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 zenithvelocity 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