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.

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