Framework for the full N-body problem in SE(3) and its reduction to the circular restricted full three-body problem

Morad Nazari, David Canales, Brennan McCann, Eric Butcher, Kathleen Howell

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


A novel, compact formalism of rigid body motion dynamics is presented in a general reference frame based on the geometric mechanics framework. This formalism, proposed on the special Euclidean space (SE(3)) of the Lie group, naturally accounts for the orbit/attitude coupling due to the gravitational moments and forces. It is demonstrated that the structure of the rigid body dynamics equations is preserved in different coordinate frames. Then, using the expression of gravitational potential, this global framework is applied to the circular restricted full three-body problem (CRF3BP) of a near-rectilinear halo orbit (NRHO) similar to that of Gateway, where the equations are uniquely provided in the body frame of the spacecraft and the barycentric rotating frame. For the sake of propagation and consistency with the classical circular restricted three-body problem (CR3BP), the equations of the CRF3BP are also presented in a non-dimensional form. Trajectories computed with different inertia tensors are compared with those obtained using the traditional equations of motion of a point-mass spacecraft subject to the gravitational fields of two larger primaries. The comparison between CRF3BP and CR3BP suggests: (a) the need for computation of customized families of halo orbits considering inertia properties of each spacecraft; and (b) the use of attitude-only control for station-keeping in future NRHO missions to reduce station-keeping costs which would otherwise be relatively large if the point-mass approximation were used.

Original languageEnglish (US)
Article number41
Issue number4
StatePublished - Aug 2023


  • Circular restricted three-body problem (CR3BP)
  • Geometric mechanics
  • Multi-body dynamical systems
  • Near-rectilinear halo orbit
  • SE(3)

ASJC Scopus subject areas

  • Modeling and Simulation
  • Mathematical Physics
  • Astronomy and Astrophysics
  • Space and Planetary Science
  • Computational Mathematics
  • Applied Mathematics


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