In 2013 a phone presentation was arranged, for me to talk for an hour with a couple dozen engineers at Raytheon. The original plan was to scrutinize the many facets and ramifications of timing in avionics. The scope expanded about halfway through, to include topics of interest to any participant. I was gratified when others raised issues that have been of major concern to me for years (in some cases, even decades).  Receiving a reminder from another professional, that I’m not alone in these concerns, prompts me to reiterate at least some aspects of the ongoing struggle — but this time citing a recent report of flight test verification

The breadth of the struggle is breathtaking. The About panel of this site offers short summaries, all confirmed by authoritative sources cited therein, describing the impact on each of four areas (satnav + air safety + DoD + workforce preparation). Shortcomings in all four areas are made more severe by continuation of outdated methods, as unnecessary as they are fundamental, Not everyone wants to hear this but it’s self-evident: conformance to custom — using decades-old design concepts (e.g., TCAS) plus procedures (e.g., position reports) and conventions (e.g., interface standards — guarantees outmoded legacy systems. Again, while my writings on this site and elsewhere — advocating a different direction — go back decades, I’m clearly not alone (e.g., recall those authoritative sources just noted). Changing more minds, a few at a time, can eventually lead to correction of shortcomings in operation.

We’re not pondering minor improvements, but dramatic ones. To realize them, don’t communicate with massaged data; put raw data on the interface. Communicate in terms of measurements, not coordinates — that’s how DGPS became stunningly successful. Even while using all the best available protection against interference, (including anti-spoof capability), follow through and maximize your design for robustness;  expect occurrences of poor GDOP &/or less than a full set of SVs instantaneously visible. Often that occurrence doesn’t really constitute loss of satnav; when it’s accompanied by history of 1-sec changes in carrier phase, those high-accuracy measurements prevent buildup of position error. With 1-sec carrier phase changes coming in, the dynamics don’t veer toward any one consistent direction; only location veers during position data deficiencies (poor GDOP &/or incomplete fixes) and, even then, only within limits allowed by that continued accurate dynamic updating. Integrity checks also continue throughout.

So then, take into account the crucial importance of precise dynamic information when a full position fix isn’t instantaneously available. Take what’s there and stop discarding it. Redefine requirements to enable what ancient mariners did suboptimally for many centuries — and we’ve done optimally for over a half-century.  Covariances combined with monitored residuals can indicate quality in real time. Aircraft separation means maintaining a stipulated relative distance between them, irrespective of their absolute positions and errors in their absolute positions. None of this is either mysterious or proprietary, and none of this imposes demands for huge budgets or scientific breakthroughs — not even corrections from ground stations.

A compelling case arises from cumulative weight of all these considerations. Parts of the industry have begun to address it. Ohio University has done flight testing (mentioned in the opening paragraph here) that validates the concepts just summarized. Other investigations are likely to result from recent testing of ADSB. No claim is intended that all questions have been answered, but — clearly — enough has been raised to warrant a dialogue with those making decisions affecting the long term.

John Bortz

In February of this year the navigation community lost a major contributor to navigation — John Bortz. To many his name is best known in connection with “the Bortz equation” which easily deserves a note here to highlight its significance in development of strapdown inertial nav. Before his work in the early 1970s, strapdown was widely considered as something with possible promise “maybe, if only it could ever come out of the lab-&-theory realm” and into operation. Technological capabilities we take for granted today were far less advanced then; among the many state-of-the-art limitations of that time, processing speed is a glaringly obvious example. To make a long story short, John Bortz made it all happen anyway. Applying the previously mentioned equation (outgrowth of an early investigation of Draper Lab’s Dr. J.H. Laning) was only part of his achievement. Working with 1960s hardware and those old computers, he made a historic mark in the annals of strapdown.  Still, importance of that accomplishment should not obscure his other credentials. For example, he also made significant contributions to radio navigation — and he spent the lst two decades of his life as a deacon.

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.

CONING in STRAPDOWN SYSTEMS

Free-inertial navigation uses accelerometers and gyros alone, unaided. For that purpose pioneers of yesteryear developed a variety of techniques, ranging from a 2-sample approach (NASA TND-5384, 1969) by Jordan to his and various others’ higher-order algorithms to reduce errors from noncommutativity of finite rotations in the presence of coning (and/or pseudoconing). The methods showed considerable insight and produced successful operation. Since it’s always good to have “another tool in the toolbox” I’ll mention here an alternative. What I describe here isn’t being used but, with today’s processing capabilities, could finally become practical. The explanation will require some background information; I’ll try to be brief.

a

A very old investigation (“Performance of Strapdown Inertial Attitude Reference Systems,” AIAA Journal of Spacecraft and Rockets, Sept 1966, pp 1340-1347) used the usual small-angle representation for attitude error expressed in the vehicle frame. With that frame rotating at a rate omega the derivative of that vector therefore contains a cross product of itself crossed with omega.  One contributor to that product is a lag effect from omega premultiplied by a diagonal matrix consisting of delays (e.g., transport lags equated to reciprocals of gyro bandwidths). Mismatch among those diagonal elements produces drift components with nonzero average, e.g., the x-component of the cross product is easily seen to be
aaaaaaaaaaa    (difference between y and z lags) times (omega_y) times (omega_z)
Even with zero-average (e.g., oscillatory) angular rates, that product has nonzero average due to rectification.  I then characterized the lags as delays from computation rather than from the gyros, with the lag differences now proportional to nonuniformities among RMS angular rate components along vehicle axes, and average products proportional to cross-correlation coefficients of the angular rate components. That was easy; I had a simple model enabling me to calculate the error due to finite gyro sampling rates producing finite rotation increments that don’t commute.

a

A theoretical model is only that until it is validated. I had to come up with a validation method with mid-1960s computational limitations. Solution came from a basic realization: performance doesn’t degrade from what’s happening but from belief in occurrences that aren’t happening. The first-ever report of coning (Goodman and Robinson, ASME Trans, June 1958) came from a gimballed platform that was believed to be stable while it was actually coning. If the true coning motion they described had been known and taken into account, then their high drift rates never would have occurred. The reason they weren’t taken into account then was narrow gimbal servo bandwidth; the gyros responded to the coning frequency but the platform servos didn’t. Now consider strapdown with the inverse problem: pseudoconing — a vehicle believed to experience coning when it isn’t. That will fall victim to the same departure of perception from reality. If you gave the same Goodman and Robinson coning motion to their strapdown gyro triad and sampled them every nanosecond, the effect from noncommutativity wouldn’t be noticeable.

a

Armed with that insight I then chose rotational dynamics with a closed form solution. Although rotations about fixed vehicle axes produced no coning, the pseudoconing was severe, with the apparent (reported-from-gyros) rotation axis changing radically within fractions of a millisecond; too fast for the 10 kHz data rate used in that computation.  The cross product formulation was then validated by making extensive sets of runs, always comparing two time histories:

* a closed form solution for a true direction cosine matrix corresponding to a vehicle experiencing a sinusoidal omega
* an apparent direction cosine matrix, obtained by brute-force but meticulous formation from processing gyro outputs at finite rates with quantization, time lags, and a wide variety of error sources.

That “bull-by-the-horns” computation allowed extended runs (up to a million attitude iterations) to be made for a wide range of angular rate frequencies, axis directions, and combinations of gyro input errors (steady, random, motion-sensitive, etc.). Deviation of apparent attitude from closed-form truth was consistently in close conformance to the analytical model, for a host of error sources. I have to admit that this “bull-by-the-horns” approach gave me an advantage of finding out answers before I understood the reasons for them. The cross-product analytical model didn’t come from my vision; it came after much head-scratching with answers computed from dozens of runs. A breakthrough came from the sensitivity, completely unanticipated, to angular acceleration about gyro output axes — clear in retrospect but not initially. After these experiences it occurred to me: if cross-axis covariances were known, the dominant contributor to errors — including noncommutativity — could be counteracted. I noted that on page 1342 of that old AIAA paper.

a

Finally I can describe the alternative means of compensating the dominant computational error. Description begins with the reason why it would be useful. Earlier I mentioned that many authors developed very good algorithms to reduce errors from noncommutativity of finite rotations in the presence of coning and/or pseudoconing. All that history, plus more detailed presentation of everything discussed here, can be found in Chapters 3 and 4 of my 1976 book plus Addendum 7.A of my 2007 book. A supreme irony upstages much of the work from those brilliant authors: without accounting for gyro frequency response characteristics, the intended benefit can be lost — or the “compensation” can even become counterproductive (Mark and Tazartes, AIAA Journal of Guidance, Control, & Dynamics, Jul-Aug 2006, pp 641-647). As if those burdens weren’t enough, the adjustment’s complexity — as shown in that paper — can be extensive. So :  that motivates usage of a simpler procedure.

 a

By now I’ve put so much explanation into preparing its description that not much more is needed to define the method. Today’s signal-processing boards enable the requisite covariances to be repetitively computed. Then just form the vector cross product already described and subtract the result from the gyro increments ahead of attitude updating. So much for coning and pseudoconing — but I’m not quite finished yet. The paper just cited leads to another consideration: even if we successfully removed all of the error theoretically arising from inexact computation, significant improvement in free-inertial performance would require more. Operation in the presence of vibrations would necessitate reduction of other motion-sensitive errors. Gyro degradations from rotations, for example, would have to be compensated — and that includes a multitude of components. For that topic you can begin with the discussion of gyro mounting misalignment following that up with the tables in Chapter 4 of my 1976 book and Addendum 4.B of my 2007 book.

LORAN REVISITED

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.

CHECK LIST for DESIGNERS

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
&ut=0XsoCju0nA5B81
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.

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.

RUNWAY INCURSIONS

The number of runway incursions, as shown on an FAA URL  was nearly a thousand in FY 2011 and 1150 for FY 2012.  A subsequent article shows renewed interest in their prevention.

A hundredfold reduction in velocity error (from meters/sec to cm/sec) was shown in flight for squitter message transmission — but with measurement-based message content, as discussed in an accompanying blog.  A publication describing highly favorable results in air (3-D) could readily extend to 2-D (surface).

Schuler cycles distorted — Here’s why

1999 publication I coauthored took dead aim at a characteristic that received far too little attention — and still continues to be widely overlooked: mechanical mounting misalignment of inertial instruments.  To make the point as clearly as possible I focused exclusively on gyro misalignment — e.g., the sensitive axes of roll, pitch, and yaw gyros aren’t quite perpendicular to one another.  It was easily shown that the effect in free-inertial coast (i.e., with no updates from GPS or other navaids) was serious, even if no other errors existed.

It’s important here to discuss why the message took so long to penetrate.  The main reason is historic; inertial navigation originated in the form of a gimbaled platform holding the gyros and accelerometers in a stable orientation.  When the vehicle carrying that assembly would rotate, the gimbal servos would automatically receive a command from the gyros, keeping the platform oriented along its reference directions (e.g., North/East/vertical for moderate latitudes).  Since angular rates experienced by the inertial instruments were low, gyro misalignment and scale factor errors were much more tolerable than they are with today’s strapdown systems.  I’ve been calling that the “Achilles’ heel” of strapdown for decades now.  The roots go all the way back to 1966 (publication #6) when simulation clearly showed how serious it is.  Not long thereafter another necessary departure from convention became quite clear: replacement of the omnipresent nmi/hr performance criteria for numerous operations.  That characteristic is an average over a period between 83 and 84 minutes.  It is practically irrelevant for a large and growing number of applications that depend on short-term accuracy. {e.g., synthetic aperture radar (SAR), inertial aiding of track loops, antenna stabilization, etc.}, Early assertions of that reality (publication #26 and mention of it in still earlier reports and publications involving SAR) were essentially lost in “that giant shouting match out there” until some realization crept in after publication #38.

Misalignment: mechanical mounting imprecision

Whenever this topic is discussed, certain points must be put to rest.  The first concerns terminology; much of the petinent literature uses the word misalignments to describe small-angle directional uncertainty components (e.g., error in perception of downward and North, which drive errors in velocity).  To avoid misinterpretation I refer to nav-axis direction uncertainty as misorientation.  In the presence of rotations, mounting misalignment contributes to misorientation.  Those effects, taking place promptly upon rotation of the strapdown inertial instrument assembly, stand in marked contrast to leisurely (nominal 84-minute) classical Schuler dynamics.

The second point, lab calibration, is instantly resolved by redefining each error as a residual amount remaining due to calibration imperfections plus post-cal aging and thermal effects — that amount is still (1) excessive in many cases, and (2) in any event, not covered by firm spec commitments.

A third point involves error propagation and a different kind of calibration (in-flight).  With the old (gimbal) mechanization, in-flight calibration could counteract much overall gyro drift effect.  Glib assessments in the 1990s promoted widespread belief that the same would likewise be true for  strapdown.  Changing that perspective motivated the investigation and publication mentioned at the top of this blog.

In that publication it was shown that, although the small-angle approximation is conservative for large changes in direction, it is not extremely so.  The last equation of its Appendix A shows a factor of (pi/2) for a 180-deg turn.  A more thorough discussion of that issue, and how it demands attentiveness to short-lived angular rates, appears on pages 98-99 of GNSS Aided Navigation and Tracking.  Appendix II on pages 239-258 of that same book also provides a program, with further supporting analysis, that supersedes the publication mentioned at the top of this blog.  That program can be downloaded from here.

The final point concerns the statistical distribution of errors.  Especially with safety involved (e.g., trusting free-inertial coast error propagation), it is clearly not enough to specify RMS errors.  For example, 2 arc-sec is better than 20 but what are the statistics?  Furthermore there is nothing to preclude unexpected extension of duration for free-inertial coast after a missed approach followed by a large change in direction.  A recent coauthored investigation (Farrell and vanGraas, ION-GNSS-2010 Proceedings) applies Extreme Value Theory (EVT) to outliers, showing unacceptably high incidences of large multiples (e.g., ten-sigma and beyond).  To substantiate that, there’s room here for an abbreviated explanation —  even in linear systems, gaussian inputs produce gaussian outputs only under very restrictive conditions.

A more complete assessment of misalignment accounts for further imperfections in mounting: the sensitive axis of each accelerometer deviates from that of its corresponding gyro.  As explained on page 72 of Integrated Aircraft Navigation, an IMU with a gyro-accelerometer combo for each of three nominally orthogonal directions has nine total misalignment components for instruments relative to each other.