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Real-time capable trajectory synthesis via multivariate interpolation methods for a moon landing manoeuvre


Conventional solutions to control design problems in complex space missions are often based on a solution of an associated optimal control problem. Due to limited onboard computing capacity, the computation of an optimal trajectory and a dedicated controller are commonly carried out offline, in advance of a mission. This approach requires the definition of a nominal set of initial conditions and system parameters, which are generally different from the conditions during the actual manoeuvre. This paper proposes an approach based on multivariate spline interpolation for generating nearly optimal solutions online, despite the difference between the nominal and actual conditions during the manoeuvre. The onboard computations are limited to simple interpolations and hence do not cause considerable additional computing effort. Compared to conventional control design approaches, the proposed method significantly increases the region of initial conditions and size of parametric uncertainties for which the manoeuvre can be successfully completed. The applicability and practical relevance of this approach are demonstrated for a Moon landing manoeuvre.

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  1. For \(x<a\) (\(x>b\)) use the first (last) \(k\) B-splines.

  2. As the main thrust must be piecewise constant, the approximated main thrust is determined by interpolation between the switching times.

  3. To better compare the results, the “downrange shifting approach” introduced in [7] is deactivated here.

  4. In fairness, the cubic interpolation methods are not tested here, because three optimal solutions in both directions are not enough to achieve better results than linear interpolation.

  5. All computations were performed on a personal computer with Dual Core CPU E6700 @3.20 GHz.


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Correspondence to E. Lockner.

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This paper is based on a presentation at the German Aerospace Congress, September 10–12, 2012, Berlin, Germany.

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Lockner, E., Oehlschlägel, T., Theil, S. et al. Real-time capable trajectory synthesis via multivariate interpolation methods for a moon landing manoeuvre. CEAS Space J 6, 107–118 (2014).

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  • Real-time trajectory synthesis
  • Optimal control
  • Moon landing
  • Multivariate spline interpolation
  • Optimal control problem