Use of a Genetic Algorithm to Assess Relative Motion in Highly Elliptic Orbits 1

Transcription

1 The Journal of the Astronautical Sciences, Vol. 55, No 3, July September 2007, pp Use of a Genetic Algorithm to Assess Relative Motion in Highly Elliptic Orbits 1 Carrie D. Olsen 2 and Wallace T. Fowler 3 Abstract This paper examines final rendezvous between two vehicles in highly elliptic orbits. The range of eccentricities addressed is 0.6 to 0.9. Due to the varying orbital speeds of the two vehicles, the relative motion during elliptic rendezvous is highly dependent on initial conditions and differs significantly from the relative motion seen in circular rendezvous. The character of the motion has important implications for operational and safety considerations. The development of relative motion targeting and propagation procedures that output relative coordinates in a suitable curvilinear coordinate system is discussed. These procedures are subsequently combined with a genetic algorithm optimization that is used to globally characterize the solution space. Results of genetic algorithm studies are presented and a fuel-optimal family of solutions is identified for further study and characterization. Introduction The ability to bring two spacecraft together in orbit has long been a necessary skill for space operations. Through the Gemini, Apollo, Space Shuttle, and International Space Station programs, rendezvous between two vehicles in nearly circular orbits has become highly standardized and routine. Rendezvous in elliptic orbit presents a new set of challenges. The velocities of the two spacecraft vary with positions in their orbits (true anomalies) and the relative motion between them is more complex. Many works in the literature have addressed various aspects of the elliptic relative motion problem, e.g. [1 6], mostly analytic formulations of the relative motion equations that are valid for various eccentricities, ranges from the target vehicle, and durations. However, very little has been published describing the complex relative motion involved in highly elliptic rendezvous or the development of operationally feasible rendezvous scenarios. Jezewski, et al. [7] noted the need for research into 1 Based on paper AAS presented at the AAS/AIAA Spaceflight Mechanics Meeting, Maui, Hawaii, Feb. 8 12, Assistant Professor of Aerospace Engineering, Mississippi State University. This work was completed while the first author was a Ph.D. candidate in the Department of Aerospace Engineering and Engineering Mechanics at The University of Texas at Austin and employed by NASA-Marshall Space Flight Center. 3 Professor of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin. 1

2 2 Olsen and Fowler highly elliptic rendezvous and the inclusion of operational constraints in rendezvous planning in their important survey paper from The objective of the current study is the systematic analysis of the types of relative motion seen between two vehicles in neighboring, highly elliptic orbits during the final rendezvous approach. The range of orbit eccentricities considered is 0.6 to 0.9 and final rendezvous is defined here as a two-burn sequence that brings the active chaser vehicle from a neighboring, elliptic orbit to the passive, target vehicle s orbit, at some final stand-off position relative to the target. It is assumed that the tasks of launching into the correct orbit plane, initial orbit insertion, and angular phasing would be accomplished using methods very similar to circular orbit rendezvous standard procedures. It is the relative motion between the two vehicles, once they occupy neighboring, elliptic orbits, which is most unusual and divergent from the circular orbit analog. This study focuses on understanding the relative motion properties and costs (propellant usage) of the many ways the chaser vehicle can achieve final rendezvous. To accomplish the systematic investigation desired, a genetic algorithm is employed to narrow the large solution space to families of desirable approaches. The genetic algorithm uses a universal Lambert formulation for targeting, standard Runge- Kutta techniques for numerical propagation of both vehicles orbits, and presents results in a curvilinear relative motion coordinate system centered at the target vehicle. The development and verification of these tools is described herein, along with some preliminary results involving particular elliptic rendezvous cases. Finally, possible tool extensions are discussed that will allow for evaluation of candidate final rendezvous scenarios in light of operationally relevant approach path and safe abort considerations. The Case for Highly Elliptic Rendezvous While the majority of Earth-orbiting satellites have orbits that are very nearly circular, a significant number of existing and future satellites do not. One example is NASA s Chandra X-Ray Observatory. This Great Observatory was launched in the summer of 1999 into an orbit with an eccentricity of approximately 0.8. Unlike the Hubble Space Telescope, this high eccentricity, high altitude astrophysics observatory was not designed for rendezvous. Its altitude and eccentricity made the option of serviceability prohibitively expensive and complex. However, as its launch date drew near, the wisdom of this design decision was questioned. The Inertial Upper Stage (IUS), responsible for taking the Chandra from the Space Shuttle s delivery orbit to an intermediate elliptic orbit, had recently experienced a failure while attempting to deliver an Air Force satellite to geostationary orbit. Such a failure on the Chandra mission would have rendered the billion-dollar instrument scientifically useless. The IUS is not, by far, the only upper stage that has left satellites stranded in useless transfer orbits. Such failures account for billions of dollars in wasted hardware. This study looks forward to the time that a robust, reusable satellite rescue tug might be used to repair and refuel such satellites and/or boost them to their intended mission orbits. In addition to Earth-orbit satellite servicing and rescue applications, elliptic orbit rendezvous has been proposed as part of the return leg of various Mars missions in the interest of mass and cost savings. This option received a favorable evaluation by Smith and Hong [8]. If elliptic rendezvous is cost-effective for a Mars return trip, it is not a huge leap to imagine its benefits for the outbound leg. One such scenario that has been proposed involves launching an unmanned Mars transfer vehicle into low

3 Genetic Algorithm to Assess Relative Motion in Elliptic Orbits 3 Earth orbit and then using highly efficient low-thrust propulsion to slowly (over the course of months) spiral it out to an extremely high energy, high eccentricity orbit. Then, when this highly elliptic orbit is achieved, a crew shuttle vehicle would be launched to rendezvous with the transfer vehicle. Once the crew had arrived, a relatively small energy change would initiate the interplanetary trajectory and the crew would have avoided months of radiation and microgravity exposure. During the early 1990s, in response to the Space Exploration Initiative, groundbreaking work in highly elliptic rendezvous was done by personnel from Draper Labs, in association with NASA s Johnson Space Center [9 11] specifically involving elliptic rendezvous in Mars orbit. It is this work that was used as a springboard for the current study. With the President s current Vision for Space Exploration, a return to the Moon and human missions to Mars, the topic of this study, is, once again, a timely one. Problem Definition and Assumptions To study relative motion, one must first decide on a suitable coordinate system in which to depict the motion. The traditional choice for a relative motion coordinate system for rendezvous studies is a target-centered, rectangular, local verticallocal horizontal (LVLH) system. The purpose of choosing a coordinate system centered at the target vehicle is to depict the way the motion of the chaser looks to the target. Merriam [10] cited the insufficiency of the rectangular LVLH system in the case of highly elliptic orbits and presented instead a set of non-analytical, curvilinear coordinates that appeared better suited to the study of highly elliptic rendezvous, providing a truer depiction of the chaser motion relative to the target. This coordinate system is used in the current study and is called the Curvilinear Relative Motion (CRM) system. Merriam s CRM system is also a local set of coordinates that moves with the target vehicle, but the along-track and radial components (X and Y) are calculated through extrapolation of the current target vehicle position along its orbit to a point where the relative vector between the two spacecraft is perpendicular to the velocity vector of the target vehicle. At this point, the magnitude of the relative vector is the magnitude of the Y coordinate. Y is defined as positive from the target vehicle away from the Earth. The accumulated arc length that the target vehicle is moved to the perpendicular point is the magnitude of the X coordinate and X is defined to be positive behind the target. The Z coordinate requires no special calculation. It is the usual LVLH cross-track value. The CRM is depicted in Fig. 1. An example of the different character of relative motion in the LVLH and CRM systems is shown in Fig. 2. In this figure, one orbital period of relative motion between vehicles in neighboring, highly elliptic orbits is plotted in both sets of coordinates. The application of the terms leading vs. trailing and above vs. below is different in the two sets of coordinates. The difference is that the CRM system, being curvilinear, better models the curvature of the orbit than the rectangular LVLH system, and gives a more useful characterization of the relative motion. The initial semimajor axis, eccentricity, and true anomaly of the target and chaser spacecraft are given at the top of the figure. After selecting a suitable coordinate system for the relative motion, several limiting assumptions had to be made involving initial orbital parameters of the two vehicles in order to reduce the solution space to a manageable size. Again, following the example of the Draper research team [9 11], two target/chaser initial orbit configurations were chosen for use in this study: the scaled elliptic and co-elliptic

4 4 Olsen and Fowler FIG. 1. Curvilinear Relative Motion Coordinate System. arrangements (Fig. 3). Scaled elliptic orbits share a line of apsides, have the same eccentricity, and the semimajor axis of the chaser is constrained only in that it is smaller than that of the target. Co-elliptic orbits also share the same line of apsides and the chaser vehicle has a smaller semimajor axis, but the product of the semimajor axis and the eccentricity is the same for both orbits. The primary difference between the two arrangements is that the altitude difference between the two orbits varies less for co-elliptic orbits than for scaled elliptic orbits. If the altitudes of the two orbits are too close, this can lead to approach characteristics that violate operationally-defined keep-out zones that may only be entered during proximity operations and docking. Therefore, the choice of initial relative orbit arrangement can affect terminal rendezvous relative motion characteristics.

5 Genetic Algorithm to Assess Relative Motion in Elliptic Orbits 5 FIG. 2. Comparison of Relative Motion in LVLH and CRM Coordinates. Parametric studies were conducted on three sample target vehicle orbits chosen to span the chosen eccentricity range. The first target orbit had a perigee similar to the radii of low Earth orbits and a high apogee. The other two target orbits were based on the initial and evolved orbits of the Chandra spacecraft. Tables 1 and 2 give details for these orbits. The software developed for the study allowed orbital parameter input for both vehicles in rectangular coordinates or orbital elements, or alternatively, the chaser vehicle position relative to the target vehicle could be specified in LVLH or CRM coordinates. The subsequent trajectories could be propagated with or without selected perturbation effects and rendezvous maneuver targeting was accomplished using a Lambert targeting algorithm developed by Klumpp [12] as an improvement to a classic algorithm by Battin [13]. The improvement involved an efficient method for addressing all possible singular cases. Clohessy-Wiltshire [14] rendezvous targeting is not valid for elliptical orbit rendezvous. Other targeting schemes [1 6], developed for elliptic orbits, were found to be either too constraining in eccentricity and/or separation distance or were considered more cumbersome to program than the Lambert method.

6 6 Olsen and Fowler FIG. 3. Scaled Elliptic and Co-Elliptic Orbits. TABLE 1. Orbital Parameters for Study Target Orbits Initial Chandra Evolved Chandra Orbit Parameters Orbit 1 Orbit Orbit Semimajor Axis (km) 34,000 81,378 81,360 Eccentricity Inclination (deg) Argument of Perigee (deg) Right Ascension of Ascending Node (deg)

7 Genetic Algorithm to Assess Relative Motion in Elliptic Orbits 7 TABLE 2. Standard Target Vehicle Orbits for Study Semimajor Radius of Radius of Orbit ID # Axis (km) Eccentricity Apogee (km) Perigee (km) 1a 16, ,800 6,700 1b 26, ,900 6,700 1c 67, ,300 6,700 2a 24, ,600 9,900 2b 39, ,300 9,900 2c 99, ,100 9,900 3a 32, ,000 13,000 3b 52, ,000 13,000 3c 130, ,000 13,000 In early, parametric studies, rendezvous cases were run one at a time or in small batches with incremented transfer times. One result of these early studies was the elimination of short transfer times from future studies. These short times, 45 to 60 minutes, are typical for circular, low-earth orbit rendezvous transfers, but result in extremely high fuel costs, and target vehicle closing rates, for highly elliptic cases. Instead, it became apparent that the transfer times that characterize good elliptic rendezvous trajectories should mimic circular orbit rendezvous in that they should be a similar fraction of an orbital period ( 1 2 to 2 3 of an orbit), although these transfers can be many hours in duration. During early investigations, the effect of perturbations on targeting and propagation were examined for a variety of orbits in the chosen eccentricity range. A wide range of errors, from tenths of meters to hundreds of meters, was seen in the prescribed final stand-off point when Lambert targeting was used, depending on the perigee height of the orbit, the transfer time, the starting and ending true anomalies and orbit inclination. When the rendezvous occurred near perigee, the largest errors were due to the J 2 gravity potential term of the Earth. However, even these errors were easily removed with one mid-course correction that constituted a 2.9% increase in the delta-velocity of the transfer. Because of this finding, and because the early studies suggested the superiority of apogee rendezvous, it was decided to continue the study without modeling perturbations. Targeting and simulation runs were made using the target vehicle orbits in Table 1, with a chosen initial chaser vehicle position 20 km behind and 10 km below the target vehicle. The desired final position was set at 1 km behind the target vehicle in the same orbit. These values were chosen to represent day of rendezvous maneuvers typical in NASA s low Earth orbit rendezvous procedures and to stop short of the region labeled as proximity operations. The initial true anomaly of the target vehicle was varied from 0 to 360. Some sample results, using transfer times of half an orbital period for Orbit 1 and the evolved Chandra Orbit, are shown in Fig. 4 and Fig. 5 below. Note that the character of the relative motion changes significantly as the true anomaly is varied. This is more pronounced in Fig. 4, which is the case of a 0.8 eccentricity orbit, than in Fig. 5, which represents a 0.58 eccentricity case. The motion seen, which agrees in character with the Draper results [9 11], is characterized by multiple changes of

8 8 Olsen and Fowler FIG. 4. Orbit 1 - Half-Period Transfers with Various Initial True Anomalies. direction, even a striking backing up feature in the 225 and 270 cases of Fig. 4 and the 225 case in Fig. 5. In addition to these qualitative efforts, quantitative studies to calculate energy requirements (delta-velocity values, Vs) for candidate rendezvous scenarios were made with transfer time as the independent variable. Final rendezvous initiation, termination, and total Vs were calculated for a full range of transfer times (80 minutes to one complete orbit revolution) for ten different cases with various starting conditions. The general result was that total V was lowest at transfer times of about 60% of an orbital period, with scaled elliptic cases tending toward slightly longer optimal transfer times than the co-elliptic cases. These limited individual runs and V studies provided some initial insight into the character of the relative motion of highly elliptic rendezvous but also helped prove the need for a more global optimization approach where a large array of starting geometries and transfer times could be systematically combined, evaluated, and analyzed. Furthermore, because there is more to an optimal rendezvous than the cost of fuel, an approach was needed that would allow for scoring of candidate rendezvous scenarios with operational constraints. Considerations such as avoidance of a prescribed keep-out zone, chaser motion in the event of an arresting burn failure

9 Genetic Algorithm to Assess Relative Motion in Elliptic Orbits 9 FIG. 5. Evolved Chandra Orbit - Half-Period Final Rendezvous Transfers with Various Initial True Anomalies. and line-of-sight to particular locations (windows) on the target vehicle are difficult to factor into calculus-of-variations formulations. However, they are manageable with a genetic algorithm optimization approach. Introduction of the Genetic Algorithm A genetic algorithm (GA) is an optimization technique that works in a manner analogous to the natural processes of genetics and evolution. They are not very efficient in finding the one true optimum in a solution space. However, for a multidimensional problem like the one at hand, the most difficult task is finding nearoptimal regions or solution families in the entire solution space. Genetic algorithms are currently in wide use in various aspects of space trajectory optimization (e.g. [15 18]), and the technique is becoming ever more refined and effective. The genetic algorithm used in this research is an adaptation of a freeware program written by Carroll [19]. For this implementation of the GA, the control parameters are the true anomalies of the chaser and target vehicles (TAC and TAT) at the initiation of the final rendezvous, the transfer time for the final rendezvous maneuver (TOF), and the

10 10 Olsen and Fowler difference in semimajor axis ( a) between the initial orbit of the chaser vehicle and the orbit of the target vehicle. For each generation of the GA execution, many chromosomes of control parameter values are evaluated. Each evaluation involves a run of the Lambert targeting routine to produce a fitness score based on the evaluation criteria. For the runs of the current study total V alone was the scoring criterion. The parameters of the target vehicle orbits under study are presented in Table 2 and the allowable ranges of the control variables used by the GA are shown in Table 3. The two true anomaly values, TAT and TAC, are not shown as they are always given a full 360 range. The initial difference in semimajor axis of the two orbits was 0 to 1000 km, except for orbits with low perigees, and time of flight (TOF) ranges varied based on the orbital period of the target vehicle orbit. The ranges on TOF listed vary between one-sixth of an orbit period to one full orbit period. Smaller transfer times were considered too fuel-expensive and dangerous from the standpoint of vehicle closing rates. The upper limit of one orbital period was selected to avoid multi-revolution transfers, which were considered out-ofscope for this research and not a likely operational choice due to the long transfer times involved in high altitude, high eccentricity orbits. With the target orbits and parameter ranges defined, the starting orbit of the chaser was assumed to be coplanar with the target orbit and separate runs were made for each of the nine target orbits with both the co-elliptic and scaled-elliptic target-chaser orbit configurations. Because a is one of the control parameters used in the optimization, it is obvious that the closer the two orbits are in semimajor axis initially, the less V will be needed to rendezvous. Therefore, if only the final rendezvous V were included in the performance evaluation the results of the optimization would be incorrectly skewed to favor low- a cases. Therefore, the calculation of V had to be adjusted such that each scenario would be judged fairly against all the others. To accomplish this, a term was added that represented the amount of fuel it would take to raise the chaser vehicle from some initial low circular orbit to the candidate rendezvous initiation orbit being evaluated. Thus, an initial circular orbit would be the same for all cases. The transfer V to the candidate rendezvous initiation orbit was calculated assuming two tangential burns performed at the apsides of the transfer orbit. In this way, what is saved in V by beginning a rendezvous close to the target orbit is paid for in the first transfer to the rendezvous initiation altitude. This also explains the large overall V values reported in the results of Tables 4, 5, and 6. TABLE 3. Range of Control Parameters Semimajor Orbital Period Range of Range of Time of Orbit ID # Axis (km) (hours) a (km) Flight TOF (hours) 1a 16, b 26, c 67, a 24, b 39, c 99, a 32, b 52, c 130,

11 Genetic Algorithm to Assess Relative Motion in Elliptic Orbits 11 TABLE 4. Orbit 1a, Co-elliptic with Micro-GA: Tuning Results Population Size Number of Generations Minimum Function Value, V ( km s) Tuning the GA The process of deciding which options to use and which values to set for the various inputs of a GA is often called tuning the algorithm. Some general guidance does exist, and Carroll provides some with his program. However, GA researchers generally admit that GA tuning can be an optimization process in its own right. The GA used in this study does not provide an indication of convergence. The program will run until it reaches the maximum number of generations specified. Generally, if tens of generations transpire without any change in the best function value, either the program has converged to the best answer or the problem has been poorly formulated and the program will not converge. To help identify this second eventuality, the V calculation for a two tangential burn transfer from the chaser orbit to the target orbit was made for each of the nine cases. This simple calculation was used to determine if the minimum V found by the GA was credible or merely the product of a poorly formulated optimization. With the GA formulation thus verified, more runs were made, with more generations, larger populations and/or different GA parameter to see if better function values could be found. The first tuning runs were made using Orbit 1a with a co-elliptic target-chaser arrangement. The micro-ga option was selected and the corresponding values for the related GA parameters suggested by Carroll were used. Several choices for the number of individuals per generation and the maximum number of generations were used and the results of these first tries are summarized in Table 4 below. For comparison s sake note that the hand-calculated minimum V objective function value for Case 1a is km s. Note from Table 4 that the combination of a 25-member population and 100 generations produced the best function value of this limited group of runs. Also note that for the 50-member population cases nothing was gained by increasing the number of generations from 50 to 100. A similar set of runs was made with the Orbit 1a-Scaled-Elliptic case. Those findings are summarized in Table 5. TABLE 5. Orbit 1a, Scaled Elliptic: Tuning Results Population Size (* indicates Minimum Function micro-ga not used) Number of Generations Value, V ( km s) *

12 12 Olsen and Fowler TABLE 6. Summary of Tuning Results Population Number of Micro-GA Theoretical Lowest Lowest V Found Case ID Size Generations (Yes or No) V( km s) by GA ( km s) 1ac Yes as No bc Yes bs Yes cc Yes cs Yes ac Yes as Yes bc Yes bs Yes cc Yes cs Yes ac Yes as Yes bc Yes bs Yes cc Yes cs Yes In this case, using a population size of 100 produced the best results. With the micro-ga method, 50 generations were enough to apparently converge while the traditional GA found a slightly better function value using 100 generations. Continuing, all 18 cases (nine different orbits and the co-elliptic and scaled elliptic arrangements for each) were investigated in this way. Table 6 gives an overall summary of the best cases. The GA was, as expected, converging on a family of very good solutions for each case it considered. Although the exact best V case might vary with changes in the population size and maximum number of generations, the same overall families were being produced. Because the study objective was to characterize elliptic rendezvous using a GA and not to optimize the performance of the GA itself, further attempts at fine-tuning were abandoned. The most common GA variable set (see Table 6), 50 members per generation and 100 generations with the micro-ga option enabled, was chosen to be used in all future GA investigations for this study. GA Study Results This tuning of the GA described above was done while performing the GA study, therefore, a number of runs was made for each of the 18 defined cases. Although one best V value and its corresponding parameter set were identified for each case, this knowledge does not tell the whole story. The GA identified, for each of the 18 cases, was a large family of rendezvous scenarios, all with Vs close to the hand-calculated best V. Within each family a variety of distinct trajectory types were found. This finding revealed the need for other evaluation factors besides V and for further, more specific investigations. Those investigations will be defined and discussed in a subsequent paper. For the remainder of this paper, the general results and trends are presented that were deduced by manually studying the output files from this first study.

13 Genetic Algorithm to Assess Relative Motion in Elliptic Orbits 13 Table 7 presents the four control parameters associated with each of the best cases whose V values were given in Table 6. An interesting picture emerges from this table. The a values tend to the minimum of 10.0 km, the true anomaly values are all in the vicinity of 0.0 (perigee) and the time of flight values, when expressed as fractions of the target vehicle orbit period, lie in the 0.46 to 0.69 range. These same statements can also be made of a randomly selected grouping of good answers from any one of the 18 cases. This family of solutions is not a surprise, but validates what is expected. Beginning at perigee and transferring to the target orbit over half a revolution and rendezvousing at apogee means that the chaser vehicle executes two tangential burns. The theoretical best V was calculated in just this way. The true anomalies of the vehicles being close together, and the a being small, assures that the maneuver will accomplish a rendezvous and not just an arbitrary orbit transfer. It assures that both vehicles will be at the target vehicle s apogee point at the same time. This rendezvous at apogee scenario is also desirable operationally, as both vehicles are traveling through the slowest part of their orbits and thus their relative positions, one to the other, are changing slowly. These findings verify intuition, as well as the entire optimization approach and software development. This provides confidence for the next phase of investigation where the target vehicle itself is no longer the target point and minimum V is no longer the sole evaluation factor. In addition, the nature of the GA optimization process itself has worked to produce a family of good answers. The families of rendezvous approaches can be seen as a menu of scenarios that a mission planner could chose from to find the one(s) that would best suit the specific mission requirements and constraints. For instance, some of these approaches begin with the chaser and target even in true anomaly, some have the chaser ahead and some have the chaser behind. These beginning geometries can have operational significance depending on the nature and location of navigation sensors, thrusters and possible humans in the TABLE 7. Summary of Parameter Values for Best Cases Initial Target True Initial Chaser True Target Orbit ID a (km) TOF (% Orbit Period) Anomaly (deg) Anomaly (deg) 1ac as bc bs cc cs ac as bc bs cc cs ac as bc bs cc cs

14 14 Olsen and Fowler control loop. Also, varying TOF and a can greatly influence the approach geometry. For each of the 18 cases run, a random sampling of solutions was compiled and their resultant approach paths were plotted. Several of these plots are included below to illustrate the variability in relative trajectories that exists. Figure 6 provides examples of geometries for Orbit 1b, for the scaled elliptic arrangement. For all three curves shown the a is small less than 20 km. However, due to the nature of the scaled elliptic arrangement the starting Y coordinate is much smaller than 20 km for approaches begun at perigee (refer to Fig. 3). This fact, when coupled with the target and chaser vehicles beginning at the same true anomaly (as is the case for Curves #1 and #2), creates a possible operational problem. The problem is that the terminal rendezvous maneuver is initiated with the chaser within 5 km of the center of the target vehicle. The margin for error is small this close to the target and errors in navigation and/or thrust magnitude and direction have the potential of causing collision or near-collision conditions. Curve #3 is the lowest- V case ( km s). Curves #1 and #2 represent cases that are more fuel expensive ( and km s, respectively) but that illustrate the kind of approach that can occur when the rendezvous initiation burn is performed directly below the target. These cases create approaches with interesting changes of direction. In Curve #2, the TOF is a little greater than that of #1 and the chaser moves ahead of the target, does a complete reverse of direction, and then loops up and above the target before it eventually approaches from the second quadrant. Curve #1, with a much smaller TOF, also changes direction but does so in a more FIG. 6. Orbit 1b, Scaled Elliptic: Examples of Possible Approaches.

15 Genetic Algorithm to Assess Relative Motion in Elliptic Orbits 15 abrupt manner. This curve, in its final stages, becomes an X-axis approach. In circular orbit terminology, this is referred to as a VBAR approach as the target vehicle s velocity vector points along the X-axis. Figure 7 depicts cases from target Orbit 2b with the co-elliptic target/chaser arrangement. (Note that Curves #3, #4, and #5 are too large to be shown from their starting points.) Several observations can be made. First, Curves #1, #2, and #3 exhibit a now familiar pattern. All cases that begin with the chaser behind and below the target end up as third-quadrant approaches. They differ mainly by amplitude of motion which is caused by the difference in a between the cases. Their similarity in geometry is even more interesting in the light of the range of starting true anomalies that they represent. Curve #3 begins with both vehicles nearly 50 degrees before reaching perigee, Curve #2 begins at perigee and Curve #1 begins almost 70 degrees after perigee passage. Also, Curves #1 and #2 have identical V costs ( km s) that represent the best value found on any of the 2b co-elliptic runs. The Vs for Curves #3, #4, and #5 are , , and km s, respectively. Curve #5 is the most fuel-expensive and provides some interesting comparisons to Curves #1 #3. Curve #5 also begins behind and below the target and actually has the same transfer time as Curve #2, the curve FIG. 7. Orbit 2b, Co-Elliptic: Examples of Possible Approaches.

16 16 Olsen and Fowler with the most compact geometry. The reason for the massive difference in the curves is obvious. Curve #5 represents a case where a is km, as opposed to 10.0 km for Curve #2. Curve #5 has to travel a much greater distance in the same time. Therefore, more energy must be expended and a larger, more exaggerated curve results. Even though the general shape of Curve #5 is not unlike Curves #1 #3 its large, exaggerated nature causes it to cross the negative X-axis and approach the target from the second quadrant. Curve #4, however, is completely different. With Curve #4 the chaser vehicle begins ahead of the target in true anomaly and at the lowest a value of 10 km. In addition, this is a long transfer time case of an orbit revolution in length. This combination of parameters creates a novel relative motion curve. The chaser vehicle begins 210 km ahead of the target and 10 km below. The chaser then backs up toward the target almost on a straight line until it is only 60 km ahead of the target. It then begins a loop up across the X-axis and eventually to the Y axis (RBAR) down which it travels to the target. Conclusion The initial study considered 18 cases. That is, there were nine target orbit cases run with each of the initial target-chaser geometries co-elliptic and scaled elliptic. The final targeted position for the chaser vehicle was the target vehicle itself (i.e., no stand-off distance specified) and the evaluation function was the total V. The genetic algorithm identified for each case a large family of rendezvous scenarios, all with Vs close to the hand-calculated best V. This expected result serves to validate the approach and the genetic algorithm implementation. Within each family of low- V cases a variety of distinct trajectory types were found. An examination of these results indicate that a good value for the initial difference in the semimajor axes of the chaser and target orbits was 10 km, that initial true anomaly values should be near perigee (within 40 degrees), and that good flight times ranged between 46% and 69% of the orbital period of the target orbit. These parameters produced, in general, a perigee to apogee type of rendezvous, which is also expected, as it mimics a Hohmann-type, minimum energy transfer. In addition to these generalized results it is notable that a large variance in the character of approach relative motion is achievable while keeping V values nearoptimal. This is encouraging from a mission planning and operations perspective. As is the case with current (Shuttle and International Space Station) rendezvous experience, multiple viable approach scenarios are desirable in order to accommodate the various operational constraints that are encountered, including but not limited to navigation, lighting, and vehicle/crew safety considerations. With the genetic algorithm method developed herein for the study of final rendezvous relative motion it is possible to further characterize the broad regions of good solutions identified. The targeted final position of the chaser can be generalized to any stand-off point (on the RBAR, VBAR or elsewhere) that is operationally desirable. A safety zone of exclusion can easily be defined about the target vehicle and candidate scenarios can be penalized if the zone is entered during final rendezvous transfer or in the event of a failed arresting burn. In short, the method described above shows promise as a preliminary design tool for operationally feasible rendezvous in highly elliptic orbit.

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