Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Coordinate transformations
- 3 Manipulator kinematics
- 4 Forward kinematic analysis
- 5 Reverse kinematic analysis problem statement
- 6 Spherical closed-loop mechanisms
- 7 Displacement analysis of group 1 spatial mechanisms
- 8 Group 2 spatial mechanisms
- 9 Group 3 spatial mechanisms
- 10 Group 4 spatial mechanisms
- 11 Case studies
- 12 Quaternions
- Appendix
- References
- Index
12 - Quaternions
Published online by Cambridge University Press: 12 September 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Coordinate transformations
- 3 Manipulator kinematics
- 4 Forward kinematic analysis
- 5 Reverse kinematic analysis problem statement
- 6 Spherical closed-loop mechanisms
- 7 Displacement analysis of group 1 spatial mechanisms
- 8 Group 2 spatial mechanisms
- 9 Group 3 spatial mechanisms
- 10 Group 4 spatial mechanisms
- 11 Case studies
- 12 Quaternions
- Appendix
- References
- Index
Summary
Rigid-body rotations using rotation matrices
In Chapter 2 it was shown how to represent the position and orientation of one coordinate system relative to another. Further, it was shown how to transform the coordinates of a point from one coordinate system to another.
The techniques introduced in Chapter 2 can also be used to define the rotation of a rigid body in space. Any rigid body can be thought of as a collection of points. Suppose that the coordinates of all the points of a body are known in terms of a coordinate system A. The body is then rotated γ degrees about a unit vector m that passes through the origin of the A coordinate system. The objective is to determine the coordinates of all the points in the rigid body after the rotation is accomplished (see Figure 12.1).
This problem is equivalent to determining the coordinates of all the points in a rigid body in terms of a coordinate system B that is initially coincident with coordinate system A but is then rotated – γ about the m axis vector (see Figure 12.2). This problem was solved in Chapter 2 using rotation matrices.
An alternate solution using quaternions will be introduced in this chapter. Quaternions in many instances may represent a more computationally efficient method of computing rotations of a rigid body compared to the rotation matrix approach. An increase in computational efficiency implies that fewer addition and multiplication operations are required.
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- Information
- Kinematic Analysis of Robot Manipulators , pp. 381 - 399Publisher: Cambridge University PressPrint publication year: 1998