these maneuvers in order to sample the (u; v) plane for a collection of stars. Interferometry .... which means that any algorithm for finding the optimal tour will have a ... Figure 5 gives a description of a map that will be used as graphical repre- sentation of the tour order used in the 1-D test case. A polar plot idea is used,.
Submitted to the 2000 AIAA Guidance, Navigation, & Control Conference
Fuel Saving Strategies for Separated Spacecraft Interferometry Christopher A. Bailey Hernandez Engineering, Houston, Texas 77058 Timothy W. McLainy Randal W. Beardz Brigham Young University, Provo, Utah 84602 Abstract
Separated spacecraft interferometry missions will require spacecraft to move in a coordinated fashion to ensure minimal and balanced consumption of fuel. This paper develops strategies for determining interferometry mission plans that result in signi cant fuel savings over standard approaches. Simulation results demonstrate that valuable reductions in fuel consumption can be realized by combining the retargeting and imaging maneuvers required to image multiple stellar sources. Fuel-optimal imaging strategies have been developed for twospacecraft interferometry missions similar to the proposed Space Technology 3 mission using chained local optimization methods. Based on these strategies, sampling pattern guidelines for space-borne interferometry missions have been developed.
1 Introduction Space-based optical interferometry has been identi ed by NASA as one of the key technologies in furthering the scienti c exploration of the universe in the Engineer, Flight Dynamics Department Assistant Professor, Department of Mechanical Engineering. Member AIAA. Assistant Professor, Department of Electrical and Computer Engineering. Member AIAA. y z
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next century. NASA's Origins Program will use the fundamentals of interferometry to form a sophisticated space-based telescope as the primary tool for future space missions to image stars and distant planetary systems with new levels of accuracy. This paper develops fuel saving schemes for separated spacecraft stellar interferometry missions. The objective is to minimize the amount of propellant mass required to perform the mission objectives while ensuring that no spacecraft in the formation is left with insucient fuel to complete the mission. Interferometers image stellar targets in a signi cantly di erent way than conventional telescopes, which use a single large primary mirror. An optical interferometer samples parallel wavefronts at two or more locations and combines the light to produce an interference pattern. From these interference patterns or fringes, high angular resolution images of the source can be constructed. Optical interferometry requires a precisely controlled separation between the light collecting apertures. The resolution of the interferometer is proportional to the largest possible baseline separation. An overview of long-baseline stellar interferometry is given in Ref. 1. Shao and Colavita give a comprehensive technical review of ground based interferometry and the advantages of space based interferometry. Joshi provides an excellent introduction to interferometry and o ers a technical discussion on how images are constructed from interferometric instrument interference patterns. Figure 1 shows an example of the relationship between the source and observation plane. The spacecraft collect observations in the form of interference fringes from the source at locations given by ( ; ) and ( ; ) in the observation plane. Joshi explains that the van Cittert-Zernike result is a two-dimensional Fourier transform and is the basis for interferometric imaging. An irradiance pattern I (x; y), that forms the image of the source S can be constructed from this transform by 2
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I (x; y ) = (u; v ) is
Z 1Z 1
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(u; v )ei2(ux+vy) du dv:
called the complex mutual coherence function. Points in the (u; v)
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η
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Figure 1: Synthetic aperture imaging scenario plane are mapped from locations of the spacecraft in the (; ) plane by 4 ? 2 u= 1 z 4 1 ? 2 v= ; z
where is the wavelength of emitted light and z the distance to the source. The mutual coherence function is calculated from amplitude and phase measurements made in the observation plane (; ) for each point in the (u; v) plane. Due to the symmetry properties of Fourier transforms, celestial targets that are spherical (e.g., a single star) only require the sampling of a radial line in the (u; v) plane. These targets are called 1D targets. Alternatively, non-spherical targets (e.g., a binary star) require sampling of half of the (u; v) plane, and are termed 2D targets. Throughout this paper stellar targets are assumed to be referenced in the celestial sphere. Separated spacecraft formations o er an intriguing platform for spacebased interferometry. As illustrated in Figure 2, interferometry requires three basic formation maneuvers. The rst is retargeting. This involves realigning the light collecting apertures with a new celestial target. To do so, the 4
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a) Retarget
c) Resize
b) Reorient
Figure 2: Repositioning maneuvers for space-based optical interferometry: a) Retarget formation to a new star, b) Reorient formation about observation direction, c) Resize the formation. spacecraft are rotated about some point in space to redirect the optical bore sight of the interferometer at the new source. The next is reorienting the formation about the bore sight of the interferometer. This provides a degree of freedom used for sampling points in the (u; v) plane at a constant baseline separation between spacecraft. The last is resizing, where the formation maintains the same observation direction while spacecraft move to change the baseline separation. A typical mission will involve a combination of these maneuvers in order to sample the (u; v) plane for a collection of stars. Interferometry requires that the baseline separation for the spacecraft be maintained to within very tight tolerances (fraction of a wavelength of light). Laser metrology systems are used to measure the spacecraft separations to this level of accuracy. Initializing the sensor can be a costly procedure in terms of fuel and time. Therefore it is sometimes desirable to perform spacecraft formation maneuvers that maintain the formation throughout the maneuver. Another strategy is to allow the spacecraft to break formation during the maneuver, thus requiring a laser metrology re-initialization. In this paper, we will assume that the spacecraft can break formation during each maneuver, however the techniques extend easily to the sensor-lock case. Retarget, reorient, and resize maneuvers all require active thrusting and therefore expend on-board propellant, making fuel the major limitation to the length of the mission. It is therefore imperative that formation maneu4
vers be performed in a way that minimizes the expended fuel. An additional complication however, is that the useful lifetime of the mission is limited by the rst spacecraft to run out of fuel. It may turn out that in performing fuel minimal retarget maneuvers, one of the spacecraft will burn fuel at a signi cantly faster rate than the others. An additional goal is therefore to perform formation maneuvers such that the fuel on each spacecraft is equalized. It turns out that fuel minimization and fuel equalization are competing objectives. For individual retarget, resize, and reorient maneuvers, it is possible to derive fuel minimizing/equalizing maneuvers. The general fuel minimization/equalization problem has been solved for individual retarget and reorientation maneuvers in Ref. 5 for constrained (i.e., sensor-lock) maneuvers, and in Ref. 6 for unconstrained maneuvers. In Ref. 7 a solution to the fuel minimization/equalization problem for resize maneuvers in the context of the planned ST3 mission is given. Equalizing or minimizing the fuel for individual maneuvers may be thought of as a local fuel optimization problem. In this paper we are primarily concerned with the global fuel optimization problem, i.e., given that each individual formation maneuver can be performed in a fuel minimizing/equalizing manner, how should the maneuvers be sequenced to minimize/equalize the overall fuel usage? We will explore two reasonable alternatives: The rst alternative is to nd the fuel optimal sequence of retarget maneuvers, assuming that once the formation has been retargeted to a single star, that all (u; v) points will be sampled for that star (through a sequence of reorient and resize maneuvers), before retargeting to the next star. We call this approach the targeting maneuver optimization (TMO) strategy. As will be shown in Section 5, if the angular separation between two stars is small, then considerable fuel saving can be obtained by mixing (u; v) points from one star with those of another. Therefore the second approach that we will pursue will be to solve the optimization problem allowing arbitrary mixing between reorient, resize, and retarget maneuvers. In other words, from a fuel point of view it may be better to collect a couple of (u; v) samples from star #1, retarget, collect a couple (u; v) samples from star #2, and then retarget back to star #1, to collect more (u; v) samples for that star. Our results in Section 5 show cases when this is true. We call this approach the targeting and imaging maneuver optimization (TIMO) strategy. Unfortunately, by allowing retarget, reorient, and resize maneuvers to be 5, 6
5, 6
5
mixed, the solution space of the problem is increased considerably. Since the resulting optimization problem is NP-hard, this can present a problem. Fortunately however, our simulation results show some general patterns for how retarget, reorient, and resize maneuvers can be mixed to achieve considerable fuel savings. These patterns can be used to derive simple search heuristics to aid the optimization algorithm.
2 Literature Review A signi cant level of research activity on space-based interferometry has been maintained in recent years. This section o ers a brief survey of the current literature on spacecraft formation control with speci c application to spacebased interferometry. An overview of the New Millennium separated spacecraft interferometer Deep Space 3 (DS3) mission is given in Refs. 8{10. Budget limitations required changes to be made to the Deep Space 3 mission. The changes re ected in the new concept of the renamed Space Technology 3 mission are described in Ref. 11. Sensors designed for space-based interferometry are described in Refs. 12{ 15. A general review of emerging thruster technologies is given in Ref. 16. Preliminary requirements for the propulsion system with respect to required time to complete maneuvers, propellant mass, and capability to perform ne attitude control are addressed in Refs. 17,18. Various strategies have been explored for formation control. Leaderfollowing strategies have been developed in Refs 19{21. A behavioral-based approach appears in Ref. 22, 23. In Ref. 24 formation rotation controls are developed using adaptive control, considering the presence of actuator saturation and small changes in the mass properties of the spacecraft. A new architecture for the coordinated control of spacecraft formations is introduced in Ref. 25 which includes leader-following, behavioral, and virtual structure approaches. In Ref. 26 formation control laws are derived for spacecraft with nite on/o thrusters. A framework for linear control of relative spacecraft position based on adaptive control is given in Refs. 27.
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3 Space Technology 3 While the methods and results of this paper are applicable to general spacebased interferometry missions, to be concrete, we will develop results with respect to the planned Space Technology 3 (ST3) mission. This section gives a brief overview of the ST3 mission. ST3 will be a separated spacecraft interferometer consisting of two spacecraft: a combiner and a collector, which will be launched together into an earth-trailing, heliocentric orbit. The mission will consist of three phases, a formation ying experiment mode, a short baseline monolithic interferometer mode, and a separated spacecraft observation mode. During this last mode, extending over the period of three months, the spacecraft will be separated by distances ranging from 50 meters up to one kilometer with interferometer baselines ranging from 40 to 200 meters. The combiner will will combine and collect fringe data for the deconvolution of astrophysical images. The collector will act solely as a light collecting aperture. It is currently estimated that the system will be designed to image 50 stars during this period. Optical interferometry requires the path lengths of incoming star light to be equal within fractions of the wavelength of visible light. Since only two spacecraft are available to collect light from a single source and then accurately combine the light to make fringe patterns, the spacecraft must be located such that the light path lengths are equal. Therefore the position of the spacecraft will be constrained such that the combiner is at the focus of a parabola, with the collector located on the parabola. Figure 3 shows the layout of the ST3 formation geometry. Notice the incoming starlight paths and where the spacecraft are located with respect to the reference parabola. The combiner with an internal delay line of two times the focal length is positioned at the focus of a parabola and the collector is located along the parabola to provide equal starlight path lengths. The baseline is the distance between the collector and the path of incoming starlight to the combiner. The baseline is always parallel to the stellar wavefront or perpendicular to the incoming starlight path. The system parameters in Table 1 have been de ned for the ST3 separated spacecraft interferometry mission. Notice that the two spacecraft have equal thrust capabilities but the combiner has more mass. 11
7
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Figure 3: Layout of the ST3 two-spacecraft interferometer concept
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Table 1: ST3 physical system and optical interferometry parameters interferometry focus length maximum interferometry baseline minimum interferometry baseline mass of combiner mass of collector maximum thrust of combiner maximum thrust of collector speci c impulse
10 meters 200 meters 40 meters 350 kg 250 kg 0.009 N 0.009 N 65 seconds
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4 Technical Approach This section describes our approach to the the fuel optimization problem. A formation con guration corresponds to a particular relative position and orientation between the combiner and the collector, and a direction of the bore sight of the interferometer in the celestial sphere. For each star, there is a formation con guration for each (u; v) point that needs to be imaged. A nite amount of fuel is required to transition between any two con gurations. In this paper we have assumed that the spacecraft are allowed to break formation during maneuvers. To compute the fuel cost between two maneuvers, we have used the results reported in Ref. 6. Alternatively, we could have used the results reported in Ref. 5 to compute the fuel costs for the case when the spacecraft are not allowed to break formation during the maneuver. Fuel costs between every possible pair of con gurations were computed and tabulated in a cost matrix. The results in Refs. 5, 6 allow a tradeo between performing fuel minimizing and fuel equalizing maneuvers for each transition. We have chosen to maneuver in such a way that fuel is equalized during each maneuver. This ensures that fuel remains equally distributed across the spacecraft throughout the planned mission. An alternative would have been to minimize the fuel at each maneuver and then run the global optimization such that the overall fuel is equalized at the end of the mission. However, this approach would preclude changes to the mission after it had begun, so we chose the other approach. After the cost matrix has been computed, it is required to nd a mini9
mizing sequence, or tour, between the di erent con gurations. This problem is a traveling salesman problem (TSP), which is well-known to be NP-hard, which means that any algorithm for nding the optimal tour will have a worst-case running time that is greater than any polynomial on the order of the number of tour stops. There is a tremendous literature devoted to solving the TSP. The book written by Lawler, Lenstra, Rinnooy Kan & Shmoys provides a good introduction to the TSP and some of the methods applied to its solution. Reinelt o ers excellent insight into choosing heuristics for nding solutions, provides a good review of present and developing methods, and gives case study comparisons for chosen TSP instances. Ref. 31 is a book on the subject of local search combinatorial optimization heuristics addresses a broad range of local search methods and associated topics. To solve the TSP problem, we have used and algorithm developed by Martin, Otto & Felten called chained local optimization (CLO). The CLO algorithm combines simulated annealing with local search heuristics which have been shown to produce good results for the TSP. Of course, any TSP solver could have been used. 29
30
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33{36
5 Results Optical interferometry missions involve three primary types of maneuvers (retarget, resize, and reorientation) to ll out the (u; v) plane for each stellar source. One-dimensional imaging involves only retarget and resize maneuvers, while two-dimensional imaging involves all three. A simple fuel saving strategy is to separate retargeting moves from imaging moves (resize and reorient) and to optimize the order in which stars are imaged. The objective is to nd the best tour order so that fuel use is minimized. In this process, only the order of retargeting moves is considered. It is assumed that all imaging moves are performed one star at a time. This targeting maneuver optimization (TMO) strategy is a traveling salesman problem and is amenable to solution by discrete optimization algorithms such as CLO. Figure 4 shows an optimal tour through 15 stars with their observation directions depicted on the celestial sphere. Orthographic views are presented to aid in visualization. As would be expected for an optimal tour, the path between stars winds along the surface of the celestial sphere rather than jumping through the interior of the sphere. Several strategies for 10
determining the optimal observation order of stellar sources are described in Ref. 7. 11 12
Scheme I top 5 13
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Figure 4: Observation order results for TMO
Targeting and Imaging Maneuvers
Optimizing the observation tour order can yield signi cant fuel-saving bene ts over an ad-hoc approach. Clearly, a fuel saving strategy for an interferometry mission considering both targeting maneuvers and imaging maneuvers could result in greater reductions than considering targeting maneuvers alone. In this section, we consider not only the targeting maneuvers for pointing the interferometer, but also all of the imaging maneuvers used to ll the (u; v) plane. These fuel minimization strategies are termed targeting and imaging maneuver optimization (TIMO) strategies. Two types of imaging operations 11
are considered with TIMO: 1-D tours involve retargeting and resizing moves, while 2-D tours involve retargeting, resizing, and reorienting moves.
1-D Imaging
Table 2 shows the di erent 1-D imaging scenarios. In each case three stars, separated by an angle , are imaged. Three di erent values for are considered (=16, =8, and =4 radians). Baseline separation distances of 10 and 20 meters are considered. Table 2: Scheme IV 1-D test case data
scenario 1D1 1D2 1D3 1D4 1D5
[radians] baselines [meters] =16 =8 =16 =8 =4
40, 50, 60, 70 40, 50, 60, 70 40, 60, 80, 100 40, 60, 80, 100 40, 60, 80, 100
Figure 5 gives a description of a map that will be used as graphical representation of the tour order used in the 1-D test case. A polar plot idea is used, where the three radial lines represent the three observation directions. White circles along the lines represent interferometer baseline separation distance, with the baseline increasing moving away from the origin. The CLO algorithm was used to determine the tour order of pointing and resizing moves that would result in minimal fuel consumption for the 1-D test cases. The cost of the tour produced by each scenario has been compared to a benchmark tour through the same scenario. For the benchmark tour, each star is fully imaged before moving to the next star. Results for scenarios 1D1 and 1D2 are given in Figure 5. A polar plot is used depict the results, where the three radial lines represent the three observation directions. White circles along the lines represent interferometer baseline separation distance, with the baseline increasing moving away from the origin. The left-hand plot shows the fuel usage results from the benchmark tour, while the right-hand plot shows the best CLO tour with its fuel consumption for each scenario. For scenario 1D1, the number of more costly baseline moves is minimized by the CLO algorithm. For scenario 1D2, the 12
result is the same as the benchmark where it is better to minimize the number of retarget operations. The di erence between the two scenarios is the size of the retarget angle. These results suggest that when making 10 meter baseline changes, targeting moves will be mixed with sampling moves when the retarget angle is =16 or less. Benchmark 0
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Figure 5: Tour map for scenarios 1D1 and 1D2 Results for scenarios 1D3, 1D4, and 1D5 are given in Figure 6. These scenarios require baseline changes of 20 meters. Again the di erence between the scenarios is the size of the retarget angle. These results suggest that when making 20 meter baseline changes, targeting moves will be mixed with sampling moves when the retarget angle is =8 or less. Notice that the greatest fuel savings is realized for closely clustered observations such as the 1D3 scenario. This is expected since the pattern used in the benchmark tour is optimal for large angles between observation directions as can be seen in the map of scenario 1D5. From these results, it is clear that when separation angle between stars 13
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Figure 6: Tour map for scenarios 1D3, 1D4, and 1D5
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is large, it is best to point at a star and perform all of the baseline resizing moves before moving on to the next star. Conversely, when the separation angle between stars is relatively small, less fuel is consumed when the baseline is held xed and all of the stars are targeted before changing the baseline. Guidelines for how these 1-D moves can be sequenced in the best way can be developed by comparing the maximum retarget and resize costs. The maximum retarget cost, RTmax , is de ned as the cost to retarget to the next closest star when the formation is at its maximum baseline. The maximum resize cost, RSmax , is de ned as the cost to resize the formation from its maximum baseline to the next smallest baseline. Because the spacecraft are constrained to move along the reference parabola to resize the formation, the maximum resize cost will result when the spacecraft are at the largest baseline separation. The maximum retarget and resize cost for scenarios 1D1-5 have been calculated and are shown in Table 3 with the cost inequalities. These cost inequalities suggest the priority of maneuvers to construct the fuel optimal tour. Table 3: 1-D test case maximum retarget and resize costs
scenario 1D1 1D2 1D3 1D4 1D5
RTmax RSmax
0.0099 0.0139 0.0138 0.0194 0.0272
0.0112 0.0112 0.0185 0.0185 0.0185
priority
RSmax > RTmax RTmax > RSmax RSmax > RTmax RTmax > RSmax RTmax > RSmax
Figure 7 shows an example of how a 1-D imaging scenario might be constructed using only the maximum retarget and resize costs (RTmax and RSmax ) as a guide. The numbers inside of the circles represent the tour order. Gray shaded circles indicate that the next maneuver is a retarget. By comparing, RTmax and RSmax , a priority ranking can be given to the lower cost maneuver. When it is more costly to retarget than resize (RTmax > RSmax ), as shown in illustration 1 of this gure, a fuel minimal tour should resize until it must retarget. In this case, a fuel optimal tour results when the number of more costly moves (retargets) is minimized. There must be at least two 15
retargets to complete the mission. Alternatively, when resizing is more expensive than retargeting (RSmax > RTmax ) as shown in illustration 2, the tour should retarget to each star in the cluster before changing baselines. A minimum of three resize maneuvers are required as shown in illustration 2 of Figure 7. 1)
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Figure 7: Sampling pattern guide for 1-D imaging When the sampling pattern guidelines are applied to the ve 1-D test cases, the results are identical to those obtained by the CLO algorithm with the exception of case 1D4. The guide tour from scenario 1D4 was more costly than the tour found with CLO. From Table 3 it can be seen that for scenario 1D4, RTmax is only slightly larger than RSmax. Thus, the sampling guidelines suggest that the number of baseline resize moves should be minimized. Because the guidelines only consider the maximum retarget or resize cost and not the total cost of all of the possible retarget or resize moves, it is possible that when RTmax and RSmax are close in value, the guidelines may not lead to the optimal fuel savings. It should be noted that under these conditions, the CLO tour and guideline tour fuel usage di er only slightly.
2-D Imaging
The 2-D test case scenarios were designed to investigate fuel saving strategies when retargeting, reorienting, and resizing moves are involved. The test case scenarios investigated are shown in Table 4. 16
Table 4: Scheme IV 2-D test case data
scenario 2D1 2D2 2D3 2D4
[radians] baselines [meters] orientations [radians] =8 =4 3=8 =2
40, 60, 80 40, 60, 60 40, 60, 80 40, 60, 80
0, =8, =4, 3=8, =2 0, =8, =4, 3=8, =2 0, =8, =4, 3=8, =2 0, =8, =4, 3=8, =2
The CLO algorithm was used to nd tours for each of these four scenarios. Following the run of each scenario, the mission was simulated using the CLO-suggested tour to determine the amount of fuel expended. Figures 8 and 9 show CLO results for scenarios 2D1 and 2D2. In each gure, the resultant CLO tour and its fuel consumption is shown with a benchmark tour and its fuel consumption for comparison. These gures use a polar plot similar to those used for the 1-D tours, however the representation is di erent. Here, two plots are given, where each plot represents an observation in the directions shown by the bold arrows in the upper left corner of the gure. The radial lines in each plot represent the orientation of the spacecraft pair about the di erent observation directions. The white circles along the radial lines represent the baseline separation distance as before. For the benchmark tour, each star is fully imaged before moving to the next star, with baseline resize moves being carried out at each orientation before moving to the next orientation. The mission durations for the CLO tours are adjusted to take the same amount of time as the benchmark tour to execute the maneuvers required to perform the mission objectives. Scenario 2D1 shown in Figure 8 images two stars separated by a angle of =8. The CLO tour samples at all orientations with a single baseline for both stars before moving to the next baseline. Note the fuel savings realized by the CLO tour. The gure demonstrates that for stars closely clustered together, mixing targeting and sampling maneuvers provides a large reduction in fuel consumption compared to the benchmark tour. The 2D2 scenario shown in Figure 9 images two stars separated by a angle of =4. Like the baseline tour, the CLO tour also entirely samples one star but samples at all orientations of that star with a single baseline before changing to the next baseline. Note the fuel savings by the CLO tour. Even though each star is imaged separately, the CLO tour is still more fuel 17
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0
ecient. The maximum retarget, reorient, and resize cost for scenarios 2D1 through 2D4 have been calculated and are shown in Table 5 with the priority cost inequalities. These cost inequalities suggest the priority of maneuvers to construct the fuel optimal tour. RTmax and RSmax are calculated the same way as for 1-D imaging. The maximum reorient cost, ROmax , is de ned as the cost to reorient to the next closest orientation when the formation is at its maximum baseline. This will be the maximum reorient cost since the spacecraft must travel further to reorient at larger baselines. By examining the priority cost inequalities for scenarios 2D2, 2D3, and 2D4, it is expected that the optimal sampling patterns for scenarios 2D3 and 2D4 would be the same as optimal pattern for scenario 2D2. This has been con rmed by CLO results for these scenarios. Table 5: 2-D test case maximum retarget, reorient, and resize costs
scenario 2D1 2D2 2D3 2D4
RTmax ROmax RSmax
0.0158 0.0221 0.0266 0.0300
0.0108 0.0108 0.0108 0.0108
0.0164 0.0164 0.0164 0.0164
priority
RSmax > RTmax > ROmax RTmax > RSmax > ROmax RTmax > RSmax > ROmax RTmax > RSmax > ROmax
Using these priority cost inequalities, guideline tours can be constructed. For example, in case 2D1 reorient moves use the least amount of fuel. Accordingly, the tour remains on the same target and baseline until no further reorient moves can be made (tour stop 5 of Figure 8). At this point, the priority suggests that it is cheaper to retarget than to resize, so a retarget move is made from stop 5 to stop 6. The tour then reorients from stop 6 through stop 10, where the rst baseline change is made. Since RSmax has the lowest priority for this scenario, the objective is to make the minimum number of resize moves while completing the mission requirements. Using such reasoning, guideline tours can be developed for every possible cost priority. The guideline tours in Figure 10 result in the same sampling patterns as the CLO algorithm for scenarios 2D1 through 2D4. The guideline tours address all of the possible scenarios for 2-D imaging of multiple stars and provide systematic methods for sweeping out the sampling pattern in a fuel ecient way. The 2-D guide tours are not the result of a formal 20
optimization process, however, for all of the test cases considered they gave results identical to those from the CLO algorithm.
3)
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3π/8
13
4)
3π/8
18
12 7 8
25
4
8
27
3
2
0
1
28
29
0
30
π/2
π/2
3π/8
5
11
6)
7
13
0
1
π/2
30
π/2
11
21
8)
29
14
28
2
1
0
3π/8
26
23
4
π/4
13
27
21
π/8
7
16
5
6
0
5
7 4
1
Observation Direction 1
ROmax > RTmax > RS max
π/8
11 10
18
8
Observation Direction 2
15 30 19
9
17
14 22
π/8
12
29 20
8 9
Observation Direction 1
0
2
19
π/4
17 16
10 3
π/2
24 18
28 21
11
15
22
Observation Direction 2
3π/8
25
20
π/8
12
27 22
π/2
π/4
13
0
1
20
Observation Direction 1
19
23
18
26 23
RSmax > ROmax > RTmax
π/4 26
3
30 27
17
π/8
15
π/8
4
0
10
3π/8
30
24
14
24
Observation Direction 2
3π/8
25
6
11
16 25
29 20
π/4
5
RSmax > RTmax > ROmax
7)
7
29 28
12
28
Observation Direction 1
3π/8
10
13
π/8
9
26 27
22 21
π/2
3π/8
9
12
14
π/8
2 19
23
0
30
21
π/4
8
15
18 25 24
π/2
π/4
3
20
Observation Direction 2
8
π/4 17
0
1
10
RTmax > RSmax > RO max
3π/8
6
4
16
19
Observation Direction 1
RTmax > ROmax > RS max
5)
π/8
29 22
18
12
Observation Direction 2
Observation Direction 1
23 16 17
9
13
24
π/8
2
15 14
26
22
28
25
7
π/8
23 16 21
π/4
3
6
20
π/8
6 5
9
27
π/4
24
17
3π/8
26
π/4
11
15 10
π/2
3π/8 4
π/4 14
π/2 5
19
0
6
3
2
0
Observation Direction 2
ROmax > RSmax > RTmax
Figure 10: Sampling pattern guide for 2-D imaging Illustrations 3 through 8 of Figure 10 show how a 2-D observation sequence might be constructed by comparing the maximum retarget, reorient, and resize costs. Scenario 2D1 discussed above corresponds to illustration 5. Guide tours for each of the other combinations of maximum repositioning costs, RTmax , ROmax , and RSmax are determined following the same priority 21
path system. For example, in illustration 8, since ROmax > RSmax > RTmax , the number of reorient moves should be minimized followed by the number of resize moves. The minimum number of reorient moves is four and the minimum number of resize moves is ten. Clearly, Figure 10 can help determine how the sampling pattern should be mapped out to ll the (u; v) plane in the most fuel ecient way. The guidelines can be used to form nal sampling plan for a mission or as an initial condition for further optimization studies. Table 6 summarizes the minimum number of basic maneuvers that must be performed for each of the six priority paths to construct fuel minimal tours. Since there are 30 tour stops, and only basic maneuvers can be used to travel between stops, 29 maneuvers must be performed. Therefore, the sum of all retarget, reorient, and resize maneuvers will be 29. Notice in this table that the number of basic maneuvers for each illustration, matches the priority inequality. Table 6: 2-D priority guide to the minimal number of basic maneuvers for fuel minimal tours
illustration 3 4 5 6 7 8
priority
RTmax > ROmax > RSmax RTmax > RSmax > ROmax RSmax > RTmax > ROmax RSmax > ROmax > RTmax ROmax > RTmax > RSmax ROmax > RSmax > RTmax
retarget reorient resize 1 1 3 15 5 15
8 24 24 12 4 4
20 4 2 2 20 10
For similar baseline-change sampling, the results of the 2-D scenarios are consistent with the results from the 1-D scenarios. Notice that the CLO algorithm chose to mix targeting and sampling maneuvers when the retargeting angle was =8 as in scenarios 1D4 and 2D1. For angles of =4, the tour images each star separately as shown by scenarios 1D5 and 2D2.
Fuel Savings Table 7 shows the percentage of savings of the guide tour below the benchmark tour for each of the TIMO scenarios tested. Where the percent savings is 0.0, the benchmark tour is the same as the guide tour. Mixing of targeting 22
Table 7: Percent Savings of TIMO test cases below benchmark
scenario percent savings 1D1 1D2 1D3 1D4 1D5 2D1 2D2 2D3 2D4
36.6 0.0 75.5 0.0 0.0 84.9 79.1 76.7 75.7
and sampling moves took place on scenario 2D1 where the savings was the greatest. Notice that even when no mixing occurs in scenarios (cf. scenarios 2D2, 2D3, and 2D4), signi cant savings below the benchmark tour are still realized. This is accomplished by optimizing the order of reorient and resize moves. This suggests that even if the stars are not tightly clustered and only one star is imaged at a time, the sampling pattern is still an important consideration for fuel eciency.
6 Conclusions Signi cant fuel savings can be realized by combining retargeting moves used to point the formation towards the stellar source with imaging moves used to position the formation about the observation direction to take the multiple samples necessary for optical interferometry. As examples of the potential fuel savings that can be realized, fuel optimal tours were compared with a benchmark tour for imaging two stars. Simulation results showed fuel savings up to 85% lower than the the benchmark tour. The fuel savings were most signi cant for test cases involving closely clustered stars. For stars with large separation angles, targeting moves are carried out independent of the imaging moves. Even so, signi cant fuel savings can be achieved by considering the sequence of the imaging maneuvers alone. These fuel-optimal tour strategies have been found using chained local optimization, a standard 23
approach for solving the traveling salesman problem. Based on these optimal strategies, sampling pattern guidelines for minimizing fuel consumption for optical interferometry missions have been developed. These guidelines are based on the targeting and imaging constraints of the interferometry mission.
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