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Axis angle

Terminology of rigid body dynamics

This entry is made by Compilation and application of scientific encyclopedia entries of "Science Popularization China" to examine.

Axis angle, rigid body dynamics term, is crystal Midknot Crystal axis Between included angle , generally expressed as α, β and γ.

Chinese name: Axis angle
Foreign name: Axis-Angle

Disciplines involved: mechanical engineering
Theory involved: Rigid body dynamics

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Schematic diagram of shaft angle

The English name of shaft angle is: axial angle; optic(-axial) angle

Rotation axis angle^[1] Means parameterized with two values rotate : One axis Or straight line, and one that describes the amount of rotation around this axis horn 。 It is also called the exponential coordinate of rotation.

Sometimes called Rotation vector Because these two parameters (axis and angle) can be used in the model Is a vector of the rotation angle.

Axis angle is being processed Rigid body dynamics The time is convenient. It is useful for characterizing rotation and for transforming between different representations of rigid body motion.

3 crystal axes in the crystal (triangular or Hexagonal system When four crystal axes are selected, their d-axis is not included) the intersection angle between each other^[2] 。 Represented by α, β and γ, they correspond to the included angle between the positive ends of b-axis and c-axis, c-axis and a-axis, and a-axis and b-axis in turn. According to the symmetry characteristics, only Monoclinic system β angle and Triclinic system All three axis angles of the crystal vary with the crystal and are characteristic constants of the crystal; The axial angles of other crystal systems and α and γ of monoclinic crystal systems are determined special angles

Sometimes called Rotation vector Because these two parameters (axis and angle) can be used in the model Is a vector of the rotation angle.

example

If you stand on the ground, choose the direction of gravity as negative z Direction. If you turn left, you will go around z Axis rotation radians

(or 90 degrees). In the axis angle representation, this will be

This can be expressed as an indication z The module of direction is

The rotation vector of.

Contact with other representations

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Indicates rotation^[3] There are many ways. It is important to understand the differences between them and how to transform them.

Exponential mapping

From the axis angle of rotation to Rotation matrix Transform Usage for Exponential mapping 。

Essentially, by using Taylor expansion You can draw a closed form relationship between these two representations. Give an axis

And angle

, the equivalent rotation matrix is given as:

R here is 3x3 Rotation matrix and Cap operator Give and Cross product Corresponding to multiplicand Antisymmetric matrix Operator.

Logarithmic mapping

To obtain Rotation matrix The axis angle of is expressed, and the angle of rotation is calculated

And then use it to find the shaft

Quaternion

To transform from an axis coordinate to a quaternion, use the following expression:

Given a unit quaternion, the following expressions can be used to extract axis angular coordinates:

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