EULER angles

In principle, three different independent rotations around frame axes are necessary in order to describe any orientation of a body in space. As a first example of an orientational description by means of three rotation angles, we introduce the EULER angles concept here.

So let us define EULER angles as follows:

$$\mbox{\boldmath$R$}_{Euler} \,:=\, \mbox{\boldmath$R$}_{Eule... ...lpha)\mbox{\boldmath$R$}(y,\beta)\mbox{\boldmath$R$}(z,\gamma)$$ (2.11)

Thus we receive after multiplication:

$$\mbox{\boldmath$R$}_{Euler} \,=\, \left( \begin{array}{ccc} ... ...box{s}\beta\,\mbox{s}\gamma & \mbox{c}\beta \end{array}\right)$$ (2.12)

This notation includes the following abbreviations:

$\mbox{c}\alpha \,:=\, \cos(\alpha)$ and $\mbox{s}\alpha \,:=\, \sin(\alpha)$

The same applies for $\beta$ und $\gamma$.

So a given frame is transformed into a desired frame by rotation of the original frame around the $z$-axis first with angle $\gamma$, followed by a rotation around the $y$-axis with angle $\beta$, and finally another rotation around the actual $z$-axis with angle $\alpha$.

Equivalent to this is a rotation around the $w$-axis with angle $\alpha$, followed by a rotation around the just produced $u$-axis with angle $\beta$ and with a final rotation around the newly created $w$-axis with angle $\gamma$ ^2.1.

Derivating EULER angles from the rotation matrix $\mbox{\boldmath$R$}_{Euler}$ according to equation 2.12 leads to:

$$\begin{array}{lcl} \sin\beta & = & \sqrt{1 - R_{Euler,33}^2}... ... \frac{\pm R_{Euler,32}}{\mp R_{Euler,31}} \right) \end{array}$$ (2.13)

$R_{Euler,ij}$ can be found as matrix element in the $i$-th row and $j$-th column of matrix $\mbox{\boldmath$R$}_{Euler}$.

Because only the absolute value of '$\sin\beta$' can be determined from the given rotation matrix, and because the matrix elements in the lower row of $\mbox{\boldmath$R$}_{Euler}$ still depend on '$\sin\beta$', two alternative solutions per angle are valid within the interval of $]-\pi,\pi]$. The problem remains ambigious.

The two alternative solutions depend on each others according to the following relation:

$$\mbox{\boldmath$R$}_{Euler}(\alpha,\beta,\gamma) \,=\, \mbox{\boldmath$R$}_{Euler}(\alpha-\pi,-\beta,\gamma-\pi)$$ (2.14)

It is a special case, if $\sin\beta = 0$, because this is similar to two consecutive rotations around the $z$-axis. The rotation matrix thus becomes:

$$\begin{array}{ccl} \mbox{\boldmath$R$}_{Euler}(\alpha,0,\gam... ...alpha+\gamma) & 0 \\ 0 & 0 & 1 \end{array}\right) \end{array}$$

Rotation angles $\alpha$ und $\gamma$ describe rotations around the same axis in this special case. So they can be summed up.

Any rotation described by $\sin\beta = 0$ is called singular.

The following VRML-Example is usefull to try the behavior of the Euler angles. Note that your WWW browser must be equipped with a Java Virtual Machine.

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