# Rotation Kinematics

• Reza N. Jazar

## Abstract

Consider a rigid body B with a local coordinate frame Oxyz that is originally coincident with a global coordinate frame OXYZ. Point O of the body B is fixed to the ground G and is the origin of both coordinate frames. If the rigid body B rotates a degrees about the Z-axis of the global coordinate frame, then coordinates of any point P of the rigid body in the local and global coordinate frames are related by the following equation
$${}^Gr = Q_{Z,\alpha } {}^Br$$
(2.1)
where
$${}^Gr = \left[ {\begin{array}{*{20}c} X \\ Y \\ Z \\ \end{array} } \right]$$
(2.2)
$${}^Br = \left[ {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right]$$
(2.3)
and
$$Q_{Z,\alpha } = \left[ {\begin{array}{*{20}c} {\cos \alpha } & { - \sin \alpha } & 0 \\ {\sin \alpha } & {\cos \alpha } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right].$$
(2.4)

## Keywords

Rigid Body Coordinate Frame Rotation Matrix Euler Angle Local Frame
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

1. Buss, S. R., 2003, 3-D Computer Graphics: A Mathematical Introduction with OpenGL, Cambridge University Press, New York.
2. Cheng, H., and Gupta, K. C, 1989, A historical note on finite rotations, Journal of Applied Mechanics, 56, 139–145.
3. Coe, C. J., 1934, Displacement of a rigid body, American Mathematical Monthly, 41(4), 242–253.
4. Denavit, J., and Hartenberg, R. S., 1955, A kinematic notation for lowerpair mechanisms based on matrices, Journal of Applied Mechanics, 22(2), 215–221.
5. Hunt, K. H., 1978, Kinematic Geometry of Mechanisms, Oxford University Press, London.
6. Mason, M. T., 2001, Mechanics of Robotic Manipulation, MIT Press, Cambridge, MA.Google Scholar
7. Murray, R. M., Li, Z., and Sastry, S. S. S., 1994, A Mathematical Introauction to Robotic Manipulation, CRC Press, Boca Raton, Florida.Google Scholar
8. Nikravesh, P., 1988, Computer-Aided Analysis of Mechanical Systems, Prentice Hall, New Jersey.Google Scholar
9. Niku, S. B., 2001, Introduction to Robotics: Analysis, Systems, Applications, Prentice Hall, New Jersey.Google Scholar
10. Paul, B., 1963, On the composition of finite rotations, American Mathematical Monthly, 70(8), 859–862.
11. Paul, R. P., 1981, Robot Manipulators: Mathematics, Programming, and Control, MIT Press, Cambridge, Massachusetts.Google Scholar
12. Rimrott, F. P. J., 1989, Introductory Attitude Dynamics, Springer-Verlag, New York.
13. Rosenberg, R., M. 1977, Analytical Dynamics of Discrete Systems, Plenum Publishing Co., New York.
14. Schaub, H., and Junkins, J. L., 2003, Analytical Mechanics of Space Systems, AIAA Educational Series, American Institute of Aeronautics and Astronautics, Inc., Reston, Virginia.Google Scholar
15. Suh, C. H., and Radcliff, C. W., 1978, Kinematics and Mechanisms Design, John Wiley & Sons, New York.Google Scholar
16. Spong, M. W., Hutchinson, S., and Vidyasagar, M., 2006, Robot Modeling and Control, John Wiley & Sons, New York.Google Scholar
17. Tsai, L. W,, 1999, Robot Analysis, John Wiley & Sons, New York.Google Scholar

## Copyright information

© Springer Science+Business Media, LLC 2007

## Authors and Affiliations

• Reza N. Jazar
• 1
1. 1.Department of Mechanical EngineeringManhattan CollegeRiverdale