Body rotation and force: the next adapter check
Component integration is not enough when the rotation axis changes. This prerequisite compares mean-rate midpoint increments with an independently integrated body-frame differential equation. It does not fit an aircraft or assume an Earth-rate target.
For one interval of length h, suppose angular rate and specific force are
linear between their two reported point samples. Define their means w, f,
angular-rate slope b, force slope d, and maximum endpoint force norm F.
The midpoint approximation uses exp(skew(w)*h) and
exp(skew(w)*h/2)*f*h. These are exactly the body-only operations that the
constructed midpoint increments imply.
An operator-norm bound for the rotation discrepancy is
To obtain it, factor out the constant mean rotation. The remaining generator
is (t-h/2) times a rotated skew(b). Its fixed midpoint contribution
integrates to zero; the changing conjugation contributes at most the commutator
term above. The second-order integral remainder is bounded by half the squared
integral generator norm, giving ||b||²*h⁴/32. Exact rotations have discrepancy
at most two. No assumption about small angle is needed for this mathematical bound.
The corresponding body-force integral discrepancy is bounded by
The first term integrates the variation-of-constants rotation bound against force. The second is the midpoint quadrature bound for a constant mean rotation acting on linear force. This is a local body calculation: geographic transport, gravity, Coriolis, antenna motion and cross-interval accumulation are separate.
Tests must compare changing-axis and changing-force paths with an independent matrix/force ODE solution, as well as exact constant and zero-rate cases. A finite generated study must freeze cases and work limits before execution. Keep all existing sources and studies unchanged.
These bounds assume linear point samples. They do not identify the actual instrument's averaging, filtering or sampling phase. They are mathematical approximation bounds, not a complete floating-point enclosure or calibrated observational uncertainty. No protected data or archive fits enter the check.