John V. Breakwell

Optimization and Control of Nonlinear Systems Using the Second Variation

A feedback control scheme is described that maximizes a terminal quantity while satisfying specified terminal conditions, in the presence of small disturbances. The scheme can also be used in a rapidly converging computation technique to find exact solutions to the nonlinear two-point boundary value problems occurring in the calculus of variations. The scheme is based on a linear perturbation from a nominal optimum path and, as such, involves the second variation of the calculus of variations. A simple analytical example is given for thrust direction control to place a vehicle in orbit. Numerical examples of both the control scheme and the optimization technique are given for a lifting vehicle re-entering the earth’s atmosphere at parabolic speed.

The ‘Halo’ family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem

The Halo orbits originating in the vicinities of both,L1 andL2 grow larger, but shorter in period, as they shift towards the Moon. There is in each case a narrow band of stable orbits roughly half-way to the Moon. Nearer to the Moon, the orbits are fairly well-approximated by an ‘almost rectilinear’ analysis. TheL2 family shrinks in size as it approaches the Moon, becoming stable again shortly before penetrating the lunar surface. TheL1-family becomes longer and thinner as it approaches the Moon, with a second narrow band of stable orbits with perilune, however, below the lunar surface.

Station-keeping for a translunar communication station

A translunar communication station is to be kept close to a nominal unstable periodic ‘Halo’ orbit, visible at all times from Earth. The analytically computed nominal orbit is not perfect, requiring an average control acceleration of about 10−6gs for tight control. An adjustable quadratic combination of position deviation and control acceleration is minimized to provide an (adjustable) control law with period feedback gains and a periodic bias. The average control acceleration can be reduced to less than 10−8gs with an error settling time of less than 21/2 months. The resulting limiting motion provides, in turn, an improved nominal, permitting the same low control cost with much tighter control, corresponding to settling times of the order of one day.

The gravity-probe-b relativity gyroscope experiment: Development of the prototype flight instrument

The Gravity-Probe-B Relativity Gyroscope Experiment (GP-B) will measure the geodetic and frame-dragging precession rates of gyroscopes in a 650 km high polar orbit about the earth. The goal is to measure these two effects, which are predicted by Einsteins General Theory of Relativity, to 0.01% (geodetic) and 1% (frame-dragging). This paper presents the development progress for full-size prototype flight hardware including the gyroscopes, gyro readout and magnetic shielding system, and an integrated ground test instrument. Results presented include gyro rotor mass-unbalance values (15–86 nm) due the thickness variations of the thin niobium coating on the rotor, interior sphericities (163–275 nm peak-to-valley) of fused-quartz gyro housings produced by tumble lapping, gyro precession rates (gyroscopes at 5 K) which imply low mass-unbalance components parallel to the gyro axis (23–62 nm), and demonstration of a magnetic shielding factor of 2×1010 for the gyro readout system with one shielding component missing (the gyro rotor). All of these results are at or near flight requirements for the GP-B Science Mission, which is expected to be launched in 1995.

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional ‘halo’ periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L3-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L1-family and L2-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.

Matched Asymptotic Expansions, Patched Conics, and the Computation of Interplanetary Trajectories

Abstract Heliocentric ellipses, corrected for the influences of planetary attraction, are matched in the vicinity of a planet with local hyperbolas. It is found that the attraction of the destination planet alone causes a displacement of the arrival asymptote and a time-of-arrival correction which consists of two parts: a “gross” time bias and a “local” time bias, the latter depending only on the eccentricity of the arrival hyperbola. Similar corrections, including velocity corrections, are due to the planet of departure. The advantages of trajectories are pointed out.

Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions.

Resonances Affecting Motion Near the Earth-Moon Equilateral Libration Points

Abstract The “restricted problem of three bodies” for the earth-moon system is extended to include direct and indirect influence of the sun on a particle near the L 4 (or L 5 ) point. The problem is made to include perturbations up to fourth order in the displacements from the Lagrange point L 4 ; comparable solar effects are included. The Von Zeipel perturbation method is carried to the second approximation and the slow variations of parameters are studied. In the restricted problem the dominant effect is caused by the faster natural frequency being three times the slower one. The sun adds an additional near-resonance because its twice-monthly perturbation frequency is twice the (monthly) faster natural frequency. These effects are studied and the nonlinear effects of initial conditions noted. Some conclusions on “perturbative stability” are drawn.

Minimum fuel lunar trajectories for a low-thrust power-limited spacecraft

Minimum fuel trajectories from a low Earth parking orbit to a low Moon orbit are obtained for a low-thrust power-limited spacecraft with thrust acceleration levels of the order of 10−3G. The trajectories are found by matching an Earth spiral to a Moon spiral at some intermediate distance. Results are given for the planar case and for the three dimensional case where the Moon orbit is polar.

An asymptotic expansion for an optimal relaxation oscillator

For certain reduced-order optimization problems where assumed fast dynamics are neglected, chattering optimal solutions occur. The chattering optimal solution is represented by some of the variables alternating between distinctively different values at an infinite rate. For a simple and somewhat transparent periodic optimal control problem, the neglected dynamics are included by an asymptotic expansion about the chattering solution. The periodic chattering arc is approached as a weighting parameter, associated with the control penalty in the performance index, goes toward zero. This weighting parameter is used as the expansion parameter to form an asymptotic expansion about the chattering arc. In particular two time scales are used in the expansions. A time scale proportional to the period is used to transform the problem to one similar to that of a relaxation oscillator where the problem is characterized by slow, almost equilibrium motions connected by fast, jump type transitions. The asymptotic expansion is divided into two parts, an outer part at the time scale of the period and an inner part characterized by an even faster time scale which captures the fast transitions. These two solutions are matched together to obtain the resulting asymptotic solution in which the performance index and the optimal period are obtained up to third order, and the states and the control are obtained up to second order.