Robust Reentry Guidance of a Reusable Launch Vehicle Using Model Predictive Static Programming

Introduction A new entry guidance concept is developed for an unpowered lifting vehicle. The two-dimensional longitudinal plane entry guidance scheme used by the U.S. Space Shuttle Orbiter is extended to a threedimensional entry guidance scheme. The guidance task is posed as planning and tracking a reference trajectory. This paper concerns the design and analysis of the tracking law. In the new guidance scheme, the complete ground-track is planned as a function of energy and tracked by commanding both the angle of attack and angle of bank. This extension to the Shuttle entry guidance scheme is carried out within the theoretical framework of differential flatness and feedback linearization. Approximately flat outputs for the system dynamics are identified which are then used to construct a state-variable transformation which together with a static state feedback control transformation renders the system approximately input-output, as well as input-state linear. In the transformed state and control variables, the system dynamics can be represented by two approximately decoupled linear systems. A feedback tracking law is then designed by taking advantage of the linear structure of the system dynamics in the transformed coordinates. This tracking law is shown to achieve bounded tracking of the flat outputs. Simulation results indicate that, if provided with an appropriate reference trajectory, the tracking law guides the vehicle to within one nautical mile of the target in downrange and cross-range during the hypersonic entry phase. 'Graduate Student, Student Member AIAA. t Research Assistant, AIAA member, presently at The Aerospace Corporation, 2350 E. El Segundo Blvd., El Segundo, CA 90245. ^Associate Professor, Associate Fellow AIAA. § Copyright ©by Kenneth D. Mease, Published by AIAA with permission Current efforts to develop a reusable launch vehicle (RLV) and a space station crew return vehicle provide particular motivation for improving the entry guidance capability for a lifting unpowered vehicle. In this paper we develop a natural extension to the longitudinal entry guidance scheme used by the U.S. Space Shuttle Orbiter. The new entry guidance scheme follows from the property of differential flatness of a A controlled system is differentially flat if there exist output functions, termed flat outputs, equal in number to the control inputs, such that all the state and control functions can be determined from the output functions and their derivatives through purely algebraic relations. A differentially flat system offers the possibility of reduced-order trajectory planning in terms of the flat outputs. Moreover, the flat outputs often permit feedback linearization of the controlled system dynamics, i.e., they lead to state and control transformations that globally linearize the dynamics. Thus guidance law design and analysis may become greatly simplified for a differentially flat system. Following this paradigm, the guidance task is divided naturally into a trajectory planning subtask of determining the reference output functions consistent with a trajectory that meets all the guidance objectives, and a trajectory tracking subtask of determining the control commands to track the reference output functions. The longitudinal-plane entry guidance logic used for the Shuttle can be viewed as a product of the approach just described. Drag is the flat output used in trajectory planning and the magnitude of the bank angle is commanded to track the reference drag trajectory. The reference drag trajectory is planned as a function of velocity such that it achieves the desired down-range and satisfies the vehicle thermal, normal load and dynamic pressure constraints, as well as the controllability constraint imposed by the

Reusable launch vehicle (RLV) coordinated reentry guidance is one of the key conditions to determine the effect of coordinated flight. The existing RLV reentry guidance methods often take a single spacecraft as the research object and do not have the ability of coordinated reentry flight. To solve these problems, a coordinated reentry guidance scheme is designed. In this paper, a time coordination strategy is designed aiming at the predictability of reentry flight time. The pseudospectral method is used to predict the flight time of RLVs, and the coordinated flight time coordination of multiple RLVs is realized. Based on receding horizon control, a coordinated reentry guidance law is designed. In each guidance cycle, RLV flies according to the guidance instructions generated in the previous guidance cycle. At the same time, taking the previous reentry trajectory as the initial value and the coordinated flight time determined by the time coordination algorithm as the constraint condition, the pseudo-spectral method is used to generate the flight trajectory quickly. The angle of attack and inclination are used as guidance instructions for the next guidance cycle of RLV, so as to guide RLV to the target. Finally, the simulation results verify the performance of the reentry coordinated guidance law and the effectiveness of the whole reentry guidance scheme.

Optimal control of grinding mill circuit using model predictive static programming: A new nonlinear MPC paradigm

The Kistler K-l is designed to be a fullyreusable, two stage launch vehicle for economical delivery of small satellite ,payloads to low earth orbit. Greatest efficiency and, hence, lowest cost are achieved by flyback of both vehicle stages to the near vicinity of the launch site. The second stage Orbital Vehicle (OV), after deploying the payload and performing necessary phasing maneuvers, performs a deorbit burn to achieve the desired trajectory conditions at atmospheric entry. After entering the atmosphere, the OV is steered aerodynamically until reaching the deployment point . for a stabilization parachute. Subsequently, drogue and main parachutes complete deceleration of the vehicle for landing orrairbags. This paper presents the design of an atmospheric guidance algorithm for bank-to-turn steering of the OV prior to deployment of the stabilization parachute. Reentry guidance targets a desired geographic position for deployment of the drogue parachute. The algorithm employs a numerical predictor/corrector technique to compute the bank angle and the start time of a single bank reversal required to null the predicted target position miss. Aerodynamic loads and heating are limited implicitly by selection of deorbit target conditions for reentry trajectory shaping. Results obtained using the Draper Laboratory K-l Integrated Vehicle Simulation illustrate guidance performance under nominal and dispersed conditions. * Senior Member of the Technical Staff, Member AIAA. Introduction The K-l Orbital Vehicle (OV) is cylindrical in shape, with a blunt nose and a flared skirt on the aft end (Figure 1). The vehicle trims at an angle of attack of approximately 11 to 12 degrees over the Mach range experienced during reentry. This results in a lift-todrag ratio ranging from 0.1 to 0.15. The only control effecters available are 12 small attitude control system (ACS) jets that are used to provide three-axis attitude stabilization and control. Due .to jet size limitations, alteration of the vehicle angle of attack from the trim condition for purposes of trajectory control is not feasible. Thus, active control of the vehicle trajectory is possible via bank maneuvers only. Given the vehicle mass characteristics and ACS jet properties, the bank acceleration is limited to approximately 2.5 deg/sec/sec. A maximum sustainable bank rate of approximately 15 deg/sec is possible; however, 10 deg/sec is used as a practical limit to avoid excessive Euler coupling (and, hence, fuel usage). Figure 1. Kistler K-l Orbital Vehicle (OV) Copyright 01999 by The Charles Stark Draper Laboratory, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission: 1275 (c)l999 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization. When the Mach number decreases to 2.5, a stabilization parachute is --deployed to stabilize the vehicle about the pitch and yaw axes in the low supersonic and transonic regimes. When a specified altitude is reached a drogue parachute and main parachutes are deployed to decelerate the vehicle for subsequent landing. on airbags. The landing site may be located either at the Nevada Test Site (NTS) or in Woomera, Australia. The key requirement levied on the reentry guidance is to steer the vehicle to the proper conditions for parachute deployment to achieve the landing target position with a given accuracy. This must be accomplished without exceeding specified aerodynamic loading and heating rate limits (8 g longitudinal, 4 g lateral, and 58 BTU/sq ft/sec, respectively). ACS fuel usage must be limited in the process. These requirements must be satisfied in the presence of potentially significant known and unknown dispersions in vehicle mass and aerodynamics properties, deorbit trajectory, and atmospheric conditions. An adaptive atmospheric reentry guidance algorithm has been designed which, as part of an overall guidance, navigation, and control subsystem, addresses the above vehicle characteristics and mission design requirements. The algorithm uses a numerical predictor/corrector technique to solve for a bank angle magnitude and the start time of a single bank reversal that will null the predicted target position miss. Numerical predictor/corrector guidance has been suggested previously for application to aerobraking and aerocapture.“2’3 -While computationally intensive, this technique allows the use of relatively high-fidelity models of vehicle and environmental characteristics, with ,correspondingly increased accuracy over analytic techniques. This paper presents a discussion of the decision’ process leading to the guidance design in addition to a detailed description of the key elements of ‘. the guidance algorithm. Details of the flight software implementation of the guidance algorithms are included where beneficial. Finally, simulation results are supplied to illustrate guidance performance. under nominal and dispersed-conditions. Reentry Dynamics Figure 2 is a free-body diagram of the OV during atmospheric reentry. The lift and drag accelerations experienced by the vehicle are expressed as where. . p f atmospheric density Vd = velocity relative to air mass C, , C, = vehicle drag and lift coefficients

The present paper describes a simple guidance strategy for the reentry mission of a generic Reusable Launch Vehicle. The guidance computation starts with a nominal feasible trajectory, which gets updated during actual reentry based on onboard measurements, using both incremental angle of attack and bank angle parameters. An analytical formulation for a modified predictor-corrector algorithm is developed, using angle of attack profile and bank angle reversal strategy. The guidance solution is arrived at under the heating rate and equilibrium glide constraints and the control actuation is implemented through the aerodynamic control surfaces. A scheme to improve the overall performance of the predictor-corrector algorithm is evolved, which is based on the determination of optimum perturbation levels of control variables. In addition, the control vector formulation is provided with a weighting option, for improving the overall terminal interface accuracy. Numerical results are obtained for the time histories of control variables and performance parameters, for a typical nominal trajectory, which show that proposed guidance scheme works well for a specified reentry mission. Results are also obtained for the optimum perturbation levels for the control parameters, which are used for estimating the sensitivity coefficients required in corrector algorithm. Lastly, the results are obtained for the trajectory parameters, by considering dispersions in the aerodynamic drag and lift coefficients, for establishing its robustness. The results clearly bring out the fact that the proposed reentry guidance algorithm is able to achieve the required terminal conditions, even in the presence of large perturbations in the aerodynamic coefficients.

An effective predictor-corrector guidance algorithm of high-lifting vehicles is presented in this paper. For the convenience of designing the bank angle profile, the path constraints are transformed into the constraints of the bank angle by using the quasi-equilibrium glide condition (QEGC). The differential equations of motion and range-to-go of hypersonic vehicle are used to predict the terminal target point. The main problem of the guidance of high-lifting vehicles is the bounces of vehicle in the trajectory. For suppressing the trajectory oscillations caused by their high ratio of lift and drag in the guidance, the current bank angle, which is calculated from the predictor-corrector guidance approach, will be improved by the reference flight path angle. What is more, a method is presented to correct the terminal bank angle. Simulations are provided to verify the effectiveness of the proposed method.

The entry-glide guidance strategy for hypersonic vehicles that can satisfy both terminal and path constraints is investigated in this paper. We propose a quasi-equilibrium glide adaptive guidance methodology based on the quasi-equilibrium glide condition (QEGC), which innovatively utilizes the quasi-equilibrium glide phenomenon in lifting entry. With the aid of QEGC, both range and terminal velocity can be predicted analytically with high precision. The path constraints are converted into angle of attack constraints, which has been difficult to realize by using traditional predictive guidance methods. The algorithm is independent of the standard trajectory. All the guidance commands, including the bank angle and the angle of attack, are calculated analytically in real time, which endows the algorithm with sufficient adapbility. The results of a CAV-H vehicle guidance test show that the algorithm leads the vehicle along a quasi-equilibrium glide trajectory satisfying both the terminal and path constraints and has sufficient flexibility for occasional mission changes. Furthermore, the robustness of the guidance algorithm under disturbances is validated through a Monte Carlo simulation.

Hypersonic reentry trajectory planning by using hybrid fractional-order particle swarm optimization and gravitational search algorithm

In this paper, a footprint determination algorithm for entry vehicles based on three-dimensional acceleration profile is proposed, which solves the problem of fast determination of the entry coverage with complex multi-constraints. The aerodynamic adjustment capability is greatly improved since both the angle of attack and bank are regarded as primary variables, and in this way larger entry footprint is achieved. Furthermore, due to the introduction of quasi-equilibrium glide condition (QEGC), the altitude of the entry trajectory changes smoothly without jumping. Firstly, based on the QEGC, a new type of longitudinal acceleration corridor and lateral lift to drag corridor are designed, and the profiles are chose as the interpolation results of the upper and lower boundaries. Secondly, based on PID theory, a three-dimensional profile tracker is designed to realize the corresponding entry trajectories. Finally, the traversal of the interpolation parameters in the corridors is carried out, and the entry footprint satisfying various complex constraints is obtained. The approach is tested using Common Aero Vehicle-H model, and the results demonstrate that the proposed algorithm can determinate the footprint of entry vehicles rapidly while satisfying all the path and terminal constraints.

Ascent Trajectory Optimization of Launch Vehicles with Air-Breathing Propulsion

The design of a linear fractional transformation gain-scheduled controller, scheduled on angle of attack, for the F-14 aircraftlateral-directionalaxisis presented.Thecontrolleris designed forthe powered approach e ight phase, during which the angleof attack and corresponding airspeed varies from 2 deg and 182 kn to 14 deg and 126 kn. A linearfractional modelof the lateral dynamics is constructed based on four linearized models that correspond to 2, 6, 10.5, and 14 deg angle of attack. Using parameter-dependentfunctions, a controller is designed that dependsin a linear fractional manner on angle of attack and delivers uniform handling quality over angle-of-attack variations that lie between 2 ‐14 deg. The resulting controller performs well when implemented in a nonlinear simulation model of the F-14 aircraft. Nomenclature p = roll rate r = yaw rate v = lateral velocity yac = lateral acceleration a = angle of attack b = sideslip angle d dsp = differential spoiler dee ection d dstab = differential stabilizer dee ection d lstk = lateral stick input d rud = rudder dee ection d rudp = rudder pedal input u = bank angle

The objective of the study described in this paper was to develop an efficient uncertainty quantification framework capable of analyzing uncertainty in integrated spacecraft system models. Specifically, this paper discusses the capabilities of the developed framework and the res ults when applied to the multidisciplinary analysis of a reusable launch veh icle (RLV). This particular framework is capable of efficiently propagating mixed (inherent and epistemic) uncertainties through complex simulation codes. The Second-Order Probability Theory utilizing a stochastic response surface obta ined with Point-Collocation NonIntrusive Polynomial Chaos was used for the propagation of the mixed uncertainties. This particular methodology was applied to the RLV analysis, and the uncertainty in the output parameters of interested was obtained in terms of intervals at various probability levels. This study has also demonstrated the feasibility of the developed uncertainty quantifica tion framework for efficient propagation of mixed uncertainties in the analysis of complex aerospace systems. I. Introduction ncertainties are generally ubiquitous in the analys is and design of highly complex engineering systems. Uncertainties can arise from the lack of k nowledge in physical modeling (epistemic uncertainty), inherent variations in the systems (a leatory uncertainty), and numerical errors in the computational procedures used for analysis. It is i mportant to account for all of these uncertainties in applications such as robust and reliable design of multi-disciplinary aerospace systems. A reusable la unch vehicle (RLV) is a highly complex aerospace system, which represents a cost viable option for access t o space missions due to its reusability aspect. Since an RLV system is composed of various subsystems which must work together to serve an over-arching p urpose or mission objective/constraint, a multidisciplinary approach for the analysis and des ign of RLV systems is required. Uncertainties are generally present in the models used in each discipline of an RLV multidisciplinary analysis framework. It is important to account for all of these uncertainties for accurate and reliabl e estimates of the RLV system performance. The primary purpose of this paper will be to demonstrate an efficient approach for uncertainty quantification in the multidiscipl inary analysis of a RLV, which has a mixture of aleatory (inherent) and epistemic input uncertainties. Uncertainty quantification will be performed on various output variables of interest f or an RLV system such as the weight and vehicle crossrange. For the propagation of mixed (aleatory-epist emic) uncertainty, Second-Order Probability Theory utilizing Point-Collocation Non-Intrusive Polynomia l Chaos (NIPC) 1 will be used. 2-3 In general, the NIPC methods, which are based on the spectral representa tion of uncertainty, are computationally more effic ient than traditional Monte Carlo methods for moderate number of uncertain variables and can give highly 1 Aerospace Engineer, 4020 Long Beach Blvd, Long Beach, CA, 90807, Member AIAA

Constrained predictor-corrector guidance via bank saturation avoidance for low L/D entry vehicles

An adaptive guidance system suitable for the terminal phase trajectory of a hypersonic strike weapon is optimized using reinforcement meta learning. The guidance system maps observations directly to commanded bank angle, angle of attack, and sideslip angle rates. Importantly, the observations are directly measurable from radar seeker outputs with minimal processing. The optimization framework implements a shaping reward that minimizes the line-of-sight rotation rate, with a terminal reward given if the agent satisfies path constraints and meets terminal accuracy and speed criteria. This paper shows that the guidance system can adapt to off-nominal flight conditions, including perturbation of aerodynamic coefficient parameters, actuator failure scenarios, sensor scale factor errors, and actuator lag, while satisfying heating rate, dynamic pressure, and load path constraints, as well as a minimum impact speed constraint. This paper demonstrates precision strike capability against a maneuvering ground target and the ability to divert to a new target, the latter being important to maximize strike effectiveness for a group of hypersonic strike weapons. Moreover, this paper demonstrates a threat evasion strategy against interceptors with limited midcourse correction capability, where the hypersonic strike weapon implements multiple diverts to alternate targets, with the last divert to the actual target.

Several minimum-fuel, aerodynamically controlled, orbital plane change problems are formulated and solved as optimal control problems. A gradient projection algorithm is used to iteratively modify both the control functions, angle of attack and bank angle, and two control parameters to obtain the optimal trajectory. The atmospheric flight profile is combined with two Keplerian (two-body vacuum flight) arcs so that a complete orbit-to-orbit analysis results. The vehicle used in this investigation is the Space Transportation System Shuttle Orbiter. The effects of heat load constraint level and plane change angle are analysed.