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The objective of this report is to discover a skyhook’s ability to change the plane of another spacecraft’s orbit while ensuring that each vehicle’s orbital energy remains constant. Skyhooks are a proposed momentum exchange device in which a tether is attached to a counterweight at one end and at the other, a capturing device intended to intercept rendezvousing spacecraft. Trigonometric velocity vector relations, along with objective comparisons to traditionally proposed uses for skyhooks and gravity-assist maneuvers were responsible for the ultimate parameterization of the proposed energy neutral maneuver. From this methodology, it was determined that a spacecraft’s initial relative velocity vector must be perpendicular to, and rotated about the skyhook’s total velocity vector if it is to benefit from an energy neutral plane change maneuver. A quaternion was used to model the rotation of the incoming spacecraft’s relative velocity vector. The potential post-maneuver spacecraft orbits vary in their inclinations depending on the ratio between the skyhook and spacecraft’s total velocities at the point of rendezvous as defined by the parameter called the alpha criterion. For many cases, the proposed maneuver will serve as a desirable alternative to currently practiced propulsive plane change methods because it does not costly require a substantial amount of propellant. The proposed maneuver is also more accessible than alternative methods that involve gravity-assist and aerodynamic forces. Additionally, by avoiding orbital degradation through the achievement of unchanging total orbital energy, the skyhook will be able to continually and self-sustainably provide plane changes to any spacecraft that belong to orbits that abide by the identified parameters.
We propose parallel algorithms for stability analysis of two classes of systems: 1) Linear systems with a large number of uncertain parameters; 2) Nonlinear systems defined by polynomial vector fields. First, we develop a distributed parallel algorithm which applies Polya's and/or Handelman's theorems to some variants of parameter-dependent Lyapunov inequalities with parameters defined over the standard simplex. The result is a sequence of SDPs which possess a block-diagonal structure. We then develop a parallel SDP solver which exploits this structure in order to map the computation, memory and communication to a distributed parallel environment. Numerical tests on a supercomputer demonstrate the ability of the algorithm to efficiently utilize hundreds and potentially thousands of processors, and analyze systems with 100+ dimensional state-space. Furthermore, we extend our algorithms to analyze robust stability over more complicated geometries such as hypercubes and arbitrary convex polytopes. Our algorithms can be readily extended to address a wide variety of problems in control such as Hinfinity synthesis for systems with parametric uncertainty and computing control Lyapunov functions.