A.6.2.2.3 Ψ3 Steering Law Angle Trade Studies1
A.6.2.2.3Ψ3 Steering Law Angle Trade Studies
We perform many trade studies to obtain the correct steering law. Once selected, the steering law needs the ending orientation of the vehicle at certain intervals to be defined. These angles are defined for our launch configuration as the required orientation of the vehicle at the end each stage during the launching trajectory.
The resulting orbit is very sensitive to the angles we choose. Also, for each launch, the initial conditions vary significantly from vehicle to vehicle. Examples of some conditions that change from vehicle to vehicle are GLOM, thrust for each stage, and burn times. The sensitivity of the trajectory to the initial conditions proved to be very difficult to choose the correct steering angles for each launch configuration.
Particularly, the final angle, denoted as Ψ3 is of great importance. Ψ3 is the angle that represents the orientation of the vehicle at then end of the third stage. We also have found the Ψ3 angle is used to burn off excess radial velocity making the final orbit more circular. A few trade studies were done to better understand how changing the angle of Ψ3 affects the resulting orbit the payload is inserted into.
The goal of the first of two trade studies done was to understand how to choose the angle for Ψ3 was to simply observe the trends, if any, this angle has on the resulting orbit. Finding the trends were done by changing Ψ3 from +90º to – 90º and observing the resulting orbit parameter trends. For this study the same launching configuration was used, and Ψ1 and Ψ2 were held constant. At each variation of Ψ3 the following were recorded; radial velocity, periapsis, apoapsis, eccentricity, ΔV drag, ΔV total, ΔV to circularize, energy of the resulting orbit, and the maximum acceleration. Some very useful data and trends were observed from this study. Figures A.6.2.2.3.1 through A.6.2.2.3.4 show plots of the most prevalent resulting data.
Figure A.6.2.2.3.1: Affect of changing Ψ3 on periapsis for a ground launch.
(Allen Guzik)
Figure A.6.2.2.3.2: Affect of changing Ψ3 on apoapsis for a ground launch.
(Allen Guzik)
Figure A.6.2.2.3.3: Affect of changing Ψ3 on eccentricity for a ground launch.
(Allen Guzik)
Figure A.6.2.2.3.4: Affect of changing Ψ3 on radial velocity for a ground launch.
(Allen Guzik)
These figures show some expected and unexpected trends. For example, Fig. A.6.2.2.3.1 shows decreasing Ψ3 caused the periapsis to decrease and the radial velocity to decrease. Unexpectedly, the behavior was complex and showed unpredictable responses from one angle to the next.
The most important result of this study is the effect Ψ2 has on Ψ3. The study showed the resulting orbit is only as good as the chosen Ψ2 angle. Also, on the whole, the most desirable orbit. A circular orbit with a periapsis close to 300 km from the surface of the Earth), occurs when Ψ3 equals Ψ2. The revelation of the best orbit is when Ψ3 equals Ψ2is particularly important because it supports using a spin stabilized third stage. Spin stabilizing the final stage is desired because it reduces the weight and cost of the vehicle by not requiring a control system to control the rocket motor.
The second trade study was done to observe the effect of error in the actual flight’s trajectory to the nominal case. The DC group needed to know the allowable range of angles that the error in Ψ3 can be and still have the payload be inserted into an acceptable orbit. For this study the same balloon launch configuration was used with only changing the Ψ3 angle. Ψ3 was changed 1º at a time off of the nominal angle both positively and negatively until the propagation of the orbit hit the surface of the Earth. The periapsis, apoapsis, and eccentricity were recorded for each Ψ3 angle change. The resulting trends are plotted and shown in Fig.A.6.2.2.3.5 through A.6.2.2.3.10.
Figure A.6.2.2.3.5: Eccentricity percent error from changing Ψ3.
(Allen Guzik)
Figure A.6.2.2.3.6: Eccentricity sensitivity from changing Ψ3.
(Allen Guzik)
Figure A.6.2.2.3.7: Apoapsis percent error from changing Ψ3.
(Allen Guzik)
Figure A.6.2.2.3.8: Apoapsis sensitivity from changing Ψ3.
(Allen Guzik)
Figure A.6.2.2.3.9: Periapsis percent error from changing Ψ3.
(Allen Guzik)
Figure A.6.2.2.3.10: Periapsis sensitivity from changing Ψ3.
(Allen Guzik)
These figures show that changing the Ψ3 angle off the nominal value greatly influences the periapsis. Figure A.6.2.2.3.9 shows that for every 1º change the percent error of the resulting change in periapsis is about 10%. Also shown is the orbit is much more sensitive to angles that have negative error. Figure A.6.2.2.3.10 shows after an error of less than -6º, the vehicle will have a resulting orbit that hits the surface of the Earth. While, for positive error angles, the error can be as much as +21º off of the nominal before the vehicle hits the Earth. In general, for an acceptable orbit the error can be no less than -1º and no more than +17ºoff of the nominal angle. A large allowable positive error shows if error is a problem, to guarantee an acceptable orbit the nominal case may need to be chosen to be well above the required 300km periapsis requirement.
Author:Allen Guzik