www.dlr.de Chart 1 Analysis of Relative Motion of Collocated Geostationary Satellites with Geometric Constraints SFFMT2013, München F. de Bruijn & E. Gill 31 May 2013
www.dlr.de Chart 2 Presentation Outline Introduction Relative motion Constraint analysis Relative orbit design problem Results & example case Conclusion & outlook
www.dlr.de Chart 3 The Problem Collocated satellites are equipped with sensors Payload Star sensors Etc. Sensor interference is caused by other satellites moving in field of view Geometric constraint: stay out of the field of view of sensors on other satellites Can this be dealt with through relative orbit design?
www.dlr.de Chart 4 Motivation Motivated by one of the leading satellite operators: Obstruction of star sensors can cause loss of attitude information Payload sensor interference is another problem that has not been dealt with Results can be applied to missions with cluster flying satellites Collocated GEO satellites Fractionated spacecraft
www.dlr.de Chart 5 Domain of Validity This research has been subject to some simplifying assumptions: Near-circular orbits Perturbation-free dynamics In reality perturbations affect the dynamics significantly BUT: differential perturbations are small Nadir pointing satellites: attitude fixed in the Hill frame These assumptions are valid for most GEO satellites
www.dlr.de Chart 6 Definition of Geometric Constraint Geometric constraint results from a sensor Sensor s pointing direction defined by e c Field of view defined through e c and β/2 Vector e 01 (t) points towards satellite #1 Angle between e 01 (t) and e c is α(t) Constraint is given by: α(t) > β/2
www.dlr.de Chart 7 Relative Motion Description Orbit described in equinoctial orbital elements Relative orbit described in orbital element differences Relative motion governed by Clohessy-Wiltshire equations a λ e x e y i x i y = a M + ω + Ω e cos (ω + Ω) e sin (ω + Ω) i cos (Ω) i sin (Ω) Δa Δλ Δe x Δe y Δi x Δi y
www.dlr.de Chart 8 Relative Orbit Assumptions: Δe = Δe x Δe y = δδ cos φ sin φ No along-track drift: Δa = 0 Centered orbit: Δλ = 0 Δi = Δi x Δi y = δδ cos θ sin θ Orbits lie on cylinder: aδδ: size of elliptical base aδδ: height of cylinder Angle between Δe and Δi determines orientation of orbit ψ = φ θ Three sample orbits shown to lie on an elliptical cylinder. Reference satellite is located in the center of the figure.
www.dlr.de Chart 9 Relation Relative Orbital Elements and Constraint Unit vector in direction of relative angular momentum: e e is completely defined through Δe and Δi (analytic relation derived) Constraint satisfied over the entire orbit if: α min > β 2 e e c > sin or β 2 Analytic relation derived between relative orbital elements and constraint!
www.dlr.de Chart 10 Relative Orbit Design Problem Feasibility problem: Find some angle ψ, between relative eccentricity and inclination vectors, that satisfies the geometric constraints Optimization problem: Find an angle ψ that maximizes α mmm The problem is generally subject to other constraints, e.g.: Bounds on tolerable separation between eccentricity and inclination vectors Bounds on the minimum and/or maximum separation distance between satellites
www.dlr.de Chart 11 Design Process 1. Identify constraints: Geometric constraints Bounds on relative orientation of eccentricity and inclination vectors Tolerable minimum and maximum separation distances 2. Formulate the problem mathematically 3. Solve the problem: find ψ (and optionally δe and δi) 4. Specify eccentricity and inclination vectors
www.dlr.de Chart 12 Example Case Fleet of 4 collocated satellites Each satellite equipped with: a Nadir pointing sensor γ cc = 0, χ cc = 0, β 1 2 = 9 North looking star sensor γ c2 = 45, χ c2 = 75, β 2 2 = 15 Constraint on separation of eccentricity and inclination vectors: ψ 0, 30 150, 210 330, 360 Constraint on size of eccentricity and inclination vectors: ae = 12 km, ai = 12 km
www.dlr.de Chart 13 Solutions including constraints on ψ Achievable α min Corresponding ψ 90 90 80 Star sensor 80 80 Star sensor 175 70 70 60 70 60 170 c 50 40 60 min c 50 40 165 30 50 30 160 20 40 20 10 Nadir sensor 0 0 10 20 30 40 50 60 70 80 90 30 10 Nadir sensor 0 0 10 20 30 40 50 60 70 80 90 155 c c The figures suggest feasibility!
www.dlr.de Chart 14 Specification of ψ Constraint on separation of eccentricity and inclination vectors: ψ 0, 30 150, 210 330, 360 Dotted lines show variations in λ of +/- 8 km Design point: ψ = 150
www.dlr.de Chart 15 Placement of e / i vectors Constraint on size of eccentricity and inclination vectors: a e = 12 km, a i = 12 km Relative phase angle from design: ψ = 150 For maximum relative separation the solutions are squares ψ
www.dlr.de Chart 16 Resulting Relative Orbits Constraints are satisfied Orbits share the same relative orbital plane
www.dlr.de Chart 17 Conclusions & Outlook Conclusion: An analytic relation between relative orbital elements and geometric constraints was defined The analysis resulted in a method for designing relative motion orbits that satisfy geometric constraints Next steps: What is the effect of perturbations and uncertainty on the relative orbit and geometric constraints? What is a suitable guidance and control method for geometry constrained collocated satellites?
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www.dlr.de Chart 19 Back-up slides
www.dlr.de Chart 20 Largest α mmm and corresponding ψ for constraint in positive octant for δe = δδ Achievable α min Corresponding ψ Solutions in positive octant in the unconstrained case