Ground-Based Testbed for Replicating the Orbital Dynamics of a Satellite Cluster in a Gravity Well David W. Miller Raymond J. Sedwick AFRL Distributed Satellite Systems Program MIT Space Systems Laboratory Hill’s Equations F Governing equations where ‘n’ is orbital frequency in rad/sec: — accelerations account for non-central forces (drag, thrust, etc.). — x-axis in zenith, y-axis in frame’s velocity, and z-axis in transverse directions. F Free orbit solution where ‘A’ and ‘B’ are lengths and ‘α’ and ‘β’ are phase angles. ? ? x ? ? y ? ? z ? ? ? ? ? ? ? ? ? ? + 0 ?2n 0 2n 0 0 000 ? ? ? ? ? ? ? ? ? ? ? x ? y ? z ? ? ? ? ? ? ? ? ? ? + ?3n 2 00 000 00n 2 ? ? ? ? ? ? ? ? ? ? x y z ? ? ? ? ? ? ? ? ? ? = a x a y a z ? ? ? ? ? ? ? ? ? ? x = Acos(nt +α) + x o y =?2A sin(nt +α) ?(3/2)nx o t + y o z = Bcos(nt +β) Closed Cluster Solution F There exist free orbits that cause a S/C to follow a closed and periodic motion with respect to the Hill’s frame as well as other S/C of the same period. F the S/C must follow a two-by-one ellipse in the Hill’s frame’s zenith- velocity plane. — transverse displacement is independent and oscillatory. F The parameters A, B, α, β, and y o can be selected for each spacecraft in the cluster. — based upon the projection of some ground track motion. — to allow natural orbital dynamics to most uniquely sweep out aperture baselines. — to make the array appear “rigid” from some perspective. x = Acos(nt +α) y =?2Asin(nt +α)+ y o z = Bcos(nt +β) Consider a Pendulum in 1-G F Parameterize pendulum motion in terms of azimuth (θ) and elevation (φ) angles: φ θ Dynamics of a Pendulum F Define the Lagrangian as the difference between the kinetic and potential energies: F Nonlinear dynamic equations found using Lagrange’s Equation: F Results in the following equations L= T? V = 1 2 m(r ? φ ) 2 + (r ? θ sinφ) 2 [] ? mgr 1?cosφ [ ] d dt ?L ?? q ? ? ? ? ? ? ? ?L ?q = 0 where q = generalized DOF φ [] :mr 2 ? ? φ ? m(r ? θ ) 2 sinφcosφ+mgrsinφ= 0 θ [] : m(rsinφ) 2 ? ? θ +2mr 2 ? θ ? φ sinφcosφ=0 Perturbed Pendulum Motion F Perturb motion about a nominal elevation angle and azimuthal angular rate: F Substitute into nonlinear equations and zero higher order terms: F Notice that forcing term zeroes about equilibrium motion: φ=φ o +δφ , ? θ = ? θ o +δ ? θ where φ o , ? θ o = const φ [] : δ ? ? φ ?[ ? θ o 2 (cos 2 φ o ? sin 2 φ o )? g r cosφ o ]δφ?2 ? θ o sinφ o cosφ o δ ? θ = ( ? θ o 2 cosφ o ? g r )sinφ o θ [] : δ ? ? θ +2 ? θ o cosφ o sinφ o δ ? φ =0 ? θ o 2 = 1 cosφ o g r Comparison with Hill’s Equations F Two DOF Linearized Pendulum Equations: F Evaluated at = 64 o F Two DOF Linearized Hill’s Equations: φ o δ ? ? φ δ ? ? θ ? ? ? ? ? ? = 02 g r sinφ o cosφ o ?2 g r cosφ o sin φ o 0 ? ? ? ? ? ? ? ? ? ? ? ? δ ? φ δ ? θ ? ? ? ? ? ? + ? g r sin 2 φ o cosφ o 0 00 ? ? ? ? ? ? ? ? ? ? δφ δθ ? ? ? ? ? ? δ ? ? φ δ ? ? θ ? ? ? ? ? ? = 01.8n ?2.2n 0 ? ? ? ? ? ? δ ? φ δ ? θ ? ? ? ? ? ? + ?4.2n 2 0 00 ? ? ? ? ? ? δφ δθ ? ? ? ? ? ? where n = g r cosφ o ? ? x ? ? y ? ? ? ? ? ? = 02n ?2n 0 ? ? ? ? ? ? ? x ? y ? ? ? ? ? ? + 3n 2 0 00 ? ? ? ? ? ? x y ? ? ? ? ? ? General Solutions: Secular & Periodic F Pendulum Equations: F Hill’s Equations: δφ= Acos(nρt +α) +δφ o δθ=? 2A ρsinφ o sin(nρt +α) + n(ρ 2 ?4) 2sinφ o δφ o t +δθ o where n = g r cosφ o and ρ= 4 + sin 2 φ o cos 2 φ o x = A cos(nt +α) + x o y =?2A sin(nt +α)?(3 /2)nx o t + y o Periodic Solutions F Pendulum Equations: F Hill’s Equations: δφ= Acos(nρt +α) δθ= ? 2A ρsinφ o sin(nρt +α)+δθ o where n = g r cosφ o and ρ= 4 + sin 2 φ o cos 2 φ o x = Acos nt +α ( ) y =?2Asin nt +α () + y o Eigenvalues F Pendulum Equations: F Hill’s Equations: s =±in 4 + sin 2 φ o cos 2 φ o where n = g r cosφ o s = ±in where i = ?1 Perturbed Motion About 63 Degree Elevation F Single pendulum system — at 63 degrees elevation, S/C oscillates slightly less than three cycles per revolution F Douple pendulum system — higher elevation S/C moves slower and falls behind — lower elevation S/C moves faster and moves ahead — similar to Hill’s equations -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 x-axis Nominal Elevation Angle of 63 Degrees y-axis -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 x-axis Nominal Elevation Angle of 63 Degrees y-axis Perturbed Motion at Other Elevation Angles F Elevation angle of 25 degrees — number of oscillations per revolution decreases with decreasing nominal elevation angle F Elevation angle of 45 degrees — speed increases with increasing nominal elevation angle -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 x-axis Nominal Elevation Angle of 25 Degrees y-axis -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 x-axis Nominal Elevation Angle of 45 Degrees y-axis Design Parameters r (m) φ o (deg) n= Y θ o (rad/s) Circum (m) Speed (m/s) T (s) 10 25 1.040 26.55 4.40 6.03 45 1.178 44.43 8.33 5.33 63 1.470 55.98 13.10 4.27 85 3.355 62.59 33.42 1.87 20 25 0.736 53.11 6.22 8.54 45 0.833 88.86 11.78 7.54 63 1.039 111.97 18.52 6.05 85 2.372 125.19 47.26 2.65 n = ? θ 0