Introduction to linear and nonlinear programming

This review presents an overview of the thermal properties of mesoscopic structures. The discussion is based on the concept of electron energy distribution, and, in particular, on controlling and probing it. The temperature of an electron gas is determined by this distribution: refrigeration is equivalent to narrowing it, and thermometry is probing its convolution with a function characterizing the measuring device. Temperature exists, strictly speaking, only in quasiequilibrium in which the distribution follows the Fermi-Dirac form. Interesting nonequilibrium deviations can occur due to slow relaxation rates of the electrons, e.g., among themselves or with lattice phonons. Observation and applications of nonequilibrium phenomena are also discussed. The focus in this paper is at low temperatures, primarily below $4\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, where physical phenomena on mesoscopic scales and hybrid combinations of various types of materials, e.g., superconductors, normal metals, insulators, and doped semiconductors, open up a rich variety of device concepts. This review starts with an introduction to theoretical concepts and experimental results on thermal properties of mesoscopic structures. Then thermometry and refrigeration are examined with an emphasis on experiments. An immediate application of solid-state refrigeration and thermometry is in ultrasensitive radiation detection, which is discussed in depth. This review concludes with a summary of pertinent fabrication methods of presented devices.

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Opportunities for mesoscopics in thermometry and refrigeration: Physics and applications

A new state transition matrix is described for the nonlinear problem of relative motion on an arbitrary elliptical orbit. A linearization is performed, leading to a set of linear differential equations with time-dependentcoefficients. A new and simpler solution to those equations is represented in a convenient state transition matrix form. This new state transition matrix is valid for arbitrary elliptical orbits of 0≤e<1. The state propagation using the new state transition matrix shows good agreement with numerical results.

New State Transition Matrix for Relative Motion on an Arbitrary Elliptical Orbit

Formation eying is a key technology for both deep-space and orbital applications that involve multiple spacecraft. Many future space applications will beneet from using formation e ying technologies to perform distributed observations (e.g., synthetic apertures for Earth mapping interferometry) and to provide improved coverage for communication and surveillance. Previous research has focused on designing passive apertures for these formation e ying missions assuming a circular reference orbit. Those design approaches are extended and a complete initialization procedure for a large e eet of vehicles with an eccentric reference orbit is presented. The main result is derived from the homogenous solutions of the linearized relative equations of motion for the spacecraft. These solutions are used to end the necessary conditions on the initial states that produce T-periodic solutions that have the vehicles returning to the initial relative states at the end of each orbit, that is, v(t0)=v(t0+T). This periodicity condition and the resulting initialization procedure are originally given (in compact form) at the reference orbit perigee, butthis is alsogeneralized to enable initialization atanypoint around thereference orbit. In particular, an algorithm is given that minimizes the fuel cost associated with initializingthe vehicle states (primarily the in-track and radial relative velocities) to values that are consistent with periodic relative motion. These algorithms extend and generalize previously published solutions for passive aperture forming with circular orbits. The periodicity condition and the homogenous solutions can also be used to estimate relative motion errors and the approximate fuel cost associated with neglecting the eccentricity in the reference orbit. The nonlinear simulations presented clearlyshowthatignoringthereferenceorbiteccentricitygeneratesanerrorthatiscomparabletothedisturbances caused by differential gravity accelerations.

http://acl.mit.edu/papers/ecc_orb_jgcd.pdf

Relative Dynamics and Control of Spacecraft Formations in Eccentric Orbits

A precise analytic solution that includes the effects of the reference orbit eccentricity and differential perturbations is needed for the relative motion of formation-flying satellites. As a result of the spherical Earthand circular reference orbit assumptions, Hill's equations, which have often been used for describing relative motion, are insufficient for the long-term prediction of the relative motion. A new approach, called the geometric method, is developed to obtain the state transition matrix for the relative motion that includes the effects caused by the reference orbit eccentricity and the differential gravitational perturbations. The geometric method uses the relationship between the relative states and differential orbital elements to obtain the state transition matrix instead of directly solving the complex relative motion differential equations. The state transition matrices are derived for both mean and osculating elements with the primary gravitational perturbation that results from the equatorial bulge term J 2 . Although the results are based on the J 2 effects, the approach can be extended easily to include other perturbing forces.

State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit

Theclassictreatmentofrendezvousmechanicsandotherproblemsinvolvingtherelativemotionoftwospacecraft assumesa circularreferenceorbit, allowing a simpleclosed-form description ofthemotion. Asolution isdeveloped using an elliptical reference orbit, expanding the state transition matrix in powers of eccentricity, while retaining the explicit time dependence of the three-dimensional motion. The solution includes separate matrix elements for e rst-and second-order terms in eccentricity and for both Cartesian and cylindrical coordinates. Assessment of the maximum errors in position and velocity components over one complete revolution of the reference satellite shows that the solution is accurate for practical purposes with eccentricities in the range 0 ‐0.3. An example application is given for the proposed laser interferometer space antenna gravity wave experiment.

Robert G. Melton

Papers

Time-Explicit Representation of Relative Motion Between Elliptical Orbits

Hybrid methods for determining time-optimal, constrained spacecraft reorientation maneuvers

Comparison of direct optimization methods applied to solar sail problems

Inverse dynamics particle swarm optimization applied to constrained minimum-time maneuvers using reaction wheels

Differential evolution/particle swarm optimizer for constrained slew maneuvers