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NASA NTRS2021

Waypoint Following Dynamics of a Quaternion Error Feedback Attitude Control System

Closed-loop attitude steering can be used to implement alternative attitude maneuvers by using a conventional attitude control system to track a non-standard attitude profile sampled as a series of sequential commands. The idea has been employed in practice to perform zero-propellant maneuvers on the International Space Station and minimum time maneuvers on NASA’s TRACE space telescope. A challenge for operational implementation of the idea is the finite capacity of a space vehicle’s command storage buffer. One approach to mitigate the problem is to downsample-and-hold the attitude commands so that the attitude control system to follows a set of waypoints. This paper explores the waypoint following dynamics of a quaternion error feedback control law for downsample-and-hold. It is shown that downsample-and-hold induces a ripple between downsamples that causes the satellite angular rate to significantly overshoot the desired limit. Analysis in the z-domain is carried out in order to understand the phenomenon. An interpolating Chebyshev-type filter is proposed that allows attitude commands to be encoded in terms of a small set of filter coefficients. Using the interpolating filter, commands can be issued at the ACS rate but with significantly reduced memory requirements. The attitude control system of NASA’s Lunar Reconnaissance Orbiter is used as an example to illustrate the behavior of a practical attitude control system.

Mark Karpenko↗
NASA NTRS1979

Application of singular perturbation techniques /SPT/ and continuation methods for on-line aircraft trajectory optimization

The singular perturbation theory (SPT) approximation of optimal feedback control laws is presented and methods for on-line application of these approximations are discussed. It is demonstrated that SPT control laws break down when the current state is near the terminal target state. The use of continuation methods to improve the accuracy of the SPT approximation and to obtain global solutions of two-point boundary value problems is also discussed. As an illustration, consideration is given to the minimum-time control of a supersonic aircraft for a three-dimensional intercept problem.

Washburn, R. B.↗
NASA NTRS1972

Perturbation guidance for minimum time flight paths of spacecraft.

The problem of transferring a rocket vehicle from a given circular orbit to a larger coplanar circular orbit in minimum time, using a constant low-thrust rocket engine, is considered. Parameters are chosen to correspond to a transfer from the earth's orbit in heliocentric space to the orbit of Mars. A path satisfying the first order necessary conditions of variational calculus is shown to be locally minimizing by application of a set of second order conditions. A physical explanation is offered to justify the retrothrust period occurring during the flight. A neighboring optimum feedback control law, based on estimated time-to-go, is applied to this problem. State variable and terminal constraint feedback gains are calculated while one of the second order conditions, involving the backward integration of a matrix Riccati equation, is being tested.

Wood, L. J.↗
NASA NTRS1989

A real-time guidance algorithm for aerospace plane optimal ascent to low earth orbit

Problems of onboard trajectory optimization and synthesis of suitable guidance laws for ascent to low Earth orbit of an air-breathing, single-stage-to-orbit vehicle are addressed. A multimode propulsion system is assumed which incorporates turbojet, ramjet, Scramjet, and rocket engines. An algorithm for generating fuel-optimal climb profiles is presented. This algorithm results from the application of the minimum principle to a low-order dynamic model that includes angle-of-attack effects and the normal component of thrust. Maximum dynamic pressure and maximum aerodynamic heating rate constraints are considered. Switching conditions are derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another. A nonlinear transformation technique is employed to derived a feedback controller for tracking the computed trajectory. Numerical results illustrate the nature of the resulting fuel-optimal climb paths.

Calise, A. J.↗
NASA NTRS1972

The 4-D guidance of STOL aircraft in the terminal area

Advanced STOL aircraft for the improvement of the nation's air transportation system by the elimination of delays and congestions associated with today's air travel are discussed. A new guidance technique, referred to as 4-D guidance, is being developed for STOL aircraft. The 4-D guidance technique synthesizes complex three-dimensional flight paths from a minimum set of input and files the aircraft along the paths according to a prespecified time schedule. The two major elements of a 4-D guidance system are the trajectory synthesizer and the control law. Inputs to the trajectory synthesizer are the three-dimensional coordinates of way points, the turning radii, the speed ranges, the acceleration limits, and the arrival times at time control way points. First, the three-dimensional trajectory is computed by using circular arcs and straight lines. Then the airspeed profile, compensated for wind, is calculated to achieve the desired arrival times. The synthesized trajectory is stored as a time sequence of reference states which the aircraft is forced to track by using a linear feedback law.

Pecsvaradi, T.↗
NASA NTRS1972

Four-D guidance of STOL aircraft in the terminal area

The primary objective of advanced STOL aircraft is the improvement of the nation's air transportation system by the elimination of delays and congestions associated with today's air travel. A new guidance technique, referred to as 4-D guidance, is being developed to achieve this objective. The 4-D guidance technique synthesizes complex three-dimensional flight paths from a minimum set of input data and flies the aircraft along the paths according to a prespecified time schedule. The two major elements of a 4-D guidance system are the trajectory synthesizer and the control law. Inputs to the trajectory synthesizer are the three-dimensional coordinates of way points, the turning radii, the speed ranges, the acceleration limits, and the arrival times at time control way points. First, the three-dimensional trajectory is computed by using circular arcs and straight lines. Then the airspeed profile, compensated for wind, is calculated to achieve the desired arrival times. The synthesized trajectory is stored as a time sequence of reference states which the aircraft is forced to track by using a linear feedback law.

Pecsvaradi, T.↗
NASA NTRS1971

A neighboring optimal feedback control scheme for systems using discontinuous control.

The calculation and implementation of the neighboring optimal feedback control law for multiinput, nonlinear dynamical systems, using discontinuous control, is discussed. An initialization procedure is described which removes the requirement that the neighboring initial state be in the neighborhood of the nominal initial state. This procedure is a bootstrap technique for determining the most appropriate control-law gain for the neighboring initial state. The mechanization of the neighboring control law described is closed loop in that the concept of time-to-go is utilized in the determination of the control-law gains appropriate for each neighboring state. The gains are chosen such that the time-to-go until the next predicted switch time or predicted final time is the same for both the neighboring and nominal trajectories. The procedure described is utilized to solve the minimum-time satellite attitude-acquisition problem.

Foerster, R. E.↗
NASA NTRS1990

Gamma guidance of trajectories for coplanar, aeroassisted orbital transfer

The optimization and guidance of trajectories for coplaner, aeroassisted orbital transfer (AOT) from high Earth orbit (HEO) to low Earth orbit (LEO) are examined. In particular, HEO can be a geosynchronous Earth orbit (GEO). It is assumed that the initial and final orbits are circular, that the gravitational field is central and is governed by the inverse square law, and that at most three impulses are employed: one at HEO exit, one at atmospheric exit, and one at LEO entry. It is also assumed that, during the atmospheric pass, the trajectory is controlled via the lift coefficient. The presence of upper and lower bounds on the lift coefficient is considered. First, optimal trajectories are computed by minimizing the total velocity impulse (hence, the propellant consumption) required for AOT transfer. The sequential gradient-restoration algorithm (SGRA) is used for optimal control problems. The optimal trajectory is shown to include two branches: a relatively short descending flight branch (branch 1) and a long ascending flight branch (branch 2). Next, attention is focused on guidance trajectories capable of approximating the optimal trajectories in real time, while retaining the essential characteristics of simplicity, ease of implementation, and reliability. For the atmospheric pass, a feedback control scheme is employed and the lift coefficient is adjusted according to a two-stage gamma guidance law. Further improvements are possible via a modified gamma guidance which is more stable with respect to dispersion effects arising from navigation errors, variations of the atmospheric density, and uncertainties in the aerodynamic coefficients than gamma guidance trajectory. A byproduct of the studies on dispersion effects is the following design concept. For coplaner aeroassisted orbital transfer, the lift-range-to-weight ratio appears to play a more important role than the lift-to-drag ratio. This is because the lift-range-to-weight ratio controls mainly the minimum altitude (hence, the peak heating rate) of the guidance trajectory; on the other hand, the lift-to-drag ratio controls mainly the duration of the atmospheric pass of the guidance trajectory.

Miele, A.↗
NASA NTRS1985

Energy management of three-dimensional minimum-time intercept

A real-time computer algorithm to control and optimize aircraft flight profiles is described and applied to a three-dimensional minimum-time intercept mission. The proposed scheme has roots in two well known techniques: singular perturbations and neighboring-optimal guidance. Use of singular-perturbation ideas is made in terms of the assumed trajectory-family structure. A heading/energy family of prestored point-mass-model state-Euler solutions is used as the baseline in this scheme. The next step is to generate a near-optimal guidance law that will transfer the aircraft to the vicinity of this reference family. The control commands fed to the autopilot (bank angle and load factor) consist of the reference controls plus correction terms which are linear combinations of the altitude and path-angle deviations from reference values, weighted by a set of precalculated gains. In this respect the proposed scheme resembles neighboring-optimal guidance. However, in contrast to the neighboring-optimal guidance scheme, the reference control and state variables as well as the feedback gains are stored as functions of energy and heading in the present approach. Some numerical results comparing open-loop optimal and approximate feedback solutions are presented.

Kelley, H. J.↗
NASA NTRS1990

Singular perturbation analysis of AOTV-related trajectory optimization problems

The problem of real time guidance and optimal control of Aeroassisted Orbit Transfer Vehicles (AOTV's) was addressed using singular perturbation theory as an underlying method of analysis. Trajectories were optimized with the objective of minimum energy expenditure in the atmospheric phase of the maneuver. Two major problem areas were addressed: optimal reentry, and synergetic plane change with aeroglide. For the reentry problem, several reduced order models were analyzed with the objective of optimal changes in heading with minimum energy loss. It was demonstrated that a further model order reduction to a single state model is possible through the application of singular perturbation theory. The optimal solution for the reduced problem defines an optimal altitude profile dependent on the current energy level of the vehicle. A separate boundary layer analysis is used to account for altitude and flight path angle dynamics, and to obtain lift and bank angle control solutions. By considering alternative approximations to solve the boundary layer problem, three guidance laws were derived, each having an analytic feedback form. The guidance laws were evaluated using a Maneuvering Reentry Research Vehicle model and all three laws were found to be near optimal. For the problem of synergetic plane change with aeroglide, a difficult terminal boundary layer control problem arises which to date is found to be analytically intractable. Thus a predictive/corrective solution was developed to satisfy the terminal constraints on altitude and flight path angle. A composite guidance solution was obtained by combining the optimal reentry solution with the predictive/corrective guidance method. Numerical comparisons with the corresponding optimal trajectory solutions show that the resulting performance is very close to optimal. An attempt was made to obtain numerically optimized trajectories for the case where heating rate is constrained. A first order state variable inequality constraint was imposed on the full order AOTV point mass equations of motion, using a simple aerodynamic heating rate model.

Calise, Anthony J.↗
NASA NTRS1992

Design definition of the Laser Atmospheric Wind Sounder (LAWS), phase 2. Volume 1: Executive summary

The LAWS phase 1 and phase 2 studies have been completed on schedule and have led to significant advances in CO2 laser development. The Phase 2 Design Definition Study has shown that a large scanning mirror/high pulse energy laser LAWS Instrument is feasible and within the existing technology. The capability to monitor wind velocities with backscatter ratios of 10(exp 11) m(exp -1) SR(exp -1) is feasible. The weight budget allocated for the baseline LAWS is adequate, and sufficient reserves exist with the potential downsized configuration. With the possible decrease in available power from the baseline of 2.2 kW guideline, power and shot management is critical for the baseline configuration (15 to 20 J). This is particularly true during the 100 day occultation period each year. With the downsized configurations (5 to 7 J), power management is still necessary during the occultation but is primarily limited to shot management over the polar regions. The breadboard effort has produced significant laser advances for a tight 18 month schedule and the minimum budgets available from NASA, Lockheed, and TDS. Using the NASA funds and Lockheed and TDS fixed assets budgets, the breadboard was designed, fabricated, and brought on-line with first laser light within 16 months after ATP. First laser beam was obtained on 21 April 1992 at a 5 J power level. Tests since then have been conducted at sustained, repetitive pulse levels of over 7 J and 20 Hz. This is an increase of over two or three times greater than any system previously developed from this type laser. Increased power levels and additional life tests will be accomplished in the next LAWS phase. The Lockheed LAWS design will operate in the gravity gradient mode on-orbit, and all possible instrument vibration and jitter modes have been considered. Adequate pointing stability and control is state-of-the-art technology for the critical time periods, frequency rates, and control responses required by LAWS. Lockheed recommends a 6-1/2 year phase C/D program for LAWS to provide adequate feedback from the engineering unit and the qualification unit to the final flight unit. Assuming a one year period for LAWS integration to the spacecraft, followed by a six-month period for launch vehicle integration, LAWS could be successfully developed and launched in eight years. Our baseline design or downsized design can be accommodated by either the Atlas 2AS or the Delta launch vehicles. Lockheed's recommendation is that, based on the successful phase 2 design study and breadboard program, a follow-on 18 month extended breadboard testing program and additional system engineering studies, primarily in interfacing with a to be defined platform, be initiated. This should be immediately followed by the phase C/D program, leading to a LAWS launch in late 2001 or early 2002.

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