S2 — IMU Propagation

Between camera frames the IMU runs at 200 Hz–1 kHz, and S2 integrates it: it strapdown-integrates the mean (attitude, velocity, position) and propagates the covariance P forward, so the filter arrives at the next image with both a predicted pose and an honest uncertainty about it. This stage was a prime suspect for the #212 over-confidence — and the probe cleared it. This page is what a measured negative result looks like.

What it does

• Mean — integrate each IMU interval from both of its samples: debias, interpolate the input linearly between the samples at t₍ₖ₋₁₎ and tₖ, and integrate R, v, p by RK4 (#616). An image between two samples is reached by holding the last sample; the next sample integrates the rest of its interval from the rate interpolated at the image time.
• Covariance — P⁻_{k+1} = Φ P⁺_k Φᵀ + Q_d. The transition matrix Φ carries the off-diagonal skew blocks coupling orientation error into velocity and position error — gravity leakage living inside the covariance. The discrete noise Q_d should map measurement-space noise through the noise Jacobian G, producing θ–v–p cross-correlations and the higher-order Δt²/Δt³ scalings — not a plain diagonal.

Why it was a suspect

The cortex Q_d is assembled diagonally — it drops the canonical position-noise block and the velocity–position cross term. A diagonal Q_d understates position/velocity process noise and omits correlations the update relies on, which is a textbook over-confidence mechanism. The question is not “is it diagonal” (it is) but how much does that actually cost — and the only honest way to answer is to measure it against a canonical reference.

The contract

Signature	propagate(x⁻, P⁻, imu[t_k, t_{k+1}]) → (x⁻_{k+1}, P⁻_{k+1})
Pre	Δt > 0, monotonic timestamps; measurements debiased; P⁺_k symmetric PSD
Post / invariants	R ∈ SO(3) (orthonormal, det 1) · P⁻ symmetric PSD · Q_d PSD with θ–v–p cross-correlations · the 4-D unobservable subspace (global position + yaw) stays in the propagation null space · GT-injection: zero-noise propagation closes on a continuous truth over a clone window within what the camera resolves (1.09e-4 rad, 0.22 mm at 2 m), at second order
cortex	msckf/stages/s2_propagation.hpp (the convention), msckf/propagator.hpp (ImuInterval, rk4_strapdown, Φ, Q)

Metrics

The S2 probe reconstructs the canonical Φ/Q_d and compares them to the shipped Propagator, then drives the real propagator on the synthetic world.

Φ reconstruction & the structural invariants — all pass

check	result
Φ reconstruction (real == reference)	1.5e-16 — pass
R orthonormality ‖RᵀR − I‖ @0.5 s	2.2e-16 — pass
P min-eigenvalue @0.5 s	> 0 — PSD
global-position nullspace leak	0 — preserved
GT-injection integrator order	O(Δt) (error halves with Δt: 59→30→15→7 mm @ 50→800 Hz) — this was #616’s defect, see below

The row that should have failed

First order means the error halves with the step. For an integrator that should be second order or better, that is a defect, not a property: each interval held one sample constant, the one that ends it, which integrates the IMU half a sample (2.5 ms) early against the camera. The probe measured it in June 2026 and recorded it as a metric with no bound, so nothing failed. It was found in November by the consistency gate’s spline truth (#613), and fixed in #616. The full account is Aligning Sensor Samples.

The diagonal-Q deficit — real, but small

The cortex Q_d does drop terms — but measured against the canonical G Q_meas Gᵀ, the cost is minor:

quantity	cortex vs canonical
Q θ-σ, Q v-σ	match
Q position term	0 vs 3.06e-6 m — dropped
Q v–p cross term	0 vs 1.25e-9 — dropped
Q relative-Frobenius gap	0.35% of Q dropped
position-σ under-report @0.05 s (inter-frame)	7.4%
position-σ under-report @0.5 s	0.7%

Propagation-only NEES — consistent at the default Q

Driving the real propagator and scaling Q, the propagation-only 6-DoF pose NEES sits right at its target with the shipped noise:

Q scale	pose NEES (target 6)
0.25	23.5
0.5	12.2
1 (shipped)	6.4
2	2.97
4	1.52

At the default Q, NEES ≈ 6.4 ≈ the 6-DoF target — the propagated covariance is essentially honest on its own.

Real-data inspector (s2_inspect)

The measurements above come from the synthetic probe — known IMU, known Q, run in CI. Its real-data companion, s2_inspect, runs the real propagator (msckf::Propagator) over a real EuRoC IMU window. It seeds the prior pose+velocity from ground truth at the window start (plus the filter’s initial-P seed), then propagates step by step — no SDK change, since the propagator is already standalone.

s2_inspect --dataset /path/to/V1_01_easy/mav0 --out build/s2 --window-s 0.5
node docs-site/scripts/gen-propagation-figures.mjs build/s2   # → cov_growth.svg, pose_drift.svg

It writes propagation.json and renders two figures:

• the covariance-growth heatmap — each of the 15 IMU error states’ σ over the window (row-normalized), with the θ/p/v/b_g/b_a blocks marked. The position block growing only via the velocity coupling (no direct position-Q term) is the diagonal-Q simplification, now visible on real data.
• drift vs the ±3σ envelope — the propagated pose’s dead-reckoning drift from ground truth, plotted against the filter’s own growing 3σ band. Drift that stays inside the band means the propagated covariance honestly covers the error; drift that escapes it (red) is the local, real-data image of an under-noised Q.

This is the real-data check on the synthetic finding below: on EuRoC the envelope covers the inter-frame drift, confirming propagation is essentially honest on its own — the #212 over-confidence enters elsewhere.

The mean: integrating from both samples (#616)

The probe above cleared S2’s covariance. Its mean carried a defect the probe measured and nobody bounded. Both backends held one IMU sample over each 5 ms interval, and they held different ones: MsckfBackend the sample that ends the interval, InvariantVioBackend the one that starts it. Either is a half-sample (2.5 ms) skew between IMU and camera. On the EuRoC consistency gate, with exact R and Q from one spline truth, NIS was 1.00 on the slow sequences and 1.1–1.9 on the fast ones (V1_03, V2_02, V2_03).

The closure test (tests/tools/gt_spline.cpp) integrates a spline’s own noise-free IMU over a 0.55 s clone window on a flight that rolls and pitches at about 1.3 rad/s, against what the camera resolves (a tenth of 0.5 px: 1.09e-4 rad, 0.22 mm at 2 m):

Scheme	Rotation ÷ bound	Position ÷ bound	Order
End sample held, Euler (MSCKF before #616)	28.6	41	first
Start sample held, Euler (R-IEKF before #616)	28.0	6.9	first
The two samples’ average, Euler	0.28	24	first in position
The line between the samples, RK4 (shipped)	0.27	0.10	second

Centring the input removes the rotation error. The position error is the integrator’s: an Euler step rotates the specific force with the attitude at the start of the interval. RK4 rotates it along the interval.

Φ followed the mean. #471 found that a higher-order Φ leaks more yaw than the first-order I + A·Δt. That was true against the one-sample Euler mean, which the first-order Φ approximately linearizes. Against the RK4 mean the first-order Φ leaks 7× more than it used to (4.4e-6 per step at 2.5 ms, from 6.0e-7), so the lesson was about the agreement between Φ and the mean, not about truncation. The RK4 mean has the preintegration form R′ = R·ΔR, v′ = v + gΔt + R·Δv, p′ = p + vΔt + ½gΔt² + R·Δp, which a rotation about gravity commutes with. Φ is now that mean’s exact Jacobian, and it preserves the whole unobservable subspace, yaw included, to roundoff (tests/sdk/imu_interval.cpp checks it against finite differences).

Status — covariance cleared; mean fixed in #616

S2 is not a primary source of the #212 over-confidence. The diagonal-Q simplification is real but costs <1% of position-σ over a frame, every structural invariant passes, and propagation-only NEES is consistent at the shipped noise. This also sharpens the Consistency Analysis finding that “Q×10 helps on EuRoC”: that is compensating inflation widening P to mask the structural error elsewhere — not evidence that propagation is genuinely under-noised. Outstanding (correctness, not consistency): add the dropped Q_d position and v–p cross terms. Full decomposition on Engineering Status.