlier book explained in detail recently came up in another discussion — http://www.linkedin.com/groupAnswers?viewQuestionAndAnswers=&discussionID=44646633&gid=160643&commentID=68798460&trk=view_disc
For example, slow (“W” radian/sec) oscillations with “W” corresponding to the Schuler period (between 83 and 84 minutes). In that case position error from accelerometer bias, propagating as (1 – cos Wt), rises much sooner than gyro drift, propagating as (t – sin Wt/W). Page 80 of that book sketches an example of behavior over a cycle.  Development offered beyond there expands as far as many analysts wish to go (other natural frequencies of error propagation, rectification of vibration-sensitive errors, etc.).
Not long after that first book appeared, GPS became operational — and I was a newcomer to that.  By the time I understood it there were many experts.  Once again I had to catch up, and the process was gradual.  With an exceptionally strong client interested in my inertial background, a synergism was formed. That led to a flight test producing state-of-the-art accuracy in dynamics; see the table describing several innovations also resulting from the work just described.
That second book, after a review chapter, begins where the first (pre-GPS) one left off.  It also (1)is used in tutorials and (2)has received testimonials from other instructors, as the URL shows.  Sources cited here, plus an online 1.5-hr tutorial, free to Inst-of-Navigation members, plus a “try-before-you-buy” 100-page excerpt available from this site, should be helpful to many.


At ION GNSS 2011 in Portland OR, Javad Ashjaee, James L. Farrell and others participated in a panel discussing the U.S. Dept. of Homeland Security’s concerns on the effects of GPS jamming and spoofing on our national critical infrastructure.


 As Dr. Todd Humphreys noted, U.S. Dept. of Homeland Security recently completed a risk assessment of the effects of GPS jamming and spoofing on national critical infrastructure. Some of us participated as subject matter experts in this assessment.


The DHS report, which is the most thorough one to date on this topic, has left many people saying “Yes, it’s a problem. Now what?”

This panel addressed the question “Now what?”


Topic: How do we secure civil GNSS?



    • 8:30: Welcome and introduction: Moderator introduces topic, format, and ground rules
    • 8:40: Moderator introduces panelists
    • 8:45: Moderator frames the central question: “How do we secure civil GNSS?”
    • 8:50: Logan Scott
    • 9:00: Panel/audience response to Logan’s remarks
    • 9:10: Javad Ashjaee
    • 9:20: Panel/audience response to Javad’s remarks
    • 9:30: Mark Psiaki
    • 9:40: Panel/audience response to Mark’s remarks
    • 9:50: Questions from audience, discussion among panelists

10:05 — 10:35: Morning break

  • 10:35: Moderator welcomes audience and panel back, summarizes morning discussion
  • 10:40: Oscar Pozzobon
  • 10:50: Panel/audience response to Oscar’s remarks
  • 11:00:James Farrell
  • 11:10: Panel/audience response to James’s remarks
  • 11:20: Felix Kneißl
  • 11:30: Panel/audience response to Felix’s remarks
  • 11:40: Questions from audience; discussion among panelists
  • 12:10: Moderator and panelists offer concluding remarks
  • 12:15: Panel concludes


September 19-23, 2011 (Tutorials: September 19-20)
Oregon Convention Center, Portland, Oregon





A subsequent experiment conducted in Texas, attracting national attention at that time, became the topic of eMail communications among several professionals in the satnav community.  That sequence of communications resulted in a summary published in GPSWorld.

Life before GPS

Before GPS took over so many operations by storm (e.g., navigation,tracking, timing, surveying, etc.), designers had access to other — far less capable — provisions.  That condition forced our hands; to derive maximum benefit from what was available, we had to extract full information content from those provisions.  Now that GPS is subjected to challenges (aging, jamming, spoofing, etc.), some of those older methods are receiving increased scrutiny.  Recently I’ve received renewed interest in areas I analyzed decades ago.  Old publications from two of those areas are discussed here: 1) attitude determination and 2) nav integration.

“Attitude Determination by Kalman Filtering” is the title of three documents I had published.  In reverse sequence they are:
1) Automatica (IFAC Journal), v6 1970, pp. 419-430,
2) my Ph.D. dissertation (Univ. of Maryland, 1967),
3) NASA CR-598, Sept., 1966.
As indicated by the last reference, the work was the result of a contractual study sponsored by NASA (specifically Goddard Space Flight Center – GSFC – in Greenbelt Maryland).  I was working for Wetinghouse Defense and Space Center at the time.  The proposal I had written to win this contract cited my work prior to then, in both modern estimation (“Simulation of a Minimum Variance OrbitalNavigation System,” AIAA JSR v 3 Jan 1966 pp. 91-98) and attitude computation (“Performance of Strapdown Inertial Attitude Reference Systems,” AIAA JSR v 3 Sept 1966, pp. 1340-1347).  Let me hasten to explain the dates of those Journal publications: each followed its inclusion at an AIAA-sponsored conference, about a year earlier.

By the mid-1960s there was an appreciable amount of validation for Kalmen filtering applied to determination of orbits (that track record was convincing) but not yet for attitude.  A GSFC-sponsored investigation was then planned — the very first one for attitude using modern estimation methods.  GSFC management understandably wanted that contractual investigation to be performed by someone with demonstrable experience in both Kalman filtering and rotational dynamics.  In those days that combination was rare; the Westinghouse proposal was chosen as the winner.  At the time of that study, provisions realistically available for attitude updating consisted of mediocre-accuracy items such as magnetometers and horizon scanners– not bad but not spectacular either.
All that was of course before GPS weighed in, with its opportunity to reveal attitude from phase differences between antennas spaced at known distances apart.  That vastly superior capability effectively reduced earlier crude measurements to relative obscurity.  A directly parallel situation occurred in connection with navigation; the book that first tied together several facets of advancement in that field (integration, strapdown inertial, modern estimation with  acceptance of all data sources, multimode operation, extension to tracking, clear exposition of all commonly used representations of attitude, etc.) was”pre-GPS” (1976), and consequently regarded as less relevant. Timing can be decisive — that’s no one’s fault.

The item just noted — attitude representation — is worth further discussion here.  Unlike many other sources, the 1976 book offered an opportunity to use quaternion properties without any need to learn a specialized quaternion algebra.  A literature search, however, will point primarily to various sources (of necessity, later than 1976).that benefit from the superior performance offered through GPS usage. Again, in view of GPS as a game-changer, that is not necessarily improper.  Most publications on attitude determination don’t cite the first-ever investigation, sponsored by GSFC, for that innocent reason.

The word beginning that last sentence (“Most”) has an exception.  One author, widely quoted as an authority (especially on quaternions), did cite the original work — dismissing it as “ad-hoc” — while using an exact copy of the sensitivity matrix elements pubished in my original investigation (the three references cited at the start of this blog).
While I obviously didn’t invent either quaternions or the Kalman filter, there was another thing I didn’t do: fail to credit, in my publications, pre-existing sources that contributed to my findings. Publication of the material cited here, I’ve been told, paved the way for understanding and insight to many who followed. No one owes me anything for that; an analyst’s work, truthfully and realistically presented, is what the analyst has to offer.

It is worth pointing out that both the attitude determination study and the 1976 book cover another facet of rotational analysis absent from many other related publications: dynamics — in the sense of physics.  Whereas modern estimation lumps time-variations of the state together into one all-encompassing “dynamic” model, classical physics makes a separation: Kinematics defines the relation between position, rates, and accelerations.  Dynamics determines translational accelerations resulting from forces or rotational accelerations resulting from torques.

Despite absence of GPS from my early (1960s/70s) investigations, one feature that can still make them useful for today’s analysts is the detailed characterization of torques acting — in very different ways — on spinning and gravity-gradient satellites, plus their effects on rotational motion. Many of the later studies focused on the rotational kinematics, irrespective of those torques and their consequences. Similarly, the “minimal-math”approach to explaining integrated navigation has enabled many to grasp the concepts.  Printed testimony to that effect, from courses I taught decades ago, is augmented by more recent source noted near the end of another page shown on this site.


Tracking acceleration dynamics by GNSS, radar, imaging

My 2007 book on GPS and GNSS (GNSS Aided Navigation & Tracking), as its title implies, involves both navigation and tracking. This discussion describes the latter, covered in the longest chapter of the book (Chapter 9).  In addition to the flight-validated algorithms for navigation (processing of inertial sensor data, integration with GPS/GNSS, integrity, etc.), this text offers extensive coverage of tracking. Formulations are given for a variety of modes, in 2-D (e.g., for runway incursion prevention or ships) and 3-D (in-air), using GPS/GNSS and/or other sensors (e.g., radar, optical).  Position and velocity vectors are formed, in some operations joined by some or all components of acceleration.

This author was fortunate to be “at-the-right-places at-the-right-times” when a need arose to address each of the topics covered.  As a result, the words of one reviewer — that the book is

…………….. “teeming with insights that are hard to find or unavailable elsewhere.”

applies to tracking as well as to navigation.  The book identifies subtleties that arise in specific applications (aircraft, ships, land vehicles, satellites, long-range or short-range projectiles, reentry vehicles, missiles, … ). In combination with a variety of possible conditions affecting sensor suite and location (air-to-air; air-to-ground; air-to-sea surface; surface-to-air, etc. — with measurements associated with distance or direction or both; shared or not shared among participants who may communicate from different positions), it is not surprising that striking contrasts can arise, influencing the characterization and approaches used.  The array of formulations offered, while fully accounting for marked differences among operations, nevertheless exploits an underlying commonality to the maximum possible extent.

Tracking dynamics of aircraft, missiles, ships, satellites, projectiles, …

Formulations described in Chapter 9 were used for tracking of both aircraft and missiles, concurrently, through usage of an agile beam radar.  For another example, air-to-surface operations subdivide into air-to-ground and vessel tracking from the air.  That latter case constrains tracked objects’ altitudes to mean sea level — a substantial benefit since it obviates the need for elevation measurements, which are subject to large errors from refraction (bearing and range measurements, much less severely degraded, suffice). Air-to-ground tracking, by contrast, further subdivides into stationary and moving targets; the former potentially involves imaging possibilities (by real or synthetic aperture) while the latter — if not being imaged by inverse SAR — separates its signature from clutter via doppler.

Reentry vehicles, quite different from other track operations, present a unique set of “do’s” and “don’ts” owing to high-precision range measurements combined with much larger cross-range errors (because of proportionality to extreme distances involved).  Pitfalls from uncertain axial direction of “pancake” shaped one-sigma error ellipsoids must be avoided.  A counterexample, having angle observations only (without distance measurements), is also addressed.  Orbit determination is unique in still another way, often permitting “patched-conic” modeling for its dynamics.  A program based on Lambert’s theorem provides initial trajectories from two position vectors with the time interval separating them.

Those operations and more are addressed with most observations from radar or other (e.g., infrared imaging) sensors rather than satellite measurements.  That of course applies to tracked objects carrying no squitters. Friendlies tracking one another, however, open the door for using GNSS data.  Those subjects plus numerous supporting functions are discussed at some length in Chapter 9.  Despite very different dynamics applicable to various operations, the underlying commonality (Chapter 2) connects the error propagation traits in their estimation algorithms and also — though widely unrecognized — short-term INS error propagation under cruise conditions (Chapters 2 and 5).  Support operations such as synthetic aperture radar (SAR) and transfer alignment are described in the chapter Addendum.