CR3BP Setup Tool
Many spacecraft trajectory design approaches involve preliminary modeling and analysis for which simplified models are appropriate. The circular restricted three-body problem (CR3BP) is one such simplified model that you can use to describe the motion of a spacecraft under the simultaneous gravitational influence of two massive bodies. An example would be a spacecraft operating in cislunar space with motion governed by both the gravity of the Earth and of the Moon, each to relatively comparable effect. The associated mathematics of this model yields well-known solutions such as the libration (or Lagrange) points and orbits in their vicinities. The inherent simplifications of the model are:
- The two massive bodies orbit on circular orbits about their common center of mass (barycenter).
- The third body is restricted to be of infinitesimal mass.
The name of the model, the circular restricted three-body problem, is derived from these simplifications.
STK provides a modeling environment that incorporates high-fidelity motion models, time systems, and general physics-based behaviors for solar system bodies. Even in situations where a three-body model yields the fundamental behavior for spacecraft motion, STK’s default modeling is consistent with a fidelity level beyond that of the simplified CR3BP. In order to use such a model for preliminary design and analysis within STK, you must integrate the CR3BP assumptions into the computational environment.
To effectively utilize the CR3BP, you must appropriately define a massive secondary body, orbiting on a circular orbit about the other massive primary body. This special secondary body reflects an idealized case of orbital motion and is denoted the ideal secondary. Further, coordinate systems and calculation objects are required to define spacecraft position and velocity states and accomplish other operations. STK supports the process of defining a custom central body, vector systems, and calculation objects. This process requires that you have advanced understanding of the problem and extensive familiarity with STK tool sets to appropriately define the custom central body as an ideal secondary. You also need substantial, manual software system configuration. The CR3BP Setup design tool helps offset some of the tedium of such configuration activities.
The mathematics associated with the implementation of the CR3BP within STK Astrogator is discussed in the following reference: “Technical Implementation of the Circular Restricted Three-body Model in STK Astrogator”, C. Short, A. Haapala, and N. Bosanac, Proceedings of the 2020 AAS/AIAA Astrodynamics Specialist Conference.
Input parameters
To access the CR3BP Setup Tool:
- Open the properties of your Astrogator satellite.
- Click the Component Browser icon (
). - Click Design Tools in the left column, select the CR3BP Setup Tool template, and click the Duplicate Component icon (
). - Enter a unique component name for your CR3BP Setup Tool instantiation.
- Double-click your new component to open the dialog box for your CR3BP Setup Tool.
In the dialog box, under Configure Idealized Secondary, you can enter values for the following CR3BP input parameters:
| Parameter | Description |
|---|---|
| Central Body | Enter the name of the body to act as the primary massive body in the CR3BP formulation as well as the central body associated with other STK operations when using the objects constructed by the design tool. |
| Source Secondary | Select a child body of the designated Central Body from which the general motion for the Ideal Secondary will be appropriately patterned. Since the Source Secondary conforms to higher-fidelity motion models, the Ideal Secondary will necessarily not reflect the same motion. |
| Initial Epoch |
Specify the epoch from which to derive associated parameters for the CR3BP. |
| Ideal Orbit Radius |
Select one of the following options:
|
| Ideal Secondary | Enter a name for the resulting idealized secondary body, which will serve as the secondary for CR3BP analysis. |
Informational output
On the right side of the CR3BP Setup Tool dialog box, you will see the following read-only parameters:
| Parameter | Description |
|---|---|
| Mass Parameter | This is the CR3BP mass parameter, μ=GM2/(GM1+GM2). The associated GM values are taken from those in use for the Central Body and the Source Secondary. |
| Characteristic Distance | This is the radius of the Ideal Secondary’s circular orbit. This value, as well as the other Characteristic values, are employed to dimensionalize and non-dimensionalize states for internal use in the traditional nondimensional coordinates associated with the CR3BP and to reproduce the associated dimensional states for other STK systems to use. |
| Characteristic Time | This is the characteristic time derived from the characteristic distance and the GM values of the Central Body and Source Secondary. |
| Characteristic Velocity | Equals the Characteristic Distance divided by Characteristic Time. |
| Characteristic Acceleration | Equals the Characteristic Velocity divided by Characteristic Time. |
Associated objects
Given particular settings for the five input parameters, you can create an Ideal Secondary within the tool by clicking . Once created, the ideal secondary body will appear in the Associated Objects table and will exist as a nonduplicable Central Body in the Component Browser that you cannot directly edit. Rather, changing the input parameters in the design tool will enable the options to either reset (by clicking ) the parameters to values consistent with the Ideal Secondary or update (by clicking ) the Ideal Secondary to reflect new selections. You can only create one Ideal Secondary for a given CR3BP Setup design tool instantiation.
When you create an Ideal Secondary, you can select a Coordinate System from the available options in the associated drop-down menu. You can then create associated Analysis Workbench points, axes, and systems by clicking for that Coordinate System.
When you click for a Coordinate System, associated Analysis Workbench (AWB) objects will appear in the table. These may include a center point for the system, the set of rotating axes defined by the Ideal Secondary’s motion, and, of course, a coordinate system comprised of the center point and axes. Some selections for Coordinate Systems utilize previously existing center points and, consequently, a center point will not be created; these include the Primary- and Secondary-centered selections. All selections use the same set of rotating axes. The resulting AWB objects will also appear in the AWB tool as noneditable components associated with the Primary Central Body filter for the Ideal Secondary. You can remove the AWB objects if you have not yet created Calculation Objects that depend on them and the AWB objects have no other dependent objects. For example, since all systems share the same rotating axes, they will not be removed for any given Coordinate System selection if you create more than one Coordinate System. To delete a Coordinate System from the list, highlight it and click .
You can create Calculation Objects for a selected Coordinate System that you have already created by using . When you create Calculation Objects, they will appear in the table and will also exist in the Cartesian Elems folder of the Calculation Objects library in the Component Browser. These Calculation Objects are the Cartesian position and velocity components of a state vector given in the primary-secondary rotating reference system. You can use them as results for an Astrogator segment, targeter controls, and in the definition of stopping conditions, as well as for all other regular usages. You can remove the generated Calculation Objects for a given system at any time by highlighting the item and clicking .
After you create an Ideal Secondary, in the Configure Propagator section becomes available. Click to make a Propagator component in the Component Browser. The CR3BP Setup Tool will use this propagator for its three-body system formulation. Select the Include State Transition Matrix check box to add a state transition matrix to the propagator. You can click to delete the propagator you created. is only available when the propagator is not in use.
Restrictions of associated objects
The CR3BP Setup Tool is limited by the following restrictions:
- You cannot use an Ideal Body as a Source Secondary for another CR3BP Setup Tool.
- STK will not save Associated Objects in the scenario since they are read-only components. This follows regular save and load behaviors of other components. STK will save your version(s) of the CR3BP Setup Tool and will recreate the Associated Objects on scenario load. This makes a given CR3BP Setup Design Tool component file entirely self-contained and modular.
- A user-created CR3BP Setup Tool, produced by duplicating the default built-in tool, may itself not be duplicated and appears using the Component Browser icon that indicates the component cannot be duplicated. You may edit your CR3BP Setup Tool version(s), and STK will save any new instantiation(s) with the scenario.
Using associated objects
After using your CR3BP Setup Tool to create various other components, you can then use these components in standard Astrogator operations. You can use the Ideal Body in conjunction with the CR3BP Propagator Function and associated Propagator in the Component Browser to numerically propagate an Astrogator satellite. You can use the coordinate systems and calculation objects to define initial states and stopping conditions, report on values, and accomplish any other regular activities. Given the objects produced by this tool, STK will be ready to accomplish CR3BP-related trajectory design and analysis.