Orbit Determination of Satellite Formations. Terry Alfriend 9 th US Russian Space Surveillance Workshop

Transcription

1 Orbit Determination of Satellite Formations Terry Alfriend 9 th US Russian Space Surveillance Workshop

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3 Outline What is a Satellite Formation Types of Formations Proposed Approach for Orbit Determination Relative Motion Theory Formation Orbit Determination Issues Summary

4 Example Formation

5 Definitions Distributed Space Systems (DSS) - an end-to-end system including two or more space vehicles and a cooperative infrastructure for science measurement data acquisition, processing, analysis and distribution (other issues: power, storage, thermal, ) Formation - multiple spacecraft with a desired position and/or orientation relative to each other or to a common target. Formation Flying - the maintenance of a relative separation, orientation or position between or among spacecraft. Drift - Secular change in the separation distance between satellites in a formation

6 Why Satellite Formations Replace a single, large, monolithic satellite with a system of small satellites Enable new missions; Long baseline interferometers, Virtual antennas, distributed apertures" u Increased angular resolution,increased sensitivity,increased temporal resolution" Enhance existing missions" u Reduce cost and complexity, improve performance" Survivability"

7 Problem is the orbit determination of N satellites in a close formation in Earth orbit. Close means the formation size is no more than a few kilometers.

8 Magnetospheric Multi-scale (MMS) How do small-scale processes control largescale phenomenology, such as magnetotail dynamics, plasma entry into the magnetosphere, and substorm initiation? 4 identical spacecraft in a variably spaced tetrahedron ( 1 km to several earth radii ) 4 orbit phases, orbit adjust 2 year in-orbit (minimum) mission life Interspacecraft ranging and communication Advanced instrumentation, integrated payload Attitude knowledge < 0.1, spin rate 20 rpm Phases 1-3, Equatorial - Phase 4, Polar - Determination of Spatial Gradients

9 Laser Interferometer Space Antenna (LISA)" Mission: 3 spacecraft separated by 5,000,000 km form a three-arm Michelson Interferometer to observe gravitational waves in a 10-4 to 10-1 Hz bandwidth Approach: Each spacecraft payload includes two freely falling proof masses which serve as arm end mirror optical references Test masses must be free of non-gravitational forces (geodesically pure) Gravitational waves cause change in optical path in one arm of interferometer relative to other arm Distance changes measured with picometer precision to detect gravitational wave strains down to Disturbance Reduction System (DRS) uses proof mass displacement sensor outputs to drive low-noise micro-newton thrusters for dragfree system operation

10 Equations of Motion x 2 θ y θy θ 2 x = y + 2 θ x + θx θ 2 y = z = µz [(r c + x) 2 + y 2 + z 2 ] 3 2 µ(r c + x) [(r c + x) 2 + y 2 + z 2 ] 3 2 µy [(r c + x) 2 + y 2 + z 2 ] J 2 f z ( e,x) + u z + µ r c 2 + J 2 f x + J 2 f y ( e,x) + u y ( e,x) + u x y z ρ x r c r c = r c θ 2 C µ r J 2 f r c ( ) e C ( ) θ C = 2 r cθ C r 1 + J 2 f θ e C c

11 Hill s (Clohessy-Wiltshire) Equations Assumptions - Spherical Earth - Circular reference orbit - Equations can be linearized ( ) + O( e) + O( J 2 ) ( ) + O( e) + O( J 2 ) ( ) + O( e) + O( J 2 ) ( ) 3x y 0 / n ( ) 3( 2x 0 + y 0 / n)ψ + 2 x 0 / n x 2n y 3n 2 x = u x + O ρ / R y + 2n x = u y + O ρ / R z + n 2 z = u z + O ρ / R x = 2 2x 0 + y 0 / n ( )cosψ + ( x 0 / n)sinψ y = y 0 2 x 0 / n ( )cosψ + 2( 3x y 0 / n)sinψ z = z 0 cosψ + ( z 0 / n)sinψ ψ = nt y z ρ x R y

12 Periodic Solutions Circular Chief Orbit Leader-Follower In Plane 2x1 ellipse y x x Circular relative orbit y Horizontal Plane Circular Orbit z x y z

13 Periodic Relative Motion Eccentric Reference Orbit a=26,608 km, de=0.0001, vary e 3000 Radial Displacement - m 2000 e=0 e=.1 e= e=0.4 e= In-Track Displacement - m

14 Periodic Relative Motion Eccentric Reference Orbit a=26,608 km, de=0.0001, di=0.0002, vary e 8000 Out of Plane Displacement - m e=0 e=.1 e=0.2 e=0.4 e= In-Track Displacement - m

15 To achieve sufficient accuracy of the relative motion one must include the reference satellite eccentricity and J 2 effects and should use the curvilinear coordinate system, not Cartesian system to minimize the nonlinear effects.

16 Proposed Approach Rather than trying to estimate directly the orbit of each satellite we propose to determine the orbit of the Chief satellite and then to determine the state of each satellite relative to the Chief. Use relative measurements, or measurement differences, if possible. To achieve this a good theory of the relative motion is needed. We have such a theory. 1,2 1. Gim, D-W and Alfriend, K.T., The State Transition Matrix of Relative Motion for the Perturbed Non-Circular Reference Orbit, AIAA J. of Guidance, Control and Dynamics, Vol. 26, No. 6 Nov-Dec 2003, pp Alfriend, K.T, Vadali, S.R., Gurfil, P., How, J.P. and Breger, L., Spacecraft Formation Flying: Guidance, Control and Navigation, Elsevier, 2010

17 Geometric Method x T = ( x, x, y, y,z, z ),e T = ( a,θ,i,q 1,q 2,Ω);θ = f + ω,q 1 = ecosω,q 2 = esinω x( t) = A( e c )δ e = A( e c )Φ e ( t)δ e( t 0 ) = A( e c )Φ e ( t)a 1 ( t 0 )x( t 0 ) x( t) = Φ x ( t)x 0,Φ x ( t) = A( e c )Φ e ( t)a 1 ( t 0 ) Φ e = DΦ em, D = e osc e m R d = R c + ρ = R + x R C d = T CE T ED D R d R d C = T CE R C d = ( ) e xc + y e yc + z e zc ( T EC + δt EC ) R + δ R c c 0 = I + T CE δt EC 0 R c + δ R c R T CE c δt 11 δt 12 δt 13 = ( ) R + δ R c c 0 R c + δ R c R c δθ + δωcosi c cosθ c sini c δω + sinθ c δi

18 Results a=7100 km, e=0.005, i=70 deg Circular projected orbit, r=500 m

19 For a spherical Earth for there to be no along-track secular relative motion the semi-major axes have to be equal. This is not the case for a non-spherical Earth. For no secular along track drift. δ a = 0.5J 2 ( ) η 0 2 = 1 e 0 2 R e a 0 3 3η η 0 5 ( 1 3cos 2 i 0 )δη ( η 0 sin2i 0 )δi

20 Formation OD Issues Space surveillance systems will need to determine the orbits of the formation satellites independent of any formation relative navigation system of the formation. u Primary relative navigation system is Differential Carrier Phase GPS. For LEO formations depending on the radar parameters and the formation size one may not get observations of each satellites but some, or all, radar reflections may get combined. Can we use telescopes (angles only) for observations of LEO formations? For angles only observations when all the formation satellites are in the same field of view observation differences can be used for estimating the relative motion. u We have had some success with this approach for maneuver detection of GEO satellites. Correlated observations. Observations of close satellites tend to be highly correlated, but we usually model observations as uncorrelated. Observations differences can reduce the correlation.

21 Formation OD Issues (cont) Observation/object association. Associating the observation to the correct satellite will be a challenge. Standard US methods will not work because the objects are too close. If the out of plane motion is created by an inclination difference this will cause differential nodal precession and out of plane secular growth. The result will be frequent maneuvers for maintenance of LEO formations.

22 Summary Determining the orbits of satellites in a formation and the type of formation will be a challenge for space surveillance systems. u Radars may not provide observations of each satellite. u Correlated observations u Track/object association u Frequent formation maintenance maneuvers Much work remains to be done.