Periodic attitudes and bifurcations of a rigid spacecraft in the second degree and order gravity field of a uniformly rotating asteroid

  • Hanlun Lei
  • Christian Circi
  • Emiliano Ortore
  • Bo Xu
Original Article


In this work, periodic attitudes and bifurcations of periodic families are investigated for a rigid spacecraft moving on a stationary orbit around a uniformly rotating asteroid. Under the second degree and order gravity field of an asteroid, the dynamical model of attitude motion is formulated by truncating the integrals of inertia of the spacecraft at the second order. In this dynamical system, the equilibrium attitude has zero Euler angles. The linearised equations of attitude motion are utilised to study the stability of equilibrium attitude. It is found that there are three fundamental types of periodic attitude motions around a stable equilibrium attitude point. We explicitly present the linear solutions around a stable equilibrium attitude, which can be used to provide the initial guesses for computing the true periodic attitudes in the complete model. By means of a numerical approach, three fundamental families of periodic attitudes are studied, and their characteristic curves, distribution of eigenvalues, stability curves and stability distributions are determined. Interestingly, along the characteristic curves of the fundamental families, some critical points are found to exist, and these points correspond to tangent and period-doubling bifurcations. By means of a numerical approach, the bifurcated families of periodic attitudes are identified. The natural and bifurcated families constitute networks of periodic attitude families.


Equilibrium attitude Families of periodic attitudes Tangent bifurcations Period-doubling bifurcations 



This work was accomplished during the first author’s visit to Sapienza University of Rome. This work is performed with the financial support of the National Natural Science Foundation (No. 11603011, 41774038), the Natural Science Foundation of Jiangsu Province (No. BK20160612) and the Visiting Scholar Program of China Scholarship Council (No. 201706195002), the Fundamental Research Funds for the Central Universities (No. 020114380024), the National Basic Research Program 973 of China (2015CB857100) and National Defense Scientific Research Fund (No. 2016110C019). The authors are much obliged to the anonymous reviewers for their insightful comments that substantially improved the quality of this paper.

Compliance with ethical standards

Conflicts of interest

The authors declared that they have no conflicts of interest to this work.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Astronomy and Space ScienceNanjing UniversityNanjingChina
  2. 2.Department of Astronautical, Electrical and Energy EngineeringSapienza University of RomeRomeItaly
  3. 3.School of Aeronautics and AstronauticsSun Yat-Sen UniversityGuangzhouChina

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