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Analysis of stability and Hopf bifurcation of delayed feedback spin stabilization of a rigid spacecraft

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Abstract

The stability and bifurcation of delayed feedback spin stabilization of a rigid spacecraft is investigated in this paper. The spin is stabilized about the principal axis of the intermediate moment of inertia using a simple delayed feedback control law. In particular, linear stability is analyzed via the exponential-polynomial characteristic equations and then the method of multiple scales is used to obtain the normal form of the Hopf bifurcation. Bifurcation diagrams and the dynamics of the delayed closed-loop system are verified using continuation software and with numerical simulations.

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Acknowledgement

Financial support of from the National Science Foundation under Grant No. CMMI-1131646 is gratefully acknowledged. The authors would also like to thank Dr. Young S. Lee for his valuable help and support.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM, 88003, USA

    Morad Nazari & Eric A. Butcher

Authors
  1. Morad Nazari
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  2. Eric A. Butcher
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Correspondence to Morad Nazari.

Appendix

Appendix

1.1 Critical frequency of the characteristic equation in Sect. 3.1

Substituting for k p from Eq. (8) into Eq. (10a) on the stability boundary, the term \(\kappa_{c}^{2}\cos 2\gamma_{c} \tau\) appears which, using trigonometric identities, can be expressed as

$$\begin{aligned} \kappa_c^2\cos 2 \gamma_c \tau =&\kappa_c^2\bigl(1-2 \sin^2\gamma_c \tau\bigr) \\ =&\kappa_c^2-2(\kappa_c\sin \gamma_c\tau)^2. \end{aligned}$$
(A.1)

Solving Eq. (10b) for κ c gives

$$\begin{aligned} \kappa_c=\frac{J_2+J_3}{2}\frac{\gamma_c}{\sin\gamma_c\tau}. \end{aligned}$$
(A.2)

Substituting Eq. (A.2) into Eq. (A.1), one can obtain

$$\begin{aligned} \kappa_c^2\cos2 \gamma_c\tau =&\frac{(J_2+J_3)^2}{4}\frac{\gamma_c^2}{\sin^2\gamma_c\tau}-2 \frac{(J_2+J_3)^2}{4}\gamma_c^2 \\ =&\frac{(J_2+J_3)^2}{4}\gamma_c^2 \biggl(\frac{1}{ \sin^2\gamma_c\tau}-2 \biggr). \end{aligned}$$
(A.3)

Substituting Eqs. (A.2) and (A.3) into Eq. (10a), it becomes

$$\begin{aligned} &-\gamma_c^2+\gamma_c^2 \biggl(\frac{J_2+J_3}{J_2J_3} \biggr)\frac{J_2+J_3}{2} \\ &\qquad{}+\frac{(J_2+J_3)^2}{4J_2J_3}\gamma_c^2 \biggl(\frac{1}{ \sin^2\gamma_c\tau}-2 \biggr) \\ &\quad{}=\frac{(J_3-J_1)}{(J_1-J_2)}{J_2J_3}\varOmega_1^2, \end{aligned}$$
(A.4)

where Eq. (8) is used to substitute for α, k p , and k d in terms of κ and the parameters of the system. Therefore, the critical frequency corresponding to κ c can be obtained via

$$\begin{aligned} \gamma_c =&\varOmega_1\sqrt{\frac{(J_3-J_1)(J_1-J_2)}{-1+\frac{1}{2}(J_2+J_3)^2 [1+\frac{1}{2} (\frac{1}{\sin^2\gamma_c\tau}-2 ) ]}} \\ =&\varOmega_1\sqrt{\frac{(J_1-J_3)(J_1-J_2)}{1-\frac{(J_2+J_3)^2}{4\sin^2\gamma_c\tau}}}, \end{aligned}$$
(A.5)

or, after using Eq. (A.2), the critical frequency can be expressed in terms of κ c as

$$\begin{aligned} \gamma_c^2=\frac{1}{J_2J_3} \bigl[\kappa_c^2+\varOmega_1^2(J_3-J_1) (J_2-J_1) \bigr]. \end{aligned}$$
(A.6)

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Nazari, M., Butcher, E.A. Analysis of stability and Hopf bifurcation of delayed feedback spin stabilization of a rigid spacecraft. Nonlinear Dyn 74, 801–817 (2013). https://doi.org/10.1007/s11071-013-1006-5

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