4.2.2 Nominal Trajectory1

4.2.2Nominal Trajectory

Describes in this section is the nominal trajectory selected for the 1kg payload launch configuration. The trajectory given is the best case that coincides with the vehicle designed by the team. The final decision to use a high altitude balloon to launch from proved to be beneficial. A balloon launching configuration reduces the amount of drag from the atmosphere which intern reduces the ΔV necessary to get into Low Earth Orbit (LEO). Table 4.2.2.1 shows the results of the most pertinent orbit parameters and other data used to describe the final orbit andtrajectory the vehicle is inserted into.

Table 4.2.2.1Orbit Parameters and Other Important Results
Variable / Value / Units
Periapsis* / 406 / km
Apoapsis* / 481 / km
Eccentricity / 0.0054 / --
Inclination / 28.5 / deg
Semi-Major Axis / 6,819 / km
Period / 5,604 / s
Footnotes:*Altitudes are from the surface of the Earth.

Some special notes about Table 4.2.2.1 need to be stated. The requirement for the team is to insert the vehicle into a 300km orbit. The table shows the periapsis of the orbit is well above this. We choose a trajectory that allows for errors that might propagate throughout the flight that lowers the resulting periapsis. An important characteristic is the nominal trajectory is very circular with an eccentricity of 0.0054. Finally, a specific value for the inclination is not requested of the team; therefore the inclination is not of great importance to the resulting orbit.

Noted in table 4.2.2.2 in the table is the ΔV budget necessary for the trajectory. The parameter ΔVtotal is the amount of ΔV the launch vehicle needs to deliver to obtain the stated orbit. ΔVtotal is the value used by the other groups of the design team to size the vehicle.

Table 4.2.2.2ΔV Breakdown
Variable / Value / Units / Percent
ΔVtotal / 9,379 / m/s / --
ΔVdrag / 6 / m/s / 0.043
ΔVgravity / 2057 / m/s / 21.745
ΔVEarth assist / 411 / m/s / 4.394
ΔVleo / 7727 / m/s / 82.606

Figure 4.2.2.1 shows a plot of the resulting trajectory and orbit for the 1 kg payload.


Figure 4.2.2.1: Nominal trajectory and orbit for the 1 kg payload.

(Allen Guzik)

Besides the resulting ΔV the trajectory predicts, the other important parameter other groups require is the steering law coefficients. The D&C group needs these values to match the nominal ascending path the Trajectory group calculates. These steering coefficients are found by optimizing the ending orientation of the vehicle. The orientation is defined by three angles, Ψ1, Ψ2, and Ψ3, where they represent and define the orientation of the vehicle at the end of the first, second, and third stages respectively. Figure 4.2.2.2 depicts how Ψ1, Ψ2, and Ψ3 define the orientation of the vehicle during the flight. Table 4.2.2.3 shows the steering angles defined for the nominal trajectory.


Figure 4.2.2.2: Ψ steering law angle orientation definition.

(Amanda Briden)

Table 4.2.2.3Angles from the Steering Law
Variable / Value / Units
End of 1st stage / 30 / deg
End of 2nd stage / -10 / deg
End of 3rd stage / -10 / deg

From these steering angles the linear tangent steering law coefficients can be defined. Equation 4.2.2.1 defines the linear tangent steering law Trajectory uses.

Eq. 4.2.2.1

The D&C group uses these coefficients to control the launching vehicle. Table 4.2.2.4 shows the coefficients used for this launching scenario. Figure 4.2.2.3 shows a close up, view of the ascending trajectory.

Table 4.2.2.4Coefficients for Steering Law
Variable / Value / Units
a1 / - 1.9900e-1 / --
b1 / 2.8636e1 / --
a2 / - 4.0000e-3 / --
b2 / 1.1700e0 / --
a3 / 8.6720e-20 / --
b3 / - 1.1760e-1 / --
Footnotes: Values are coefficients so no units.


Figure 4.2.2.3: Close up view of the ascending trajectory for the 1kg launch configuration.

(Allen Guzik)

In conclusion, the trajectory group is very pleased with the resulting nominal trajectory of the 1 kg launch configuration. Our periapsis is above the required 300km, and the orbit is very circular. The trajectory also allows for error to be tolerable and still meet the required orbit.

Author:Allen Guzik; (Amanda Briden, Brad Ferris, Daniel Chua

Elizabeth Harkness, Jun Kanehara, Kyle Donahue, Scott Breitengross)