Higher-order sensitivity matrix method for probabilistic solution to uncertain Lambert problem and reachability set prob
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(2020) 132:50
ORIGINAL ARTICLE
Higher-order sensitivity matrix method for probabilistic solution to uncertain Lambert problem and reachability set problem Zach Hall1
· Puneet Singla1
Received: 27 January 2020 / Revised: 21 July 2020 / Accepted: 14 October 2020 © Springer Nature B.V. 2020
Abstract This paper presents a derivative-free method for computing approximate solutions to the uncertain Lambert problem (ULP) and the reachability set problem (RSP) while utilizing higher-order sensitivity matrices. These sensitivities are analogous to the coefficients of a Taylor series expansion of the deterministic solution to the ULP and RSP, and are computed in a derivative-free and computationally tractable manner. The coefficients are computed by minimizing least squared error over the domain of the input probability density function (PDF), and represent the nonlinear mapping of the input PDF to the output PDF. A nonproduct quadrature method known as the conjugate unscented transform is used to compute the multidimensional expectation values necessary to determine these coefficients with the minimal number of full model propagations. Numerical simulations for both the ULP and the RSP are provided to validate the developed methodology and illustrate potential applications. The benefits and limitations of the presented method are discussed. Keywords Lambert problem · Reachability sets · Uncertainty propagation · Stochastic systems · Conjugate unscented transform · Higher-order sensitivity
1 Introduction The classical Lambert problem is a well-known two-point boundary value problem, where the solution is the initial velocity vector that connects two known position vectors for a given time of flight. This problem and its many variants have a rich history in academic research and has been solved in an equally diverse range of methods. A good overview of the Lambert problem is given in Battin (1999), Battin and Vaughan (1984), Prussing and Conway (2013) and Vallado (2001). A complimentary but equally important problem in astrodynamics is the Reachability Set Problem (RSP). The RSP is the determination of the set
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Zach Hall [email protected] Puneet Singla [email protected]
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The Pennsylvania State University, University Park, PA 16802, USA 0123456789().: V,-vol
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of all possible final positions of a satellite given some range of initial positions and velocities. Similar to the Lambert problem, reachability sets have a robust background in the literature of mathematics, dynamics, optimization, control and game theory (Kurzhanski and Varaiya 2001; Bicchi et al. 2002; Patsko et al. 2003; Lygeros 2004; Mitchell et al. 2005; Hwang et al. 2005). The reachability set problem has been investigated in many different contexts, often interpreting the reachability set as an envelope of possible future states generated by some analytically derived boundaries (Vinh et al. 1995; Dan et al. 2010; Li et al. 2011; Gang et al. 2013; Wen et al. 2018). The Lambert problem and the reachability set pro
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