More Kinematics

Let us take stock of the progress we have made since laying out our require­ments toward expressing Eq. (9.4) in component form. Equation (9.5) provides the transformation matrix [T]ul, and Eqs. (9.8) and (9.11) deliver [T]B for the skid-to-turn missile and bank-to-tum aircraft, respectively. We build [T]UG from

[T]UG = [T]U,[T]GI (9.14)

with [T]GI, the transformation matrix of the geographic wrt inertial coordinates, given by the longitude and latitude angles (see Chapter 3).

To come to grips with the [ TBU transformation matrix, we string it out

[T]BU = [T]BV[T]VG[T]UG (9.15)

The challenge is to calculate [T]VG, the TM of the geographic velocity wrt geo­graphic coordinates. This will take several steps. We first calculate the geographic velocity v’l from the definition of the inertial velocity vf using the inertial posi­tion of the vehicle s Bi and the Euler transformation

vf = D! sBi = DesBi + ftE1sBi = vf + nE! sBi

Solve for vf and express it in geographic coordinates

[,f f = mYK]7 – [я£/] W)

From [uf]G we calculate the geographic heading and flight-path angles jrvc and 6vg, recognizing the fact that they are the polar angles of the velocity vector in geographic axes (CADAC utility MATPOL). Finally, from these angles we obtain [T]v’c (CADAC utility MAT2TR). We are now able to calculate Eq. (9.15).

In a moment we also will need

which we construct from Eqs. (9.15) and (9.5).

Back to Eq. (9.4), the angular velocity [Q. UI]U or its vector counterpart [ыи1]и is given by Eq. (9.7). Furthermore, we also need the body rates шв,]в for various modeling tasks. We build them up from

[o)B!]B = [шву]в + [a)VU]B + [coUI]B (9.17)

[шви]в is given by Eq. (9.10) for skid-to-turn missiles and by Eq. (9.13) for bank – to-turn aircraft. The second term [covu]B is the angular velocity vector of the geographic velocity frame wrt the inertial velocity frame. These two frames differ by the Earth’s angular velocity coEI. Expressed in inertial coordinates

[CO™]1 = [oF]1 =

Using Eqs. (9.16), (9.6), and (9.15), we can calculate the body rates

[(‘a)b,]b = [coBV]B + [T]BI[(oEI]’ + [T]BV[coul]u (9.18)

Rejoice! Our kinematic construction set is complete, and we can turn to the more profitable task of formulating the equations of motions.