Modelica Modeling: Optimization of Spacecraft Orbital Parameters

Transcription

1 Modelica Modeling: Optimization of Spacecraft Orbital Parameters ME6105Q, Fall 2008 Homework #3 Drew Winder Daniel Lim

2 Goals and Problem Domain Based on the design scenario described in the last assignment, the goal of this assignment is to create and energy-based models of the satellite systems orbit around the Earth for IRSAT-1, 3, and 5 and their respective subsystems, such as thrust controller and image recorder. These models are designed to provide the ability to experiment with several parameters such as thruster burn duration, direction, timing, etc. with relation to each satellite s orbital location to provide inputs for the decision makers to determine the best course of action to take, with respect to the resultant IR image resolution and amount of fuel consumed during orbital change maneuvers. All in all, the characterization of the image resolution and the mass of fuel required for orbital changes is of main concern for these experiments, since these two parameters are the main inputs for the decision maker to decide which courses of action to take. The decision maker must balance the benefits of increasing the image resolution for IRSAT-5 by the degradation of the satellite operational lifetime and operationally responsive flexibility (i.e. ability to make future orbital adjustments to cover future infrared (IR) targets of interest) because of the amount of onboard fuel expended to correct the altitude of the satellite. Also, the total fuel consumption from an inclination adjustment to meet the operational demand of coverage of the North Korean missile facility will be a key determining factor for whether the decision maker will authorize an inclination plane change. The objective of this exercise is to address the following questions with the models created: 1. What are each satellite s orbit parameters at a given time as it travels around the world? a. What is each satellite s altitude? b. What is each vehicle s inclination? c. What is the orbital period of each satellite? d. What is each satellite s orbital location? e. What are the relative positions of each satellite with respect to each other? 3. When are the optimal times to apply thrust to perform an altitude change for IRSAT-5 to bring it back down to its optimal altitude? 4. When are the optimal times to apply thrust to perform an inclination plane change for IRSAT-5 to be aligned with the inclinations of IRSAT-1 and 3? 5. What is the image resolution of the IR imagers onboard IRSAT-1, 3, and 5? 6. How much fuel is consumed for various orbital maneuver options? a. How much fuel is required to correct IRSAT-5 s altitude? Page 2 of 36

3 Goals and Problem Domain- cont d b. How much fuel is required to align IRSAT-5 s inclination to match IRSAT-1 and 3? c. How much fuel is required to adjust both the altitude and inclination? Due to the complex scope of modeling the IRSAT orbital parameters, the scope of the design project has been narrowed in two main areas: First, due to limitation of time and modeling experience, the option of leaving IRSAT-5 alone and compensating for the loss of coverage by IRSAT-2 going not mission capable (NMC) by adjusting the distances between IRSAT-1 and 3 by orbital spread maneuvers has been eliminated. Also, the capability to adjust IRSAT-5 s orbital location to align with IRSAT-1 and 3 to minimize coverage gaps (once IRSAT-5 has been placed in the same inclination and altitude of IRSAT-1 and 3) has not been covered in the subsequent experiments. Overall, the determination of the coverage times and the resulting gaps and overlaps have been neglected in our experiments. Page 3 of 36

4 System and Simulation Specification The system contains two key components: The satellites and the Earth. The Earth component is a body located at the origin of the coordinate system. The body is given a rotation equal to that of the Earth. The gravity of the Earth not included in this component, but is rather modeled using the built-in point gravity function of the world component. This point gravity is the only gravitational force in the model. This gravitational force is idealized. It does not account for the real variances in Earth s true gravitational field. The satellite model is based on the built-in point mass model. Each satellite orbits the earth based on its initial position and velocity vectors. The gravity of the earth keeps the satellites in orbit. No atmospheric drag is modeled. The satellites are independent, and do not interact with each other. In addition to these two components, two additional components of the system are needed: A tracking system and a thruster controller. A thrust controller model is used to make changes to the satellite s orbital parameters. The thruster model applies a force at the times and in the direction determined by its internal logic controls. Individual thrusters are not modeled. The thrust can be applied in any direction. While the mass of fuel burned is calculated by the thruster, the mass of the satellite during the burn is assumed to be constant for simplicity. A tracking model is used to measure the relationship of the satellite and the earth. This component outputs values such as the satellites altitude, latitude and longitude. It also calculates the maximum resolution of the satellite. The altitude and resolution are based on sea level. The surface of the earth is not, of course, a perfect sphere. This would lead to differing altitudes and resolutions at differing points over the earth. This has been neglected for simplicity. Page 4 of 36

5 Dymola Models The overall model is shown in Figure 1. The model includes a world component which provides the gravity force. It also includes an Earth body located near the origin of the coordinate system. (An error occurred when the earth was located at exactly the origin, so the world was relocated. The one meter error is not significant.) Also included are three satellites. Each satellite has a separate tracking, thrust control, and fuel burn calculation module. Figure 1: Overall Model World The world component is from the Modelica standard library. It produces a point gravity field centered at the origin (0,0,0) with a gravity field constant of x m 3 /s 2. Page 5 of 36

6 Dymola Models- cont d Earth The Earth body model is fixed to a location. It then has an actuated revolute joint from the standard Modelica library as shown in Figure 2. The joint is actuated by a speed object from the standard Modelica library with a source input that sets a constant speed of rotation equal to the required Earth rotational speed of 360 deg per day or radians per second. Speed_of_Ro... k= speed w _... fixed1=0 a b earth frame_a1 n={0,0,1} actuatedrev... m=1 frame_b Figure 2: Earth Body Model The rotating side of the revolute joint is attached to a body. This body is given a radius equal to the radius of the Earth for animation purposes. An output frame is also attached to the output of the joint. This is then attached outside the model to a sensor from the Modelica standard library so that the relative orientation of the Earth can be measured by the tracking model. Tracking The tracking model inputs position data from the satellite as well as orientation data from the sensor attached to the earth body to calculate parameters of the satellite. These parameters include altitude, the height of the satellite above the surface of the earth, as well as latitude and longitude. The resolution (i.e. the smallest object detectable by the satellite) is also calculated using the following formula. Resolution = 2.44 Wavelengt of Interest Altitude Aperture Diameter Page 6 of 36

7 Dymola Models- cont d For the purposes of the simulation, the systems are assumed to have an optimal resolution of 1 m (at an operational altitude of 282 km), detecting a wavelength of 1 x 10-4 m (which is in the middle of the IR spectrum) and an equivalent instrument aperture size of 68.8 m. Spacecraft The spacecraft model is a derivative of the point mass model from the Modelica standard library. It has a mass of 2100 kg. It is given a starting position and velocity to locate the spacecraft in the correct starting orbit at the beginning of the simulation. Outputs of the position and velocity vector data were added to the base point mass model to allow tracking and thrust control. An input for the thrust force vector was also added. Thrust Controller The thrust controller provides for modification of the satellites orbit. The thrust controller can modify the altitude and the inclination of the orbit. It also provides for the ability to execute re-circularization burns to remove eccentricity from the orbit brought about from error in the burns. Figure 3 shows the input parameters of the model. Figure 3: Thrust Controller Controls Page 7 of 36

8 Dymola Models- cont d The parameter thrust_coeff provides the magnitude of the thrusters. In the simulation, the IRSATs were equipped with six Kaiser-Marquardt 22 N (5 lbf) hydrazine monopropellant engines (one engine per positive and negative axial directions), which deliver a 235 sec specific impulse per engine. Thruster cutoff and start times for various phases of the burn are also input. To change the altitude, the thrusters are fired at times determined by the thrust control algorithm, maximizing the advantage of gravity to assist in lowering the altitude. Once the altitude transfer burns are accomplished, a series of re-circularization burns are accomplished to re-circularize the orbits, which have gained some eccentricity during the transfer burns. The values for these are manually set to remove the eccentricity produced by the altitude change maneuver. To change the inclination, the thrusters are fired in a direction normal to both the position and velocity vectors to push the satellite into a higher or lower inclination. Fuel Burn The fuel burn model calculates the fuel expended by the firing of the thrusters for maneuvering. It is based on a hydrazine monopropellant engine built by Kaiser-Marquardt. The input parameters of this model are shown in Figure 4. Using the thrust force vector, the specific impulse of the thrusters and the thruster force, this model calculates the fuel consumed in maneuvering. Figure 4: Parameters of Fuel Burn Model Page 8 of 36

9 Dymola Models- cont d This model calculates the fuel required for the burns by determining the total time the thruster was fired and the amount of fuel the thruster uses per second. The total time is calculated by determining the magnitude of this thrust force vector and dividing it by force produced by the thruster (22 N). This is then integrated with respect to time to determine the amount of time the thruster is fired. The mass of fuel burned can be calculated once the time of the burn is known. The mass flow rate of fuel is given by the following equation: Specific Impulse = Trust Force Mass Flow Rate Gravitational Acceleration Constant Multiplying the time of the burn by the mass flow rate then gives the fuel burned during the maneuver. Although not formally used in the modeling, it was assumed that 300 kg of propellant was available for IRSAT-5, which was an upgrade from the previous series of IRSAT spacecraft. This becomes a factor in understanding the feasibility of accomplishing the orbital maneuvers. Page 9 of 36

10 Verification Three verification tests were created to verify that the models worked as intended. These tests verified the operation of the tracking model, the thrust control model, and the fuel burn model. Tracking Test Model The test model shown for the tracking model is shown in Figure 5. The model consists of a single satellite orbiting the Earth. The satellite has a constant force applied to it in each of the three cardinal directions. For a correctly operating model, this should lead to the tracking model outputting varying values for the latitude, longitude, altitude, and resolution. y world const2 x const1 k=1 const k=1 fixed earth_b... a resol... k=1 f_in ve... r={1,1,1} absolutese... IRS... m=100 po... ea... Trac... p... Figure 5: Tracking Test Model The applied forces result in an orbit that would be impossible in reality, but provides for an excellent review of the output of the tracking model. The satellite is pushed out of its initial stable circular orbit into an orbit that becomes more and more eccentric. Its perigee dips below the surface of the earth, while its apogee grows larger and larger. Figure 6 shows the components of the position vector of the satellite. Page 10 of 36

11 Verification- cont d Figure 6: Tracking Test Model Satellite Position Results As can be seen in Figure 7 and Figure 8, the results reported by the tracking model are generally what are expected. The altitude matches expectations, with the altitude starting out at the correct level for the starting orbit and then experiencing increasing oscillations. As shown in the figure, the altitude drops below zero, indicating that the satellite would be below the surface of the earth. As shown in Figure 8 the resolution also performs expected. The resolution is directly related to altitude, and the plot shows this direct relationship. The longitude also behaves as expected, starting at the correct value and generally traveling smoothly from -180 to 180. The curve grows more irregular as expect as the simulation progresses. The longitude curve shows several spikes, but these are intermittent and can be easily ignored when reviewing the data. Latitude also performs as expected starting at the correct value and then oscillating more and more as the simulation progresses. Page 11 of 36

12 Verification- cont d Figure 7: Tracking Test Model Altitude, Latitude, and Longitude Results Figure 8: Tracking Model Altitude and Resolution Results Page 12 of 36

13 Verification- cont d Thrust Controller Test Model The next test model is designed to test the thrust controller. The model is shown in Figure 9. The thrust controller in the model is intended to first drive the satellite to a lower orbit, burn to re-circularize, and then change the inclination. y world po... m=2100 fixed x earth_b... a resol... earth a... Tracking pos_in vel... f_in IRSAT... r={1,1,1} absolutese... p_in v_in f_out Figure 9: Thrust Controller Test Model thrust ctrl Figure 10 shows a plot of the altitude of the satellite during the simulation. The plot shows the satellite losing altitude at the beginning of the simulation, as was predicted. The thrust controller was intended to drop the satellite from a circular orbit with an altitude of approximately 580 km to a final altitude of approximately 280 km. The altitude change burns did introduce eccentricity to the orbit. Note the range of the oscillations before and after the re-circularization burns at approximately 4 x 10 5 seconds into the simulation. The re-circularization burns did reduce the oscillations, through some eccentricity remains. Page 13 of 36

14 Verification- cont d Figure 10: Thrust Control Test Model Altitude Plot The next phase of the thrust is to change from the starting inclination of deg to a final inclination of deg. Figure 11 shows a plot of the desired final inclination in blue and the actual inclination in red. The figure shows that the thrust controller correctly changed the inclination to the correct final value. This is verified by the plot of latitude also shown in the figure. The latitude oscillation increases, and does end with a stable oscillation from deg to deg latitude. This is correct for an inclination of deg. Figure 11: Thrust Control Test Model Latitude and Inclination Plot Page 14 of 36

15 Verification- cont d Figure 12 shows the traces of the force specified by the thrust controller. The x- and y-axes in the model are the equatorial plane while the z-axis is in the north-south pole axis. The thrust force matches expectations. The altitude and re-circularization burns are primarily in the y- and x- direction. The inclination change burn includes a large z- component. These results indicate that the thrust controller is working as expected. It also shows that a much larger burn is required for the inclination change of approximately 30 deg than for the altitude change of approximately 300 km. Figure 12: Thrust Control Test Force Plot Fuel Burn Test Model Calculation of the fuel burn during maneuvers is very important to the study of the potential orbital changes. The trade off of fuel expenditure for increased detection capability is the main tradeoff of the orbital changes. The fuel burn test model is shown in Figure 13. This test model consists of three pulse source blocks from the Modelica standard library. These signals are fed into the fuel burn calculation. The force inputs for the test are shown in Figure 14. The calculated value for the amount of fuel burned is shown in Figure 15. This test model used the same values of specific impulse and force magnitude discussed previously. The final calculated burn for the simulation is kg. This can be checked by performing simple hand calculations. Page 15 of 36

16 Verification- cont d f_input_x period=100 f_input_y period=100 f_in f_input_z period=100 Fuel Burn Figure 13: Fuel Burn Test Model The amount of time the fuel burned is 100 seconds from the plot. The rate of fuel mass consumption can be calculated from: Mass Flow Rate = Trust Force Gravitational Acceleration Constant Specific Impulse 22 N Mass Flow Rate = 9.81 m s 2 235s Mass Flow Rate = kg s Multiplying this mass flow rate by the thruster burn time gives us: Mass Flow Rate Time Burned = kg s 100 s = kg This agrees with the simulation test model output. This shows that the model appears to be working correctly. Page 16 of 36

17 Verification- cont d Figure 14: Fuel Burn Test Force Inputs Figure 15: Fuel Burn Test Calculated Fuel Burn Page 17 of 36

18 Experimentation and Interpretation For the experiments, each satellite was placed in the following starting positions and conditions: IRSAT-1 and 3 are in the same inclination plane of deg at an orbiting altitude of 282 km. At that altitude, the orbital period of IRSAT-1 and 3 is ninety minutes, and the corresponding velocity in direction of flight (tangential to the orbital curve) is 7.74 km/s for that altitude. IRSAT-3 is clocked 120 deg from IRSAT-1, as would be typical in a three constellation system. IRSAT-5 was placed in an altitude of 582 km (per our problem scenario), which has a corresponding velocity of 7.57 km/s. This spacecraft was given an initial inclination of deg. Experiment 1: Baseline Scenario The first experiment was to run the simulation without any change to the satellite orbits. This sets context of the current state of the spacecraft constellation. This was accomplished by setting all of the thrust parameters in a three thrust controllers to zero, as shown in Figure 16. Figure 16: Experiment 1 Fuel Burn Model Parameters Page 18 of 36

19 Experimentation and Interpretation- cont d The resulting orbital parameters over time are presented in the subsequent figures. Figure 17 on the next page plots the altitude, latitude, and longitude of IRSAT-1, 3 and 5, in meters. The inclination angle, IR image resolution, in meters, and the fuel consumption, in kilograms are plotted in Figure 18. Figure 17: Experiment 1 Spacecraft Altitude, Latitude and Longitude vs. Time Notice that the output of Figure 17 and Figure 18 indicate that the model correctly represents the orbital characteristics and IR image resolution with respect to computed hand calculations. Each satellite was placed in the correct inclination based on calculated velocity vectors adjusted to the Cartesian coordinate system utilizing a tool created in Microsoft Excel to transform the velocity vectors into x, y, and z directions for the velocity vectors. As shown above, these vectors have been correctly inputted, resulting in the desired orbits. Also, although it seems that the trend for fuel consumption seems to be rising at a steep rate at first glance of Figure 18, notice that the vertical scale indicates that a negligible amount of fuel is being expended by the system. No thruster burns have been accomplished during this experiment; thus the rising trend of the fuel consumption is assumed to be a legacy of the modeling versus a real loss of fuel mass. Page 19 of 36

20 Experimentation and Interpretation- cont d Figure 18: Experiment 1 Orbital Inclination, Image Resolution and Fuel Consumption vs. Time Therefore, the results of Experiment 1 serve as the baseline expectations for the subsequent experiments. It serves as a double checking of the overall spacecraft model s ability to correctly represent a three satellite constellation with one spacecraft with differing orbital parameters. Experiment 2: IRSAT-5 Altitude Correction Next, the scenario tested was one that only corrects the altitude of IRSAT-5, which is 300 km above the optimal operating altitude. As reference, in earlier prototypes of the thrust controller model, the thruster application logic was based on hand calculations assuming that an instantaneous velocity change could be applied in order to execute a Holmann transfer to correct IRSAT-5 s altitude. However, experimentation during the verification stage of this report indicated that the 22 N thrusters would not produce enough change in velocity required to come anywhere near an instantaneous velocity change. Therefore, it was plainly evident that a Holmann transfer could not be accomplished to adjust the satellite altitude. A Holman transfer consists of an initial braking maneuver to instantaneously slow down the spacecraft velocity in the direction of flight to put the vehicle in an elliptical transfer orbit. Once the spacecraft reaches perigee (which happens to be the radius of the new lower orbit), a re- Page 20 of 36

21 Experimentation and Interpretation- cont d circularization burn is accomplished to speed up the space vehicle to the correct steady-state orbit at the new radius. Since the thrusters were unable to put the spacecraft into the correct transfer orbit, the thrust was applied in the direction of travel instead of the braking direction until the spacecraft reached its desired lower altitude. This resulted in an extremely large amount of fuel expended, on the order of 700 kg! However, further experimentation during the verification stage resulted in the final thrust algorithm, which maximized the advantage of gravity assistance to determine when the thrust maneuvers would be applied. Therefore, thrust was applied periodically when gravity was assisting the burn, thus introducing a significant fuel cost savings. In this second experiment, to determine the duration of the transfer thruster burns, a very long duration was inputted for the t_burn_cutoff1 parameter, and time was measured when the spacecraft reached the correct orbital radius ( km, which is equivalent to the desired altitude of 282 km). A snapshot of IRSAT-5 s radius is shown in Figure 19. A cutoff time of secs was chosen, and the simulation was run for secs. Figure 19: Experiment 2 Determining Correct Transfer Burn Cutoff Time Page 21 of 36

22 Experimentation and Interpretation- cont d As shown in Figure 19, the system approximately reaches x 10 6 m (6653 km) at secs. Thus this value becomes the input for the parameter. Inputting this value in t_burn_cutoff1, the simulation was rerun to determine if the spacecraft reaches the correct stead state mean radius. This result is presented in Figure 20 below. Figure 20: Experiment 2 Spacecraft Radius with Correct Transfer Burn Cutoff Time As shown in Figure 20, IRSAT-5 s radius oscillates roughly around the same radius as IRSAT-1 and 3. As expected with orbital maneuvering, the transfer burns produced a slightly elliptical orbit, which is shown by the oscillations of approximately 900 km from crest to trough. Therefore, another set of thruster burns, shorter in duration and once in the braking and once in the accelerating directions must be applied to re-circularize the orbit, shown by a dampening out of the oscillations. An arbitrary re-circularization burn start time was chosen after the transfer burn cutoff time, chosen optimally when the velocity of IRSAT-5 is at its peak, as shown in Figure 21. The time was arbitrarily chosen at least two oscillations after the transfer burn cutoff time, per Figure 21. Page 22 of 36

23 Experimentation and Interpretation- cont d Figure 21: Experiment 2 Time Chosen to Initiate Re-Circularization Burns As indicated in Figure 21, the time of secs was arbitrarily chosen per the criteria expressed in the previous paragraph. This time was inputted within the t_burn_start2 parameter, and then an arbitrary burn time is chosen, with an opposite burn with the same duration immediately applied afterwards. After several iterations of experimentation with several start and stop times and burn durations, the parameters entered as shown in Figure 22 below results in the plot in Figure 23, with a burn duration of 1500 secs in the retro- and pro- directions. Figure 23 shows that the oscillations of the radius for IRSAT-5 have been reduced by at least a third of the original oscillations, thus indicating a near-circular orbit. Further experimentation could be accomplished to further circularize IRSAT-5 s orbit, but this experiment was concluded prior to further dampening of the oscillations. Page 23 of 36

24 Experimentation and Interpretation- cont d Figure 22: Experiment 2 Parameters for Optimized IRSAT-5 Re-Circularization Burn Times Figure 23: Experiment 2 Final Results of IRSAT Orbital Radius After IRSAT-5 Altitude Change Page 24 of 36

25 Experimentation and Interpretation- cont d Since the primary driver for correcting the altitude for IRSAT-5 to increase the satellite IR resolution to approximately 1 meter, Figure 24 was generated to verify the increase of IRSAT-5 s IR image resolution. Figure 24: Experiment 2 Final Results of IRSAT IR Resolution After IRSAT-5 Altitude Change Further, another main factor for the decision maker would be the amount of fuel consumed by IRSAT-5 in order to correct its altitude error. Figure 25 shows both the magnitude and times of the thruster burns in this experiment, as well as the graph of the total amount of fuel consumed by the altitude reduction maneuvers. As shown in Figure 25, the total amount of fuel expended for the orbital maneuvers was kg, which is over half of the total amount of fuel initially loaded on IRSAT-5, which was indicated above as 300 kg. The decision maker must weigh the benefit of increasing the IR image resolution with drastically reducing the operational lifetime for IRSAT-5. Figure 25 also clearly depicts that only eight burns were accomplished for the transfer maneuver stage of this flight. This is a drastic reduction in time of burn and fuel consumption than the case of continuously burning thrusters to reach the desired altitude. Note that the ninth burn indication contains both the burns in the retro- and pro- directions. Page 25 of 36

26 Experimentation and Interpretation- cont d Figure 25: Experiment 2 IRSAT-5 Thruster Burns and Fuel Consumption After Altitude Change Experiment 3: IRSAT-5 Inclination Change Only For this scenario tested, the inclination change portion of the thrust controller is exercised to model the inclination change maneuver to move IRSAT-5 from its initial inclination of deg to be co-planar with IRSAT-1 and 3 at and inclination angle of deg. In order to isolate the inclination plane change maneuver from the thrust controller logic, all of the thruster burn start and cutoff times were zeroed out in IRSAT-5 s thrust controller parameter window. Also, the desired final inclination angle of deg was input into the i_final parameter field, as indicated in Figure 26. Figure 27 displays the result of the simulation run for a duration of secs. Notice the extremely large amount of time required to encapsulate the results of a 22 N thruster burn applied directly to IRSAT-5 to achieve the same inclination as IRSAT-1 and 3. One can already surmise that it would be not only undesirable, but outright impossible to achieve such a drastic inclination change. The thruster application magnitude (22 N) and duration are also shown in Figure 27. Page 26 of 36

27 Experimentation and Interpretation- cont d Figure 26: Experiment 3 IRSAT-5 Parameters for Inclination Change Only Case Figure 27: Experiment 3 IRSAT-5 Inclination and Thruster Application of Inclination Change Only Page 27 of 36

28 Experimentation and Interpretation- cont d Obviously, the models created outputs results that indicate that it is possible to accomplish an orbital change maneuver. However, the amount of time it takes to execute is on the order of secs, which is more than a magnitude longer than accomplish the altitude change case. Lack of time and understanding of the orbital physics behind rocket science precluded further experimentation of the inclination thrust logic to explore whether or not gravity could assist in making an orbital plane change, such as was discovered for the altitude change case as described in Experiment 2. With more time, modeling tweaks could conceivably reveal ways to further optimize the inclination change maneuver to minimize burn times. Knowing that the decision maker would be most interested in the amount of fuel required for an inclination change, Figure 28 displays the total amount of fuel consumed in an inclination change maneuver utilizing a mere 22 N thruster. Figure 28: Experiment 3 IRSAT-5 Fuel Consumption for Inclination Change Case Only As indicated in Figure 28, the an inclination change of deg requires kg of propellant, which is more than an order of magnitude larger than the initial hydrazine propellant load of 300 kg. Also, the bottom graph of Figure 28 shows that the total burn duration is secs! Page 28 of 36

29 Experimentation and Interpretation- cont d Figure 29 plots out the inclination change and fuel consumption on the same figure. As shown below, IRSAT-5 would, without a doubt, run out of fuel before an inclination change of 2 deg was accomplished. Figure 29: Experiment 3 IRSAT-5 Inclination and Fuel Consumption for Inclination Change Case Only This result seems reasonable because gravity does not seem to be of much assistance in the direction of thruster burns required to accomplish an inclination plane change as was defined in the thruster control model generated for this experiment. A much more efficient method of accomplishing a change in inclination angle may exist, but is unknown to the modelers at this time. It is obvious why large orbital inclination changes are not typically accomplished for assets already on orbit due to the sheer magnitude of energy required to change the inclination plane, unless the spacecraft was specifically designed to routinely make these orbital changes. For the sake of completeness, IRSAT-5 s IR resolution was generated in Figure 30 to verify that the inclination change does not affect the resolution. Comparing Figure 30 with the baseline case shown in the middle graph of Figure 18, it is clear that the resolution of IRSAT-5 remained unchanged, as expected. Page 29 of 36

30 Experimentation and Interpretation- cont d Figure 30: Experiment 3 IRSAT-5 IR Resolution for Inclination Change Case Only Experiment 4: Combined IRSAT-5 Altitude and Inclination Change Although Experiment 3 yielded results that indicate that it is infeasible to accomplish an inclination plane change, for the sake of exercising the model s capability to perform an altitude and inclination change, this experimental case was investigated. The parameters for IRSAT-5 s thrust controller were chosen as follows: The altitude change thruster burn start and cutoff times were used from Experiment 2. Then an arbitrary time was chosen after the final re- circularization thrust maneuvers to start the inclination burn phase of flight. The parameters are displayed in Figure 31 on the following page. Page 30 of 36

31 Experimentation and Interpretation- cont d Figure 31: Experiment 4 IRSAT-5 Thrust Controller Parameters for Combined Orbital Changes Much deliberation was made to whether or not to create the ability for the model to simulate a simultaneous altitude and inclination plane change maneuver. However, it was decided to create the thrust control algorithm to accomplish the two orbital changes separately since it seemed intuitively logical that there would be negligible physical (i.e. gravitational) advantage to accomplishing them together since a Holmann transfer was not able to be accomplished because of the relatively small thrust thrown by the onboard rockets, which were designed to make small orbit adjustments and spacecraft attitude changes. The result of the combined altitude and inclination test case is presented in Figure 32. As shown in the figure, the models correctly depict the altitude change maneuvers accomplished prior to the inclination change maneuver. Figure 32 shows that it is theoretically possible to correct IRSAT-5 s altitude and adjust its inclination angle without the constraint of fuel limitations. However, it is clear that there is a tremendous amount of fuel required to accomplish the inclination change portion of the experiment, as reflected in Experiment 3. Page 31 of 36

32 Experimentation and Interpretation- cont d Figure 32: Experiment 4 IRSAT Altitude and Inclination for IRSAT-5 Combined Orbital Changes Figure 32 also shows that the inclination change maneuver does not observably affect the orbital radius of IRSAT-5, i.e. the altitude doesn t change during the inclination maneuver phase of flight. For completeness, the fuel required for both maneuvers and the total time of IRSAT-5 thruster burns are plotted versus time in Figure 33. The figure shows that the total amount of fuel required is 5480 kg hydrazine, which correlates to the results from Experiments 2 and 3. Again, it is impossible for IRSAT-5 to accomplish both the altitude and inclination change with the physical constraints of the fuel load. Perhaps if the IRSATs had a nuclear powered thrust system, the space vehicles could accomplish these orbital maneuvers. Figure 33 also shows that the total time the thrusters were applied was sec. Also, Figure 34 was generated to depict the IR resolution for each IRSAT as a result of the orbital maneuvers. The results mirror those shown in Figure 24 created in Experiment 2. These are the expected results, showing that the inclination change has no observable affect on the IR image resolution. Page 32 of 36

33 Experimentation and Interpretation- cont d Figure 33: Experiment 4 IRSAT-5 Fuel Consumption and Burn Time Combined Orbital Changes Figure 34: Experiment 4 IRSAT IR Resolution for IRSAT-5 Combined Orbital Changes Page 33 of 36

34 Experimentation and Interpretation- cont d As a final experimentation note, Figure 35 shows a screenshot of the animation of the satellite simulation when zoomed way out to capture the size of the Earth. IRSAT-1 is the green sphere, IRSAT-3 is the pink sphere, and IRSAT-5 is the grey sphere. This snapshot is when all IRSATs are coplanar and at the same altitude, with the view tilted to show all three birds in orbit. Figure 35: Experiment 4 IRSATs In Orbit Around the Earth With more time, the axis lines for the Earth could have been added to show the orientation of the satellites with respect to latitude and longitude. Page 34 of 36

35 Lessons Learned There were several noteworthy lessons learned from the modeling of the orbital parameters of a spacecraft. First of all, it was difficult to model the satellite orbits in the Cartesian plane since the satellite positions and velocities change from positive to negative depending on the location of the spacecraft around the Earth. Several coordinate system transformations were necessary to get the models to behave correctly, especially when trying to deal with the inclination change maneuvers. It was extremely difficult to get the inclination change algorithm correct because an incorrect directional unit vector was used to determine the direction to apply the thrust to perform an inclination plane change. Extensive understanding was gained in programming in Modelica through the creation of these models. Dealing with a programming language that is acausal took some getting used to. At one point, the thrust controller model was outputting a thrust force in the initial direction and was not updating according to the new direction that the spacecraft travelled. Although the output of the system looked correct, it was found that this was not a correct representation of the real physics of orbital mechanics. After much investigation, it was found that the source of the non-updating force application direction was the use of a when statement in the algorithm, which takes the initial direction of the clause and doesn t update it throughout the course of when the when is true. Converting the when statement to if statements resolved the issue, which outputted a more realistic result that correctly matches true orbital dynamics. Another source of long hours of wasted time troubleshooting the models was the incorrect use of units. During the creation of the altitude change algorithm, a lot of time was wasted troubleshooting the reason why the model would not shut off the thrust although the correct algorithms were being utilized. Everything was tried, from converting the statements to utilize Boolean variables, and extensive experimentation of deciding what state variables would be used to trigger thrust events. After long hours of frustration, it was found that the reason why the thrust would not cutoff was because the trigger clause in the algorithm was using kilometers, whereas the data was being read in meters. From that point onward, careful review of the magnitude and units were incorporated in the programming of the models. A large lesson learned was finding out that gravity could be used to assist in making more fuel efficient altitude change maneuvers. Embedded in this lesson was the fact that Holmann transfers could not be accomplished with such small thrusters that are typically onboard most satellites. After discovering that the satellite had to thrust directly towards its final destination, it was a good find that the thrust could be applied when the velocity was increasing due to the gravitational pull of the Earth. A lot was learned about orbital mechanics, to include the lesson that inclination changes require a large amount of energy. Finally, as alluded earlier in this section, in experimentation with the thrust control logic, it was found that the easiest way to program logic of when to start and cut of thruster burns was to use time as the trigger variable. Utilizing radius (or altitude) resulted in additional unwanted thruster burns because of the oscillations of the radius due to the increased eccentricity introduced in accomplishing some Page 35 of 36

36 Lessons Learned- cont d maneuvers. Utilizing velocities had the same result. The best triggers were the burn start and stop times that were found through experimentation with orbital models. This was found to be the best method of controlling the system without utilizing more complete control systems and controller algorithms. Page 36 of 36