Elsevier

Automatica

Volume 38, Issue 4, April 2002, Pages 571-583
Automatica

Halo orbit mission correction maneuvers using optimal control

https://doi.org/10.1016/S0005-1098(01)00279-5Get rights and content

Abstract

This paper addresses the computation of the required trajectory correction maneuvers for a halo orbit space mission to compensate for the launch velocity errors introduced by inaccuracies of the launch vehicle. By combining dynamical systems theory with optimal control techniques, we are able to provide a compelling portrait of the complex landscape of the trajectory design space. This approach enables automation of the analysis to perform parametric studies that simply were not available to mission designers a few years ago, such as how the magnitude of the errors and the timing of the first trajectory correction maneuver affects the correction ΔV. The impetus for combining dynamical systems theory and optimal control in this problem arises from design issues for the Genesis Discovery Mission being developed for NASA by the Jet Propulsion Laboratory.

Section snippets

The Genesis Mission

The Genesis Discovery Mission is a solar wind sample return mission (see Lo et al., 1998). It is one of NASA's first robotic sample return missions and is scheduled for launch in the summer of 2001 to a halo orbit in the vicinity of the Sun–Earth L1 Lagrange point; L1 is one of the five equilibrium points in the circular restricted three-body problem (CR3BP). Fig. 1 shows a three dimensional view of the Genesis trajectory.

In the standard convention, L1 is the unstable equilibrium point between

Optimal control for trajectory correction maneuvers

We now introduce the general problem of optimal control for the spacecraft trajectory planning problem. We start by recasting the TCM problem as a spacecraft trajectory planning problem. Mathematically they are exactly the same. We discuss the spacecraft trajectory planning problem as an optimization problem and highlight the formulation characteristics and particular solution requirements. Then the loss in fuel efficiency caused by possible perturbation in the launch velocity and by different

Numerical results

Circular restricted three-body problem: As mentioned earlier, we use the e.o.m. derived under the CR3BP assumption as the underlying dynamical model. In this model, it is assumed that the primaries (the Earth and Sun in our case) move on circular orbits around the center of mass of the system and that the third body (the spacecraft) does not influence the motion of the primaries. We write the e.o.m. in a rotating frame, as in Fig. 3.

Using nondimensional units, the e.o.m. in the CR3BP model areẋ

Conclusions and future work

This paper explores new approaches for automated parametric studies of optimal trajectory correction maneuvers for a halo orbit mission. Using the halo orbit insertion approach, we found optimal recovery trajectories for all the launch velocity errors and TCM1min considered. The cost functions (fuel consumption in terms of ΔV) are within the allocated budget even in the worst case (largest launch velocity error and TCM1min).

Using the stable manifold insertion approach, we obtained similar

Acknowledgements

This work was carried out at the Computational Science and Engineering Group at the University of California, Santa Barbara, the Jet Propulsion Laboratory and the California Institute of Technology. The work was partially supported by the Caltech President's fund, the NASA Advanced Concepts Research Program, The Genesis Project, NSF grant KDI/ATM-9873133 and AFOSR Microsat contract F49620-99-1-0190.

Radu Serban was born in Sibiu, Romania in 1968. He received his M.Sc. in Aerospace Engineering from the Polytechnic Institute in Bucharest, Romania in 1992. In 1998 he received his Ph.D. in Mechanical Engineering from the University of Iowa. From 1998 to 2001 he worked as a postgraduate researcher in the Computational Science and Engineering group at the University of California Santa Barbara. In July 2001 he joined the Center for Applied Scientific Computing at Lawrence Livermore National

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Radu Serban was born in Sibiu, Romania in 1968. He received his M.Sc. in Aerospace Engineering from the Polytechnic Institute in Bucharest, Romania in 1992. In 1998 he received his Ph.D. in Mechanical Engineering from the University of Iowa. From 1998 to 2001 he worked as a postgraduate researcher in the Computational Science and Engineering group at the University of California Santa Barbara. In July 2001 he joined the Center for Applied Scientific Computing at Lawrence Livermore National Laboratory. His research topics include optimal control and optimization, sensitivity analysis, and multibody dynamics.

Wang Sang Koon is a Senior Postdoctoral Scholar of Control and Dynamical Systems at the California Institute of Technology. He received his Ph.D. in 1997 at the University of California, Berkeley and was awarded the Bernard Friedman Memorial Prize for the best Ph.D. Thesis in Applied Mathematics. He has done research in the area of geometric mechanics such as optimal Control for holonomic and nonholonomic mechanical systems with symmetry. His current research interests are in applied dynamics and control theory, especially in the study of formation flight for micro-satellites, and the design of optimized space missions which employ solutions of the three or more body problem.

Dr. Martin W. Lo is a member of the technical staff in the Navigation and Mission Design Section of the Jet Propulsion Laboratory, California Institute of Technology. Lo received his Ph.D. from Cornell University and his B.S. from the California Institute of Technology in mathematics. As Mission Design Manager, he led the development of the trajectory for the Genesis Mission which recently launched and is on its way to L1. He is currently leading the development of LTool, JPL's newest mission design tool which uses dynamical systems techniques to design highly nonlinear trajectories. The Genesis Mission is the first user of LTool. He is the organizer of the Lagrange Group, an international group of researchers and engineers from universities, NASA centers, and industry whose focus is on the development of nonlinear astrodynamics techniques with applications to space missions. His interests includes mission design, the three body problem, dynamical systems theory, satellite constellation coverage analysis, dynamical astronomy, and computational mathematics.

Jerrold E. Marsden is a Professor of Control and Dynamical Systems at the California Institute of Technology. He received his B.Sc. at Toronto in 1965 and his Ph.D. at Princeton University in 1968, both in Applied Mathematics. He has done research in the area of geometric mechanics, with applications to rigid body systems, fluid mechanics, elasticity theory, plasma physics and general field theory. He is one of the founders of reduction theory for mechanical systems with symmetry. His current research interests are in applied dynamics and control theory, especially how these subjects relate to mechanical systems and systems with symmetry.

Dr. Linda R. Petzold is currently Professor in the Departments of Mechanical and Environmental Engineering, and Computer Science, and Director of the Computational Science and Engineering Program, at the University of California Santa Barbara. She was awarded the Wilkinson Prize for Numerical Software in 1991, and the Dahlquist Prize, for numerical solution of differential equations, in 1999. She is currently Vice President at Large of SIAM, the Society for Industrial and Applied Mathematics. Her research interests include numerical ordinary differential equations, differential-algebraic equations, and partial differential equations, dynamic optimization, nonlinear model reduction, mathematical software and scientific computing.

Shane D. Ross earned a B.S. in physics from the California Institute of Technology in 1998. He is currently pursuing a Ph.D. in Control and Dynamical Systems from the same university. His research interests include material transport in the solar system and the application of dynamical systems and optimal control methods to the design of fuel optimized space missions which employ solutions of the three or more body problem.

Dr. Roby S. Wilson is a member of the technical staff in the Navigation and Mission Design section of the Jet Propulsion Laboratory, California Institute of Technology. He received his B.S. in 1991, MS in 1993, and Ph.D. in 1998, all in Aeronautical and Astronautical Engineering from Purdue University. He has been involved in the development of the Genesis trajectory since 1997, and has worked extensively on the design and re-design of the mission. His interests involve trajectory design in the three and four body problems, especially those involving multiple lunar encounters. He is currently the trajectory analyst for the Genesis spacecraft operations at JPL.

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Kok Lay Teo under the direction of Editor Tamer Basar. This research was supported in part by the NSF-KDI grant number NSF ATM-9873133, and by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.

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