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A direct formula for the conversion from a quaternion to Euler angles in any of the 12 possible sequences exists. [2] For the rest of this section, the formula for the sequence Body 3-2-1 will be shown. If the quaternion is properly normalized, the Euler angles can be obtained from the quaternions via the relations:
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
The Euler parameters can be viewed as the coefficients of a quaternion; the scalar parameter a is the real part, the vector parameters b, c, d are the imaginary parts. Thus we have the quaternion = + + +, which is a quaternion of unit length (or versor) since
In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T 3 to the real projective space RP 3, which is the same as the space of rotations for three-dimensional rigid bodies, formally named SO(3)) is not a local homeomorphism at every point, and thus at some points the rank (degrees ...
In Van Elfrinkhof's formula in the preceding subsection this restriction to three dimensions leads to p = a, q = −b, r = −c, s = −d, or in quaternion representation: Q R = Q L ′ = Q L −1. The 3D rotation matrix then becomes the Euler–Rodrigues formula for 3D rotations
Rotation formalisms are focused on proper (orientation-preserving) motions of the Euclidean space with one fixed point, that a rotation refers to.Although physical motions with a fixed point are an important case (such as ones described in the center-of-mass frame, or motions of a joint), this approach creates a knowledge about all motions.
The equation presented for conversion from Euler angles to Quaternion has several discontinuities that are not necessarily present in the Quaternions themselves. For instance, for the Euler angles (0,0,-180) and (0,0,180), the conversion would produce the quaternions (0,0,0,1) and (0,0,0,-1).
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