AAS 04-238 A Variable-Step Double-Integration Multi-Step Integrator

A variable-step double-integration multi-step integrator is derived using divided differences. The derivation is based upon the derivation of Shampine-Gordon, a single-integration method. Variable-step integrators are useful for propagating elliptical orbits, because larger steps can be taken near apogee. As a double-integration method, the integrator performs only one function evaluation per step, whereas Shampine-Gordon requires two evaluations per step, giving the integrator a significant speed advantage over Shampine-Gordon. Though several implementation issues remain, preliminary results show the integrator to be effective. INTRODUCTION Use of numerical integration in space surveillance has grown in recent years as accuracy requirements have increased. Numerical integration requires a great deal of computation time compared to the analytic propagators previously used. An upgrade planned for the Navy’s Space Surveillance System (known as the Fence) will greatly increase the number of objects being tracked, and hence significantly increase the amount of computation time required. Numerical integration methods requiring less computation time than those currently employed while maintaining or improving accuracy requirements are needed to reduce this burden. The Gauss-Jackson method (Ref. 1) is frequently used for orbit propagation in space surveillance applications. It is a predictor-corrector, multi-step integrator. Multi-step integrators have a considerable advantage over single-step integrators (such as Runge-Kutta) because multi-step integrators can take larger step sizes to yield a given accuracy, and also have fewer evaluations per step. The Gauss-Jackson integrator performs double integration, meaning that the algorithm computes position directly from acceleration, without first integrating to find velocity. Because velocity is also needed to compute the state of the satellite and to compute atmospheric drag, Gauss-Jackson is generally implemented alongside an Adams integrator, which performs single integration. The Adams integrator is also a multi-step integrator, and can be implemented with Gauss-Jackson without requiring additional evaluations. Because Gauss-Jackson is fixed step, it cannot efficiently handle highly elliptical orbits (Ref. 2). In order to achieve a given accuracy at perigee, more steps are taken at apogee than needed. One way to remedy this problem is with s-integration, which involves changing the independent variable using a generalized Sundman transformation (Ref. 3) to another variable, s. This transformation spreads the integration points more evenly about the orbit and can be considered an analytical step size adjustment based on advance knowledge of the orbit and not on error analysis. A disadvantage ∗Graduate Assistant, Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, Virginia 24061. E-mail: [email protected]. †Research Physicist, Naval Research Laboratory, Code 8233, Washington, DC 20375-5355. E-mail: [email protected].