1998 - Draper Laboratory

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BiographyChristopher D’Souza is a Senior Member of Technical Staff in the Guidance and ControlDirectorate. Since joining Draper in 1996, Dr. D’Souza’s technical interests have included guidancelaw design for entry and landing and GPS navigation filters, most recently for rendezvous anddocking. He received BS and MS degrees in Aeronautical and Astronautical Engineering from theUniversity of Illinois in Urbana and a PhD in Aerospace Engineering from the University of Texas atAustin.e-mail: cdsouza@draper.comChristopher D’SouzaAn Optimal Guidance Law for Planetary Landing / Biography1
An Optimal Guidance Law forPlanetary LandingChristopher N. D’Souza The Charles Stark Draper Laboratory, Inc.Based on the paper published in the Proceedings of the AIAA Guidance, Navigation, and Control ConferenceAbstractA guidance law that minimizes the commanded accelerationalong with the (weighted) final time is developed. This guidancelaw is a linear function of the states (relative to the landing point)and a nonlinear function of the time-to-go. The time-to-go isobtained as a solution to a quartic equation that is solvedanalytically. The advantage of this guidance law is that it doesnot involve any iterations whatsoever. It is the exact solution tothe two-point boundary-value problem associated with the firstvariation necessary conditions. It also satisfies the secondvariation necessary conditions for a minimum. An example of alunar landing is given to demonstrate the optimality of thisguidance law.IntroductionThere has been renewed interest in planetary exploration withthe announcement of signs of life on Mars and trapped water atthe south lunar pole. In order to confirm these scientificfindings, detailed in situ analyses will have to be performed.This will necessitate sending spacecraft equipped with landers tocarry out these inquiries. It is desirable to land a spacecraft at aparticular site of interest using the least amount of fuel. Itfollows that carrying less propellant on board would allow for alarger scientific payload.Previous investigations, including many of them at DraperLaboratory, have obtained guidance laws that are based onposing a problem that approximates the optimal control problem(Refs. [1], [2]). One of the difficulties has been solving for thetime-to-go. These studies were used to evaluate the targeting anddivert capabilities of Mars and lunar landers. They analyzed thecases of specified acceleration profiles as well as unspecifiedacceleration profiles. The landing phase was broken up intothree subphases: braking, pitchup, and vertical descent/terminallanding. The algorithms were structured similar to the ShuttlePowered Explicit Guidance scheme, which is a predictor/corrector algorithm.This paper will develop a guidance law for a soft planetarylanding at a desired target location that minimizes theacceleration (and hence the propellant required). A closed-form,easily mechanized optimal guidance law will be obtained usingthe Calculus of Variations. This law is a function of the time-togo,which is also obtained in a closed-form analytic solution.This guidance law, which solves the associated two-pointboundary value problem, does not involve any iterations.This paper presents the following:• A discussion of the dynamics and assumptionsassociated with the problem at hand.• The optimal control problem formulation.• Derivation of the first variation necessary conditionsand a closed-form guidance law dependent on time-togoand the current states.• Derivation of time-to-go from the transversalitycondition on final time, which involves the solution ofa quartic equation.• The closed-form solution of the quartic equation.• Derivation of the second variation necessaryconditions.• An example that involves a lunar landing to adesignated site, and is compared with a numericaloptimal solution.• Concluding remarks.The DynamicsFor this analysis, the following assumption will be made:h / R s
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Letter from thePresident and CEO,Vi
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Information TechnologyMilton AdamsE
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BiographyMilton Adams has been at D
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Figure 1 represents a functional de
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Programs. In effect, these controll
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Although the terminal area traffic
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Table 2. ATFM performance evaluatio
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In the experiments, a nominal capac
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[3] Wambsganss, Michael C. “Colla
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Guidance, Navigation, and Control A
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A Control Lyapunov FunctionApproach
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x( 0) ∈ X and w(t) ∈Wfor all t
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(b) Select a quadratic RCLF V i (x)
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at each grid point. In the case w 1
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References[1] Ball, J.A. and A.J. v
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Relative and Differential GPSData T
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The first term on the right in the
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H R# δρ R,GPS -H A# δρ A,GPSThi
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selection; and (3) shown that the a
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Segmentation of MR ImagesUsing Curv
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(3)where ν now represents a contin
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Experimental ResultsThe results of
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Table 1. A summary of segmentation
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Guidance, Navigation,and ControlJim
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BiographyGeorge SchmidtGeorge Schmi
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clock and ephemeris errors, as well
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maintained in a rigid structure, wh
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Table 5. “Typical” absolute GPS
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performed, then the target location
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tightly-coupled system, however, ca
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Concluding RemarksRecent progress i
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As real-time systems evolve into th
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Advanced Fault-TolerantComputing fo
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The Viking and Voyager were both in
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Containment Regions (FCRs). There a
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well as reversing the whole process
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As real-time systems evolve into th
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Automated Station-Keepingfor Satell
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Figure 2. Minimum elevation angles
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anomaly M and/or the ascending node
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However, since optimization and rec
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is maintained in the Northern Hemis
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autonomy. It must have the ability
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[31] Neelon, Joseph G., Jr., Paul J
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Draper’s primary goal is to Drape
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)Rotordynamic Modelingof an Activel
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Eq. (9) becomes:λ[ R ] { Φ } = [
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chosen to be 24, for a total of 48
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Page 131 and 132: BiographyAnthony Kourepenis is an A
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1998 - Draper Laboratory

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