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A general purpose FORTRAN program for simulating and optimizing point mass trajectories (POST) of aerospace vehicles is described. The equations and the numerical techniques used in the program are documented. Topics discussed include: coordinate systems, planet model, trajectory simulation, auxiliary calculations, and targeting and optimization.
Information pertinent to users of the program to optimize simulated trajectories (POST) is presented. The input required and output available is described for each of the trajectory and targeting/optimization options. A sample input listing and resulting output are given.
A trajectory analysis is presented which shows the advantage of lifting ascent and the penalties associated with increased wing mass due to increases in aerodynamic loading. The vehicle analyzed is made up of an orbiter, an expendable liquid fuel tank, and two solid rocket motors. Results of the tradeoff are illustrated in a graph of the total vehicle mass plotted against the trajectory constraint. Results indicate that even when structural mass penalties are considered lifting ascent provides a payload delivery.
The two methods which are suitable for use in a 4-body trajectory optimization program are both multiconic methods. They include an approach due to Wilson (1970) and to Byrnes and Hooper (1970) and a procedure developed by Stumpff and Weiss (1968). The various steps in a trajectory optimization program are discussed, giving attention to variable step integration, the correction of errors by quadrature formulas, questions of two-impulse transfer, three-impulse transfer, and two examples which illustrate the implementation of the computational approaches.
Five pocket mice (Perognathus longimembris) were flown on Apollo XVII, each with a solid-state (plastic) nuclear track detector implanted beneath its scalp. The subscalp detectors were sensitive to HZE cosmic ray particles with a LET greater than or approximately equal to 0.15 million electron volts per micrometer (MeV/micron). A critical aspect of the dosimetry of the experiment involved tracing individual particle trajectories through each mouse head from particle tracks registered in the individual subscalp detectors, thereby establishing a one-to-one correspondence between a trajectory location in the tissue and the presence or absence of a lesion. The other major aspect was the identification of each registered particle. An average of 16 particles with Z greater than or equal to 6 and 2.2 particles with Z greater than or equal to 20 were found per detector. The track density, 29 tracks/sq cm, when adjusted for detection volume, was in agreement with the photographic emulsion data from an area dosimeter located next to the flight package.
The selection process for interfacing a nonreal time trajectory program with the procedures and performance program is outlined; the interface provides summary data timelines for any desired trajectory profile. Consideration was given to two separate digital programs for satisfying capabilities. One was the CDC 6400 digital program BANDITO, and the second was the UNIVAC 1110 SVDS program.
A computational procedure is described which numerically integrates the equations of motion of an unguided rocket. Three translational and two angular (roll discarded) degrees of freedom are integrated through the final burnout; and then, through impact, only three translational motions are considered. Input to the routine is: initial time, altitude and velocity, vehicle characteristics, and other defined options. Input format has a wide range of flexibility for special calculations. Output is geared mainly to the wind-weighting procedure, and includes summary of trajectory at burnout, apogee and impact, summary of spent-stage trajectories, detailed position and vehicle data, unit-wind effects for head, tail and cross winds, coriolis deflections, range derivative, and the sensitivity curves (the so called F(Z) and DF(Z) curves). The numerical integration procedure is a fourth-order, modified Adams-Bashforth Predictor-Corrector method. This method is supplemented by a fourth-order Runge-Kutta method to start the integration at t=0 and whenever error criteria demand a change in step size.
The results are presented of a study to show the effect on flyback trajectories of constant inertial attitude during the fuel dissipation phase of a return-to-launch-site abort. It is shown that the value of the constant inertial attitude can be chosen to shape the flyback trajectory.
Commands are found to drive a linear system to optimally track a class of prescribed trajectories, each of which contains a point of discontinuity. The paper focuses on the guidance problem of command generation, to be implemented in a feedforward fashion; the necessary additional control or feedback regulation structure is not studied in detail, but assumed to be provided in applications as a conventional error-feedback regulator. Solutions are found for the infinite-time problem which are optimal with respect to a quadratic performance criterion; suboptimal controls which satisfy a continuity condition are also found. The controls have applications to certain problems in aircraft guidance where command trajectories are piecewise continuous. Several examples are worked out in detail, with comparisons with conventional, nonfeedforward solutions to the problem, and a brief discussion of a simpler, suboptimal solution.
The paper describes a numerical simulation of electron trajectories in weak random electric fields under conditions that are approximately true for Langmuir waves whose wavelength is much longer than the Debye length. Two types of trajectory calculations were made: (1) the initial particle velocity was made equal to the mean phase velocity of the waves, or (2) it was equal to 0.7419 times the mean velocity of the waves, so that the initial velocity differed substantially from all phase velocities of the wave spectrum. When the autocorrelation time is much greater than the trapping time, the particle motion can change virtually instantaneously from one of three states - high-velocity, low-velocity, or trapped state - to another. The probability of instantaneous transition from a high- or low-velocity state becomes small when the difference between the particle velocity and the mean phase velocity of the waves becomes high in comparison to the trapping velocity. Diffusive motion becomes negligible under these conditions also.
Equations are derived by using branched trajectory optimization techniques and the maximum principle to maximize the payload capability of a reusable tug/expendable kickstage vehicle configuration for planetary missions. The two stages and the payload are launched into a low earth orbit by a single space shuttle. The analysis includes correction for precession of the orbit. This correction is done by the tug. The tug propels the payload and the kickstage to an energy beyond earth escape and returns within a specified time to the precessed orbit. After separating from the tug, the kickstage accelerates the payload to the required injection conditions. Planetary injection conditions are specified by the mission energy and a fixed declination and right ascension of the outgoing asymptote. The multipoint boundary value problem resulting from the analysis is solved by a Newton-Raphson iteration technique. Partial derivatives of the boundary conditions are obtained by perturbing the initial conditions one at a time, integrating the trajectory and adjoint equations, and observing the changes in boundary conditions. Maximum payload capability is derived for two typical mission energies. In addition, the variations of several mission and stage parameters are also examined.
The flight plans of the Venera 9 and Venera 10 spacecraft are discussed in terms of trajectory control and guidance. Results obtained from real time analysis to determine and predict the satellites' trajectories show good agreement between the actual and the design orbit evolution.
A state-of-art survey of computational techniques employed in design and optimization of trajectories for the Atlas/Centaur launch vehicle is presented. Attention is focused on the constrained optimization technique, related to Hestenes' (1969) method of multipliers, with various formulas used in updating the multipliers. Advantages of applying multiplier method to trajectory optimization, with gains in computing speed, are argued, and optimization of the HEAO-A (high energy astronomical observatory) is discussed. Open-loop atmospheric guidance strategy and closed-loop exoatmospheric pitch and yaw guidance equations are dealt with, and the full range of constraints to be observed during the flight is discussed.
Flight-mechanics problems associated with large-scale transport of lunar mass to a space colony or manufacturing facility are discussed. The proposed transport method involves launch of payloads from a mass-driver on the lunar surface, onto ballistic trajectories to a passive mass-catcher located near the L2 libration point, with the caught mass subsequently being transported to the colony. Arrival velocities at L2, sensitivities in arrival dispersion due to launch errors, and effects of launch site location are treated, via numerically integrated orbits in the restricted three-body problem. From any launch site it is possible to define a target point reached with zero dispersion due to errors in a selected component of launch velocity. Effects of lunar geometrical librations and of obliquity, as well as the conditions for biasing a trajectory away from L2 so as to reduce stationkeeping costs, are dealt with along with transfer orbits from L2 to the colony. The theory of capture and the theory of resonance lead to a colony orbit, with period approximately two weeks, reached from L2 with velocity increment as low as 9.02 m/sec.
A derivation of an explicit solution to the two point boundary-value problem of exoatmospheric guidance and trajectory optimization is presented. Fixed initial conditions and continuous burn, multistage thrusting are assumed. Any number of end conditions from one to six (throttling is required in the case of six) can be satisfied in an explicit and practically optimal manner. The explicit equations converge for off nominal conditions such as engine failure, abort, target switch, etc. The self starting, predictor/corrector solution involves no Newton-Rhapson iterations, numerical integration, or first guess values, and converges rapidly if physically possible. A form of this algorithm has been chosen for onboard guidance, as well as real time and preflight ground targeting and trajectory shaping for the NASA Space Shuttle Program.
Necessary conditions of optimality are derived for trajectories whose structure is limited to climb, steady cruise, and descent segments. The performance function consists of the sum of fuel and time costs, referred to as direct operating cost (DOC). The state variable is range to go and the independent variable is energy. In this formulation a cruise segment always occurs at the optimum cruise energy for sufficiently large range. At short ranges (500 n. mi. and less) a cruise segment may also occur below the optimum cruise energy. The existence of such a cruise segment depends primarily on the fuel flow vs thrust characteristics and on thrust constraints. If thrust is a free control variable along with airspeed, it is shown that such cruise segments will not generally occur. If thrust is constrained to some maximum value in climb and to some minimum in descent, such cruise segments generally will occur. Computer calculations of typical short-range trajectories obtained about a 1% cost penalty for constraining the thrust.
The only ballistic trajectory mode feasible for a close solar probe or for an orbit inclined approximately 90 degrees to the ecliptic is the Jupiter gravity assisted mode. A comparison of the trajectories of the Solar Polar and the Solar Probe Mission for 1983 launches is shown. The geometry of the solar encounter phase is practically the same for the 4.3 year orbit achieved by a Jupiter gravity assist and for a one year orbit. Data describing the geometry of an orbit with perihelion at 4 solar radii and aphelion at Jupiter are listed. The range of apparent directions of the solar wind if it is flowing radially outward from the Sun with a speed of either 150 or 300 km/sec is shown. The minimum sun-earth-probe angle during the solar encounter as a function of the earth-node angle and the orbital inclination is also shown. If the inclination is 60 degrees or more, the minimum SEP angle is not greatly different from the 90 degree value.
In February 1977, the Viking 1 Orbiter made repeated flybys of the Martian satellite Phobos at distances near 100 kilometers. These close encounters allowed a detailed scientific investigation of the nature and origin of Phobos. A sequence of three maneuvers was required to achieve the encounters. This paper presents the trajectory analysis performed to accommodate the scientific objectives and the spacecraft propulsive maneuver strategy used to achieve the desired trajectory while satisfying operational constraints. The actual maneuvers executed and the resultant Phobos encounter conditions are presented.