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A.3.2.7 Simulation Results

In order for the Monte Carlo simulation to be run we used the nominal values for each case to set up a nominal run of the simulation. For this nominal run we output a wide range of data for analysis. Using the steering law modification process outlined in Section A.3.2.5 we found a nominal solution that achieved a final periapsis altitude much higher than the required 300 kilometers in order to account for the losses that would occur in the Monte Carlo process. The relevant results presented for each case will be the final trajectory, the steering and pointing angle history, and thrust vector angle histories. Standard two body orbit parameters will also be provided.

A.3.2.7.2 200 g Case

We successfully solved a nominal solution using the six degree of freedom simulator setup for the 200 g payload case by iterating between simulator runs and the spline process. The mission requirements are for the payload to achieve an orbit with a periapsis altitude greater than 300 kilometers. For the successful nominal 200 gram case the periapsis altitude was 486 kilometers. This overshoot of more than 186 kilometers is there for two reasons. First of all the 200 gram case was over designed to deliver a ΔV 145% higher than the initial calculated requirement. The second cause of this is that a certain amount of overshoot is desired in order to account for variances that will cause losses in the Monte Carlo process.

The final launch vehicle launch path and orbit is output by the simulator code. We compare the nominal launch path and orbit from the trajectory code to the results achieved by the simulators run of the nominal case. Examining the launch path shows that the results from the simulator run result in a higher orbit insertion. This is a result of the differences in the steering law caused by the steering law modification process outlined in Section A.3.2.5 and by losses of up to a maximum of 10% caused by vectoring the thrust off of the center axis of the launch vehicle.

[pic]

FigA.3.2.7.1.1: Launch Path.

(Michael Walker, Alfred Lynam, Adam Waite)

We also propagate the final orbit of the vehicle after burning is stopped and compared it to the output from the trajectory code. 1 As can be seen the final propagated orbit has a much higher eccentricity than the nominal orbit. This is caused because of inaccuracies in control as well as the fact that the steering law modification process, which produces a spline of the steering law, does not exactly follow the steering law. It should be realized that this case was over designed. Due to this over design the nominal steering law requires the rocket to burn up to thirty degrees down for a significant portion of the flight in order to burn off radial velocity. As we will show later in this section the spline does not do that exactly which results in a higher eccentricity than the nominal orbit.

[pic]

FigA.3.2.7.1.2: Propagated Orbit.

(Michael Walker, Alfred Lynam, Adam Waite)

The orbital elements associated with the above orbit are shown in Table A.3.2.7.1.1: 1

|Table A.3.2.7.1.1 Orbital Elements |

| |

| |

|Variable |Value |Units |

|a |12453.39248105163 |Km |

|e |0.450580217388334 |-- |

|i |26.68600785143881 |Deg |

|Ω |13.92864739182941 |Deg |

|ω |267.3823541431028 |Deg |

| | | |

|Footnotes: Definitions below |

Where a is the semi-major axis, e is the eccentricity, i is the inclination, Ω is the right ascension of the ascending node and ω is the argument of periapsis. These values were used to generate the propagated orbit in Fig A.3.2.7.1.2 using standard two body orbit equations. 1

Examining the steering angle time history and comparing it to the nominal and modified nominal results help to explain the differences between the nominal and the actual orbit. We can see in Fig A.3.2.7.1.3 that the first stage is controlled almost exactly to the spline and the steering law and the rest of the time the controller works with low angular error in the steering angle. This conclusion obviously does not apply for the third stage which is spin stabilized and has no thrust vectoring capability.

[pic]

FigA.3.2.7.1.3: Steering Angle

(Michael Walker, Alfred Lynam, Adam Waite)

An orbit with lower eccentricity could be found by further optimizing the spline created in the steering law modification process. However, this was not done because of time limitations in the design process coupled with the fact that we had achieved a solution that reached the required periapsis and withstand the Monte Carlo process. If trying to optimize to a lower eccentricity the goal would be to change the steering law spline so the launch vehicle burns while pointing below horizontal for a longer period of time.

Further investigation of the orbit reveals a difference in inclination from the nominal orbit. This difference can be clearly seen below in Fig A.3.2.7.1.3

[pic]

FigA.3.2.7.1.4: Inclination Differences

(Michael Walker, Alfred Lynam, Adam Waite)

We also analyzed the results of the pointing and steering angle histories to try and explain this occurrence as well as to make sure that the results seem reasonable. We realized from early on in the design process that precise control of the steering angle would come at the cost of a slight pointing angle error. This is due to the simplification of our controller design. With this simplification in place it was not possible for us to easily attain complete angular control with a constant gain matrix. Optimally the pointing angle would remain zero but errors do occur for reasons stated above.

[pic]

FigA.3.2.7.1.5: Pointing and Steering Angle History

(Michael Walker, Alfred Lynam, Adam Waite)

. The pointing error leads to the change in inclination as shown in Fig A.3.2.7.1.4 however, this error acceptable because there are no limitations on inclination outlined in the mission requirements. Ways of possibly eliminating this error are outlined in the Control Theory Section A.3.2.2.

Analyzing the thrust vector angle history is also important to determining if the thrust vector control (TVC) system has been over or under designed. As can be seen from Fig A.3.2.7.1.6 for most of the flight the thrust vector deflection angle (δ) is very small, however during the ‘pitch over’ maneuver between 100 and 150 seconds the full range of thrust vectoring is employed in order to maintain the desired steering angle. The direction thrust vector angle (κ) is the angle that defines the direction the thrust vector deflects in as defined in the simulator design section A.3.2.6. From the plot of the TVC angles we can see that in this case the controller puts a great deal of emphasis on steering angle control as during the pitch over maneuver The thrust vector rotation angle (κ) is almost exclusively at +90 or -90 degrees. This signifies that the controller is exclusively trying to handle the steering angle.

[pic]

FigA.3.2.7.1.6: Thrust Vector Angle History

(Michael Walker, Alfred Lynam, Adam Waite)

We can also deduce from the plot of the TVC deflection angle (δ) that the maximum TVC angle of 5 degrees is not really required to provide adequate control of the launch vehicle. While there are a few points that the maximum TVC angle is used it is a reasonable assumption to say that the controller would be able to still work with a smaller maximum TVC deflection angle. While this may or may not directly affect the cost of the launch vehicle it does suggest that we could get away with using a thrust vectoring method with less capability for the 200g case. Another possible interpretation of this data is that we could employ a more aggressive steering law for the pitch over maneuver for this launch vehicle.

It is hard to analyze the TVC deflection angle during the time when it is only maintaining stability of the system. It is hard to see because the angle is so much greater during the pitch over maneuver than during the rest of the time of flight. For this purpose Fig. A.3.2.7.1.7 is provided in order to see the magnitude of the deflection angle during stability control.

[pic]

FigA.3.2.7.1.7: Zoomed Thrust Vector Angle History

(Michael Walker, Alfred Lynam, Adam Waite)

As Fig. A.3.2.7.1.7 illustrates the maximum deflection angle reached by the TVC system, excluding the pitch over maneuver is 0.01 degrees. This shows that stability outside the atmosphere is quite easy to control and that the real challenge in controllability of a launch vehicle is in the pitch over maneuver portion of flight.

Looking back at the design process for this launch vehicle’s control system it is obvious that more time would have been greatly beneficial in shaving off the large eccentricity of the orbit. Also with more time to do a complete design process we would have suggested picking a case that is less over designed as this case can supply 145% of the ΔV required to reach the orbit we want. With that over design being the case we would suggest picking one of the less over designed cases and rerunning the optimization process to see if similar results could be achieved. Unfortunately there was not enough time in the design process to effectively do this. This is a case that could definitely benefit for some more optimization before moving to the next phase of the design.

A.3.2.7.2 1 Kg Case

We successfully solved a nominal solution using the six degree of freedom simulator for the 1 kg payload case by iterating between simulator runs and the spline process. The mission requirements are for the payload to make it into an orbit with a periapsis altitude greater than 300 kilometers. For the successful nominal 1 kilogram case the periapsis altitude was 366.96 kilometers. This overshoot of more than 60 kilometers was intentional in order to leave enough room for the variances in the final orbit that would be caused by uncertainties in vehicle mass and propellant properties, these are later simulated in the Monte Carlo process.

The final orbit is output by the simulator code along with the launch path of the vehicle. We compared this to the predicted launch path from the 3 degree of freedom trajectory code that generates the nominal steering law. As can be seen below the trajectories are somewhat different and the 6 degree of freedom results makes a lower orbit insertion than the 3 degree of freedom results. This is expected because losses are incurred on the order of 10% of the thrust when thrust vectoring is used for steering.

[pic]

FigA.3.2.7.2.1: Launch Path.

(Michael Walker, Alfred Lynam, Adam Waite)

The final orbit is also propagated and compared to the final orbit from the projected orbit from the trajectory code. 1 As can be seen the results from the simulator return an orbit with a higher eccentricity than that from the trajectory code. This is also expected because, as explained in the section on steering law modification, the nominal steering law does not have a continuous derivative, this makes following this steering law exactly virtually impossible. Also with the use of a spin stabilized third stage modifications needed to be made so that the steering angle would change in a desired manner when thrust vectoring is unavailable. We believe that with time to do a more complete design trade between the trajectory and dynamics and controls group the eccentricity of the orbit from the 6 degree of freedom model could be lowered and that the results would converge to something closer to what was predicted from the trajectory code.

[pic]

FigA.3.2.7.2.2: Propagated Orbit

(Michael Walker, Alfred Lynam, Adam Waite)

The orbital elements associated with the above orbit are shown in Table A.3.2.7.2.1: 1

|Table A.3.2.7.2.1 Orbital Elements |

| |

| |

|Variable |Value |Units |

|a |8080.831608032714 |km |

|e |0.1680355167471291 |-- |

|i |26.72402772687530 |Deg |

|Ω |15.23159608924024 |Deg |

|ω |26.17267939863041 |Deg |

| | | |

|Footnotes: Definitions below |

Where a is the semi-major axis, e is the eccentricity, i is the inclination, Ω is the right ascension of the ascending node and ω is the argument of periapsis. These values were used to generate the propagated orbit in Fig A.3.2.7.2.2 using standard two body orbit equations. 1

We examined the results of the plot of the steering angle time history in order to help explain the differences in the ascent of the launch vehicle. Referencing Figure A.3.2.7.2.3 we see that the controller produces an actual steering angle that is different from what the spline requests. It can also be seen that the spline is also different from the steering law provided by trajectory.

[pic]

FigA.3.2.7.2.3: Steering Angle History

(Michael Walker, Alfred Lynam, Adam Waite)

An orbit with lower eccentricity could be achieved by further optimizing the controller or the spline being used. However, further optimization on this case has not yet been completed due to time constraints on the design process. The main goal of the optimization we completed was to attain the desired periapsis outlined in the mission requirements.

Further comparison between the nominal and the achieved orbit show a slight difference in inclination as is illustrated in Fig A.3.2.7.2.4 as shown below.

[pic]

FigA.3.2.7.2.4: Inclination Difference

(Michael Walker, Alfred Lynam, Adam Waite)

We also analyzed the time histories of the steering and pointing angles. Throughout the course of the design process we realized that due to our simplification of the controller design the gain matrix could not be easily defined as a constant achieve complete angular control. Since this is the case a decision had to be made on which angle required the most precise control. This angle was obviously the steering angle as small steering angle changes lead to big changes in the final orbit. For this reason the pointing angle could not always be controlled as much as we would have liked. Optimally the pointing angle would always be close to zero; however a maximum pointing error of 10 degrees is reached.

[pic]

FigA.3.2.7.2.5: Steering and Pointing Angle History

(Michael Walker, Alfred Lynam, Adam Waite)

This pointing angle error is the cause of the slight offset in orbit inclination from the projected orbit. This is not necessarily a problem because currently the only requirement on the orbit is the periapsis altitude. With that being the case we decided it was acceptable to allow the pointing error to remain in order to achieve better steering angle control. Ways of possibly eliminating this error are outlined in the Control Theory Section A.3.2.2.

It is also important to analyze the thrust vector angle histories in order to see if we are exploiting all of our possible thrust vectoring or if a more sparse thrust vector system could be employed. As can be seen from Fig A.3.2.7.2.6 for most of the flight the thrust vector deflection angle (δ) is very small, however during the ‘pitch over’ maneuver between 100 and 150 seconds the full range of thrust vectoring is employed in order to maintain the desired steering angle. The thrust vector rotation angle (κ) is the angle that defines the direction the thrust vector deflects. As can be seen the controller places a much greater emphasis on the steering angle. This is especially true during the pitch over maneuver when κ jumps almost exclusively between 90 and -90 degrees signifying it is only working on correcting the steer angle.

[pic]

FigA.3.2.7.2.6: Thrust Vector Angle History

(Michael Walker, Alfred Lynam, Adam Waite)

We can see from the plot of thrust vector deflection angle (δ) that we employ the maximum allowance of 5 degrees. This full use of capability is in fact required. Looking back at Fig A.3.2.7.2.3 we can see that at 120 seconds, which is where the large TVC angles begin, there is a reasonably error that the control system is trying to correct. Based on these TVC angle results it is safe to assume that we could not handle a steering law that is much more aggressive than the one we are currently using.

Employing a less aggressive steering law would probably lead to better controllability of the system. Exploiting a less aggressive steering law, however, would not necessarily lead to a more optimal orbit. Further studies should be done to determine which is more optimal, having a system that can not exactly follow an aggressive steering law or having a less aggressive steering law that can be followed exactly. As already stated conclusions on that subject can not be drawn at this time.

Unfortunately the large magnitude of the thrust deflection angle during the pitch over maneuver makes it difficult to see the thrust vectoring done during the less intense section of the steering law. Since this is the case a zoomed in plot of the deflection angle is shown. This plot shows a more detailed view of the amount of thrust vectoring used during the period of flight that is not during the pitch over maneuver.

[pic]

FigA.3.2.7.2.7: Zoomed Thrust Vector Angle History

(Michael Walker, Alfred Lynam, Adam Waite)

We can see in Fig. A.3.2.7.2.7 during most of the flight the maximum value of the thrust vector deflection angle is about 0.02 degrees. This shows that if the maximum thrust vector deflection angle was reduced from 5 degrees because of subsystem limitations we would need to have a more gradual pitch over in the steering law or accept an increased error during the maneuver. Another possible solution to a needed reduction in the maximum thrust vector deflection angle is changing the controller gain matrix or using a different control solution entirely.

In retrospect the design process for this case went smoothly, however it is not quite as optimal as it could be as the final orbit still maintains some eccentricity. For this case especially a longer trade study between the nominal steering law and the steering law modification process is required in order to provide a converged solution. Any continuation of work on this launch vehicle would greatly benefit from starting with this case as it is the most converged solution we achieved.

A.3.2.7.3 5 kg Case

We successfully solved a solution for full control of the 5 kg payload case using the six degree of freedom simulator by iterating between the steering law modification process and simulator runs. The mission requirements include a periapsis altitude of 300 kilometers. For this successful case we achieved a periapsis altitude of 513 kilometers. This overshoot is directly caused by two contributing factors. First of all there is a certain amount of overshoot desired because we know there will be losses during the Monte Carlo process. Secondly, this case has been over designed and delivers a greater ΔV than optimally required.

The final launch path of the vehicle is shown in Fig A.3.2.7.2.1. We compared the controlled path to the path predicted by the trajectory code which generates the steering law. Examining the launch path we can see that the controlled path inserts into orbit slightly higher than the nominal case. This is a result of the differences in the steering law caused by the steering law modification process outlined in Section A.3.2.5 and by losses due to thrust vectoring.

[pic]

FigA.3.2.7.3.1: Launch Path

(Michael Walker, Alfred Lynam, Adam Waite)

We also propagate the final orbit of the payload after separation from the launch vehicle.1 As can be seen in Fig A.3.2.7.3.2 the orbit after the controlled ascent has a much higher eccentricity than that of the nominal orbit. This is due to inconsistencies in the steering law caused by the steering law modification process or from controllability concerns. Also maintaining a constant steering angle for the third stage is nearly impossible as it is spin stabilized and has no thrust vector control to deal with changes in the steering angle. We believe that with this case if more time was available in the design process these results could be converged to an orbit with a much lower eccentricity. The main issue in this case as will be demonstrated later is that the third stage does not burn far enough down in the third stage of the controlled ascent. It should also be realize that this case is slightly over designed. With that being realized it may have also been beneficial to investigate another case that delivers a lower ΔV.

[pic]

FigA.3.2.7.3.2: Propagated Orbit

(Michael Walker, Alfred Lynam, Adam Waite)

The orbital elements associated with the above orbit are shown in Table A.3.2.7.2.1: 1

|Table A.3.2.7.3.1 Orbital Elements |

| |

| |

|Variable |Value |Units |

|a |12606.78457215642 |km |

|e |0.4550982437909168 |-- |

|i |26.89402855748951 |Deg |

|Ω |15.79966635880931 |Deg |

|ω |280.3284035254887 |Deg |

| | | |

|Footnotes: Definitions below |

Where a is the semi-major axis, e is the eccentricity, i is the inclination, Ω is the right ascension of the ascending node and ω is the argument of periapsis. These values were used to generate the propagated orbit in Fig A.3.2.7.2.2 using standard two body orbit equations. 1

We also examined the plot of the steering angle throughout the course of the flight. Referencing Fig A.3.2.7.3.3 we can see that the controller follows the steering law almost exactly during the first stage. During the second stage however the controller less exactly follows the spline of the steering law. The difference between the nominal steering law, the steering law spline, and the actual controlled steering angle obviously account for the differences in nominal and controlled orbits.

[pic]

FigA.3.2.7.3.3: Steering Angle

(Michael Walker, Alfred Lynam, Adam Waite)

A more optimal final orbit could be achieved by further modifying the steering law spline to burn down for a longer period of the flight. This was not done because of time constraints in our design process. Despite the sub optimum nature of the steering law spline, this vehicle still meets the periapsis requirements for our mission and seemed to withstand several Monte Carlo runs so we chose this case.

Further comparison between the nominal and the achieved orbit show a slight difference in inclination as is illustrated in Fig A.3.2.7.3.4 as shown below.

[pic]

FigA.3.2.7.3.4: Inclination Difference

We analyzed the time histories of our steering and pointing angles in order to attempt to find the cause of this orbit inclination difference. As in the other two cases there is a decent amount of pointing error incurred during the flight. This is caused by the way we designed the control matrix. The control matrix is defined as a constant diagonal matrix; this coupled with the control theory simplification outline in Section 3.2.2.2 causes it to be difficult to control both angles completely. We also purposely put a large emphasis on the steering angle and only a small emphasis on the pointing angle because of how sensitive the periapsis is to steering angle control.

[pic]

FigA.3.2.7.3.5: Steering and Pointing Angle

(Michael Walker, Alfred Lynam, Adam Waite)

. The pointing error leads to the change in inclination as shown in Fig A.3.2.7.3.4 however, this error acceptable because there are no limitations on inclination outlined in the mission requirements. Ways of possibly eliminating this error are outlined in the Control Theory Section A.3.2.2.

It is also important to analyze the thrust vector control (TVC) angle time histories in order to see if we require full use of the thrust vector capability or if a more sparse thrust vectoring capability could be implemented. As can be seen in Fig A.3.2.7.3.6 the magnitude of the thrust deflection angle (δ) is very small for most of the flight. This does not hold true during the pitch over maneuver where δ rises to between 1 and 5 degrees. This is expected as the pitch over maneuver requires a large amount of control in order to maintain the required angular velocity. We can also see that the TVC rotation angle (κ), which defines the direction of deflection angle is pointing, goes through a variety of different angles showing an attempt to control both pointing and steering angles. This is not true, however, during the pitch over maneuver. During the pitch over maneuver δ jumps to values between 1 and 5 degrees in order to properly control the steering angle. Also during this time period we can see that κ is almost exclusively jumping between +90 and -90 degrees signifying that the controller is only commanding control of steering angle.

[pic]

FigA.3.2.7.3.6: Steering and Pointing Angle

(Michael Walker, Alfred Lynam, Adam Waite)

As previously discussed, and can be seen in Fig A.3.2.7.3.6 the full capacity of the TVC system is hardly ever used. This lack of use suggests that we could possibly get away with using a more sparse thrust vectoring system. A further trade study using smaller thrust vector angles would be beneficial to any further design done on this vehicle, especially since there are hardly any points above 3 degrees deflection. This would not necessarily translate into a lower cost for the vehicle; however it should be investigated further in a full design. Another solution to this lack of thrust vectoring would be looking into a more aggressive steering law. It would most likely not be beneficial but would be an interesting study.

It is beneficial to see the thrust deflection angle history more closely during the portion of the flight that is not associated with the pitch over maneuver. To effectively see δ during this section Fig A.3.2.7.3.7 shows a zoomed in view. This view gives a better idea of the deflection angles needed for the less aggressive portion of the steering law.

[pic]

FigA.3.2.7.3.7: Zoomed Thrust Deflection angle

(Michael Walker, Alfred Lynam, Adam Waite)

It can be seen in Fig A.3.2.7.3.7 that the maximum thrust vector deflection angle during stability control is about 0.05 degrees. This suggests, as we expected, that large thrust vector deflection angles are only required to do a pitch over maneuver, and that stability can be maintained with relatively minimal TVC. This also shows that if the maximum thrust vector deflection angle was reduced from 5 degrees because of subsystem limitations we would need to have a more gradual pitch over in the steering law or accept an increased error during the maneuver. Another possible solution to a needed reduction in the maximum thrust vector deflection angle is changing the controller gain matrix or using a different control solution entirely.

In retrospect this is probably the case that needs the most work. The least iterations between the simulator and the spline process were run for this case. It is obvious that more eccentricity could be shaved off of the final orbit by changing the steering law spline. Trade studies should also be done between Dynamics & Controls and Trajectory generation in order to find a more optimal solution.

References:

1K Howell “AAE 532 Course Notes” Purdue University, West Lafayette, Indiana 2007

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