COMPLETE SETS OF EOM

Using Equation 3.11 and Equations 3.15 through 3.19, the following nine coupled differential equations describe the aircraft motion in flight.

Force equations:

m(u + qw — rv) — qSCx — mg sin U + T cos sT

m(v + ru — pw) — qSCy + mg sin f cos U (3.20)

m(w + pv — qu) — qSCZ + mg cos f cos U — T sin sT

Moment equations:

qSbCi — p Ix — q Ixy — r/xz + qr(/z — Zy) + (r2 — q2)/yz — pq/xz + rp/xy qScCm — —_/xy + <q/y — r/yz + rp(/x — Iz) + (p2 — r2)/xz — qr/xy + pq/yz (3.21)

qSbC„ — —_ /xz — ^/yz + Г/z + pq(/y — /x) + (q2 — p2)/xy — rp/yz + qr/xz

Euler angles and body angular velocities:

f — p + q tan U sin f + r tan U cos f U — q cos f — r sin f (3.22)

fi — r cos f sec U + q sin f sec U

The above set of equations represents the 6DOF motion of the aircraft in flight. These can rearranged and expressed in two equivalent forms of equations.

3.5.1 Rectangular Form

When the velocity is represented in a rectangular coordinate system, Equations 3.20 through 3.22 can be rearranged to yield the following set of equations in rectangular form:

p = l_ 2 {qSb(IzCi + IxO — qr(I2xz + l2 — Iylz) + pqlxzdx — Iy + Iz)} lxlz 1xz

q = J {qScCm — (p2 — r2)Ixz + pr(Iz — Ix) + T(itx sin St + itz cos St)} (3.23)

Iy

r = l_ 2 {qSb(IxCn + IxzCi) — qrIxz(Ix — Iy + Iz) + pq(12 + iI — Uy)}

IxIz — Ixz

f = p + q tan U sin f + r tan U cos f U = q cos f — r sin f C = r cos f sec U + q sin f sec U

Note that the terms in the rectangular boxes in the expressions for U, v, and w are the linear accelerations. This gives a set of kinematic equations, which can be used to check data consistency and estimate calibration errors in the data. The model equations normally used for kinematic consistency check are

U = —qw + rv — g sin U + ax v = —ru + pw + g cos U sin f + ay w = —pv + qu + g cos U cos f + az

f = p + q tan U sin f + r tan U cos f (3.24)

U = q cos f — r sin f C = r cos f sec U + q sin f sec U h = u sin U — v cos U sin f — w cos U cos f