9712

Spacecraft Flying in Formation

This Demonstration shows the relative orbit of one satellite, called the follower, as observed from a Cartesian frame fixed at the center of another satellite, called the leader. Both vehicles orbit a central body like the Earth. The shape of the relative orbit depends on the relative initial position and velocity of the satellites. This Demonstration shows two views of the orbit, circular (the actual relative orbit is a circle, its projection on the - plane is an ellipse) and projected circular (the projection on the - plane is a circle). The user can choose which one to view.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The relative orbit between satellites can be described by the Hill–Clohessy–Wiltshire (HCW) equations. In this model, a so-called reference satellite is considered that orbits the Earth in a circular trajectory. The motion of one or more follower satellites is studied from a reference frame fixed at the center of the reference satellite. The orbital radius of the reference satellite is assumed to be much larger than the relative separation between the satellites. This set of coordinate axes is called the Local Vertical Local Horizontal Frame. The HCW equations are as follows:
Here, is the mean motion of the reference satellite and , , and are the radial, along-track, and cross-track directions, respectively. In general, uncontrolled HCW dynamics lead to unbounded relative motion, causing the formation to disperse. The disintegration of the formation can be avoided by matching the semi-major axes of the orbits of the two satellites around the Earth, giving them equal energies. The initial condition
accomplishes this. In circular formation, the leader and the follower maintain a constant relative separation, satisfying
.
The motion takes place in a plane ±30° to the - plane and is achieved by deploying the satellites with the following relative initial conditions:
where is the formation radius. In projected circular formation, the two satellites maintain a fixed separation in the - plane:
.
The following initial conditions must be satisfied:
.
Reference: P. Ghosh, "A Critical Study of Linear and Nonlinear Satellite Formation Flying Control Methodologies from a Fuel Consumption Perspective," Master of Science Thesis, University of Cincinnati, 2007.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+