Atmospheric Trajectories
4.0 Trajectories
Tamara Roark
4.1 Nomenclature
cd = Coefficient of drag
cl = Coefficient of lift
D = Drag force
L = Lift force
vmass = Spacecraft mass
r = Distance from center of planet
S = Reference area
T = Thrust
v = Velocity in a Mars fixed frame
( = Spacecraft angle of attack with respect to velocity vector
( = Gravitational parameter (mass of planet . universal gravitational constant)
( = Bank angle
( = Angle between velocity vector and thrust
( = Angular velocity of Mars
[pic] = Flight path angle
( = Latitude
[pic] = Heading angle
4.2 Introduction
Our vehicle is to land on Mars. In order to perform several passes by the planet, the vehicle must be captured into an elliptical orbit. Once in orbit about the destination planet, we present three options. If nothing is done, the spacecraft orbits the planet once, and then contacts the planet’s surface on the succeeding pass through the atmosphere. Aerobraking is a second option and is performed by doing a maneuver at apoapsis to raise the periapsis to a level such that repeated passes through the atmosphere will eventually cause the spacecraft to contact the planet’s surface. A third option is to have the vehicle land on the Martian surface on its first pass. This is achieved by implementing an angle of attack controller to prevent the vehicle from reaching its g-loading limits, and bank angle controller to fly at a constant altitude. The advantages of these certain schemes compared to the other schemes will be discussed while keeping the g-loading of the vehicle below its predetermined limits, and heating of the vehicle within acceptable limits.
4.3 Analysis of the Algorithm
We solve the second order equations governing the motion of the spacecraft (Eqs. 4.1.1-4.1.7) in FORTRAN using the Runga-Kutta 4 method. The equations of motion are also found in Reference 1 on page 27.
[pic]T[pic]2[pic] (4.1.1)
[pic]N[pic]2[pic] [pic]2[pic] (4.1.2)
[pic]N [pic]2 [pic] +
[pic]2 [pic] (4.1.3)
where:
[pic] (4.1.4)
[pic]2 [pic]dS (4.1.5)
[pic] (4.1.6)
[pic]2 clS (4.1.7)
The goal of the mission is to land on Mars starting from Project PERForM’s initial design. The parameters that were specified was the range of entry flight path angle had to be -9(to -11(, and an initial velocity of 8390 m/s. Performing small maneuvers far from the planet can change the arrival flight path angle. The propellant required in doing these angle changes are budgeted in the en route maneuvers section of propellant. Therefore, the parameters that were considered for our design were the shape and size of the vehicle, the location of the center of gravity, the trajectory scheme, the control surfaces, and the thermal protection that was used.
In order to get the spacecraft captured around the Martian planet, the bank angle (() is set to 180( prior to its first pass through the atmosphere. By changing this angle, the lift vector will be pointed toward the planet causing drag instead of lift. This prevents the spacecraft from “skipping off” the planet and traveling on a hyperbolic trajectory, and does not require any kind of retroburn to slow the vehicle down, prior to entering the atmosphere.
Because this is a manned mission, a g-loading limit was set to not exceed 6 g’s. On the first pass through the atmosphere, the spacecraft creates as much drag as possible without crashing. A controller is implemented to control the g-loading. If the spacecraft is experiencing less than 5 g’s, then the angle of attack is changed to the maximum value that will still trim, and if the g-loading is more than 5 g’s, the angle of attack is lowered to the minimum value that will still trim.
Option 3 as described above is to land the spacecraft on Mars on its first pass by the planet. In order to achieve this, a target altitude is set to have the spacecraft aim towards. Once at this altitude, another controller is implemented that controls the bank angle ((), using a proportional-derivative control. By keeping the spacecraft at a specified altitude, the atmosphere of Mars will produce drag on it, and cause the velocity to decrease. The bank angle is varied in order to keep the spacecraft from crashing. If the bank angle is 0(, then the lift vector is pointing up, causing lift. If the bank angle is 180(, the lift vector will point down, causing more drag. In varying the bank angle, the spacecraft is able to reach Mach 3, where the supersonic parachutes will then deploy.
4. Using trajectory codes
In order to run the trajectory files, the user only needs to change a few parameters, which can be found in a block in the beginning of the traj.f FORTAN program. The parameters that can be modified are the length of the cylinder, the radius of the cylinder, the vehicle mass, the length of the leading edge of the fins, the radius of the leading edge of the fins, and the sweep angle of the fins. The axial and vertical center of gravity location also needs to be specified. The cruising altitude is the target altitude that the vehicle is supposed to aim towards. If this value is set to approximately 20,000 meters for this design, the vehicle will land on its first pass by the planet. If it is set to approximately 35,000 meters, the vehicle will begin aerobraking. The range also needs to be specified. This determines how far away from the target altitude the controllers turn on.
If aerobraking is desired, the correction factor also can be changed. This correction factor determines what percentage of the atmosphere to aim towards each time.
The initial flight path angle can also be varied. It was found that it had to be less than -10.5 degrees in order to keep the trajectory on an elliptical orbit. It was arbitrarily chosen to have an initial flight path angle to –10.75 degrees.
4.5 Trade study comparison
The main concern of the trajectory designer, is to design a trajectory that will not create an excess of –6 g’s on the astronauts, while maintaining heating on the vehicle that will not cause the astronauts to burn up, and create a vehicle that is capable of reaching Mach 3 at an acceptable altitude in order for the parachutes to have enough time to deploy. Many trade studies were completed and compared in order to decide on the best design. The main question in the design was how big to make the spacecraft. The smaller the spacecraft, the less mass, and the cheaper it is to launch. But the smaller the vehicle is, the less drag it will produce while flying through the Martian atmosphere. Analysis was completed to optimize the size of the vehicle. Figures 4.1-4.7 show a comparison of the spacecraft’s behavior with two different sized vehicles, a 13 meter diameter and a 9 m diameter. For the 13 meter diameter case, an overall mass of 61952.52 kg was used, with a center of gravity location at 6.8 meters in the x-direction and 0 m in the y-direction (see Section 5.2 for coordinate definitions). For the 9 meter diameter case, an overall mass of 52352 kg was used, with a center of gravity location at the same location as the 13 diameter case. The length of both vehicles was 10 meters. These graphs are for a trajectory in which the spacecraft lands on the first pass by Mars. Figure 4.1 is a plot of the velocity in the Mars fixed frame. It is shown that for both cases, the initial velocity is 8390 m/s, and the two vehicles behave quite similarly continually until the vehicle reaches Mach 3, where the supersonic parachutes will take over. Figure 4.2 displays the vehicle’s flight path angle history. Both vehicles began at an initial flight path angle of –10.75(. The 13-meter diameter vehicle has two gradual changes, while the 9 meter diameter vehicle is quite oscillatory in behavior. The oscillations are of small magnitude, and would barely be felt by the astronauts. Figure 4.3 shows the heading angle that is initialized to 1(. This plot shows that for the 13-meter diameter case it remains almost constant, by only varying by approximately one degree. The 9 meter diameter case; however, appears to change linearly until it reaches approximately –15 degrees at which it then remains constant. Figure 4.4 displays the altitude history for both vehicles. Both vehicles follow similar trends, but when the vehicle reaches Mach 3, the 9-meter diameter vehicle is lower in the atmosphere. The final altitude for the 13-meter diameter case is 10,487 meters, while for the 9 meter case the final altitude is 6388 meters. Figure 4.5 shows the g-loading on the vehicles. As shown, the vehicles do not exceed the –6 g-load limit, and as expected, the 9-meter diameter vehicle does not produce as many g’s as the larger diameter vehicle. Figure 4.6 illustrates the changing angle of attack for the vehicle. The initial difference between the two vehicles is because the different vehicles have different angles of attack that allow the vehicle to be trimmed. In the main trajectory FORTRAN program, traj.f, the range of angle of attacks is found through an iterative process. The initial drop in the plot is when the vehicle is approaching Mars. As described previously, the angle of attack is dropped to its minimum in order to create as much drag as possible. Once the vehicle is near the specified target altitude, the angle of attack is then set to its maximum value again to prevent the vehicle from crashing onto the Martian surface. The two vehicles demonstrate the same behavior, with just a shift due to the different angles of attack that will trim the vehicle. As mentioned before, the bank angle is controlled to keep the spacecraft near the target altitude. Figure 4.7 demonstrates this behavior. As shown, the 13-meter diameter vehicle only changes the bank angle once from -180( to 0( while the 9 meter diameter vehicle is quite jumpy with several oscillations that could cause the astronauts onboard to be sick.
Figure 4.8 and Figure 4.9 show the angle of attack and g-loading plots on the same graph. It is shown that during the period of maximum acceleration on the vehicle, the angle of attack is set to its maximum value. In comparing Figure 4.4 to Figure 4.7 it is shown that once the spacecraft has reached an altitude smaller than the target altitude set to 40,000 meters, the bank angle is changed. If the bank angle is smaller than -180( than the bank angle is controlled to be -180(. Or if the bank angle is greater than 0(, than it is controlled to 0(. This analysis shows that both vehicles could be used for our optimized design.
Not only was size an issue, but which trajectory scheme that would be used was also a concern. Further analysis was completed that compared a trajectory with aerobraking to one that would land the vehicle on its first pass by Mars.
The 9 meter diameter vehicle was examined using both schemes. The mass of the vehicle was refined to a more realistic mass of 47324 kg, and the center of gravity location was located at 6.554755 meters in the x-direction, and 0.133519 in the y-direction. When the spacecraft went through an aerobraking scheme, it went to apoapsis 25 times before reaching Mach 3. At Mach 3, the vehicle was at an altitude of 11167 meters, and a final flight path angle of –9.97 degrees. The spacecraft experienced a maximum g-load of –4.3, and required a delta V of 11.893 meters/sec. The behavior of the vehicle during the aerobraking scheme is shown in Figures 4.10-4.15. In doing apoapsis maneuvers, to lower the periapsis altitude, the change in velocity at each apoapsis is delta V. The value of delta V described above is the total delta V used for the 25 times the spacecraft is at apoapsis. Figures 4.16- 4.23 show the trajectory for when the vehicle lands on its first pass. For this case, the final altitude at Mach 3 was 12124 meters, with a final flight path angle of –9.4130 degrees. The maximum g-load that the spacecraft experienced was –5.3. The bank angle has 6 oscillations caused by controlling the altitude to the specified target altitude. The change in the bank angle is approximately 7 degrees per second. This rate is rather high, and could be altered by changing the controller gains. Further analysis should be done to try to make the change in bank angle less dramatic.
Table 4.1 summarizes the comparison of both trajectory schemes for the 9 meter diameter vehicle.
|Diameter |Final Altitude (meters) |Final Flight Path Angle (deg) |Maximum g-load |Delta V required (m/s) |
|(meters) | | | | |
| | | | | |
|9 |11167 |-9.97 |-4.3 |11.893 |
| | | | | |
|9 |12124 |-9.413 |-5.3 |0 |
Table 4.1: Comparison of aerobraking to landing on 1st pass.
In examining Table 4.1, it is shown that aerobraking does not help the ending result very much. The final altitude is lower than that found when landing on the 1st pass, and the final flight path angles are approximately equal. The only difference is the g-loading. In aerobraking, the maximum g-load is –4.3, while the maximum g-load for landing is –5.3. Both of these values are below the limit of –6 g’s. Using aerobraking, adds more overall mass to the vehicle because of the extra propellant needed to do apoapsis corrections. The estimated 11.893 m/s delta V that is required would add an additional 223.53 kg to the overall mass. Also to get these final results using aerobraking, it took four days to complete; while landing on the first pass only takes approximately 400 seconds. This shows that it would be safer to land on the first pass, due to unexpected encounters that could happen while the spacecraft is aerobraking in space for those additional four days. Landing on the first pass, is within the design limits, and will be implemented into the optimal design.
4.6 Optimal design
The diameter used for the optimal design was 9 meters. Using this size of vehicle, the trajectory scheme that was used was to land on the first pass. The total vehicle mass was found to be 47042.7 kg. The center of gravity location is 6.552806 in the x direction and 0.093672 in the y direction. It was found that the final altitude the vehicle reached Mach 3 was at 12138 meters. The maximum g-load the vehicle experiences is –5.4. This value is close to the limit of –6 g’s. If an aerobraking trajectory was implemented instead of landing on the first pass, it would take longer to complete, approximately 4 days, and it would only decrease the g-loading by about 1 g. Since the g-loading for landing on the first pass was below the limit, this was decided on the best scheme for the design.
Figures 4.24-4.30 show the behavior of the vehicle as it enters the atmosphere of Mars and reaches Mach 3. Figure 4.24 shows the velocity in a Mars fixed frame. Figure 4.25 shows the flight path angle. There are slight oscillations of approximately 1 degree, which do not impose any kind of problem on the spacecraft or astronauts. Figure 4.26 demonstrates the behavior of the heading angle. It shows that the spacecraft begins at an initial angle of 1 degree and decreases almost linearly to around –13 degrees. Figure 4.27 shows the altitude history. Figure 4.28 displays the g-loading on the vehicle. Figure 4.28 shows how the Mach number decreases to a value of 3. Figure 4.29 shows the angle of attack history. Figure 4.30 displays the bank angle versus time. Again, this is somewhat oscillatory and changes at a rate of approximately 10 degrees per second. Further work is needed in getting this to be less erratic.
4.7 Conclusions
We successfully designed a vehicle that prevented it from reaching the predetermined limits in g-loading, while keeping the heating within acceptable limits. In addition, we showed that by adjusting two constraints (bank angle, angle of attack), we could control the spacecraft to remain in the atmosphere on the first pass and land, without having to do multiple passes with aerobraking. The trajectories involve no propulsive maneuvers and no intense g-loading, thereby significantly reducing the weight of the vehicle necessary to perform such a mission.
4.8 Tables and Figures
|Diameter (meters) |Length (meters) |X location of c.g. |Y location of c.g. |Overall mass (kg) |
| | |(meters) |(meters) | |
|9 |10 |6.8 |0 |52352 |
|13 |10 |6.8 |0 |61952.52 |
Table 4.2: Parameters of both vehicles used in Figures 4.1-4.7.
[pic]
Figure 4.1: Velocity in Mars fixed frame versus time.
[pic]
Figure 4.2: Flight path angle versus time.
[pic]
Figure 4.3: Heading angle versus time.
[pic]
Figure 4.4: Altitude versus time.
[pic]
Figure 4.5: G-loading versus time.
[pic]
Figure 4.6: Angle of attack versus time.
[pic]
Figure 4.7: Bank angle versus time.
[pic]
Figure 4.8: Angle of attack and G-loading versus time for 13 meter diameter vehicle.
[pic]
Figure 4.9: Angle of attack and G-loading versus time for 9 meter diameter vehicle.
|Diameter (meters) |Length (meters) |X location of c.g. |Y location of c.g. |Overall mass (kg) |
| | |(meters) |(meters) | |
|9 |10 |6.554755 |0.133519 |47324 |
Table 4.3: Parameters used in Figures 4.10-4.23.
[pic]
Figure 4.10: Velocity in Mars fixed frame versus time for 9 meter diameter vehicle aerobraking.
[pic]
Figure 4.11: Flight path angle versus time for 9 meter diameter vehicle aerobraking.
[pic]
Figure 4.12: Heading angle versus time for 9 meter diameter vehicle aerobraking.
[pic]
Figure 4.13: Altitude versus time for 9 meter diameter vehicle aerobraking.
[pic]
Figure 4.14: G-loading versus time for 9 meter diameter vehicle aerobraking.
[pic]
Figure 4.15: Changing mass versus time due to burns performed at apoapsis for 9 meter diameter vehicle aerobraking.
[pic]
Figure 4.16: Velocity in a Mars fixed frame versus time for 9 meter diameter vehicle landing on 1st pass.
[pic]
Figure 4.17: Flight path angle versus time for 9 meter diameter vehicle landing on 1st pass.
[pic]
Figure 4.18: Heading angle versus time for 9 m diameter vehicle landing on 1st pass.
[pic]
Figure 4.19: Altitude versus time for 9 m diameter vehicle landing on 1st pass.
[pic]
Figure 4.20: G-loading versus time for 9 meter diameter vehicle landing on 1st pass.
[pic]
Figure 4.21: Mach number versus time for 9 meter diameter vehicle landing on 1st pass.
[pic]
Figure 4.22: Angle of attack versus time for 9 diameter meter vehicle landing on 1st pass.
[pic]
Figure 4.23: Bank angle versus time for 9 meter diameter vehicle landing on 1st pass.
|Diameter (meters) |Length (meters) |X location of c.g. |Y location of c.g. |Overall mass (kg) |
| | |(meters) |(meters) | |
|9 |10 |6.552806 |0.093672 |47042.7 |
Table 4. 3: Parameters used in Figures 4.24-4.31.
[pic]
Figure 4.24: Velocity fixed in Mars fixed frame versus time for optimized design.
[pic]
Figure 4.25: Flight path angle versus time for optimized design.
[pic]
Figure 4.26: Heading angle versus time for optimized design.
[pic]
Figure 4.27: Altitude versus time for optimized design.
[pic]
Figure 4.28: G-loading versus time for optimized design.
[pic]
Figure 4.29: Mach number versus time for optimized design.
[pic]
Figure 4.30: Angle of attack versus time for optimized design.
[pic]
Figure 4.31: Bank angle versus time for optimized design.
References
R.4.1. “Hypersonic and Planetary Entry Flight Mechanics”, by Vinh, Busemann, and Culp
R.4.2. “The Mars Reference Atmosphere”, by Hansen, Hubbard, James, Kieffer, Leovy, Martin, Miner, Owen, Pollack, Seiff, and Stewart
R.4.3. “Fundamentals of Astrodynamics”, by Bate, Mueller, and White
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