SPACECRAFT FORMATION FLYING A SIMULATION TOOL TO DESIGN SATELLITE FORMATIONS Aykut Kutlu1 and Prof.Dr. Ozan Tekinalp2 Middle East Technical University Ankara, Turkey [1PhD Student, Aerospace Engineering Department, Email: e150515@metu.edu.tr [2]Professor, Aerospace Engineering Dept., Email: tekinalp@metu.edu.tr ICATT-2016

SPACECRAFT FORMATION FLYING Contents Introduction, Definitions Simulation Tool Details Graphical User Interface: Main and Sub Windows Modeling Relative Motion Using the Keplerian Formulation Modeling Relative Motion Using Orbital Elements Modeling Desired Formation Flight: ICs Computation Methods Projected Circular Orbit based ICs computation [1] Geometrical Relative Orbit Modeling based ICs Computation[2] Optimization based ICs Computation Analyses and Results

SPACECRAFT FORMATION FLYING Introduction, Definitions Definition done by NASA’s Goddard Space Flight Center (GSFC) : “an end-to-end system including two or more space vehicles and a cooperative infrastructure for science measurement, data acquisition, processing, analysis and distribution”. Approaches defined for FF Orbit Tracking Each satellite is controlled to a particular predetermined desired orbit. Leader/Follower Leader spacecraft is controlled to a reference orbit and the other follower spacecraft in the formation control their relative stated to that leader. Virtual Structure Virtual structure and virtual center approaches fit a set of desired states to a formation in a way that minimizes the overall state error of the formation.  Swarming Simple heuristic control laws for arranging arbitrarily large numbers of vehicles into regular arrangements based on local information.

Simulation Tool

Modeling Relative Motion Keplerian Formulation Equations of Relative Motion in the unperturbed case:

Modeling Relative Motion Chief’s orbit propagation The chief’s position is computed using an orbit propagator including the variation of the mean classical orbital elements. The perturbations due to non-spherical earth (J2), due to moon and sun are included on the computations. As known, Kepler’s equation states that: J2 perturbations:

Modeling Relative Motion Keplerian Formulation, Simulation Block Diagram

Simulation Tool

Simulation Tool

Simulation Tool

Modeling Relative Motion Orbital Elements Formulation

Formation Design via OEs Orbital Elements Formulation Block Diagram Initial distance & phase angle Initial Orbital Elements

Comparaison of the modeling methods Kepler Formulation VS Orbital Elements Formulation These two different modeling methods give different solution for the relative motion analyses. For example for initial relative conditions given as

Comparaison of these two modeling methods Kepler Formulation VS Orbital Elements Formulation it is seen that the obtained motion dynamics is critically different, especially for y axis relative motion. For a very specific the initial condition values, it is possible to obtain the similar dynamic for both the modeling method, for:

Modeling Desired Formation Flight Initial Conditions Computation: Along Track and Cross Track ICs for Projected Circular Orbit (PCO-ICs) [1] An approximate solution to determine the initial conditions for near circular orbit Formation Design with desired distance and angular position: the along track and the cross track initial conditions

Modeling Desired Formation Flight Initial Conditions Computation: Geometrical Relative Orbit Modeling (GROM) [2] Spherical geometrical representation of the relative motion A set of equation to determine the orbital parameters of the deputy satellite for desired relative orbit size (along track (Ay) and cross track (Az) distance) and the relative phase angle ψ [2]: Figure 8. Geometrical Relative Orbit Modeling [2]

Modeling Desired Formation Flight Initial Conditions Computation: Optimal ICs, performs optimization to determine the Initial Conditions (OPTICs) Question: Is it possible to find ICs set that ensure formation flight scheme for any arbitrary desired formation scheme, by using an OPTIMIZATION method ? A minimization function based on the Energy Matching Approach, that theoretically guaranties the formation flight, is written: The initial relative positions are described by a desired azimuth, desired elevation angle and desired distance. The initial position in Cartesian coordinate system according to the desired azimuth and elevation

Analyses and Results 3 Diferent Cases are taken into consideration Orbital elements values of the Chief satellite

Orbital Elements based Modeling Analyses and Results PCO-ICs’ Results PCO-ICs, Case-1 Results Results using Nonlinear Modeling Results using Orbital Elements based Modeling

Orbital Elements based Modeling Analyses and Results PCO-ICs’ Results PCO-ICs, Case-2 Results Results using Nonlinear Modeling Results using Orbital Elements based Modeling

Orbital Elements based Modeling Analyses and Results PCO-ICs’ Results PCO-ICs, Case-3 Results Results using Nonlinear Modeling Results using Orbital Elements based Modeling

Orbital Elements based Modeling Analyses and Results GROM’s Results Case-1 Results Results using Nonlinear Modeling Results using Orbital Elements based Modeling

Orbital Elements based Modeling Analyses and Results GROM’s Results Case-2 Results Results using Nonlinear Modeling Results using Orbital Elements based Modeling

Orbital Elements based Modeling Analyses and Results GROM’s Results Case-3 Results Results using Nonlinear Modeling Results using Orbital Elements based Modeling

Analyses and Results OPT-ICs’ Results Case-1 Results Case-2 Results Results using Nonlinear Modeling Results using Nonlinear Modeling

Analyses and Results OPT-ICs’ Results Case-3 Results Results using Nonlinear Modeling

Analyses and Results Long-time (10 days) simulations results Case-1 Results The relative distances computed at the end of the 10th day are approximately 4 km for Case-1.

Analyses and Results Long-time (10 days) simulations results Case-2 Results The relative distances computed at the end of the 10th day are approximately 10 km for Case-2.

Analyses and Results Long-time (10 days) simulations results Case-3 Results The relative distances computed at the end of the 10th day are approximately 7 km for Case-3. These results can be considered as a efficiency measure of the OPT-ICs.

Analyses and Results OPT-ICs’ Results For a desired elevation and azimuth: x_dot_0 = 1.918504e+01 m/sec y_dot_0 = -5.547504e+02 m/sec z_dot_0 = 0.000000e+00 m/sec x_0 = 2.499989e+05 m y_0 = 2.500000e+05 m z_0 = 3.535534e+05 m

Simulation Tool – Post Processing

Simulation Tool – Post Processing

Conclusion In this study, the common methods used for satellite formation flight are presented. The main goals were to present the developed simulation tool and to present the results obtained according to the approaches and models used According to the results, it seen that that the optimization based method, OPTICs ensures the desired formation flight for any arbitrary flight scheme (relative distance and initial phase angle) and provides more stable, long-term flight compared to the other two methods. OPTICs provides reliable ICs for an initial relative position having both elevation and azimuth angles. It means that is it possible to define desired initial relative position in three dimensions, in XYZ axis frame. OPTICs provides ICs that ensure the formation flight for any desired initial position.

References [1] Alfriend K.T., Vadali S.R., Gurfil P., Jonatan P.H., Breger L.S. , Spacecraft Formation Flying, Dynamics, Control and Navigation, Elsevier, Bulrington, MA, USA, 2010. [2] S.Sub Lee, Dynamics and Control of Satellite Relative Motion: Designs and Applications, PhD. Dissertation, Blacksburg, Virginia March 20, 2009 [3] Chris Sabol, Rich Burns, and Craig A. McLaughlin., “Satellite Formation Flying Design and Evolution”, Journal Of Spacecraft And Rockets, Vol. 38, No. 2, March–April 2001 [4] H.Cui, J.Li, Y.Gao. (2006), “An Orbital Design Method for Satellite Formation Flying", Journal of Mechanical Science and Technolojy, Vol.20, No.2,pp177-184,2006. [5] Mark B. Milam, Nicolas Petit, and Richard M. Murray., “Constrained Trajectory Generation For Micro-Satellite Formation Flying”, AIAA 2001 -4030, California Institute of Technology.

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