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(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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 39E
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(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.
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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(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
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
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