A.3.2.7 Simulation Resultspg. #
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. We shot for a higher periapsis 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. Below we provide a free body
diagram of a rocket that helps describe the forces acting on our launch vehicle.
Fig. 3.2.1.1: Forces acting on the launch vehicle during ascent.
(Jeffery Stuart)
Author:Michael Walker
A.3.2.7 Simulation Resultspg. #
A.3.2.7.2 200 g Case
We create 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 higher than the initial calculated requirement. The second reason 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 simulator 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 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.
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 compare it to the output from the trajectory code.1According to the plot the final propagated orbit has a much higher eccentricity than the nominal orbit. This is caused by 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 nominal steering law. We realize 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 follow this behavior exactly which results in a higher eccentricity than the nominal orbit.
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 ElementsVariable / 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 helps to explain the differences between the nominal and the actual orbit. We can see in Fig A.3.2.7.1.3 most of the stability concerns for this launch vehicle occur in the third stage where there is no thrust vectoring. For the rest of the flight we see that there is little error from what we ask the controller to do.Having a good modified steering law is extremely important in guaranteeing the stability of the third stage.
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 withstood the Monte Carlo process. If we want to get to a lower eccentricity the goal would be to change the steering law spline so the launch vehicle thrusts downward for a longer period of time.
Further investigation of the orbit reveals a difference in inclination of 1 degree from the nominal orbit. This difference is illustrated below in Fig A.3.2.7.1.3 we can see that the controlled and nominal orbits are not in the same plane
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 orbit inclination difference.Another reason we look at these angles is 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 the reasons stated above.
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 is 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 equations of motion. From the plot of the TVC angles we can see that the controller puts a great deal of emphasis on steering angle control, especially 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.
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 pointsthat the maximumTVC 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.
We find it hard to analyze the TVC deflection angle during the time when it is only maintaining stability of the system. The angle is hard to see during this time because it 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.
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 controlling 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 refining 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 we seein Fig A.3.2.7.2.1 the trajectories are somewhat different and the controlled ascent makes a lower orbit insertion than the nominal. This is expected because losses are incurred on the order of 10% of the thrust when thrust vectoring is used for steering.
FigA.3.2.7.2.1: Launch Path.
(Michael Walker, Alfred Lynam, Adam Waite)
We propagate the final orbit of the controlled case and compare it to the final orbit of the nominal case from the trajectory code.1The 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 non-continuous derivative makes following this steering law exactly virtually impossible. Also, with the use of a spin stabilized third stage, modifications need 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 controlled case could be lowered.We also believe that a longer process would help the results converge to something closer to what was predicted from the trajectory code.
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 OrbitalElementsVariable / 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 the spline. We can also see that the spline is also different from the steering law provided by trajectory.
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 illustrated in Fig A.3.2.7.2.4 shown below.
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 designing the simplified gain matrix and getting complete angular controlis non trivial. Since this is the case a decision had to be made on which angle required the most precise control. This angle was 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.
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 the only current 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 whenκ jumps almost exclusively between 90 and -90 degrees signifying it is only working on correcting the steer angle.
FigA.3.2.7.2.6: Thrust Vector Angle History
(Michael Walker, Alfred Lynam, Adam Waite)
We can see from the plot of the 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 large 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. The choices are 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 we already stated, conclusions on that subject can not be drawn at this time.Further studies should be done to determine the more optimal choice.
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.