What receiver samples can establish about aircraft motion

The ESA development recording has some quiet position-derived trends, but no qualifying continuous stretches yet. A fitted trend is not a measurement of every movement between receiver samples. Even perfect position and velocity samples cannot establish that instantaneous course changes stayed below a small cutoff unless additional motion assumptions are supplied.

The version-1 selector screens a ten-second median of sampled course-rate estimates. That statistic and an instantaneous-motion guarantee are different requirements.

This is a sampling limitation, not evidence of poor instruments or a claim that the aircraft actually made the unobserved movements in the example below.

An exact example

Consider receiver samples every half second and forward speed 50 m/s. Compare a straight path with the same path plus a small sideways movement within each sample interval. With interval length Δ and normalized time u between zero and one, define the sideways displacement as:

\[ y(t)=A\,u^2(1-u)^2,\qquad u=(t-t_i)/\Delta. \]

At both ends of every interval, the added displacement and velocity are exactly zero. Both paths therefore give identical receiver position and velocity samples. Their sampled course and fitted course trend are identical too.

Yet the second path has acceleration \(2A/\Delta^2\) at those endpoints. Its instantaneous course rate there is \(2A/(v\Delta^2)\) radians per second. For A = 0.004 m, Δ = 0.5 s and v = 50 m/s:

The largest sideways displacement is A/16: 0.25 mm.
Endpoint sideways acceleration is 0.032 m/s².
Endpoint course rate is approximately 0.0367°/s, above the 0.02°/s quiet-course cutoff, despite identical sampled positions and velocities.

Repeating this polynomial gives continuous position, velocity and acceleration; the ambiguity does not depend on a position jump or a missing receiver epoch. Reducing receiver noise alone cannot remove a movement that leaves its samples unchanged. A physically justified acceleration, jerk or bandwidth restriction could rule out particular examples, but that restriction is extra information.

Consequence for the ESA study

There are two valid development paths:

Keep the quiet-flight requirement. Specify and justify the additional between-sample motion bound, propagate receiver uncertainty and screen every required interval. Sparse quiet trends alone cannot certify it.
Qualify the complete motion treatment. Demonstrate that the rate-input adapter and navigation calculation correctly handle the relevant maneuvers, sampling hypotheses and timing uncertainty under all three physical candidates. A new receiver-only selection can then use the demonstrated capability domain.

The current ESA adapter constructs component integrals under explicit sampling hypotheses. It does not yet qualify complete finite rotations, coning/sculling or geographic uncertainty propagation. Those requirements remain open; this example does not establish that a less restrictive selection is ready.

Recording 187 remains development material. The zero-eligible version-1 inventory and all earlier diagnostics are preserved, and the seven confirmation groups remain protected. No new receiver inventory or Earth-model fitting is performed by this analysis.

Completed receiver assessment · Selection proposal · Current receiver-only rule · Rate-input adapter · Local body bounds · Roadmap.