Quaternion Kinematic and Dynamic Differential Equations

IEEE Transactions on Robotics and Automation

Volumn 8, Issue 1, 1992, Pages 53-64

  Chou, Jack C K ^a  

^a UNIVERSITY OF TEXAS AT DALLAS   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DYNAMICS; KINEMATICS; MATHEMATICAL TECHNIQUES - DIFFERENTIAL EQUATIONS;

DYNAMIC DIFFERENTIAL EQUATIONS; KINEMATIC DIFFERENTIAL EQUATIONS; MULTIPLICATIVE COMMUTATIVITY; QUATERNION ALGEBRA; THREE-DIMENSIONAL VECTOR SPACE; VECTOR QUATERNIONS;

MATHEMATICAL TECHNIQUES;

EID: 0026817485     PISSN: 1042296X     EISSN: None     Source Type: Journal    
DOI: 10.1109/70.127239     Document Type: Article

References (55)

1. K. W. Spring, “Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: A review,” Mechanism Machine Theory, vol. 21, no. 5, pp. 365–373, 1986.
2. J. Stuelpnagel, “On the parametrization of the three-dimensional rotation group,” SIAM Rev., vol. 6, no. 4, pp. 422–430, Oct. 1964.
3. W. R. Hamilton, Lectures on Quaternions. Dublin: Hodges and Smith, 1853.
4. __, Elements of Quaternions, 3rd ed. New York: Chelsea, 1969.
5. P. G. Tait, An Elementary Treatise on Quaternions. London: MacMillan, 1867.
6. P. Kelland and P. G. Tait, Introduction to Quaternions, with Numerous Examples London: MacMillan, 1882.
7. A. S. Hardy, Elements of Quaternions. Boston: Ginn, Health, and Co., 1881.
8. A. Cayley, “On the quaternion equation qQ-Qq' = 0,” Messenger of Mathematics, vol. XIV, pp. 108–112, 1885.
9. _, “On the matrical equation qQ — Qq' = 0,” Messenger of Mathematics, vol. XIV, pp. 176–178, 1885.
10. _, “On certain results relating to quaternions,” The Collected Mathematical Papers of Arthur Cayley, vol. I, pp. 123–126, 1889.
11. F. Klein, Elementary Mathematics From the Advanced Standpoint. New-York: Dover, 1925.
12. A. T. Yang and F. Freudenstein, “Application of dual-number quaternion algebra to the analysis of spatial mechanisms,” ASME J. Appl. Mech. E, vol. 31, no. 2, pp. 300–308, June 1964.
13. P. E. Nikravesh, “Spatial kinematic and dynamic analysis with Euler parameters,” in Computer Aided Analysis and Optimization of Mechanical System Dynamics, E. J. Haug ed. Heidelberg: Spring-Verlag, 1984, pp. 261–281.
14. G. N. Sandor, “Principles of general quaternion-operator method of spatial kinematic synthesis,” ASME J. Appl. Mech. E, vol. 35, no. 1, pp. 40–46, Mar. 1968.
15. A. C. Robinson, “On the use of quaternions in simulation of rigid-body motion,” Tech. Rep. 58–17, Aeronautical Research Lab., Wright Air Development Center, Dayton, OH, Dec. 1958.
16. P. E. Nikravesh and I. S. Chung, “Applications of Euler parameters to the dynamic analysis of three-dimensional constrained mechanical systems,” ASME J. Mech. Design, vol. 104, no. 4, pp. 785–791, 1982.
17. V. N. Koshlyakov, “Equations of a heavy rigid body that rotates about a fixed point in the Rodrigues-Hamilton parameters,” Izv. Akad. Nauk SSSR, Mekhanika Tverdogo Tela, vol. 18, no. 4, pp. 16–25, 1983.
18. Yu. N. Chelnokov, “Use of quaternions in the precessional theory of gyroscopes,” Izv. Akad. Nauk SSSR. Mekhanika Tverdogo Tela, vol. 19, no. 6, pp. 3–10, 1984.
19. J. C. K. Chou, H. K. Kesavan, and K. Singhal, “Dynamics of 3-D isolated rigid-body systems: Graph-theoretic models,” Mechanism Machine Theory, vol. 21, no. 3, pp. 261–272, 1986.
20. J. C. K. Chou, K. Singhal, and H. K. Kesavan, “Multi-body systems with open chains: Graph-theoretic model,” Mechanism Machine Theory, vol. 21, no. 3, pp. 273–284, 1986.
21. J. C. K. Chou, H. K. Kesavan, and K. Singhal, “A systems approach to 3-D multi-body systems using graph-theoretic models,” IEEE Trans. Syst. Man Cybern., vol. SMC-16, no. 2, pp. 219–230, 1986.
22. R. H. Taylor, “Planning and execution of straight line manipulator trajectories,” IBM J. Res. Develop., vol. 23, pp. 424–436, 1979.
23. R. L. Huston and F. A. Kelly, “The development of equations of motion of single-arm robots,” IEEE Trans. Syst. Man Cybern., vol. SMC-12, no. 3, pp. 259–266, May/June 1982.
24. J. C. K. Chou, G. Baciu, and H. K. Kesavan, “Graph-theoretic models for simulating robot manipulators,” inProc. IEEE Int. Conf. Robotics Automat., vol. 2 (Raleigh, NC, Mar. 31-Apr. 3, 1987), pp. 953–959.
25. B. P. Ickes, “A new method for performing digital control system amplitude computations using quaternions,” AIAA J., vol. 8, no. 1, no. 13–17, Jan. 1970.
26. G. Meyer, “Design and global analysis of spacecraft attitude control systems, NASA Tech. Rep. R-361, Mar. 1971.
27. A. C. Hendley, Quaternions for control of space vehicles,” in Proc. Inst. Navigation, National Space Meeting on Space Shuttle - Space Station - Nuclear Shuttle Navigation (Huntsville, AL), Feb. 1971.
28. J. L. Junkins and J. D. Turner, “Optimal continuous torque attitude maneuvers,” J. Guidance Control, vol. 3, no. 3, pp. 210–217, May-June 1980.
29. J. R. Werz, Ed., Spacecraft Attitude Determination and Control. Boston: Reidel, 1980.
30. T. A. W. Dwyer, “Exact nonliner control of large angle rotational maneuvers,” IEEE Trans. Automat. Control, vol. AC-29, no. 9, pp. 769–774, Sept. 1984.
31. J. T. Wen and K. Kreutz, “Globally stable control laws for the attitude maneuver problem: Tracking control and adaptive control,” in Proc. 27th Conf. Decision Control (Austin, TX, Dec. 7–9, 1988), pp. 69–74.
32. J. C. K. Chou and M. Kamel, “Quaternions approach to solve the kinematic equation of rotation, A a A x = A x Ab, of a sensor-mounted robotic manipulator,” in Proc. IEEE Int. Conf. Robotics Automat., vol. 2 (Philadelphia, PA, Apr. 24–29, 1988), pp. 2656–2662.
33. _, “Finding the position and orientation of a sensor on a robot manipulator using quaternions,” Int. J. Robotics Res., vol. 10, no. 3, pp. 240–254, June 1991.
34. J. C. K. Chou, “The solution of quadratic equations for the camera location calibration using Newton-Raphson method,” Tech. Rep. TR-91002, Computer Vision Lab., Erik Jonsson School of Engineering and Computer Sci., Univ. of Texas at Dallas, Richardson, TX, Feb. 4, 1991.
35. _, “Solving diagonalized quadratic equations for the calibration of camera orientation using Newton-Raphson method and closed-form formalae for LU factorization,” Tech. Rep. TR-91003, Computer Vision Lab., Erik Jonsson School of Engineering and Computer Sci., Univ. of Texas at Dallas, Richardson, TX, June 13, 1991.
36. B. K. P. Horn, “Closed-form solution of absolute orientation using unit quaternions,” J. Opt. Soc. Amer., vol. 4, no. 4, pp. 629–642, Apr. 1987.
37. J. Wittenburg, Dynamics of Systems of Rigid Bodies. Stuttgart: B. G. Teubner, 1977.
38. L. Brand, Vector and Tensor Analysis. New York: Wiley, 1957.
39. B. Brainerd, D. A. Clarke, M. J. Liebovitz, R. A. Rose, and G. A. Scroggie, Vector and Matrices. Toronto: Ryerson, 1967, ch. 7.
40. D. J. Foulis, Fundamental Concepts of Mathematics. Weber and Schmidt, Prindle, 1969, ch. 4.
41. O. Bottema and B. Roth, Theoretical Kinematics. Amsterdam: North-Holland, 1979.
42. R. A. Wehage, “Quaternions and Euler parameters—A brief exposition,” in Computer Aided Analysis and Optimization of Mechanical System Dynamics, E. J. Haug, Ed. Heidelberg: Spring-Verlag, 1984, pp. 147–180.
43. O. Rodrigues, “Des lois g é ometriques qui r é gissent les déplacements d'un systeme solide dans l'espace, et de la variation des coordonn é es provenant de ces déplacements consid ér és ind é pendamment des causes qui peuvent les produire,” J. Mathematiques Pures Appl., vol. 5, pp. 380–440, 1840.
44. J. W. Gibbs, Vector Analysis. New York: Dover, 1960 (republished from 2nd ed., 1909).
45. C. H. Suh and C. W. Radcliffe, Kinematics and Mechanisms Design. New York: Wiley, 1978.
46. H. Goldstein, Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, 1981.
47. R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control. Cambridge, MA: MIT Press, 1981.
48. L. T. Wang and B. Ravani, “Recursive computations of kinematic and dynamic equations for mechanical manipulators,” IEEE J. Robotics Automat., vol. RA-1, no. 3, pp. 124–131, Sept. 1985.
49. L. A. Pipes, Matrix Methods for Engineering. Englewood Cliffs, NJ: Prentice-Hall, 1963.
50. J. C. K. Chou, “A nearly-linear computational-cost scheme for the forward dynamics of an n-body pendulum,” in Proc. 3rd Ann. Conf. Aerospace Computational Control, D. E. Bernard and G. K. Man, Eds. (Pasadena, CA, Dec. 15, 1989), JPL 89–45, pp. 88–101.
51. A. R. Klumpp, “Singularity-free extraction of a quaternion from a direction-cosine matrix,” J. Spacecraft Rockets, vol. 13, no. 12, pp. 754–755, 1976.
52. J. C. K. Chou, “Graph representation and partial derivatives of successive rotations with quaternions,” Tech. Rep. TR-890011, Program in Engineering Science, Erik Jonsson School of Engineering and Computer Sci., Univ. of Texas at Dallas, Richardson, TX, Nov. 1989.
53. T. B. Huang, Engineering Mechanics. Reading, MA: Addison-Wesley, 1968.
54. K. R. Symon, Mechanics. Reading, MA: Addison-Wesley, 1971.
55. S. N. Rasband, Dynamics. New York: Wiley, 1983.