Y = m[u + ur - wq + Xq (pq + r) - yQ (p + r ) + Zq (qr - p)] Z = m[w- uq + up + XQ(pr - q) + yQ(qr + p) - Zq(p2 + q2)]
K = Ixp+(Iz- Iy)qr+Ixy(pr - q) - Iyz (q - r ) - Ixz(pq+r)+m[yQ(w - uq+up~) - zQ(u+ur - wp)]
M = Iyq+(Ix- Iz) pr -1 xy (qr+p) -1yz(pq - r) - Ixz(p2 - r 2)+m[ xq (w - uq+up) - zq (u -ur+wq)] n = Izr+(Iy - Ix) pq -1 xy(p2 - q2) - Iyz(pr + q)+Ixz (qr - p)+m[ xq (u+ur - wp) - yQ(u-ur - wq)]
where, X, Y, and Z are surge, sway, and heave force; K, M, and N are roll, pitch, and yaw moment; p, q, and r are roll, pitch, and yaw rate; u, v, and w are surge, sway, and heave velocity; x, y, and z are body fixed axes in positive forward, positive starboard, and positive down; Ix, Iy, and Iz is vehicle mass moment of inertia around the x-axis, around the y-axis, and around the z-axis; Xg, yG, and zG are longitudinal position, athwart position, and vertical position of center of gravity; (j), 0, and are roll, pitch, and yaw angle.
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