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Notes_R%20BP

# Notes_R%20BP - Euler-Hill Equations of Relative Motion Two...

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Unformatted text preview: Euler-Hill Equations of Relative Motion Two vehicles in near vicinity OR One vehicle moving in orbit slightly Perturbed from some nominal orbit Coordinate Frame Approximate relative motion: assume • two vehicles moving in neighboring orbits • near-circular orbits — radii 1 2 r ; r • Coordinate frame attached to target rotates with orbit • Unit vectors ˆ ˆ , x y radial and tangential to orbit; ˆ z normal • Relative position vector ˆ ˆ ˆ ρ = + + x x y y z z • Three-dimensional motion relative to target frame R 1 Convenient to describe relative motion Note: This ˆ ˆ ˆ , , x y z frame is called the local vertical coordinate frame. George Hill originally introduced this frame in 1878 . He was describing the motion of the Moon relative to the Earth. He modelled the motion of the Moon with respect to a known reference motion, i.e., Earth moving around the Sun. In Hill's application, distance of the Moon from the Earth is small compared to the distance of the Earth from the Sun. In aerospace engineering, this is sometimes labelled the CW frame after Clohessy-Wiltshire. Clohessy and Wiltshire rediscovered the equations in 1960 in a study of the motion of a vehicle relative to a satellite in orbit about the Earth. The two applications are different but both are investigations of small displacements relative to a known reference motion. Hill, G.W., "Researches in Lunar Theory," American Journal of Mathematics, Vol. 1, 1878, pp. 5-26. Clohessy, W.H., and Wiltshire, R.S., "Terminal Guidance Systems for Satellite Rendezvous," Journal of Aerospace Sciences, September 1960, pp. 653-658, and 674. R2 Derivation of EOM Target assumed in unperturbed orbit 1 1 3 1 μ = − ¡¡ r r r Chase may be perturbed (thrusters, drag, ........) 2 2 3 2 μ = − + ¡¡ r r f r Relative position: 2 1 ρ = − r r Then → f and 2 1 − r r assumed small Relative vector equation of motion: 2 1 ρ = − ¡¡ ¡¡ ¡¡ r r RHS Æ 3 1 2 1 1 2 3 3 1 2 μ ⎡ ⎤ − = − + ⎢ ⎥ ⎣ ⎦ ¡¡ ¡¡ r r r r r f r r replace 2 3 2 r r using 2 1 ρ = + r r 2 1 3 3 2 2 2 2 1 1 2 ρ ρ ρ + = + + ⎡ ⎤ ⎣ ⎦ i r r r r r R 3 perturbing force per unit 3 2...
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