Tutorials
Optimal Lunar Flyby using Multiple
Shooting
Audience
Length
Prerequisites
Script File
Advanced
90 minutes
Complete Simulating an Orbit, Simple Orbit Transfer, Mars B-Plane Targeting tutorial and take GMAT Fundamentals training course or watch videos
Tut_MultipleShootingTutorial_Step1.script,
Tut_MultipleShootingTutorial_Step2.script,...
Tut_MultipleShootingTutorial_Step5.script
Objective and Overview
Note
For highly elliptic earth orbits (HEO), it is often cheaper to use the Moon’s gravity to raise periapsis or to perform plane changes, than it is to use the spacecraft’s propulsion resources. However, designing lunar flyby’s to achieve multiple specific mission constraints is non-trivial and requires modern optimization techniques to minimize fuel usage while simultaneously satisfying trajectory constraints. In this tutorial, you will learn how to design flyby trajectories by writing a GMAT script to perform multiple shooting optimization. As the analyst, your goal is to design a lunar flyby that provides a mission orbit periapsis of TBD km and changes the inclination of the mission orbit to TBD degrees. (Note: There are other mission constraints that will be discussed in more detail below.)
To efficiently solve the problem, we will employ the Multiple Shooting Method to break down the sensitive boundary value problem into smaller, less sensitive problems. We will employ three trajectory segments. The first segment will begin at
Transfer Orbit Insertion (TOI) and will propagate forward; the second segment is centered at lunar periapsis and propagates both forward and backwards. The third segment is centered on Mission Orbit Insertion (MOI) and propagates forwards and backwards. See figures 1 and 2 that illustrate the final orbit solution and the
“Control Points” and “Patch Points” used to solve the problem.
To begin this tutorial we start with a several views of the solution to provide a physical understanding of the problem. In Fig. 1, an illustration of a lunar flyby is shown with the trajectory displayed in red and the Moon’s orbit displayed in yellow. The Earth is at the center of the frame.
We require that the following constraints are satisfied at TOI:
1. The spacecraft is at orbit perigee,
2. The spacecraft is at an altitude of 285 km.
3. The inclination of the transfer orbit is 28.5 degrees.
At lunar flyby, we only require that the flyby altitude is greater than 100 km. This constraint is satisfied implicitly so we will not explicitly script this constraint. An insertion maneuver is performed at earth perigee after the lunar fly to insert into the mission orbit. The following constraints must be satisfied after MOI.
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Tutorials Optimal Lunar Flyby using Multiple Shooting
1. The mission orbit perigee is 15 Earth radii.
2. The mission orbit apogee is 60 Earth radii.
3. The mission orbit inclination is 10 degrees.
Note: (Phasing with the moon is important for these orbits but design considerations for lunar phasing are beyond the scope of this tutorial)
Figure 74. View of Lunar Flyby from Normal to Earth Equator
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Figure 75. View of Lunar Flyby Geometry
Figure 3 illustrates the mission timeline and how control points and patch points are defined.
Control points are drawn using a solid blue circle and are defined as locations where the state
Optimal Lunar Flyby using Multiple Shooting
Tutorials of the spacecraft is treated as an optimization variable. Patch points are drawn with an empty blue circle and are defined as locations where position and/or velocity continuity is enforced.
For this tutorial, we place control points at TOI, the lunar flyby and MOI. At each patch point, the six Cartesian state elements, and the epoch are varied for a total of 18 optimization variables.
At the MOI patch point, there is an additional optimization variable for the delta V to
Figure 76. Definition of Control and Patch Points
Notice that while there are only three patch points, we have 5 segments (which will result in
5 spacecraft). The state at the lunar flyby, which is defined as a control point, is propagated backwards to a patch point and forwards to a patch point. The same occurs for the MOI control point. To design this trajectory, you will need to create the following GMAT resources.
1. Create a Moon-centered coordinate system.
2. Create 5 spacecraft required for modeling segments.
3. Create an Earth-centered and a Moon-centered propagator.
4. Create an impulsive maneuver.
5. Create many user variables for use in the script.
6. Create A VF13ad optimizer.
7. Create plots for tracking the optimization process.
After creating the resources using script snippets you will construct the optimization sequence using GMAT script. Pseudo-code for the optimization sequence is shown below.
Define optimization initial guesses
Initialize variables
Optimize
Loop initializations
Vary control point epochs
Set epochs on spacecraft
Vary control point state values
Configure/initialize spacecraft
Apply constraints on initial control points (i.e before propagation)
Propagate spacecraft
Apply patch point constraints
Apply constraints on mission orbit
Apply cost function
EndOptimize
After constructing the basic optimization sequence we will perform the following steps:
1. Run the sequence and analyze the initial guess.
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