A.3.2.5 Steering Law Modification p. #
A.3.2.5 Steering Law Modification
The main function of the controller for the launch vehicles is to follow a steering law derived by the trajectory sub-group. This steering law consists of a separate law for each stage. The desired pitch angle is solved by using the steering law in Eq. A.3.2.5.1 shown below.
A.3.2.5.1
Where y is the pitch angle, a is the first component of the steering law, t is time, and b is the second component of the steering law. The variables a and b are input by the trajectory sub-group and are different for each stage of the launch vehicle.
The result of this multiple part steering law was a function with a non-continuous derivative. The launch vehicle simply could not follow this original steering law function. One of the problems included the singularity in the derivative of the pitch angle between the second and third stage. Because the third stage is uncontrolled, the launch vehicle does not have enough time to achieve the final desired pitch angle rate before the third stage engine ignites. The simulation shows that the launch vehicle will eventually lose stability and crash as a result. Unfortunately, this result is not the objective of this project.
We solved this problem by modifying the steering law. This means breaking apart the steering law given to us by the trajectory group and using functions called splines. Splines are polynomials that are used to approximate a bounded, finite function.1 During our analysis of the simulations, we discovered that the main problem with the launch vehicle’s control was maneuvers occurring in the second stage and between the second and third stage. We used Excel to modify the steering laws for the upper two stages to eventually reach a steering law that allowed the launch vehicle to reach orbit and achieve a periapsis of at least three hundred kilometers.
A.3.2.5.1 Excel Steps to Modify the Steering Law
The steps for using Excel to modify the steering law require many manual steps and do not have many parts with automation. For the first step, we plotted the entire steering law as given to us by the trajectory sub-group. The corners of the nominal steering law plot were found because these locations are where the splines needed to be implemented. These corners are what cause the launch vehicle to lose stability and eventually crash back down to Earth. Next, we created three or four separate plots along the sections of the original steering law. We then used Excel’s trendline approximation to insert a polynomial equation that closely fit with every section. These approximations helped to round out the sharp changes in angle from the original steering law. The end result was a smoother function that eliminated singularities in the pitch angle. For our purposes, the angle pi or 180 degrees means the launch vehicle is pointed straight up. A plot of an early attempt at modifications is shown in Fig. A.3.2.5.1.1 below.
Fig. A.3.2.5.1.1: Early attempt at modifying trajectory’s steering law.
(Adam Waite)
As shown in Fig. A.3.2.5.1.1, the original steering law is shown by the blue, pink, and yellow points. We broke the steering law into three parts. Trendlines shown as green and black lines were then added using Excel. The black trendline in particular made the steering law continuous without an abrupt pitch angle change between the first and second stage. Further testing of the simulation using these new steering law modifications showed that there were still many problems. For one, the launch vehicle still had a hard time adjusting the thruster angle to fit this more continuous steering law. The transition between the second and third stage was also still too abrupt. Even with the gentle, smoother curve, the launch vehicle couldn’t achieve the final desired angle in time and with the uncontrolled third stage, would lose stability. It was apparent that a different approach needed to be taken to modify the original steering law.
A.3.2.5.2: New Method for Modifying the Steering Law
After much further testing, we decided to create a linear steering law starting in the second stage and continuing throughout the third stage. This would allow ample time for the nozzle to achieve the desired rate of change for the pitch angle and this eliminates any abrupt changes in final pitch angle. The linear steering law also allows for a constant change in pitch angle throughout the course of the burn rather than having instantaneous changes in pitch angle, which the launch vehicle proved not able to handle. This method involved using the same earlier method for the first stage because no problems were experienced in the first stage during the simulation runs. We also wanted to remain as close as possible to the original steering law during the initial stage to stay near the trajectory sub-group’s desired path into orbit. A change to the above method involved picking a point during the second stage and then creating a linear line from this point to the desired pitch angle end point determined by the trajectory sub-group. This allowed for a constant change in pitch angle that was much easier for the launch vehicle to handle. An example of this new method is shown in Fig. A.3.2.5.2.1 below.
Fig. A.3.2.5.2.1: Example of new method for creating a modified steering law
(Adam Waite)
In Fig. A.3.2.5.2.1, the blue line is the nominal steering law and the green line is the modified steering law. It can easily be seen that a polynomial is used to approximate the first corner in the nominal steering law to avoid problems of instantaneous pitch angle rate changes. The linear steering law for the final two stages is also shown. This linearization of the nominal steering law allows for a stable transition between the second and third stages of the launch vehicle.
The above process is extremely manually intensive. It is not a process that can be automated very easily. Small changes in the spline affect the periapsis and other orbit parameters immensely. These affected orbit parameters mean that we have different results than what the trajectory sub-group predicted. Modifications to the steering law are made with tiny adjustments between trial cases to try to eliminate drastic changes to the final periapsis of the orbit. With more time to work on this project, we predict that a better method can be reached to modify the steering law to make the process quicker and achieve better results.
A.3.2.5.3: 200g Payload Case
We analyzed the two hundred gram payload case and used a linear steering law for the second and third stages. For all three cases, the plots are output with 90 degrees as the upwards direction from the Earth’s surface. The reason for this is that we wanted to match the configuration that the trajectory sub-group used. Figure A.3.2.5.3.1 below shows the modified steering law.
Fig. A.3.2.5.3.1: Modified steering law for launch vehicle carrying 200 gram payload.
(Adam Waite, Mike Walker)
The modified steering law for the two hundred gram case is different than the other two cases as shown in Fig. A.3.2.5.3.1 above. This modification starts its linear trend later in the second stage. The reason for this is that this gives the best result from all of our test runs. This modified steering law tells the launch vehicle to spend more time during the second stage almost exactly on the original trajectory provided by the trajectory sub-group. During the middle of the second stage, the modified steering law begins its linear trend. This allows for the launch vehicle to again attain a constant pitch angle rate so that the launch vehicle doesn’t lose control during the transition between the second and third stages. The third stage again shows a steeper pitch angle change than desired but the launch vehicle does not lose stability and crash. This modified steering law gave a more eccentric orbit than the one kilogram case, but the launch vehicle still achieved a nominal periapsis of 488 kilometers. This is much higher than the desired periapsis of at least 300 kilometers.
A.3.2.5.4: 1kg Payload Case
The first model analyzed using the new linear steering law was the 1kg case. The linear steering law can be moved up or down with a simple shift that still does not affect the first stage. For the 1kg case, the modified linear steering law was shifted down by 0.12 radians to burn off more radial velocity to achieve a more circular orbit while still achieving a periapsis greater than three hundred kilometers. This shift is shown in Fig. A.3.2.5.4.1 below.
Fig. A.3.2.5.4.1: Modified steering law for launch vehicle carrying 1kg payload.
(Adam Waite, Mike Walker)
As shown in Fig. A.3.2.5.4.1, the launch vehicle does not achieve the maximum pitch angle desired but it manages to complete the required burns without losing stability. The transition between the second and third stage is handled smoothly with no sharp changes in pitch angle or the pitch angle rate. It is apparent that the launch vehicle begins to become more unstable towards the end of the burn during the third stage due to its sharper slope and increasing pitch angle. The engine cut out before the launch vehicle began to lose stability and an orbit was achieved that had a periapsis of 365 kilometers. This is a good solution given the objective for the launch vehicle is a periapsis of 300 kilometers.
Many hours were spent modifying and perfecting the steering law for the 1kg case. The orbit had a very low eccentricity and a periapsis well above the required number of three hundred kilometers. This left little time for the two hundred gram payload case and the five kilogram payload case because we weren’t able to start modifying those cases until close to deadline due to problems with other sections of the launch vehicle. The inertia matrices and propellant mass flow rates were not provided to us until two days before deadline. The method for modifying the steering law was perfected from the 1kg case though, and less time was needed to achieve a satisfactory new steering law for other cases.
A.3.2.5.5: 5kg Payload Case
The final steering law that we modified was the steering law for the 5 kg case. The result achieved a highly eccentric orbit. With more time, it is very probable that we could circularize this orbit and choose a smaller launch vehicle to fill our needs. As it stands, the desired periapsis is still achieved with the modified steering law. The modifications are shown in Fig. A.3.2.5.5.1 shown below.
Fig. A.3.2.5.5.1: Modified steering law for launch vehicle carrying 5kg payload
(Adam Waite, Mike Walker)
As can be seen in Fig. A.3.2.5.5.1 above, the modified steering law shown in green is considerably different from the nominal steering law shown in purple. The modified steering law remains equivalent to the original steering law during the first stage as is the case for the other two launch vehicles. A major difference between the two laws is the change between the first and second stage. The pitch angle rate has a singularity at this position with the original steering law. As can be seen by the green line, the modified steering law has a smooth curve around this position. This continuous steering function allows the launch vehicle to follow the controller without losing stability. After the smooth maneuver between the first and second stage, the modified steering law creates a linear line all the way to the final desired pitch angle as outlined by the original steering law. Again, this was chosen so that the launch vehicle has time to stabilize at a constant pitch angle rate before the third, uncontrolled stage ignites. Given more time, the steering law could be perfected to lower the eccentricity and create a more circular orbit. This would give a nominal periapsis value that would be less likely to change when deviations are applied to the simulation. With the current modified steering law, this launch vehicle achieves a nominal periapsis of 514 kilometers. This is far above the given objective of 300 kilometers for the periapsis.
References
1. Bartels, R.H.; Beatty, J.C.; and Barsky, B.A. An Introduction to Splines for Use in Computer Graphics and Geometric Modelling. San Francisco, CA: Morgan Kaufmann, 1998.
Author: Adam Waite