# A.3.2.7.4: Test Case Analysis

A.3.2.7 Nominal Simulation Results p. #

**A.3.2.7.4: Test Case Analysis**

During the design process of the launch vehicle, we were given a test case of a sample launch vehicle. This vehicle was not included in the final design, but it did have all the proper inputs for our simulation. The test launch vehicle had an inertia matrix derived by the structures sub-group and a propellant mass flow rate along with burn times for each stage provided by the propulsion sub-group. This test vehicle was used to develop a design process for the final launch vehicle cases. We developed efficient ways to adjust the gain matrix to achieve an orbit. We also learned quick methods for modifying the steering law so that it follows the nominal path given by the trajectory sub-group.

After completing our controller design, we inputted values for our gain matrices and modified the original steering law. We shifted the third stage steering law up by one radian to try to eliminate radial velocity so that the launch vehicle would enter into a circular orbit. At this point,no deviations were used so the case that was tested used all nominal values. We then ran the full simulation for the test case to test our controller design. The orbit of our first run is shown in Fig. A.3.2.7.4.1 below.

**Fig. A.3.2.7.4.1:** Actual path of launch vehicle(blue) and desired path of launch vehicle(red)

(Adam Waite, Mike Walker)

In Fig. A.3.2.7.4.1, the blue line shows the actual path of the launch vehicle as simulated by our controller and the red path shows the desired trajectory. From this initial run, we learned that we needed to somehow lower the eccentricity of the orbit to generate a circular, stable orbit around the Earth. A stable orbit means an orbit that doesn’t deteriorate over time with the launch vehicle not rotating upon itself very quickly. An indication of this instability can be observed in Fig. A.3.2.7.4.2 below.

**Fig. A.3.2.7.4.2:** Altitude plot for controlled launch vehicle

(Adam Waite)

By examining Fig. A.3.2.7.4.2, it is apparent that the plot of altitude has an exponential shape. This indicates that the launch vehicle isn’t circularizing at a given altitude but rather entering into a highly elliptical orbit. The eventual periapsis was a negative number indicating that the launch vehicle eventually crashed.

It was apparent that we needed to follow the steering law much closer with sharper turns in the first and second stages. This meant limiting the error between the actual pitch angle of the launch vehicle and the desired pitch angle of the launch vehicle. This was done by adjusting the gains in the gain matrices to obtain tighter control over the TVC in the nozzle.

From the above data, we determined that by adjusting the values of the gain matrix and creating a modified steering law, it would be possible to better circularize the orbit of the launch vehicle. We first changed the values of the gain matrix. We gave the highest value to the first number in the gain matrix because this told the controller to place the most emphasis on the steering angle of the launch vehicle. The other two values in the gain matrix were given smaller values because the yaw and spin rates were not as important to achieve orbit. By manipulating the gain matrix, we controlled the launch vehicle on a path much closer to the optimal trajectory to reach a more circular orbit.

The second step in the design process for this test case involved modifying the nominal steering law. We needed to modify the nominal steering law because the launch vehicle couldn’t follow the original configuration. The nominal steering law had corners that indicated an instantaneous change in the pitch angle rate. These corners caused the launch vehicle to lose stability and crash. By creating smooth, continuous functions, we discovered that the launch vehicle was more stable and controlled during ascent into orbit. Modifying these original steering laws was a long and difficult process. This process involved lots of trial and error with different polynomial expressions and linear functions. Because of this process of trial and error, we were not able to complete the test case design to an optimal level. Analysis needed to be started on the final design cases. The final run of the test case is shown in Fig. A.3.2.7.4.3 below.

**Fig. A.3.2.7.4.3:** Final orbit for test case with actual path(blue) and desired path(red)

(Adam Waite, Mike Walker)

As shown in Fig. A.3.2.7.4.3, the launch vehicle’s trajectory is much more curved in the first and second stages, which allow the launch vehicle to achieve a nearly circular orbit. The launch vehicle still deviates from the desired path in the second stage, but this is so that the launch vehicle can be controlled into a circular orbit. Since the third stage is uncontrolled, the maneuvers to circularize needed to be performed in the second stage. As can be seen, the final test case orbit is still not perfect. Time had to be devoted to the final design cases that were eventually given.

The most important idea that was learned from analyzing and manipulating the test case was the steps for the design process for the final launch vehicles. Once given the final designs, we ran them through the simulator with all of the original gain matrix values and trajectories. After this, we adjusted the gain matrix so that the launch vehicle closely followed the desired trajectory. Next, we modified the steering law so that the launch vehicle would obtain an orbit with an eccentricity close to zero. Finally, we simulated another trial launch. Small adjustments were made again until we achieved a more circular orbit with a periapsis above 300 kilometers.

Author: Adam Waite