美国麻省理工大学：《航天系统工程》（英文版） Governing equations where n' is orbital frequency in rad/sec

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

Ground-Based Testbed for Replicating the Orbital Based Testbed for Replicating the Orbital Dynamics of a Satellite Cluster in a Gravity Well Dynamics of a Satellite Cluster in a Gravity Well David W. Miller Raymond J. Sedwick AFRL Distributed Satellite Systems Program MIT Space Systems Laboratory

Hills equations 8 Governing equations where 'n' is orbital frequency in rad/sec s0-2n0|-3n200 箩}+2n00y+000 00000n a accelerations account for non-central forces(drag, thrust, etc. x-axis in zenith, y-axis in frame's velocity and z-axis in transverse di erections o Free orbit solution where Aand B are lengths anda ar phase angles X= Acos(nt+a) y=-2A Sin(nt +a)-(3/2)nxt+y Z=Bcos(nt+β)

Hill’s Equations 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 śś y śś śz ś ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ + 0 −2n 0 2n 0 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤⎦ ⎥ ⎥ ⎥ x śy śz ś ⎧⎨ ⎪ ⎩ ⎪ ⎫⎬ ⎪ ⎭ ⎪ + −3n2 0 0 0 0 0 0 0 n2 ⎡⎣ ⎢ ⎢ ⎢ ⎤⎦ ⎥ ⎥ ⎥ xyz⎧⎨ ⎪ ⎩ ⎪ ⎫⎬ ⎪ ⎭ ⎪ = ax ay az ⎧⎨ ⎪ ⎩ ⎪ ⎫ ⎬⎪ ⎭⎪ x = Acos(nt + α) + xo y = −2Asin(nt + α) −(3/ 2)nxot + yo z = Bcos(nt + β)

Closed Cluster Solution 8 There exist free orbits that cause a S/c to follow a closed and periodic motion with respect to the Hills frame as well as other S/c of the same period x= Acos(nt +a) y=-2Asin(nt +a)+ Z Bcos(nt+β) 8 the s/c must follow a two-by-one ellipse in the Hills frames zenith- velocity plane transverse displacement is independent and oscillatory 8 The parameters A, B, a,B, and y 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

Closed Cluster Solution 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 dis placement 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 orbit al dynamics to most uniquely sweep out aperture baselines. — to make the array appear “rigid” from some perspective. x = Acos(nt + α ) y = − 2 Asin(nt + α ) + y o z = Bcos(nt + β )

Consider a pendulum in 1-G 8 Parameterize pendulum motion in terms of azimuth(0)and elevation (O )angles 0

Consider a Pendulum in 1 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 8 Define the lagrangian as the difference between the kinetic and potential energies L=T-V=,m()+(re d)I-mgri-cosd 8 Nonlinear dynamic equations found using Lagranges Equation d al aI 0 where q= generalized DOF dt(as丿ac 8 Results in the following equations P: mr's-m(re) sin o cos o+mgr sin o=0 m(rsin￠)(+2mr6sin￠cosp=0

Dynamics of a Pendulum 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 2m ( rφ ś ) 2 + (rθ śsin φ ) 2 [ ] − mgr [1 − cos φ ] d dt ∂ L ∂q ś ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ∂ L ∂ q = 0 where q = generalized DOF [ ]φ : m r 2 φśś − m( r θś ) 2 sin φcosφ + mgrsin φ = 0 [ ]θ : m ( rsin φ ) 2θ śś + 2m r 2θ śφ śsin φcosφ = 0

Perturbed Pendulum motion s Perturb motion about a nominal elevation angle and azimuthal angular rate φ=p。+8,=8。+8月 whereφ。,。= const y Substitute into nonlinear equations and zero higher order terms φ6-62(cos2。-sin2。)-scos。F6-26sind。cos6 r (。cos。-=)sin。 δ{+2 COS δF=0 sIn 8 Notice that forcing term zeroes about equilibrium motion cOSΦ、r

Perturbed Pendulum Motion 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 + δ φ , θś = θś o + δ θś where φ o,θś o = const [ ]φ : δ śφ ś − [θ ś o 2(cos 2 φ o − sin 2 φ o ) − g r cos φ o ]δφ − 2θ ś o sin φ o cos φ o δθ ś = (θ ś o 2 cos φ o − g r )sin φ o [ ]θ : δ śθ ś + 2θ ś o cos φ o sin φ o δφ ś = 0 θ ś o 2 = 1 cos φ o g r

Comparison with Hills equations s Two doF Linearized Pendulum equations 2=sin corcos po gsin r 213 r cos ncos o 0 sIn y Evaluated at =649 01.8n66-42n20| where n= COS 6L-2.2n086 0060 8 Two DOF Linearized Hills Equations 02n 3n20x ssL-2n oslo oy

Comparison with Hill’s Equations Comparison with Hill’s Equations F Two DOF Linearized Pendulum Equations: F Evaluated at = 64 o F Two DOF Linearized Hill’s Equations: φo δ śφ ś δ śθ ś ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ = 0 2 g r sin φ o cos φ o − 2 g r cos φ o sin φ o 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ δφ ś δθ ś ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ + − g r sin 2 φ o cos φ o 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ δφδθ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ δφ śś δθ śś ⎧⎨ ⎩ ⎫⎬ ⎭ = 0 1.8 n −2.2n 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ δ φś δθ ś ⎧⎨ ⎩ ⎫⎬ ⎭ + −4.2n 2 0 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ δφδθ ⎧⎨ ⎩ ⎫⎬ ⎭ where n = g r cos φ o x śś y śś ⎧⎨ ⎩ ⎫⎬ ⎭ = 0 2 n −2n 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ x śy ś ⎧⎨ ⎩ ⎫⎬ ⎭ + 3n 2 0 0 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ x y ⎧⎨ ⎩ ⎫ ⎬ ⎭

General Solutions: Secular periodic s Pendulum equations δp=Acos(npt+o) +6 2A sin(npt +a)+ n(p-4) 6t+δ0。 psin o。 2sin o sin where n VCOS φ。andp=1{4+ cOS 8 Hills Equations x= Acos(nt+a) y=-2A sin(nt +a)-(3/2)nxt+y

General Solutions: Secular & Periodic General Solutions: Secular & Periodic F Pendulum Equations: F Hill’s Equations: δφ = Acos( n ρ t + α ) +δ φ o δθ = − 2A ρsin φ o sin( n ρ t + α ) + n( ρ 2 − 4) 2sin φ o δφ o t + δ θ o where n = g r cos φ o and ρ = 4 + sin 2 φ o cos 2 φ o x = A cos(nt + α ) + x o y = − 2 A sin(nt + α ) − (3 / 2 ) n x o t + y o

Periodic solutions s Pendulum equations 8o=Acos(npt+a) 2A sin(npt+)+δ psin。 sin where n=v=vcos do and p=4+ r cOS 8 Hills equations A cos(nt+a y=-2Asin(nt +a)+yo

Periodic Solutions Periodic Solutions F Pendulum Equations: F Hill’s Equations: δφ = Acos(nρt + α) δθ = − 2A ρsin φo sin(nρt + α) + δθo where n = g r cosφo and ρ = 4 + sin2 φo cos 2 φo x = Acos(nt + α) y = −2Asin( ) nt + α + yo

eigenvalues s Pendulum equations sin s=±in,4+ where n cOS COS r 8 Hills Equations s=± in where i

Eigenvalues Eigenvalues F Pendulum Equations: F Hill’s Equations: s = ±in 4 + sin2 φo cos2 φo where n = gr cosφo s = ±in where i = −1