Tech Briefs

Simulation study is an integral part of the validation of navigation algorithms for spacecraft. While it is possible to come up with an estimate of a navigation algorithm’s performance with a low-fidelity system model, the mathematical analysis is intractable for higher-fidelity models that include fuel slosh, flexible booms, sensor saturation, etc. Thus simulation study is a vital step in validating navigation algorithms before an actual satellite is launched.

High-fidelity modeling of spacecraft often focuses on stochastic (random) phenomena. Many navigation sensors, such as angular rate gyros and accelerometers, are known to exhibit stochastic processes such as random walk noise. The goal of a good navigation algorithm is to then reduce the impact of these noise sources on the navigation solution, where the solution may include attitude, angular rate, orbital position, velocity, etc. The estimate of a navigation algorithm’s performance, which is based on the accuracy of the numerical simulation, is then used to verify if the sensor and actuator hardware are sufficient to meet requirements.

At its essence, a simulation is nothing more than a set of differential equations and a numerical integrator. Those differential equations are often very complicated as they model orbital dynamics, sensor functionality, and even the inner workings of flight software, but these are still differential equations integrated by a numerical integrator to produce a simulation of the satellite.

Classic numerical integration techniques are known to work well for deterministic differential equations, but have surprisingly poor performance at preserving the statistical properties when integrating stochastic differential equations (SDEs). Preliminary study has shown that these inaccuracies can lead to gross underestimation of navigation filter performance that can lead to procurement of sensing hardware that is more accurate and more expensive than needed. The performance of numerical integrators commonly used in NASA GSFC’s dynamics analysis divisions when integrating SDEs will be studied, and their performance compared with statistically justified SDE numerical integrators.

The performance of various techniques when integrating stochastic differential equations, such as a high-fidelity differential equation model of a gyro, will be compared with several stochastic differential equation numerical integrators’ performances. The study will provide recommendations on situations where a SDE numerical integrator is necessary and when a typical numerical integrator will suffice.

This work was done by Joseph M. Galante of Goddard Space Flight Center. GSC-17017-1

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Advanced Numerical Integration Techniques for High-Fidelity SDE Spacecraft Simulation (reference GSC-17017-1) is currently available for download from the TSP library.

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