Flight dynamics conventions

Frame transformation matrices

The frame transformation matrix from a frame "Framei" to a frame "Framef" is defined as the matrix transforming vector coordinates relative to "Framei" to coordinates relative to "Framef":

For example, the transformation matrix for a rotation of angle θ around the X axis is such that:

Note: The rotation matrix that transforms the basis vectors of Framei into the basis vectors of Framef is the transpose (or inverse) of P(i->f).

Quaternions

The quaternion of the rotation from a frame "Framei" to a frame "Framef" is defined by:

q = [cos(θ/2); sin(θ/2) * axis], where:

- θ is the angle of the rotation that transforms the basis vectors of Framei into the basis vectors of Framej,

- axis is the vector of components of the rotation axis of the same rotation in Framei.

The image (v) of a vector u by the rotation defined by q is then: V = q * U * q', where q' is the conjuquate of q, and U is the quaternion [0; u], and V is the quaternion [0; v].

Offsets between time scales

The convention used is the following:

Let's assume that t1 represents the time of some event relative to time scale 1, and t2 represents the time of the same event relative to time scale 2. Then the notation t1_t2 simply represents t1 minus t2.

Examples:

• ut1_tai (function argument):

  Time of some event in UT1 time scale minus time of the same event in TAI time scale.

• %CL_TT_TREF (CelestLab predefined variable):

  Time of some event in TT time scale minus time of the same event in TREF time scale.