Power-limited spacecraft trajectory optimization in Kustaanheimo-Stiefel variables

Space technologies


Ivanyukhin A. V.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: distantstar@nm.ru


The paper concerns the power-limited spacecraft trajectory optimization in parametrical space of Kustaanheimo-Stiefel (KS) variables. There are shortly discussed the KS-variables and their main properties. The problem of optimal control is formulated as a problem with mixed constraints (equality constraints on functions of the control and state variables), which follow from the bilinear relationship and it is solved using the Pontryagin’s maximum principle.
The boundary-value problem is solved using the continuation (homotopic) method. Parameterization of the boundary value problem is presented which eliminates the need in bilinear relationship. The parameterization extends a set of admissible controls to trajectories without analogues in the physical space at the start of continuation procedure.
There is presented the numerical example showing some qualitative features of the KS-transformation such as the conformality and mapping of the physical point into a circle in the KS-space. The latter leads to the correspondence of trajectory in the physical space to the family of trajectories in the KS-space.
As an example, the problem of fixed-time transfer between fixed points of non-coplanar circular orbits is solved. There are presented trajectories both satisfying and not satisfying the bilinear relationship. Geometry of trajectories in the physical space is presented.
If an arc of initial circular orbit is assumed as an initial approximation of transfer trajectory, the use of KS-variables does not lead to the growth of computational consumptions in comparison to using the Cartesian coordinates with the same precision. In case of problems with large inclination change, the KS-variables provide better convergence in comparison with Cartesian coordinates.


Kustaanheimo-Stiefel transformation, power-limited problem, continuation (homotopic) method


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