The orientation of a body‐fixed coordinate system can be obtained by rotating the inertia coordinate system in space three times.

II.1 First Rotation

If the bases before rotation are , the bases after rotation are , the rotation axis is , and the rotation angle is θ₁, substituting b = i₁, a = i and λ = i into Equation (A.8) gives

(A.9)

where , and similarly hereinafter.

Substituting , and into Equation (A.8), we get

(A.10)

Substituting , , and into Equation (A.8), we get

(A.11)

So

(A.12)

(A.13)

II.2 Second Rotation

The bases before rotation are , the bases after rotation are , the rotation axis is , and the rotation angle is θ ₂. Similar to Equation (A.9), substituting these parameters into Equation (A.8), and considering

we get

(A.14)

namely

(A.15)

(A.16)

II.3 Third Rotation

The bases before rotation are , the bases after rotation are , the rotation axis is and the rotation angle is θ ₃. Similar to Equation (A.9), substituting these parameters into Equation (A.8) and considering

we get

(A.17)

namely

(A.18)

So we get the coordinate transformation matrix from the body‐fixed coordinate system to the inertia coordinate system

(A.19)