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# 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.

### 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.

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