NASA Technical Reports Server (NTRS) 20110002995: Rapid Calculation of Spacecraft Trajectories Using Efficient Taylor Series Integration

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by  NASA Technical Reports Server (NTRS)

Rapid Calculation of Spacecraft Trajectories Using Efficient
Taylor Series Integration
Software greatly accelerates the calculation of spacecraft trajectories.
John H. Glenn Research Center, Cleveland, Ohio
A variable-order, variable-step Taylor the step size and never requires a re- duces auxiliary variables and sets up ini-
series integration algorithm was imple- peat step. tial conditions and integrates; (2) a rou-
mented in NASA Glenn’s SNAP (Space- High-order Taylor series integration al- tine that calculates system reduced de-
craft N-body Analysis Program) code. gorithms have been shown to provide rivatives using recurrence relations for
SNAP is a high-fidelity trajectory prop- major reductions in computer time over quotients and products; and (3) a rou-
agation program that can propagate conventional integration methods in nu- tine that determines the step size and
the trajectory of a spacecraft about vir- merous scientific applications. The objec- sums the series. The order of accuracy
tually any body in the solar system. The tive here was to directly implement Taylor used in a trajectory calculation is arbi-
Taylor series algorithm’s very high series integration in an existing trajectory trary and can be set by the user. The al-
order accuracy and excellent stability analysis code and demonstrate that large gorithm directly calculates the motion
properties lead to large reductions in reductions in computer time (order of of other planetary bodies and does not
computer time relative to the code’s magnitude) could be achieved while si- require ephemeris files (except to start
existing 8th order Runge-Kutta multaneously maintaining high accuracy. the calculation). The code also runs
scheme. Head-to-head comparison on This software greatly accelerates the with Taylor series and Runge-Kutta
near-Earth, lunar, Mars, and Europa calculation of spacecraft trajectories. At used interchangeably for different
missions showed that Taylor series inte- each time level, the spacecraft position, phases of a mission.
gration is 15.8 times faster than Runge- velocity, and mass are expanded in a This work was done by James R. Scott and
Kutta on average, and is more accurate. high-order Taylor series whose coeffi- Michael C. Martini of Glenn Research Cen-
These speedups were obtained for cal- cients are obtained through efficient ter. Further information is contained in a
culations involving central body, other differentiation arithmetic. This makes TSP (see page 1).
body, thrust, and drag forces. Similar it possible to take very large time steps Inquiries concerning rights for the commer-
speedups have been obtained for calcu- at minimal cost, resulting in large sav- cial use of this invention should be addressed
lations that include J2 spherical har- ings in computer time. The Taylor se- to NASA Glenn Research Center, Innovative
monic for central body gravitation. ries algorithm is implemented prima- Partnerships Office, Attn: Steve Fedor, Mail
The algorithm includes a step size se- rily through three subroutines: (1) a Stop 4–8, 21000 Brookpark Road, Cleve-
lection method that directly calculates driver routine that automatically intro- land, Ohio 44135. Refer to LEW-18445-1.
Efficient Kriging Algorithms
Goddard Space Flight Center, Greenbelt, Maryland
More efficient versions of an interpo- communities, but is now being re- bor searching techniques were used.
lation method, called kriging, have been searched for use in the image fusion of These implementations were used
introduced in order to reduce its tradi- remotely sensed data. This allows a com- when the coefficient matrix in the lin-
tionally high computational cost. Writ- bination of data from various locations ear system is symmetric, but not neces-
ten in C++, these approaches were tested to be used to fill in any missing data sarily positive-definite.
on both synthetic and real data. from any single location. This work was done by Nargess
Kriging is a best unbiased linear esti- To arrive at the faster algorithms, Memarsadeghi of Goddard Space Flight Cen-
mator and suitable for interpolation of sparse SYMMLQ iterative solver, co- ter. For further information, contact the God-
scattered data points. Kriging has long variance tapering, Fast Multipole dard Innovative Partnerships Office at (301)
been used in the geostatistic and mining Methods (FMM), and nearest neigh- 286-5810. GSC-15555-1
Predicting Spacecraft Trajectories by the WeavEncke Method
Lyndon B. Johnson Space Center, Houston, Texas
A combination of methods is proposed combination is denoted the WeavEncke arise within that formulation, arriving at
of predicting spacecraft trajectories that method because it is based on unpub- an orbit-predicting algorithm optimized
possibly include multiple maneuvers lished studies by Jonathan Weaver of the for complex trajectory operations.
and/or perturbing accelerations, with orbit-prediction formulation of the noted In the WeavEncke method, Encke’s
greater speed, accuracy, and repeatability astronomer Johann Franz Encke. Weaver method of prediction of perturbed or-
than were heretofore achievable. The evaluated a number of alternatives that bits is enhanced by application of mod-
NASA Tech Briefs, January 2011 37