Euler Angles

Euler angles are three angles that specify a general rotation as a product of three rotation matrices. According to Euler's rotation theorem, any rotation may be described in this way. If the rotations are written as matrices , , and , then a general rotation can be written as

The rotation matrices below describe passive rotations in a right-handed coordinate system. They act on column vectors, so gives the coordinates of a fixed vector in the rotated coordinate system. The corresponding active rotation of a vector in a fixed coordinate system is described by the transpose . In the illustrations, faint axes show the coordinate system before each rotation, while solid axes show it afterward. The yellow sector marks the angle of each rotation, and subscripts 0, 1, 2, and 3 distinguish the successive coordinate systems. There are several conventions for Euler angles, depending on the axes about which the rotations are carried out. Write the matrix as

The so-called "-convention," illustrated above, is the most common definition. It uses successive rotations of the coordinate system about the -, -, and -axes. These are intrinsic rotations, meaning that each axis moves with the coordinate system after each step. In this convention, the Euler angles specify

1. the first rotation is by an angle about the z-axis using ,

2. the second rotation is by an angle about the x-axis after the first rotation using , and

3. the third rotation is by an angle about the z-axis after the second rotation using .

The equivalent extrinsic description uses rotations about fixed axes in the reverse order, with angles , , and about the -, -, and -axes, respectively.

Note, however, that several notational conventions for the angles are in common use. Goldstein (1980, pp. 145-148) and Landau and Lifschitz (1976) use , Tuma (1974) says is used in aeronautical engineering in the analysis of space vehicles (but claims that is used in the analysis of gyroscopic motion), while Bate et al. (1971) use . Goldstein remarks that continental authors usually use , and warns that left-handed coordinate systems are also in occasional use (Osgood 1937, Margenau and Murphy 1956-64). Varshalovich (1988, pp. 21-23) uses the notation or to denote the Euler angles, and gives three different angle conventions, none of which corresponds to the -convention.

Here, the notation is used. In the Wolfram Language, the passive rotation matrix is given by EulerMatrix[-psi, -theta, -phi, 3, 1, 3]. The corresponding active rotation matrix is given by EulerMatrix[phi, theta, psi, 3, 1, 3]. To apply the passive rotation to a list of three-dimensional graphics primitives g, use GeometricTransformation[g, m], where m is the matrix . In the -convention, the component rotations are then given by

so

When the Euler angles depend on time, overdots denote derivatives with respect to time. To obtain the components of the angular velocity in the body axes, note that for a matrix

it is true that

Now, corresponds to rotation about the axis, so look at the component of ,

The line of nodes corresponds to a rotation by about the -axis, so look at the component of ,

Similarly, to find rotation by about the remaining axis, look at the component of ,

Combining the pieces gives

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

The "-convention", illustrated above, describes successive intrinsic rotations about the -, -, and -axes. Its passive rotation matrix can be obtained from the -convention by the substitutions

Therefore,

The component passive rotation matrices for these intrinsic rotations are

and is given by

In the intrinsic -- (yaw-pitch-roll) convention illustrated above, is yaw, is pitch, and is roll. The successive rotations are about the initial z-axis, the y-axis after the first rotation, and the x-axis after the second rotation. The component passive rotation matrices are

and is given by

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

Using Euler parameters (which are quaternions), an arbitrary rotation matrix can be described by

(Goldstein 1980, p. 153).

If the coordinates of two sets of points and are known, with one obtained exactly from the other by a rotation about the origin, then the Euler rotation matrix can be obtained using least squares fitting. Write the points as arrays of vectors, so

Writing the arrays of vectors as matrices gives

If has matrix rank 3, then is invertible, and solving for gives

For noisy data, this formula gives an unconstrained least squares fitting solution, which need not be a rotation matrix. A rotation matrix must be an orthogonal matrix with determinant 1. However, we want the angles , , and , not their combinations contained in the matrix . Therefore, write the matrix

as a column vector

Now set up the matrices

These derivatives can be used in an iterative nonlinear least squares fitting procedure to estimate the angles , , and . Convergence depends on the initial estimates and the fitting method.