Ground-Based Testbed for Replicating the Orbital Dynamics of a Satellite Cluster in a Gravity WellHill’s EquationsClosed Cluster SolutionConsider a Pendulum in 1-GDynamics of a PendulumPerturbed Pendulum MotionComparison with Hill’s EquationsGeneral Solutions: Secular & PeriodicPeriodic SolutionsEigenvaluesPerturbed Motion About 63 Degree ElevationPerturbed Motion at Other Elevation AnglesDesign ParametersGroundGround--Based Testbed for Replicating the Orbital Based Testbed for Replicating the Orbital Dynamics of a Satellite Cluster in a Gravity WellDynamics of a Satellite Cluster in a Gravity WellDavid W. MillerRaymond J. SedwickAFRL Distributed Satellite Systems ProgramMIT Space Systems LaboratoryHill’s EquationsHill’s EquationsF 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. ś ś x ś ś y ś ś z ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ +0 −2n 02n 0 0000⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ś x ś y ś z ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ +−3n20000000n2⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ xyz⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ =axayaz⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ x = Acos(nt+α) +xoy =−2A sin(nt +α) −(3/2)nxot + yoz = Bcos(nt +β)Closed Cluster SolutionClosed Cluster SolutionF 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 yocan 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) + yoz = Bcos(nt +β)Consider a Pendulum in 1Consider a Pendulum in 1--GGF Parameterize pendulum motion in terms of azimuth (θ) and elevation (φ) angles:φθDynamics of a PendulumDynamics of a PendulumF 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 =12m(rś φ )2+ (rś θ sin φ)2[]− mgr 1− cosφ[] ddt∂L∂ś q ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −∂L∂q= 0 where q = generalized DOF φ[]:mr2ś ś φ −m(rś θ )2sinφcosφ+mgrsinφ=0θ[]: m(rsin φ)2ś ś θ + 2mr2ś θ ś φ sin φcosφ=0Perturbed Pendulum MotionPerturbed Pendulum MotionF 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=φo+δφ , ś θ =ś θ o+δś θ where φo,ś θ o=const φ[]: δś ś φ −[ś θ o2(cos2φo− sin2φo) −grcosφo]δφ − 2ś θ osinφocosφoδś θ = (ś θ o2cosφo−gr)sin φoθ[]: δś ś θ + 2ś θ ocosφosin φoδś φ = 0 ś θ o2=1cosφogrComparison with Hill’s EquationsComparison with Hill’s EquationsF Two DOF Linearized Pendulum Equations:F Evaluated at = 64oF Two DOF Linearized Hill’s Equations:φo δś ś φ δś ś θ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ =02grsin φocos φo−2grcos φosin φo0⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ δś φ δś θ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ +−grsin2φocos φo000⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ δφδθ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ δś ś φ δś ś θ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ =01.8n−2.2n 0⎡ ⎣ ⎢ ⎤ ⎦ ⎥ δś φ δś θ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ +−4.2n2000⎡ ⎣ ⎢ ⎤ ⎦ ⎥ δφδθ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ where n =grcos φo ś ś x ś ś y ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ =02n−2n 0⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ś x ś y ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ +3n2000⎡ ⎣ ⎢ ⎤ ⎦ ⎥ xy⎧ ⎨ ⎩ ⎫ ⎬ ⎭General Solutions: Secular & PeriodicGeneral Solutions: Secular & PeriodicF Pendulum Equations:F Hill’s Equations: δφ = Acos(nρt+α) +δφoδθ = −2Aρsin φosin(nρt +α) +n(ρ2− 4)2sin φoδφot +δθowhere n =grcosφo and ρ= 4 +sin2φocos2φo x = A cos(nt+α) +xoy =−2A sin(nt +α) − (3 /2)nxot + yoPeriodic SolutionsPeriodic SolutionsF Pendulum Equations:F Hill’s Equations: δφ = Acos(nρt+α)δθ = −2Aρsin φosin(nρt +α) +δθowhere n =grcosφo and ρ= 4 +sin2φocos2φo x= Acos nt+α()y =−2Asin nt +α()+ yoEigenvaluesEigenvaluesF Pendulum Equations:F Hill’s Equations: s =±in 4 +sin2φocos2φo where n =grcosφo s=±in where i=−1Perturbed Motion About 63 Degree ElevationPerturbed Motion About 63 Degree ElevationF Single pendulum system— at 63 degrees elevation, S/C oscillates slightly less than three cycles per revolutionF 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-1-0.500.51-1-0.500.5100.20.40.60.81x-axisNominal Elevation Angle of 63 Degreesy-axis-1-0.500.51-1-0.500.5100.20.40.60.81x-axisNominal Elevation Angle of 63 Degreesy-axisPerturbed Motion at Other Elevation AnglesPerturbed Motion at Other Elevation AnglesF Elevation angle of 25 degrees— number of oscillations per revolution decreases with decreasing nominal elevation angleF Elevation angle of 45 degrees— speed increases with increasing nominal elevation angle-1-0.500.51-1-0.500.5100.20.40.60.81x-axisNominal Elevation Angle of 25 Degreesy-axis-1-0.500.51-1-0.500.5100.20.40.60.81x-axisNominal Elevation Angle of 45 Degreesy-axisDesign ParametersDesign Parametersr(m)φo(deg)n=Ý θ o(rad/s)Circum(m)Speed(m/s)T(s)10 25 1.040 26.55 4.40 6.0345 1.178 44.43 8.33 5.3363 1.470 55.98 13.10 4.2785 3.355 62.59 33.42 1.8720 25 0.736 53.11 6.22 8.5445 0.833 88.86 11.78 7.5463 1.039 111.97 18.52 6.0585 2.372 125.19 47.26 2.65 n =ś θ