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NASA Contractor Report 189653, Volume IV

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Interplanetary Program To Optimize Simulated Trajectories (IPOST)

Volume W - Sample Cases

P. E. Hong, P. D. Kent, D. W. Olson, and C. A. Vallado

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Mar tin Marietta Astronautics Space Launch Systems Company Denver, Colorado

Contract NAS 1 - 18230 October 1992

NASA National Aeronautics and Space Administration

Langley Research Center Hampton, Virginia 23665-5225

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https://ntrs.nasa.gov/search.jsp?R=19930005579 2020-03-31T10:15:40+00:00Z

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TABU3 OF CONTENTS

INTRODUCTION ........................................................................ 1

APPLICATIONS ......................................................................... 6

SAMPLE CASES ......................................................................... 9 VOYAGER 2 ................................................................................ MANNED MARS ......................................................................... 36

LUNAR ORBITER ....................................................................... 80 EARTH TO JUPITER (COLLOCATION) ........................................ 77

OTHER CASES ............................................................................ 1~ TRAJECTORY SlMULATION ................................................... 118

COMPLETE VOYAGER I1 ......................................................... 1a0 H O " N TRANSFER ........................................................... 125 LOW THRUST TO ..R ....................................................... REFERENCES ............................................................................. 136

i

1.0 INTRODUCTION

m T u R E Optimization method Ontimization alporithm

The Interplanetary Program to Optimize Simulated Trajectories (POST) is intended to support many analysis phases, from early interplanetary feasibility studies through spacecraft development and operations. The POST output provides information for sizing and understanding mission impacts related to propulsion, guidance, communications, sensorlactuators, payload, and other dynamic and geometric environments.

CAPABILITY Explicit (Master/subproblems), Implicit (collocation) NPSOL

Much of the overall architecture for POST has been derived e o m the Program to Optimize Simulated Trajectories (POST). Indeed certain POST parameters and capabilities have been incorporated into POST to aid in POST-IPOST user compatibility. POST has extended trajectory capabilities to target planets and other celestial bodies with intermediate and velocity correction maneuvers. IPOST capabilities and limitations are summarized in Table 1-1.

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Maximum controls Control parameters*

25 (Master), 45 (subproblems), 1700 (collocation) Values of event rriteria, AV, arrival conditions, thrust,

Maximum targets Target parameters* Targeting method Sensitivity matrix

Maximum events

- ~- ~

Optimization Darameter*

AV magnitude, mass, time, . . .

... 25 (Master), 45 (subproblems), 1700 (collocation) Time, position, velocity, orbital conditions, . . . NPSOL, Newton-Raphson, special Onestep Finite differencing, analytic for special interplanetary targeting 100

'Event criteria* Event activities

~- -

Time, distance, speed, closest approach, . . . Info, impulsive AV, launch, orbit insertion, mass

Maximum maneuvers/ subnroblems

jettison 15

Trajectory propagation Conic, Onestep, Multiconic, Encke, Cowell, implicit Planetary bodies Sun, nine planets, Earths moon, any user-defined

Ephemeris Trajectory '

perturbations InDut/OutDut frames

- bodies Analytic, precision (JPL) Central body, perturbing bodies, radiation pressure, 52, aerodynamics, thrust EdiDtic or danet eauator. Mean 1950 or Mean 2000

I * User selectable I

Table 1 - 1. POST Featuredcapabilities

1

POST, along with members of its family, such as POST and PREP, can analyze and support almost every activity associated with space exploration (Figure 1- 1).

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POST is event driven. That is, the user defines a sequence of events which are executed in the simulation process. The events can be triggered by different criteria, such as absolute or relative time, distance from a body, or propellant consumption. At the event times, various activities can be initiated or terminated, such as employing a different thmst steering law, changing trajectory propagators or propagation step size, performing an impulsive delta velocity maneuver or jettisoning a probe or stage.

The time period between two contiguous events is called a phase. Trajectory propagation takes place in each phase. Five types of propagators are available (listed in order of increasing accuracy and decreasing computational speed): Conic, Onestep, Multiconic, Encke, Cowell. Propagator selection depends upon user needs, such as simple fast simulations for parametric feasibility analysis, or precision detailed trajectories to support subsystem design.

POST can nm a single trajectory simulation or it can run multiple simulations. For multiple simulations, one can run a parametric scan and/or an optimization mode. The search mode will vary one parameter, such as planetary amival time, over a specified interval and increment size, and perform a simulation (or optimization) for each search parameter value.

The optimization mode wi l l optimize a user cosilobjective function, such as maximum mass that can be placed in a desired orbit, subject to user-specified constraints. The constraint variables, such as periapsis altitude or orbital inclination, are called dependent variables or target parameters. The parameters which are free to vary, such as maneuver delta velocity (AV), are called independent variables or control parameters. As part of, or instead of, optimization, trajectory targeting can be performed. In this case, there is no cost b c t i o n and the POST problem reduces to finding a set of control parameter values that meet specified target parameter conditions.

Generalized targeting and optimization uses the Stanford NPSOL algorithm. For certain types of problems, a trajectory decomposition method is available. There is a master optimization process which requires that the trajectory be divided into legs or subproblems. Each subproblem is an optimization problem in itself, containing controls, constraints and an (optional) objective function. A special application of decomposition is the Interplanetary Targeting and Optimization Option (ITOO). This technique uses analytical partials generated during nominal trajectory propagation to determine minimum AV (or mass) trajectories, usually for gravity assist (swingby) missions.

In addition to the classic method of explicit optimization, there exists an option to perform implicit optimization using the collocation method. In this case, each phase is divided into independent segments which are allowed to vary subject to intersegment continuity and the equations of motion. Optimization using collocation is less sensitive to faulty initial guesses, but requires much greater CP time than explicit optimization to achieve the same level of accuracy.

4

POST input is via three namelists: $TOP, $ T W and $TAB. $TOP contains a description of the targeting and optimization problem. It must be input first. $TRAJ contains data that describes each mission evenuphase. It must follow $TOP , and there must be one $TRAJ for each event. $TAB is used to input tabular data such as thrust vs. time or drag coefficient vs. mach number and angle of attack. Input and output units are metric.

5

I 0 APPLICATIONS Examples of mission applications are shown in Table 2 - 1. They illustrate some of the diverse POST model capabilities, including lunar, interplanetary, orbital, thrusting (impulsive, low, high), and gravity assist.

The mid POST application is usually in the form of run sequences which compare various mission options. These analysis threads build on each other, culminating in a reference mission which is used to support detailed system design and analysis. Table 2 - 2 illustrates representative mission threads which exercise key POST capabilities.

For example, in the Comet Rendezvous thread, each case is actually several runs of POST to generate parametric data such that mission decisions and refinements can be made. The sequence of multiple runs per case feeds each succeeding case with each case becoming more realistic in terms of model fidelity, and encompassing more system objectives and constraints.

The first step in the Comet Rendezvous thread is to define mission requirements (necessary conditions) and goals (desired conditions), as well as any known constraints, such as technology status. In this thread, a mission requirement would be to successfdly rendezvous with a specific comet in a specific time frame. A mission goal might be to collect data on planets or bodies that are encountered during the interplanetary trajectory from Earth to the comet. A mission constraint might be the availability of a Cesium ion t h t engine powered by solar arrays, which provide limits on powerlthrust levels and on specific impulse.

The first case is an approximate impulsive AV solution using a Venus gravity assist. This examines coarse energy requirements, benefits of gravity assist, and optimum mission opportunities (launch-arrival dates with payloadllaunch mass sensitivities). The mission may be analyzed as separate phases, e.g., Earth to Venus, Venus to comet approach, rendezvous and stationkeeping.

The second case models a low thrust mission using a single thrust segment with variable steering and variable throttle. Implicit optimization (collocation) is used. This recognizes the coarseness of the initial guess, and provides rapid solution searches. A determination is made whether available technology is sufficient to provide the required payload mass at comet rendezvous.

As mission knowledge evolves, the third case introduces multiple coastlthrust segments. These added degrees of freedom provide more flexible, and more realistic, mission solutions. The optimization method can be implicit or explicit (with master-subproblems), depending upon how many, and what level, of mission decisions need to be made. This would include interplanetary and close encounter geometries, flight times, subsystem performance, etc.

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The final case provides an end-to-end precision optimized reference trajectory for system analyses and subsystem design support. Science and mission objectives can be assessed with a high degree of confidence.

Lunar Mission Thread L1 Earth departure to lunar orbit with patched conic L2 Space Station to moon with free return L3 Space Station with finite burn escape to libration point

V1 EJS portion with IT00 (analytic partials) V2 WSUN with finite differences partials

Comet Rendezvous with Solar Electric Propulsion Thread C1 Approximate impulsive AV (DSM) with Venus gravity assist. C2 Single thrust segment, simple collocation C3 Multiple thrust segments, complex collocation C4 Explicit optimization comparison

M1 EM launch/arrival date search, simple aerocapture M 2 MVE return leg optimization, Earth orbit capture M3 Mars surface ascent to orbit (MAV design) M4 EMMVE round-trip optimization

Voyager I1 Thread

Human Mission to Mars Thread

ICase 1 Bodies 1 Propagatorb) I Forces loptimization 1 I I I I method

L1 IE-Moon I Conic - IM I - ---- -.- ~- _ - ~~

L2 SS-Moon-E 1-step SB, 52 M I3 E-L1 Multiconic, DB, HT

-- I 52 = Zond(J2) SP = Solar pressure SB = Secondarv bodv

M = Master 04y I "I' = High thrust MS = Master + SuMs)

Table 2 - 2. Example Mission Threads

8

SAMPLE CASES The following sample cases illustrate detailed mission applications of IPOST. These are by no means intended to cover all capabilities nor are they realistic in every detail, but they do provide rneaningfid examples for constructing and understanding mission applications. There are 8 case discussions, four include both IPOST input and output, and the last four cases indude only IPOST input.

The first sample case is a Voyager 2 mission which illustrates a Master- Subproblem formulation. The second sample case is a manned Mars mission that shows a complete round-trip. T h e third sample case is a Lunar mission which departs from Space Station Freedom and terminates with an insertion into lunar orbit. The fourth sample case illustrates optimization using collocation for an EarthJupiter mission.

The fifth case illustrates trajectory simulation with no targeting or optimization. The sixth case is a version of Voyager 2 with subproblem optimization. The seventh case shows a classic H o h a n n transfer problem using finite thrust and collocation. The eighth case is a low thrust mission to Jupiter.

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The Voyager 2 case illustrates a master subproblem formulation and multiple planetary encounters. Only the Earth-Jupiter-Saturn phase of Voyager is performed. The simulation has 8 events, starting in Earth park orbit and ending with a Saturn flyby. Total impulsive delta V is minimized in the master problem. The two subproblems target each of two legs, Earth to Jupiter and Jupiter to Saturn.

In setting up an POST problem, the trajectory simulation is defined first, as opposed to the optimization process, because it describes the primary mission.

The first $TRAJ namelist is event 5. This utilizes parameters such as SJC mass and propulsion characteristics. The initial date of July 31, 1977 precedes the actual Voyager launch date. The SJC is placed in a circular orbit about Earth, and trajectory propagation will use conic, or two-body, equations of motion.

The second event (#lo) is triggered by a flight time of 20 days. The lSTEP propagator is activated in connection with activation of the LAUNCH mode. Hence, Voyager orbits the Earth for 20 days and then is impulsively injected onto an escape hyperbola. Using flight time as a control parameter would allow variation of initial launch date. For lSTEP, the primary body is the Sun and Earth is the secondary body.

The third event (#E) is initiated after a flight time of 20 days from the launch event. At this point, the SJC is well outside of the Earth's sphere of influence. The secondary body is now defined as Jupiter for lSTEP propagation. The reference body for trajectory calculations is also set to Jupiter.

At event 20, the triggering criteria is mission time. The intended value is Jupiter closest approach time which is not specified explicitly, but indirectly through the optimization process. A conic propagator is used with Jupiter as the primary body.

One day later, at event 23, an impulsive trajectory correction maneuver is executed. S T E P is reactivated as the propagator as the vehicle flies away from Jupiter. The combination of gravity assist and midcourse correction will set up the trajectory for the flight towards Saturn.

The fiRh event (#25) occurs 300 days after the midcourse maneuver. The secondary body for lSTEP is switched to Saturn, as is the reference body. A general rule of thumb is that planetary sphere's of influence for lSTEP (and Multiconic) are about 20 days for small inner planets (Mercury to Mars) and 200 days for large outer planets (Jupiter to Neptune).

The next event (#30) is Saturn's closest approach, on August 25, 1981. It occurs 1467 days after Earth park orbit escape.

lo

A final event (#go) is used as a "dummy" event, and coincides with the previous event. This event is needed because final event computations are done only on the "minus" side of the event. The namelist specification of "NONE" means no more input follows; that is, this is the end of POST problem specification.

We now return to the optimization definition, or $TOP namelist. The NPSOL algorithm is used with finite difference perturbations being calculated internally by NPSOL. This is the normal mode, as opposed to the user specifying perturbations. A maximum of 10 iterations is allowed for master problem optimization and 600 iterations for each subproblem targeting process. A subproblem targeting iteration is typically much shorter than a master problem iteration because the latter must solve all subproblems successfully.

The master problem controls are Jupiter closest approach time (defined as the criterion of event 20), B dot T and B dot R of the Jupiter approach, or incoming, hyperbola (defined as the 2nd and 3rd dependent variables of subproblem 1). In addition to the initial guesses for control values, the upper and lower bounds, and weighting values are important inputs. These define the performance manifold and often mean the difference between problem convergence and divergence. The objective h c t i o n is the sum of all delta V magnitudes, which in this case corresponds to the launckdescape maneuver at Earth plus the Jupiter midcourse correction.

The two subproblems are defined next. The first subproblem ends at event 20 (Jupiter closest approach) and the second subproblem ends at event 30 (Saturn closest approach). POST automatically assumes that subproblems are non- overlapping. Both subproblems use a Newton-Raphson targeting technique, as opposed to subproblem optimization with NPSOL.

The control variables for subproblem 1 are the V-infinity vector of the Earth departure hyperbola. Controls for subproblem 2 are the AV components after Jupiter flyby. As in the master problem, important inputs are the control initial guesses (USUS), bounds (INDSLB and INDSUB), and weightings (WGTS).

The constraint or target variables for subproblem 1 are time from periapsis, B dot T and B dot R at Jupiter. For subproblem 2 the constraints are time from periapsis, B dot T and closest approach distance at Saturn. The use of B dot T as a control parameter (with loose bounds) at Saturn affects orbit inclination as well as which side of Saturn the S/C flies by. B dot T and B dot R are oRen used because of their stability in the targeting and optimization process. The choice of constraint parameters for subproblem 2 reflect termination of the mission at Saturn flyby. For the actual Voyager mission, which continues on to Uranus and Neptune, the constraint parameter types at Saturn would have been identical to those at Jupiter.

Because PRINT was not input, the default value of PRINT = 0 is used. This wil l result in only summary information of the master problem, plus the final trajectory, being output. Except for well tested production runs, it is recommended that more detailed levels of print be exercised.

11

The first page of output, after the Namelist echo, summarizes POST input, including initial conditions, event summary, targeting and optimization definition, master-subproblem structure, and NPSOL options.

The next page of output completes NPSOL parameter definition and then prints an iteration summary for each major (master level) iteration. This includes the objective function value (sum of delta V) and other conditions of the optimization process. Of some interest are the number of objective b c t i o n evaluations WlXJ"FuN and the condition number of the Jacobian (COND T). This problem does not have any nonlinear constraints, only control bounds, but if it did, the value of the nonlinear constraint norm is displayed, as well as the constraint values.

As the iterations progress, certain key parameters should be monitored. These are the objective function value (which should decrease), the condition numbers of the Hessian and Jacobian (which should remain small), and the value of the nonlinear constraint norm (which should decrease). Also of interest are the convergence indicators at the right side of the summary. When all these flags are "'I"', then successfirl optimization has been achieved (according to NPSOL). One cautionary note is that reliance on a few indicators, such as the convergence flags, can be misleading. It is important to examine all measures, including the final solution.

The iteration summary continues until page 4 of the output, where a maximum iteration limit has been reached (INFORM = 4). NPSOL is exited, and the final trajectory is displayed. Conditions at each event are output, including the minus and plus sides of each impulsive maneuver. At the end of each subproblem, a final iteration summary is displayed. In this case, Subproblem 1 convergence is sllmmafized between trajectory blocks for event 15 and event 20 (Jupiter closest approach), and Subproblem 2 is summarized just before event 30 (Saturn closest approach).

The last solution has a total AV of 7.27 W s , reduced from the initial solution of 7.67 W s . Jupiter closest approach time is 711.9 days from launch, and B dot T, B dot R were selected as 1898747 km, 130453 km.

Finally, a master problem summary output is given which includes the number of iterations and CPU time. A valid solution may or may not meet all convergence criteria. Only the user, with adequate engineering experience, can make that judgment.

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35

The manned Mars case is launched from Earth park orbit, performs a Venus gravity assist, orbits into a high elliptic Mars orbit, remains in orbit for two months, then launches back to Earth where the S/C inserts into an elliptical holding orbit. The simulation has 13 events with a total mission time of 610 days. This mission is a pure targeting case with no master controls, constraints, or objective function. All targeting is done on the subproblem level. Each subproblem is full rank (number of controls = number of constraints).

To understand this case we begin with the simulation, or first $TMJ input. At event 5, the initial conditions are specified, such as a WC mass of 1000000 kg and an initial circular park orbit of 6878 radius.

The next event (#lo) performs an impulsive launch maneuver, .5 days later, to insert the WC on an Earth escape hyperbola. The lSTEP propagator is activated to account for 3 body motion (Earth-Sun-WC). The S/C will leave Earth and begin its journey toward Mars, via Venus.

At event 20,40 days after park orbit escape, the secondary body for lSTEP is changed to Venus (from Earth). The reference body (IDBODY) for target calculations is also switched to Venus.

Events 25 and 30 are simultaneous, but contiguous events. Event 25 is triggered by a mission flight time of 190 days. This is intended to correspond to Venus closest approach. Event 30 performs an impulsive midcourse correction.

At event 35,40 days after Venus closest approach, the secondary body and reference body are changed to Mars.

Event 40 and 45 are simultaneous contiguous events. Event 40 corresponds to Mars closest approach, 350 days after Earth escape. Event 45 performs an impulsive orbit insertion into a highly elliptical Mars orbit whose inclination is the same as the approach hyperbola. The propagator is switched to a simple conic. At this point subsequent events could have simulated a Mars surface descent and subsequent ascent. However, for test case simplicity, only the 60 day orbital phase is simulated.

At event 50, after a two month stay in orbit, the S/C is inserted into an escape trajectory towards Earth. The lSTEP propagator is reactivated with Mars as the secondary body (and the Sun as primary). ARer 40 days, event 60 switches the secondary body and target reference body to Earth.

36

The final 3 events (#65, #70, #75) are all simultaneous contiguous events. Event 65 corresponds to Earth closest approach after a total mission time of 610 days. Event 70 performs an impulsive orbit insertion into a highly elliptical holding orbit about Earth. The orbit might be typical of a staging orbit which transfers crew and material to and from the low Earth Space Station orbit. Event 75 exists to allow conditions aRer the orbit insertion (before event 75) to be used a~ output values.

We now return to the optimization formulation, or $TOP. No optimization algorithm was selected and only subproblem targeting wil l be done. Forward differences are specified for the finite differencing construction of the Jacobian. The maximum subproblem targeting iterations is 300.

The 3 subproblems all use the Newton-Raphson targeting algorithm. The subproblems correspond to Earth-Venus, Venus-Mars, Mars-Earth legs. The first subproblem vanes escape V-infinity to achieve desired B dot T, B dot R, TFP at Venus closest approach. The second subproblem varies Venus midcourse delta V to achieve desired closest approach distance, time, and B dot T at Mars. The third subproblem varies Mars escape V-infinity to achieve desired closest approach distance, time, and B dot T at Earth return.

The output for this case is straightforward.

The final trajectory is displayed, corresponding to each event in the specified simulation input, along with the subproblem S U ~ I X I ~ I - Y output (achieved by setting ISUB (i) = 1 in $TOP). Of particular note are the conditions before and after each maneuver. The final S/C mass at Earth return is 70404 kg, reduced from an initial mass of 1 million kg in Earth park orbit.

37

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This case examines flight from a low Earth orbit (LEO) into a low lunar orbit. It has a master-subproblem formulation, and minimizes total AV.

The simulation starts with a specified Cartesian state vector at Earth (which corresponds to circular orbit of 6878 km radius). An impulsive maneuver is identified. The conic propagator is activated with Earth as the primary body. At a mission time specified as an optimization control, an impulsive midcourse correction is applied, at which time the primary and reference bodies are switched to the Moon. 3 days after Earth orbit escape, the S/C arrives at lunar closest approach and inserts into lunar orbit.

"he optimization defines NPSOL as the optimization algorithm using finite difference perturbations computed internally by NPSOL. 50 iterations are allowed for the master problem and 200 iterations each for the 2 subproblems. The master control parameters are the Earth escape delta V components and the mission times of the midcourse maneuver and lunar closest approach.

The first subproblem varies midcourse AV to achieve desired hyperbolic conditions (closest approach distance, inclination, time fiom periapsis) at lunar arrival. The second subproblem varies lunar insertion AV, applied at hyperbolic periapsis, to achieve desired lunar orbit conditions (periapsis and apoapsis radii, and inclination).

The output echoes the namelist $TOP and all $l'RAJ namelists. A summary of the problem is given, and execution output begins with a srlmmary of each iteration of each subproblem for master iteration 0. Following the first converged iteration (13) of subproblem 2 is the s u m m a r y of master iteration 0. The sequence repeats for all master iterations. Only master iterations 0,1,2, and the final iteration (13) are shown. After the iteration summary is displayed, the final trajectory simulation is output.

The total AV was 3.968 W s , reduced fiom 4.058 WS for the initial guess. Note that the iteration summaries display the objective h c t i o n value. "he midcourse maneuver time has been moved fiom 2.39 days to 1.50 days, the lower control bound. Lunar closest approach time has been moved from 3.00 days to 3.5 days, the upper control bound.

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74

The following sample case is a portion of the Voyager mission, Earth to Jupiter phase, to illustrate collocation. Of particular interest is the difference between explicit optimization such as the test case described in Section 3.1, and implicit optimization, or collocation.

The mission starts (event 10) with a S/C mass of 2213689 kg and an input state corresponding to a post park orbit escape burn. Ideally, the initial state should produce a trajectory which leaves Earth and flies by Jupiter with the desired encounter conditions. This would be true for an accurate explicit propagator. With collocation, trajectories are defined as segments between events, and by Hermite cubic polynomials between segment end points, or nodes. In this case, the number of segments per phase is set to 1 for phases. This represents a rather poor trajectory approximation, unless the events are placed very close together. More will be said about this later. An impulsive AV maneuver is placed at this first event (ideally the AV should be zero), and at every event except the final event.

The next event (#11) occurs 14 minutes later (.01 days), where another impulsive AV is performed (again, ideally equal to zero).

4

There are 6 more events in the case (#12,13,17,18,19,20) which are planned at mission times of 0.21 days, 2.21,666.1,684.1,685.6, and 686.1 days. The last event (#20) ideally corresponds to Jupiter closest approach.

The placing of event times is an attempt to normalize accelerations within each phase. If there is a large difference in accelerations, or trajectory curvature, within a segment (= phase for this case), then the Hermite polynomials are very poor representations of a real trajectory. Inaccurate Hermites will cause divergence in the optimization process.

In this case, the optimization specifies "colloc" with "autoperts". The MXITOP array indicates that Jacobian rescaling occurs immediately after it is initially calculated by finite differencing. A maximum of 150 iterations is allowed afeer rescaling.

The mission control variables are the AV components of the impulsive maneuver at each of the first seven events. Other input control variables for the collocation formulation are specified by the U(22) array elements. These correspond to the Cartesian state vectors at each event from #11 through #20, a total of 7 (components per state) x 7 (events) = 49 elements. These state vectors were obtained from a previous explicit optimization run. In practice, the state vectors could have been obtained by a "WAG simulation run since collocation can be robust.

75

The constraints are TFP, B dot T and B dot R at Jupiter closest approach (event 20). Note that lower and upper bounds for the constraints may contain more than 3 elements each. This is not a problem because the collocation process overrides all non-mission constraints, that is, specifies upper and lower bounds for constraints introduced by the collocation process. The collocation process also computes weighting6 for the non-mission constraints.

The test case output first echoes namelist inputs. Then a summary of NPSOL parameters is displayed. The problem begins with computation of the Jacobian. POST wi l l only allow non-zero elements of the Jacobian to be numerically computed. The explicit zeroing out of null elements saves a considerable amount of CP time because a collocation Jacobian will be a sparse m a t h . In this case, there are a total of 126 controls and 108 constraint^. Of the 13608 Jacobian elements, about 9600 are theoretically non-zero.

The first (before rescaling) iteration (0) summary follows, Of critical interest is the Jacobian condition value (COND T). A value of 3.E7 before rescaling and 3000 after rescaling indicates a poorly conditioned Jacobian. The independent (control) and dependent (constraint) parameter values are displayed after each iteration summary.

The next iteration takes place and the resultant independent and dependent parameter values are displayed after the iteration 1 summary. Noteworthy items include the Jacobian condition of 40000 (a significant worsening), norm of the nonlinear constraints of 8.84 (also degraded), and an objective (total AV) of 4.40 (internal units).

Intermediate iterations are omitted to save space. The final converged iteration (#140) ie displayed. The Jacobian condition number has improved slightly, but is still bad. The norm of constraint violation is very small, indicating constraints have been met. However, this has occurred at the expense of an objective value increase to 5.19.

The final solution is displayed after convergence. In this case, a whopping 51.9 km/s total AV is required. This large AV indicates an aberrant solution. Figure 3 - la is an x-y plot of the final trajectory. This pointe out an interesting phenomenon. Even though NPSOL has found an optimal, and a "feasible", solution, as indicated by the TIT convergence flags, the solution is not realistic from an engineering perspective. A local optimum has been found. Sometimes the mathematical robustness of collocation is deceiving.

The problem in this case is the number of segments per phase. When they are increased to two segments per phase, then a solution illustrated in Figure 3 - lb occuf~. The total AV in that case is 14 W s . The input file shown can be used for two (or more) segments per phase by changing NSEGPH to more than 1. With COINTP = 'LINEAR', POST wil l interpolate linearly between the event states to supply the internal node states.

76

Unfortunately, collocation is not as simple as always increasing the number of segments per phase. As with any optimization problem, particularly one with high dimensionality such as collocation, the local performance manifold is strongly determined by weightings (independent and dependent parameters), bounds, and initial conditions. associated with collocation, and may produce misleading solutions. Caveat emptor!

These can be very tricky with the large dimensions

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Figure 3 - 1. Ecliptic View of Earth to Jupiter Mission

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117

The following cases contain only the IPOST input, and related discussion. The input corresponds to executable IPOST runs, and are shown to expand the envelope of IPOST usage.

S-

The input file described here performs explicit trajectory propagation. A Cowell, or numerical integration of the equations of motion, is emphasized. There is no targeting or optimization. This rather simple case can be used to compare propagators or effects of forces, to perform sensitivity analyses, to generate initial guesses, such as for collocation runs, or to debug a simulation.

118

C

rl u m b,

a Ll a h 0 L)

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4 4

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0)

0 . c o . . m d U U H

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n

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u d 1 6 111

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ll I1 II I1

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c u > c

.2 COMPLETE VOYAGER II This case is an extension of the Voyager 2 case (Section 3.1). The mission continues beyond Saturn, and includes flybys of Uranus and Neptune. In addition, the Newton-Raphson 111 rank subproblem targeting is replaced by NPSOL optimization for each subproblem. The master problem remains the same, although central differencing with input perturbation values is used to fonn the Jacobian.

In the subproblem setup MODELT is set to 'SUBOPT' for each subproblem. Each subproblem still has 3 controls and 3 constraints, but now target error is minimized except for the first subproblem where AV is minimized.

120

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2 m P

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-4 4

4 U U U 4 u

4 r U

P u P C

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n w . . . a w o o w 0, w o o a a w o o w C m w v ) w

c a U C

x . > . a 0 5 % 4 . r l m o .

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The Hohmann transfer problem is a classic orbital mechanics solution: The optimal two impulse solution for transferring from one circular orbit to another circular orbit in the same plane is to perform the impulses 180" apart in the direction of travel (for raising an orbit).

This case uses finite burns and collocation. Although the case assumes a low enough thrust such that "impulse" is not a good approximation, and only one segment per phase is assumed, the Hohmann transfer solution in principle is validated.

The first burn occurs in the phase between the first event (# 10) and the second event (# 20), followed by a coasting phase, and then the second burn starting at event 30 and ending at the last event (# 40). Simple thrust tables are input, in this case constant thrust (5000 newtons).

There are 7 mission controls: Thrust direction relative to velocity, throttle, and stop time of the first burn, and start time, thrust direction relative to velocity, throttle, and stop time of the second burn. Invalid guesses of the states are input at each event. There are a total of 49 additional, collocation introduced controls. The constraints are Cartesian position at the end of the second burn. Final mass is maximized.

125

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128

9 This case illustrates a nuclear powered low thrust mission from Earth to Jupiter. There is one long continuous burn. The control parameters are thrust throttle level, and pitch and yaw thrust directions (VVW relative). Constraints are TFP, and closest approach distance at Jupiter amval. S/C mass at amval is maximized.

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135

4.0 R EFERENCEe 1. "Interplanetary Program to Optimize Simulated Trajectories," Final Report,

Volumes I, 11, 111, Fitzgerald, Hong, Kent, Milleur, and Olson, Martin Marietta Corporation, March 1990.

Form Approved I REPORT DOCUMENTATION PAGE I OMB NO. 0704-0188

4. TITLE AND SUBTITLE

Tra j e c t o r i es I n t e r p l a n e t a r y Program To Optimize Simulated

Volume I V - Sample Cases

P . E . Hong, P . D . Kent , D. id. Olson, and C . A . Vallado

(IPOST)

6. AUTHOR(S)

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

5. FUNDING NUMBERS

C NAS1-18230

WU 506-49-11-02

8. PERFORMING ORGANIZATION REPORT NUMBER

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) AGENCY REPORT NUMBER

NASA CR-189653, Volume I V

N a t i o n a l Aeronaut ics and Space Adminis t ra t ion Langley Research Center Hampton, VA 23681-0001

I

11. SUPPLEMENTARY NOTES

Langley T e c h n i c a l bloni t o r : Richard IJ. Powell F i n a l Report

12a. DISTRIBUTION /AVAILABILITY STATEMENT

U n c l a s s i f i e d - Unlimited

S u b j e c t Category 16 I 13. ABSTRACT [Maximum 200 words)

12b. DISTRIBUTION CODE

IPOST is intended to support many analysis phases, from early interplanetary feasibility studies through spacecraft development and operations. The IPCSX output prosrides information €or sizing and understanding mission impacts related to propulsion, guidance, communications, sensor/actuators, payload, and other dynamic and geometric environments. IPOST models three degree of freedom trajectory events, such as launchlascent, orbital coast, propulsive maneuvering (impulsive and finite burn), gravity assist, and atmospheric entry. Trajectory propagation is performed using a choice of Cowell, Encke, Multiconic, Onestep, or Conic methods. The'user identifies a desired sequence of trajectory events, and selects which parameters are independent (controls) and dependent (targets), as well as other constraints and the coat function. algorithm. IPOST structure allows sub-problems within a master optimization problem to aid in the general constrained parameter optimization solution. optimization method uses implicit simulation and collocation techniques.

Targeting and optimization is performed using the Stanford NPSOL

An alternate

115. NUMBER OF PAGES 14. SUBJECT TERMS

C o l l o c a t i o n , t a r g e t i n g , t r a j e c t o r y propagat ion I + 16. PRICE CODE

U n c l a s s i f i e d U n c l a s s i f i e d I I Standard Form 298 (Rev 2-89) Prescr ibed by ANSI Std 239-18 298-102

NSN 7540-0 1-280-5500