Table Of ContentRapid 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
ern numerical methods. Among these ments for accuracy, reproducibility, and This work was done by Jonathan K.
methods are efficient Kepler’s-equation efficiency (and, hence, speed). Self-start- Weaver of Johnson Space Center and Daniel
time-of-flight solutions and self-starting ing numerical integration also supports R. Adamo of United Space Alliance. For fur-
numerical integration with time as the fully analytic regulation of integration ther information, contact the JSC Innovation
independent variable. Self-starting nu- step sizes, thereby further increasing Partnerships Office at (281) 483-3809.
merical integration satisfies the require- speed while maintaining accuracy. MSC-23802-1
An Augmentation of G-Guidance Algorithms
This augmented algorithm can be used in small-body proximity operations utilizing model
predictive control with a need for safety from surface-constraint uncertainty.
NASA’s Jet Propulsion Laboratory, Pasadena, California
The original G-Guidance algorithm gation of any unexpected trajectory or a control policy (feedback), along with
provided an autonomous guidance and state changes that occur during standard sensors to monitor actual spacecraft
control policy for small-body proximity mode operations. state. The feedback is designed to en-
operations that took into account uncer- In order to have the G-Guidance algo- sure that the spacecraft stays within a
tainty and dynamics disturbances. How- rithm detect an unsafe condition, it re- specified proximity to the feedforward.
ever, there was a lack of robustness in re- quired some modification. This modifi- The feedforward is designed to achieve
gards to object proximity while in cation provides a policy to safely the goals of each mode: hover for safety
autonomous mode. The modified G- maneuver the spacecraft between its cur- mode and maneuver toward target for
Guidance algorithm was augmented rent state and a desired target state while standard mode. By giving the spacecraft
with a second operational mode that al- ensuring satisfaction of thruster and tra- the ability to re-compute its trajectory
lows switching into a safety hover mode. jectory constraints, along with safety con- on-the-fly in response to local condi-
This will cause a spacecraft to hover in straints. In standard mode, this modifica- tions, minimization of fuel usage is pro-
place until a mission-planning algorithm tion brings the spacecraft from its vided. The original G-Guidance algo-
can compute a safe new trajectory. No current position closer to its target state. rithm provides robustness to uncertainty
state or control constraints are violated. In safety mode, the algorithm maintains affecting the dynamics. The safety aug-
When a new, feasible state trajectory is the spacecraft’s current state at zero ve- mentation provides a form of state-con-
calculated, the spacecraft will return to locity. Since the safety mode is designed straint robustness, which further miti-
standard mode and maneuver toward to be temporary, the destination location gates risk.
the target. The main goal of this aug- in this mode is also temporary, and once This work was done by John M. Carson
mentation is to protect the spacecraft in a new destination location is provided, III and Behcet Acikmese of Caltech for
the event that a landing surface or obsta- the spacecraft returns to standard mode. NASA’s Jet Propulsion Laboratory. For more
cle is closer or further than anticipated. The G-Guidance algorithm uses both information, contact iaoffice@jpl.nasa.gov.
The algorithm can be used for the miti- a planned trajectory (feedforward) and NPO-46452
Comparison of Aircraft Icing Growth Assessment Software
The goal is to provide software that can predict ice growth under any condition for
any aircraft surface.
John H. Glenn Research Center, Cleveland, Ohio
A research project is underway to pro- The Icing Branch at NASA Glenn has The project addresses the validation
duce computer software that can accu- produced several computer codes over of the software against a recent set of ice-
rately predict ice growth under any me- the last 20 years for performing icing shape data in the SLD regime. This vali-
teorological conditions for any aircraft simulation. While some of these tools dation effort mirrors a similar effort un-
surface. An extensive comparison of the have been collaborative projects, most dertaken for previous validations of
results in a quantifiable manner against have been developed primarily by one LEWICE. Those reports quantified the
the database of ice shapes that have person, with some assistance by others. ice accretion prediction capabilities of
been generated in the NASA Glenn The state of computing has also the LEWICE software. Several ice geom-
Icing Research Tunnel (IRT) has been changed dramatically in that time pe- etry features were proposed for compar-
performed, including additional data riod. As these codes have grown in com- ing ice shapes in a quantitative manner.
taken to extend the database in the plexity and have been accepted by users The resulting analysis showed that
Super-cooled Large Drop (SLD) regime. as production icing tools, there has LEWICE compared well to the available
The project shows the differences in ice arisen a need for the developers to ad- experimental data.
shape between LEWICE 3.2.2, Glen- here to standard software practices used The effects of super-cooled large
nICE, and experimental data. to develop commercial software. droplets in icing have been researched
38 NASA Tech Briefs, January 2011
