QuickLoris said:
Thanks! That was very helpful.
I'm still wondering though, about how the angles are derived straight from the matrix. In the book it has the example of a global roll-pitch-yaw rotation matrix
GQB = QZ,γQY,βQX,α
=
[cβcγ -cαsγ+cγsαsβ sαsγ+cαcγsβ ]
[cβsγ cαcγ+sαsβsγ -cγsα+cαsβsγ ]
[-sβ cβsα cαcβ ]
A word of caution: Your book must use a different convention than do I. For a roll-pitch-yaw (XYZ) sequence, I use
<br />
\begin{bmatrix}<br />
\cos\beta\cos\gamma &<br />
\cos\alpha\sin\gamma+\sin\alpha\sin\beta\cos\gamma &<br />
\sin\alpha\sin\gamma-\cos\alpha\sin\beta\cos\gamma \\<br />
-\cos\beta\sin\gamma &<br />
\cos\alpha\cos\gamma-\sin\alpha\sin\beta\sin\gamma &<br />
\sin\alpha\cos\gamma+\cos\alpha\sin\beta\sin\gamma \\<br />
\sin\beta &<br />
-\sin\alpha\cos\beta &<br />
\cos\alpha\cos\beta<br />
\end{bmatrix}
I'm not saying your book is wrong. It's just convention, the distinction boiling down to <br />
\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta<br />
\end{bmatrix} versus <br />
\begin{bmatrix}<br />
\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta<br />
\end{bmatrix}<br /> Using my convention (I'll just make a mess of things if I use your book's convention), let's take a look at that roll/pitch/yaw sequence but treat the upper right 2x2 submatrix as noise:
\begin{bmatrix}<br />
\cos\beta\cos\gamma & \cdots & \cdots \\<br />
-\cos\beta\sin\gamma & \cdots & \cdots \\<br />
\sin\beta & -\sin\alpha\cos\beta & \cos\alpha\cos\beta<br />
\end{bmatrix}
Here's my convention for a roll/yaw/pitch (XZY) sequence, reducing some of the elements to ellipses (...):
\begin{bmatrix}<br />
\cos\beta\cos\gamma & \cdots & \cdots \\<br />
-\sin\beta & \cos\alpha\cos\beta & \sin\alpha\cos\beta \\<br />
\cos\beta\sin\gamma & \cdots & \cdots<br />
\end{bmatrix}
Finally, here's the astronomical ZXZ sequence:
\begin{bmatrix}<br />
\cdots & \cdots & \sin\beta\sin\gamma \\<br />
\cdots & \cdots & \sin\beta\cos\gamma \\<br />
\sin\alpha\sin\beta & -\cos\alpha\sin\beta & \cos\beta<br />
\end{bmatrix}
Note that there is always some key row that involves just the first two angles of the sequence, a key column that involves the last two angles only, and a key element at the intersection of that key row and column that involves the middle angle only. The reason is pretty simple. When written as a product of three matrices, the leftmost matrix will always have one trivial row that comprises a one and two zeros, while the rightmost matrix will always have one trivial column that comprises a one and two zeros. The trivial row in the leftmost matrix and trivial column in the rightmost matrix dictate the key row, key column, and key element in the product. For example, for an XYZ sequence it is the third row, first column; for a XZY sequence it is the second row, first column; and for a ZXZ sequence it is the third row, third column. That key element is \sin\beta, -\sin\beta, or \cos\beta. The first two pertain to aerodynamics sequences (XYZ, XZY, ...), the sign depending on whether the sequence is an even or odd permutation of XYZ. The value of \cos\beta pertains to the six astronomical sequences (ZXZ, ...).