Estimation problem for impulsive control systems under ellipsoidal state bounds and with cone constraint on the control

The paper deals with the state estimation problem for the linear control system containing impulsive control terms (or measures). The problem is studied here under uncertainty conditions when the initial system state is unknown but bounded, with given bound. It is assumed also that the system states should belong to the given ellipsoid in the state space. So the main problem of estimating the reachable set of the control system is studied here under more complicated assumption related to the case of state constraints. It is assumed additionally that impulsive controls in the dynamical system must belong to the intersection of a special cone with a generalized ellipsoid both taken in the space of functions of bounded variation. The last constraint is motivated by problems of impulsive control theory and by models from applied areas when not every direction of control impulses is acceptable in the system. We present here the state estimation algorithms that use the special structure of the control system and take into account additional restrictions on states and controls. The algorithms are based on ellipsoidal techniques for estimating the trajectory tubes of uncertain dynamical systems. Numerical simulation results related to proposed procedures are also given. © 2012 American Institute of Physics

Application of dynamical systems theory to a very low energy transfer

We use lobe dynamics in the restricted three-body problem to design orbits with prescribed itineraries with respect to the resonance regions within a Hill’s region. The application we envision is the design of a low energy trajectory to orbit three of Jupiter’s moons using the patched three-body approximation (P3BA). We introduce the “switching region,” the P3BA analogue to the “sphere of influence.” Numerical results are given for the problem of finding the fastest trajectory from an initial region of phase space (escape orbits from moon A) to a target region (orbits captured by moon B) using small controls

Design of a Multi-Moon Orbiter

The Multi-Moon Orbiter concept is introduced, wherein a single spacecraft orbits several moons of Jupiter, allowing long duration observations. The ΔV requirements for this mission can be low if ballistic captures and resonant gravity assists by Jupiter’s moons are used. For example, using only 22 m/s, a spacecraft initially injected in a jovian orbit can be directed into a capture orbit around Europa, orbiting both Callisto and Ganymede enroute. The time of flight for this preliminary trajectory is four years, but may be reduced by striking a compromise between fuel and time optimization during the inter-moon transfer phases

Constructing a Low Energy Transfer Between Jovian Moons

There has recently been considerable interest in sending a spacecraft to orbit Europa, the smallest of the four Galilean moons of Jupiter. The trajectory design involved in effecting a capture by Europa presents formidable challenges to traditional conic analysis since the regimes of motion involved depend heavily on three-body dynamics. New three-body perspectives are required to design successful and efficient missions which take full advantage of the natural dynamics. Not only does a three-body approach provide low-fuel trajectories, but it also increases the flexibility and versatility of missions. We apply this approach to design a new mission concept wherein a spacecraft "leap-frogs" between moons, orbiting each for a desired duration in a temporary capture orbit. We call this concept the "Petit Grand Tour." For this application, we apply dynamical systems techniques developed in a previous paper to design a Europa capture orbit. We show how it is possible, using a gravitional boost from Ganymede, to go from a jovicentric orbit beyond the orbit of Ganymede to a ballistic capture orbit around Europa. The main new technical result is the employment of dynamical channels in the phase space - tubes in the energy surface which naturally link the vicinity of Ganymede to the vicinity of Europa. The transfer V necessary to jump from one moon to another is less than half that required by a standard Hohmann transfer

Differential games through viability theory : old and recent results.

This article is devoted to a survey of results for differential games obtained through Viability Theory. We recall the basic theory for differential games (obtained in the 1990s), but we also give an overview of recent advances in the following areas : games with hard constraints, stochastic differential games, and hybrid differential games. We also discuss several applications.Game theory; Differential game; viability algorithm;

Invariant manifolds, discrete mechanics, and trajectory design for a mission to Titan

With an environment comparable to that of primordial Earth, a surface strewn with liquid hydrocarbon lakes, and an atmosphere denser than that of any other moon in the solar system, Saturn's largest moon Titan is a treasure trove of potential scientific discovery and is the target of a proposed NASA mission scheduled for launch in roughly one decade. A chief consideration associated with the design of any such mission is the constraint imposed by fuel limitations that accompany the spacecraft's journey between celestial bodies. In this study, we explore the use of patched three-body models in conjunction with a discrete mechanical optimization algorithm for the design of a fuel-efficient Saturnian moon tour focusing on Titan. In contrast to the use of traditional models for trajectory design such as the patched conic approximation, we exploit subtleties of the three-body problem, a classic problem from celestial mechanics that asks for the motion of three masses in space under mutual gravitational interaction, in order to slash fuel costs. In the process, we demonstrate the aptitude of the DMOC (Discrete Mechanics and Optimal Control) optimization algorithm in handling celestial mechanical trajectory optimization problems

ESTIMATES OF REACHABLE SETS OF CONTROL SYSTEMS WITH BILINEAR–QUADRATIC NONLINEARITIES

The problem of estimating reachable sets of nonlinear impulsive control systems with quadratic nonlinearity and with uncertainty in initial states and in the matrix of system is studied. The problem is studied under uncertainty conditions with set – membership description of uncertain variables, which are taken to be unknown but bounded with given bounds. We study the case when the system nonlinearity is generated by the combination of two types of functions in related differential equations, one of which is bilinear and the other one is quadratic. The problem may be reformulated as the problem of describing the motion of set-valued states in the state space under nonlinear dynamics with state velocities having bilinear-quadratic kind. Basing on the techniques of approximation of the generalized trajectory tubes by the solutions of control systems without measure terms and using the techniques of ellipsoidal calculus we present here a state estimation algorithms for the studied nonlinear impulsive control problem bilinear-quadratic type

CONTROL AND ESTIMATION FOR A CLASS OF IMPULSIVE DYNAMICAL SYSTEMS

The nonlinear dynamical control system with uncertainty in initial states and parameters is studied. It is assumed that the dynamic system has a special structure in which the system nonlinearity is due to the presence of quadratic forms in system velocities. The case of combined controls is studied here when both classical measurable control functions and the controls generated by vector measures are allowed.  We present several theoretical schemes and the estimating algorithms allowing to find the upper bounds for reachable sets of the studied control system.   The research develops the techniques of the ellipsoidal calculus and of the theory of evolution equations for set-valued states of dynamical systems having in their description the uncertainty of set-membership kind.  Numerical results of system modeling based on the proposed methods are included

Advanced Solar-propelled Cargo Spacecraft for Mars Missions

Three concepts for an unmanned, solar powered, cargo spacecraft for Mars support missions were investigated. These spacecraft are designed to carry a 50,000 kg payload from a low Earth orbit to a low Mars orbit. Each design uses a distinctly different propulsion system: A Solar Radiation Absorption (SRA) system, a Solar-Pumped Laser (SPL) system and a solar powered magnetoplasmadynamic (MPD) arc system. The SRA directly converts solar energy to thermal energy in the propellant through a novel process. In the SPL system, a pair of solar-pumped, multi-megawatt, CO2 lasers in sunsynchronous Earth orbit converts solar energy to laser energy. The MPD system used indium phosphide solar cells to convert sunlight to electricity, which powers the propulsion system. Various orbital transfer options are examined for these concepts. In the SRA system, the mother ship transfers the payload into a very high Earth orbit and a small auxiliary propulsion system boosts the payload into a Hohmann transfer to Mars. The SPL spacecraft and the SPL powered spacecraft return to Earth for subsequent missions. The MPD propelled spacecraft, however, remains at Mars as an orbiting space station. A patched conic approximation was used to determine a heliocentric interplanetary transfer orbit for the MPD propelled spacecraft. All three solar-powered spacecraft use an aerobrake procedure to place the payload into a low Mars parking orbit. The payload delivery times range from 160 days to 873 days (2.39 years)