Fa2004 16.33346 gravity Forces Gravity acts through the CoM in vertical direction(inertial frame +Z) Assume that we have a non-zero pitch angle eo Need to map this force into the body frame Use the Euler angle transformation(2-15) sIn B=T()T2On①)0= g sin重cose 9」 COsΦcos日 For symmetric steady state flight equilibrium, we will typically assume that≡60,①≡Φo=0,so SIn b== mg e Use euler angles to specify vehicle rotations with respect to the earth rame =OcosΦ-Rsin Φ=P+Qsin重tan+BcosΦtan (Qin重+Rcos重)sece Note that if更≈0,then≈Q Reca:≈Rol,≈ Pitch,andy≈ HeadingFall 2004 16.333 4–6 Gravity Forces • Gravity acts through the CoM in vertical direction (inertial frame +Z) – Assume that we have a non­zero pitch angle Θ0 – Need to map this force into the body frame – Use the Euler angle transformation (2–15) ⎡ ⎤ ⎡ ⎤ 0 − sin Θ Fg = T1(Φ)T2(Θ)T3(Ψ) ⎣ 0 ⎦ = mg ⎣ sin Φcos Θ ⎦ B mg cos Φcos Θ • For symmetric steady state flight equilibrium, we will typically assume that Θ ≡ Θ0, Φ ≡ Φ0 = 0, so ⎡ ⎤ − sin Θ0 Fg = mg ⎣ 0 ⎦ B cos Θ0 • Use Euler angles to specify vehicle rotations with respect to the Earth frame Θ˙ = Q cos Φ − R sin Φ Φ˙ = P + Q sinΦtan Θ + R cos Φtan Θ Ψ˙ = (Q sinΦ + R cos Φ)sec Θ – Note that if Φ ≈ 0, then Θ˙ ≈ Q • Recall: Φ ≈ Roll, Θ ≈ Pitch, and Ψ ≈ Heading