(a) Explain the main difference between the Euler angle representation and Quaternion representation for a 3D rotation problem, and write down how a unit Quaternion representation can represent a complex rotation easily.
(a) Explain the main difference between the Euler angle representation and Quaternion representation for a 3D rotation problem, and write down how a unit Quaternion representation can represent a complex rotation easily.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(a) Explain the main difference between the Euler angle representation and Quaternion
representation for a 3D rotation problem, and write down how a unit Quaternion
representation can represent a complex rotation easily.
(b) Prove that under the assumption of very small rotations and right-hand convention,
the 3-D rotation matrix about the three fixed X-, Y-, and Z-axes can be represented as
follows:
- 0.
R= 0.
sy
1
where 0,0y,0, are the three Euler angles about the fixed X, Y, and Z - axes, respectively.
(c) Derive that the condition of geometric compatibility for a motion stage in spatial
motion is expressed as follows, under the assumption that all three Euler angles are very
small.
10 0 0
010
0 0 1
-x,
0 0 0
1
0 0 0
1
0 0 0
where the variables x, ys and z, are the translational displacements of the mass centre of
the motion stage along the fixed X-, Y-, and Z-axes, respectively, and 8sx, Osy and Os are
the rotational angles of the motion stage about the fixed X-, Y-, and Z-axes, respectively.
Also, xi, yi and z; are the translational displacements of any one point on the motion
stage along the X-, Y- and Z-axes, respectively.
Note: The local coordinates of any one point with regard to the mobile rigid body
coordinate system are x', y' and zi'. In the initial configuration, a mobile rigid-body
coordinate system O'-X'Y'Z' and a global fixed coordinate system 0-XYZ are coincident,
and both origins are at the mass centre, O', of the motion stage.
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Transcribed Image Text:(b) Prove that under the assumption of very small rotations and right-hand convention,
the 3-D rotation matrix about the three fixed X-, Y-, and Z-axes can be represented as
follows:
- 0.
R= 0.
1
1
where 0,0y,0, are the three Euler angles about the fixed X, Y, and Z- axes, respectively.
sys
Solution
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