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Symmetric stereographic orientation parameters applied to constrained spacecraft attitude control

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Abstract

The full kinematic properties of a minimal set of rigid body attitude coordinates called Symmetric Stereographic Orientation Parameters (SSOPs) are developed. These coordinates result from a stereographic projection of the Euler parameter constraint hypersphere onto a three-dimensional hyper-plane. As discussed in previous work [5], this family contains the well-known classical and modified Rodrigues parameters. Considering general SSOP projection points, transformations to the Euler parameters and the direction cosine matrix are discussed. The set of three SSOP coordinates have the unique feature that the associated singularity can be placed at a desired principal rotation angle by adjusting the projection point. In contrast to the Rodrigues parameters, the SSOP coordinates do not represent a unique orientation. The impact of this non-uniqueness on the constrained spacecraft attitude control problem is discussed. An attitude feedback control law in terms of SSOPs will inherently avoid reaching this singular attitude description, and thus constrain the attitude error response to be within a well-defined cone. Lyapunov’s direct method is used to illustrate how a SSOP-based control law can be derived to drive the spacecraft attitude away from the singularity and towards a desired orientation. This control law generalizes the previously developed classical and modified Rodrigues parameter-based attitude control laws for general stereographic projection points.

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References

  1. SCHAUB, H. and JUNKINS, J. L. Analytical Mechanics of Space Systems, AIAA Education Series, Reston, VA, first edition, 2003.

    Google Scholar 

  2. MARANDI, S. R. and MODI, V. J. “A Preferred Coordinate System and the Associated Orientation Representation in Attitude Dynamics,” Acta Astronautica, Vol. 15, No. 11, 1987, pp. 833–843.

    Article  MATH  Google Scholar 

  3. SHUSTER, M. D. “A Survey of Attitude Representations,” The Journal of the Astronautical Sciences, Vol. 41, No. 4, 1993, pp. 439–517.

    MathSciNet  Google Scholar 

  4. TSIOTRAS, P. “New Control Laws for the Attitude Stabilization of Rigid Bodies,” Proceedings of the 13th IFAC Symposium on Automatic Control in Aerospace, Palo Alto, CA, September 12–16, 1994, pp. 316–321.

    Google Scholar 

  5. SCHAUB, H. and JUNKINS, J. L. “Stereographic Orientation Parameters for Attitude Dynamics: A Generalization of the Rodrigues Parameters,” The Journal of the Astronautical Sciences, Vol. 44, No. 1, January–March 1996, pp. 1–19.

    MathSciNet  Google Scholar 

  6. AKELLA, M. R. “Rigid Body Attitude Tracking without Angular Velocity Feedback,” Systems and Control Letters, Vol. 42, 2001, pp. 321–326.

    Article  MathSciNet  MATH  Google Scholar 

  7. SPINDLER, K. “Attitude Maneuvers Which Avoid a Forbidden Direction,” Journal of Dynamical and Control Systems, Vol. 8, January 2002, pp. 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  8. TSIOTRAS, P. “A Passivity Approach To Attitude Stabilization Using Nonredundant Kinematic Parameterizations,” Proceedings of the 34th IEEE Conference on Decision and Control, December 13–15, 1995, pp. 515–520.

    Google Scholar 

  9. ROBINETT, R.D., PARKER, G. G., SCHAUB, H., et al. “Lyapunov Optimal Saturated Control for Nonlinear Systems,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 20, No. 6, November2–December 1997, pp. 1083–1088.

    Article  MATH  Google Scholar 

  10. WALLSGROVE, R. J. and AKELLA, M. R. “Globally Stabilizing Saturated Attitude Control in the Presence of Bounded Unknown Disturbances,” AIAA Journal of Guidance, Control, and Dynamics, Vol. 28, No. 5, 2005, pp. 957–963.

    Article  Google Scholar 

  11. TSIOTRAS, P. “New Control Laws for the Attitude Stabilization of Rigid Bodies,” Proceedings of the IFAC Symposium on Automatic Control in Aerospace, September 122–16, 1994, pp. 833–843.

    Google Scholar 

  12. MARANDI, S. R. and MODI, V. J. “A Preferred Coordinate System and the Associated Orientation Representation in Attitude Dynamics,” Acta Astronautica, Vol. 15, 1987, pp. 833–843.

    Article  MATH  Google Scholar 

  13. STANLEY, W. S. “Quaternion from Rotation Matrix,” Journal of Guidance and Control, Vol. I, No. 3, 1978, pp. 223–224.

    MathSciNet  Google Scholar 

  14. MUKHERJEE, R. and CHEN, D. “Asymptotic Stability Theorem for Autonomous Systems,” Journal of Guidance, Control and Dynamics, Vol. 16, September2–October 1993, pp. 961–963.

    Article  MATH  Google Scholar 

  15. CRAIG, R. R.,Jr. Structural Dynamics: An Introduction to Computer Methods, John Wiley & Sons. New York, NY, 1st edition, 1981.

    Google Scholar 

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Authors and Affiliations

  1. Aerospace and Ocean Engineering Department, Virginia Tech., USA

    Charles M. Southward II (Graduate Student) & Hanspeter Schaub (Assistant Professor)

  2. Aerospace and Ocean Engineering, Virginia Tech, National Defense Science and Engineering Graduate Fellow, USA

    Joshua R. Ellis (Graduate Student)

Authors
  1. Charles M. Southward II
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  2. Joshua R. Ellis
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  3. Hanspeter Schaub
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Southward, C.M., Ellis, J.R. & Schaub, H. Symmetric stereographic orientation parameters applied to constrained spacecraft attitude control. J of Astronaut Sci 55, 389–405 (2007). https://doi.org/10.1007/BF03256531

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