“Euler Angles”的概念、定义、习题解题方法及翻译-数学辞典

单词

Euler Angles

释义

Euler Angles

According to Euler's Rotation Theorem, any Rotation may be described using three Angles. Ifthe Rotations are written in terms of Rotation Matrices B, C, and D, then a general Rotation A can be written as

(1)

The three angles giving the three rotation matrices are called Euler angles. There are several conventions for Euler angles,depending on the axes about which the rotations are carried out. Write the Matrix A as

(2)

In the so-called ``-convention,'' illustrated above,

			(3)
			(4)
			(5)

To obtain the components of the Angular Velocity

in the body axes, note that for a Matrix

(6)

it is true that


			(7)
			(8)

Now,

corresponds to rotation about the

axis, so look at the

component of

(9)

The line of nodes corresponds to a rotation by

about the

-axis, so look at the

component of

(10)

Similarly, to find rotation by

about the remaining axis, look at the

component of

(11)

Combining the pieces gives

(12)

For more details, see Goldstein (1980, p. 176) and Landau and Lifschitz (1976, p. 111).

The -convention Euler angles are given in terms of the Cayley-Klein Parameters by

			(13)
			(14)
			(15)

In the ``-convention,''

(16)

(17)

Therefore,

(18)

(19)

(20)

(21)

			(22)
			(23)
			(24)

and A is given by

In the ``'' (pitch-roll-yaw) convention, is pitch, is roll, and is yaw.

			(25)

			(26)

and A is given by

A set of parameters sometimes used instead of angles are the Euler Parameters , , and ,defined by

			(27)
			(28)

Using Euler Parameters (which are Quaternions), an arbitrary Rotation Matrix can be described by

(Goldstein 1960, p. 153).

If the coordinates of two pairs of points and are known, one rotated with respect to the other,then the Euler rotation matrix can be obtained in a straightforward manner using Least Squares Fitting. Write thepoints as arrays of vectors, so

(29)

Writing the arrays of vectors as matrices gives

(30)

(31)

and solving for

gives

(32)

However, we want the angles

, and

, not their combinations contained in the Matrix

. Therefore, write the

Matrix

(33)

as a

Vector

(34)

Now set up the matrices

(35)

Using Nonlinear Least Squares Fitting then gives solutions which converge to .

See also Cayley-Klein Parameters, Euler Parameters, Euler's Rotation Theorem, Infinitesimal Rotation, Quaternion, Rotation, Rotation Matrix

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 198-200, 1985.

Goldstein, H. ``The Euler Angles'' and ``Euler Angles in Alternate Conventions.'' §4-4 and Appendix B in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 143-148 and 606-610, 1980.

Landau, L. D. and Lifschitz, E. M. Mechanics, 3rd ed. Oxford, England: Pergamon Press, 1976.

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