The Clohessy–Wiltshire equations describe a simplified model of orbital relative motion, in which the target is in a circular orbit, and the chaser spacecraft is in an elliptical or circular orbit. This model gives a first-order approximation of the chaser's motion in a target-centered coordinate system. It is used to plan the rendezvous of the chaser with the target.[1][2]

History

Early results about relative orbital motion were published by George William Hill in 1878.[3] Hill's paper discussed the orbital motion of the moon relative to the Earth.

In 1960, W. H. Clohessy and R. S. Wiltshire published the Clohessy–Wiltshire equations to describe relative orbital motion of a general satellite for the purpose of designing control systems to achieve orbital rendezvous.[1]

System Definition

Suppose a target body is moving in a circular orbit and a chaser body is moving in an elliptical orbit. Let be the relative position of the chaser relative to the target with radially outward from the target body, is along the orbit track of the target body, and is along the orbital angular momentum vector of the target body (i.e., form a right-handed triad). Then, the Clohessy–Wiltshire equations are

where is the orbital rate (in units of radians/second) of the target body, is the radius of the target body's circular orbit, is the standard gravitational parameter, If we define the state vector as , the Clohessy–Wiltshire equations can be written as a linear time-invariant (LTI) system,[4]

where the state matrix is

For a satellite in low Earth orbit, and , implying , corresponding to an orbital period of about 93 minutes.

If the chaser satellite has mass and thrusters that apply a force then the relative dynamics are given by the LTI control system[4]

where is the applied force per unit mass and

Solution

We can obtain closed form solutions of these coupled differential equations in matrix form, allowing us to find the position and velocity of the chaser at any time given the initial position and velocity.[5]

where:

Note that and . Since these matrices are easily invertible, we can also solve for the initial conditions given only the final conditions and the properties of the target vehicle's orbit.

See also

References

  1. 1 2 Clohessy, W. H.; Wiltshire, R. S. (1960). "Terminal Guidance System for Satellite Rendezvous". Journal of the Aerospace Sciences. 27 (9): 653–658. doi:10.2514/8.8704.
  2. "Clohessy-Wiltshire equations" (PDF). University of Texas at Austin. Retrieved 12 September 2013.
  3. Hill, G. W. (1878). "Researches in the Lunar Theory". American Journal of Mathematics. Johns Hopkins University Press. 1 (1): 5–26. doi:10.2307/2369430. ISSN 0002-9327. JSTOR 2369430.
  4. 1 2 Starek, J. A., Schmerling, E., Maher, G. D., Barbee, B. W., Pavone, M. (February 2017). "Fast, Safe, Propellant-Efficient Spacecraft Motion Planning Under Clohessy–Wiltshire–Hill Dynamics". Journal of Guidance, Control, and Dynamics. American Institute of Aeronautics and Astronautics. 40 (2): 418–438. arXiv:1601.00042. Bibcode:2017JGCD...40..418S. doi:10.2514/1.G001913. ISSN 0731-5090. S2CID 4956601.
  5. Curtis, Howard D. (2014). Orbital Mechanics for Engineering Students (3rd ed.). Oxford, UK: Elsevier. pp. 383–387. ISBN 9780080977478.

Further reading

  • Prussing, John E. and Conway, Bruce A. (2012). Orbital Mechanics (2nd Edition), Oxford University Press, NY, pp. 179–196. ISBN 9780199837700
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