Reachability Results for Nonlinear Systems with Ellipsoidal Initial Sets

Holzinger, Marcus J.; Scheeres, Daniel J.

doi:10.1109/taes.2012.6178080

Cited by 33 publications

(15 citation statements)

References 21 publications

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“…Range set was studied by Breakwell and Heine [10], Marec [11], and Holzinger [12][13][14]. Hozinger and Scheeres [12] modified traditional reachability theory by reformulating the Hamilton-Jacobi-Bellman partial differential equations using a generalized independent parameter [15].…”

Section: Introductionmentioning

confidence: 99%

Reachable domain for spacecraft with ellipsoidal Delta-V distribution

Wen

Peng

2018

Astrodyn

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Conventional reachable domain (RD) problem with an admissible velocity increment, Δv, in an isotropic distribution, was extended to the general case with Δv in an anisotropic ellipsoidal distribution. Such an extension enables RD to describe the effect of initial velocity uncertainty because a Gaussian form of velocity uncertainty can be regarded as possible velocity deviations that are confined within an error ellipsoid. To specify RD in space, the boundary surface of RD, also known as the envelope, should be determined. In this study, the envelope is divided into two parts: inner and outer envelopes. Thus, the problem of solving the RD envelope is formulated into an optimization problem. The inner and outer reachable boundaries that are closest to and farthest away from the center of the Earth, respectively, were found in each direction. An optimal control policy is then formulated by using the necessary condition for an optimum; that is, the first-order derivative of the performance function with respect to the control variable becomes zero. Mathematical properties regarding the optimal control policy is discussed. Finally, an algorithm to solve the RD envelope is proposed. In general, the proposed algorithm does not require any iteration, and therefore benefits from quick computation. Numerical examples, including two coplanar cases and two 3D cases, are provided, which demonstrate that the proposed algorithm works efficiently. KEYWORDS reachable domain maneuverability orbit uncertaintyResearch Article

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Section: Introductionmentioning

confidence: 99%

Reachable domain for spacecraft with ellipsoidal Delta-V distribution

Wen

Peng

2018

Astrodyn

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“…1,2 The reachability theory was originally derived from the optimal control theory to compute the potential state reach set (or reachable set) of a system under given constraints (terminal state distribution, control capability, etc.). 3–9 In reachability theory, two types of reach set are defined, namely, the fixed-time reach set and the free-time reach set ; the former is defined as the bounded reachable state distribution at given time t=tk, whereas the latter is defined as the set of reachable states on some time interval t∈[t0,tf]. 9 Meanwhile, a reach tube (or trajectory/reachability tube) recorded the evolution of the fixed-time reach set along the time axis.…”

Section: Introductionmentioning

confidence: 99%

“…10,11 Most of existing literature in reachability theory were focusing on the fixed-time reach set and the reach tube. 1,3,5,8,9,[12][13][14][15][16] However, research on the free-time reach set is relatively less. A free-time reach set certainly is of its value compared to a series of discrete fixed-time reach sets.…”

Section: Introductionmentioning

confidence: 99%

“…[19][20][21] In the field of orbit dynamics, the free-time reach sets described by three classic orbit elements, namely, the semimajor axis a, eccentricity e, and inclination i, were studied by Breakwell and Heine, 22 Marec, 23 and Holzinger. 8,17,24 Holzinger and Scheeres 17 used term ''range set'' to represent the free-time reach set in terms of orbital elements. They modified the traditional reachability theory by reformulating the HJB PDE using a generalized independent parameter (GIP).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Three-dimensional relative reachable domain with initial state uncertainty in Gaussian distribution

Wen

Shi²

2018

Proceedings of the Institution of Mechanical Engineers, Part G:

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A probability threshold is a confidence level defining a bound outside which the occurrence of a random variable is considered as a rare event. Upon providing such a probability threshold on the initial state uncertainty, relative reachable domain is defined as the minimum positional volume enclosing all the possible relative trajectories resulting from the initial state uncertainty. In this study, the conventional coplanar relative reachable domain with initial state uncertainty in isotropic Gaussian distribution is extended to the three-dimensional case with uncertainty in arbitrary Gaussian distribution. Positional errors in Gaussian distribution are thought to be confined within an error ellipsoid depending on the given probability threshold. Such an error ellipsoid will evolve with time and consequently sweep out a volume, which is the relative reachable domain to be determined. This paper proposed an algorithm of solving the envelope surface of the relative reachable domain for close range relative motion based on the linearized dynamical model and the mathematical definition of an envelope. Moreover, the algorithm is modified to improve the accuracy for long range relative motion. Comparisons between the solved relative reachable domain and the result of 10,000 Monte Carlo runs, which can be regarded as the true result, for different scenarios on circular reference orbits demonstrated the feasibility of the proposed method.

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“…in [12], based on support functions in [13], and by the use of ellipsoids in [15]. The reachability problem of nonlinear systems was addressed in [16], [19], [20], among others, with different techniques to propagate reachable sets, represented mainly by polyhedra, forward in time. System uncertainties were considered for linear dynamics in [6] and [9] using reach set representations by ellipsoids and polytopes, and for nonlinear dynamics in [7] with reachable sets specified by zonotopes, respectively.…”

Section: Introductionmentioning

confidence: 99%