AFRL-VA-WP-TP

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1 AFRL-VA-WP-TP NONLINEAR FEEDBACK CONTROL FOR RAPID, ON-LINE TRAJECTORY OPTIMIZATION OF REENTRY VEHICLES (PREPRINT) Dr. David B. Doman Kevin P. Bollino I. Michael Ross DECEMBER 2005 Approved or public release; distribution is unlimited. STINFO FINAL REPORT This work has been submitted to AIAA or publication in the AIAA Atmospheric Flight Mechanics Conerence proceedings. I this work is published, it is considered a work o the U.S. Government and is not subject to copyright in the United States. AIR VEHICLES DIRECTORATE AIR FORCE RESEARCH LABORATORY AIR FORCE MATERIEL COMMAND WRIGHT-PATTERSON AIR FORCE BASE, OH

2 NOTICE Using Government drawings, speciications, or other data included in this document or any purpose other than Government procurement does not in any way obligate the U.S. Government. The act that the Government ormulated or supplied the drawings, speciications, or other data does not license the holder or any other person or corporation; or convey any rights or permission to manuacture, use, or sell any patented invention that may relate to them. This report was cleared or public release by the Air Force Research Laboratory Wright Site (AFRL/WS) Public Aairs Oice (PAO) and is releasable to the National Technical Inormation Service (NTIS). It will be available to the general public, including oreign nationals. PAO Case Number: AFRL/WS , 9 Jan THIS TECHNICAL REPORT IS APPROVED FOR PUBLICATION. /s/ /s / David Doman Senior Aerospace Engineer Control Design and Analysis Branch Air Force Research Laboratory Air Vehicles Directorate Deborah S. Grismer Chie Control Design and Analysis Branch Air Force Research Laboratory Air Vehicles Directorate /s/ Brian W. Van Vliet Chie Control Sciences Division Air Force Research Laboratory Air Vehicles Directorate This report is published in the interest o scientiic and technical inormation exchange and its publication does not constitute the Government s approval or disapproval o its ideas or indings.

3 REPORT DOCUMENTATION PAGE Form Approved OMB No The public reporting burden or this collection o inormation is estimated to average 1 hour per response, including the time or reviewing instructions, searching existing data sources, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection o inormation. Send comments regarding this burden estimate or any other aspect o this collection o inormation, including suggestions or reducing this burden, to Department o Deense, Washington Headquarters Services, Directorate or Inormation Operations and Reports ( ), 1215 Jeerson Davis Highway, Suite 1204, Arlington, VA Respondents should be aware that notwithstanding any other provision o law, no person shall be subject to any penalty or ailing to comply with a collection o inormation i it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 1. REPORT DATE (DD-MM-YY) 2. REPORT TYPE 3. DATES COVERED (From - To) December 2005 Conerence Paper Preprint 12/19/ /19/ TITLE AND SUBTITLE NONLINEAR FEEDBACK CONTROL FOR RAPID, ON-LINE TRAJECTORY OPTIMIZATION OF REENTRY VEHICLES (PREPRINT) 6. AUTHOR(S) David B. Doman (AFRL/VACA) Kevin P. Bollino and I. Michael Ross (Naval Postgraduate School) 5a. CONTRACT NUMBER In-house 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER N/A 5d. PROJECT NUMBER N/A 5e. TASK NUMBER N/A 5. WORK UNIT NUMBER N/A 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER Control Design and Analysis Branch (AFRL/VACA) Control Sciences Division Air Vehicles Directorate Air Force Research Laboratory, Air Force Materiel Command Wright-Patterson AFB, OH Naval Postgraduate School Department o Mechanical and Astronautical Engineering Monterey, CA AFRL-VA-WP-TP SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING AGENCY ACRONYM(S) Air Vehicles Directorate Air Force Research Laboratory Air Force Materiel Command Wright-Patterson Air Force Base, OH DISTRIBUTION/AVAILABILITY STATEMENT Approved or public release; distribution is unlimited. AFRL/VACA 11. SPONSORING/MONITORING AGENCY REPORT NUMBER(S) AFRL-VA-WP-TP SUPPLEMENTARY NOTES Conerence paper preprint to be presented at the AIAA Atmospheric Flight Mechanics Conerence, 24 Aug 06, Keystone, CO. This work has been submitted to AIAA or publication in the AIAA Atmospheric Flight Mechanics Conerence proceedings. I this work is published, it is considered a work o the U.S. Government and is not subject to copyright in the United States. 14. ABSTRACT A new, on-line sampled-data eedback control algorithm or ast and accurate nonlinear optimal closed-loop control o a generic reentry vehicle in the presence o disturbances is presented. A direct Legendre pseudospectral method is used or the rapid re-computation o the open-loop optimal controls such that a successive optimal solution is ed back to the system to help manage uncertainties and external disturbances. The method is based on a receding-horizon control approach, but does not require and advanced knowledge o the computation time. Successul perormance o this approach, combined with a spectral algorithm, shows that rapid trajectory generation is not only achievable, but that it is ast enough to provide optimal closed-loop eedback with successive re-computations. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION 18. NUMBER 19a. NAME OF RESPONSIBLE PERSON (Monitor) OF ABSTRACT: OF PAGES SAR 28 a. REPORT Unclassiied b. ABSTRACT Unclassiied c. THIS PAGE Unclassiied Dr. David B. Doman 19b. TELEPHONE NUMBER (Include Area Code) (937) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std. Z39-18 i

4 Nonlinear Feedback Control or Rapid, On-line Trajectory Optimization Kevin P. Bollino and I. Michael Ross Naval Postgraduate School, Monterey, CA David D. Doman Air Force Research Laboratory, WPAFB, OH Abstract Rapid, on-line trajectory generation or reusable launch vehicles is gaining high interest with the explosive desire or more intelligent and autonomous systems. There are many aspects that make such achievement very appealing, such as being able to autonomously re-plan a mission on-the-ly in the case o abort guidance. Due to the constrained, highly nonlinear nature o the reentry problem, easible and computationally-eicient guidance and control methods have been a challenge since the beginning o the space age. This paper demonstrates the easibility o using a new, on-line sampled-data eedback control algorithm or ast and accurate nonlinear optimal closed-loop control o a generic reentry vehicle in the presence o disturbances. A direct Legendre psueodospectral method is used or the rapid re-computation o the open-loop optimal controls such that a successive optimal solution is ed back to the system to help manage uncertainties and external disturbances. The method is based on a receding-horizon control approach, but does not require any prior knowledge o the computation time. Successul perormance o this approach, combined with a spectral algorithm, shows that rapid trajectory generation is not only achievable, but that it is ast enough to provide optimal closed-loop eedback with successive re-computations. *This material is declared a work o the U.S. Government and is not subject to copyright protection in the United States. Ph.D. Candidate, Department o Mechanical and Astronautical Engineering, Guidance, Navigation, and Control Lab, Ph , kpbollin@nps.edu, AIAA Member. Proessor, Department o Mechanical and Astronautical Engineering, Ph , imross@nps.edu, AIAA Associate Fellow. Senior Aerospace Engineer, 2210 Eighth Street, Bldg. 146, Rm. 305, Ph , David.Doman@wpab.a.mil, Senior AIAA Member 1

5 1. Introduction With ongoing eorts to improve the saety, reliability, and cost o reusable launch vehicle (RLV) systems and operations along with the more recent ocus on a U.S Space Shuttle replacement ollowing its retirement around 2012, there is a new opportunity to implement advanced, yet simpler and more eective guidance and control methods than used in previous launch technologies. A key objective that is sought in this endeavor as well as almost all other sectors o technology is the requirement or more intelligent systems that can operate autonomously without as much human interaction. O course this has the underlying stipulation that they operate saer, more reliable, and more eiciently than beore. To accomplish this objective or the case o a RLV, the guidance and control during ascent, reentry, or landing needs to autonomously happen on-line and in real-time. The RLV intelligent control (i.e. brains ) needs to be capable o adapting to rapidly changing circumstances, handle large external disturbances, large parameter uncertainties, re-generate trajectories (i.e. re-plan how to get to a designated or alternate landing site), reconigure its controls in the advent o an unoreseen control ailure and then o course to igure out how to allocate the controls. Originally, the challenge o solving constrained, highly nonlinear dynamics and the limitations o computational speed at the time, demanded linearized, approximate equations. Oten, the control equations were driven with tediously derived gain-scheduled schemes such as the drag-based tracking techniques used in Space Shuttle entry guidance where a series o drag reerence segments required individual controller gain design [1]. With the realization o advances in computational power and numerical algorithms, recent research eorts or solving the reentry problem have ocused on real-time, on-line trajectory generation, guidance adaptation, and control reconiguration. Oppenheimer, Bolender, and Doman at the Air Force Research Lab s oice o Advanced Guidance and Control have done extensive work on reconigurable control in the orm o an optimal control allocation algorithm or such vehicles as the X-40, X-37, and X-33 [2]-[4]. Reconiguration capabilities or RLV guidance and control systems has also been rigorously pursued by others mainly in support o two program initiatives to support such work: Marshall Space Flight Center s (MSFC) Advanced Guidance and Control (AG&C) Program and the Air Force Research Lab s (AFRL) Integrated Adaptive Guidance and Control (IAG&C) Program. More on these programs can be ound in Re. [5] and [6]. Since control reconiguration and guidance adaptation will not always be enough to recover a vehicle ater control ailures, Schierman et al, has taken this one step urther by developing an architecture that allows or reshaping o the outer-loop guidance trajectory commands and retargeting an alternate landing site should the nominal or original mission be aborted [6,7]. Likewise, Shaer has integrated this need o trajectory reshaping and retargeting with the reconigurable control work o Oppenheimer et al., to demonstrate relatively ast computations o optimal trajectories under trim deicient path-constraints [8]. Moving away rom the successul, but extremely time-consuming gain-scheduling approach or light-control designs, numerous eorts have employed a orm o eedback linearization, oten reerred to as dynamic inversion. Through the use o nonlinear eedback, dynamic inversion essentially cancels the nonlinearities o the system to orm an augmented linear system that can 2

6 then be controlled using more traditional linear control techniques. Unortunately, this linearization technique requires exact knowledge o the system model, such as speciic plant parameters. For air vehicles where large uncertainties such as aerodynamics orces and moments are inevitable, this method could easily result in unstable, unrecoverable light conditions. For this reason, other nonlinear eedback methods are being investigated. Some methods include various combinations o adaptive control, backstepping, and even more robust versions o dynamic inversion. One way to address the problem is by using an optimal trajectory generator to solve or a reerence input trajectory o-line, and then use other inner-loop control means to track the desired trajectory. Schierman et al. has recently demonstrated the use o an outer-loop optimal control scheme to generate a reerence trajectory that is tracked by an LQR inner-loop controller [9]. In a similar ashion, Carson uses an outer-loop guidance scheme with LQR inner-loop eedback to manage disturbances and uncertainty [10]. Carson s work employs a method presented by Milam [11], whereby real-time trajectory generation is made possible by inding the trajectory curves in a lower dimensional space, parameterizing the curves with B-splines, and then using Sequential Quadratic Programming (SQP) to ind the spline coeicients that satisy the optimization objectives and constraints. With online approaches becoming more capable o solving highly nonlinear systems, there is really no need or o-line designs o closed-loop eedback laws. In this paper, an intelligent trajectory guidance system is designed or a generic reentry vehicle by solving the optimal control problem online without an inner-loop tracking controller. Once provided the initial and inal conditions or the reentry vehicle, the proposed algorithm computes the nonlinear optimal control. As in most control designs, eedback is required to manage uncertainties. For the reentry problem, parameter uncertainty and external disturbances such as wind gusts (or shear) cause the vehicle to deviate rom its nominal trajectory that may have been the originally predicted/computed/planned optimal trajectory. The proposed guidance and control method, adapted rom the promising work o Pooya, Fleming, and Ross on a spacecrat optimal-time slew maneuver [12], uses a sampled-data eedback law to provide an optimal trajectory in the presence o uncertainty and disturbances. Rather than track a pre-computed solution, the scheme re-solves the optimal control problem and updates the control command as soon as a new solution is calculated. This method is also similar to Schierman s Optimal-Path-to-Go (OPTG) Trajectory Reshaping Strategy presented in his early 2001 work [13]. The work presented herein diers rom Schierman s in that it is primarily used or eedback without using an inner-loop tracking controller. The original concept dates back to the early 1990 s, when Pesch discussed o-line and on-line methods or aerospace applications [14]. He introduced the combination o what he called the Neighboring Optimal Feedback Guidance with a Repeated Correction Guidance Scheme to essentially perorm the same objective as the method presented in Pooya s work [12]. The main dierence is that his method relied on a linearized indirect multiple shooting technique that requires a more complicated implementation at increased computational cost. Unlike Pesch s repeated correction method, the method presented in this paper guarantees that the constraints are satisied beore eeding back the control signal provided there is a easible solution. Also, it was claimed that the theoretical and numerical basis or the online approach is not mature enough or general optimal control problems. Although this paper 3

7 does not provide the mathematical justiication that Pesch desires, it is yet another steppingstone to making on-line trajectory optimization more practical. To solve the optimal control problem, a spectral algorithm [15-18] known as the Legendre Pseudospectral Method is employed by use o a MATLAB-based sotware package called DIDO [19]. This direct method discretizes the problem and approximates the states, co-states and control variables by use o Lagrange interpolating polynomials where the unknown coeicient values coincide with the Legendre-Gauss-Lobatto (LGL) node points. Ater this approximation step, an NLP solver (SNOPT) solves a sequence o inite-dimensional optimization problems that capture the nonlinearities o the system in the orm o an optimal control. For an extensive description o this method and its use or real-time optimal control, see reerences [15]-[21]. With a lot o interest in real-time trajectory generation, this paper intends to demonstrate that by using a ast psuedospectral optimization method, not only are rapid optimal trajectories achievable, but through successive trajectory generations as a nonlinear sampled-data eedback law, optimal guidance and control is attained in the presence o parameter uncertainty and external disturbances. 2. Problem Deinition & Formulation 2.1 Reentry Model (Kinematics and Dynamics) Using a reduced-order model is all that is required to clearly demonstrate that the nonlinear eedback technique is an eective method towards reentry trajectory optimization. Thereore, or this paper, the ull 6-DOF equations o motion are simpliied and decoupled to create a reduced-order model. The model assumes a point-mass-model over a lat, non-rotating earth such that the positional and translational equations o motion in a Cartesian local horizontal coordinate system become x = Vcosγ cosξ y = Vcosγ sinξ z = Vsin γ DV {, z} V = gsin γ m LVz {, } gcosγ γ = mv V LVz {, }sin σ ξ = mv cosγ (1) Where x (down-range), y (cross-range), and z (altitude) are the vehicle s position with respect to the ixed-earth reerence rame, V is the velocity magnitude (i.e. total airspeed), γ is the light-path-angle (FPA), ξ is the heading angle, α is the angle-o-attack (AoA), and σ is the angle-o-bank (AoB). 4

8 The lit and drag orces are represented as L and D, respectively, given by 1 1 L = ρ{} zvc { α} S = ( ρ e 2 2 ) V C { α} S 1 1 D= ρ{} zvc { α} S = ( ρ e 2 2 ) V C { α} S 2 z/ z0 2 L re 0 L re 2 z/ z0 2 D re 0 D re (2) (3) where Sre is the reerence area given as 1600 t 2 1 2, the term ρ () zv is the dynamic pressure, q, 2 z/ z0 and the term ρ0e is the commonly accepted two-parameter model or atmospheric density, ρ, as a unction o altitude. This density approximation, where ρ 0 is the reerence density taken to be kg/m 3 ( slugs/t 3 ), z is the current altitude, and z 0 is the atmospheric scale height (i.e. reerence height) taken to be 6700 m ( t), is good or closed-orm solutions and its well rom 5 to 40 km in altitude [22]. For simplicity, the coeicients o lit (C L ) and drag (C D ) are modeled as C { α} = C + C L L0 Lα α C { α} = C + C α + C α D D0 Dα Dα2 Where the coeicients are 2 C = L , C = , , , 0 L C = D C = and α 0 D C α D α 2 α in degrees [23]. 2 (4) (5) 3 = with To ully capture the state at any instant, the state vector or this reduced-order model is given as x= [ xyzvγξ] T R 6 (6) Without a thrust orce or the reentry gliding problem, the only controllable parameters or this reduced-order model are the lit and drag orces. Typically, or symmetric light (i.e. coordinated turns where β = 0 ), the lit and drag coeicients can be determined by the vehicle s angle-o-attack and Mach number, a unction o velocity and speed-o-sound at a given altitude. However, it is the physical modulation o angle-o-attack and bank angle that controls the vehicle s translational motion through x-y-z space. Thereore, a common control vector or the reentry problem is u = [ α σ] T R 2 (7) O course with these control variables deining the control vector, it is assumed that there are no command delays (i.e. lags), hence; this type o control is sometimes reerred to as inertialess control. To help compensate or this and add a little realism to the problem, as explained in reerences [8] and [23], a new control vector is ormed such that rate limits can be modeled as well. This is accomplished by using virtual controls [23] mathematically expressed as 5

9 u = [ u u ] T R α Now, since the algorithm uses these, the original state vector must include the physical controls, α and σ, to orm a new state vector: σ 2 (8) x= [ xyzvγξασ] T R 8 (9) Remark: For a real vehicle, or in a 6-DOF simulation, it is the control surace delections that create body moments and orces to augment the wind-relative angle-o-attack, bank angle, and sideslip angle. 2.2 General Problem Formulation As with any dynamical optimization problem, the cost unction (a.k.a objective unction), governing equations o motions, path constraints, boundary limits on initial/inal conditions, and any constraints (on states and/or controls) must be deined. As such, the general optimal control problem or trajectory generation is ully posed in the ollowing manner: (,,, ) = ( ( ), ( ),, ) ( ( ), ( ), ) min u ( ) ( ) J x τ u τ τ τ E x τ x t τ τ F x τ u τ τ dτ subject to x = ( xuτ,, ) τ τ 0 l (,, τ ) h h x u h u (10) ( ( τ ), 0 ( τ ), τ, 0 τ ) e e x x e l u l ( τ ) x x x l ( τ ) u u u u u { } The goal is to ind a state-control unction pair, x( ), u( ), or sometimes time, τ, that minimizes the perormance index represented by the Bolza orm, J ( ), consisting o either a Mayer term, E ( ), a Lagrange term, F ( ), or both as stated above. 6

10 Summarizing the previous reentry equations, the speciic optimal control ormulation or this RV x, y, z, velocity magnitude problem is stated as ollows: Given an initial position vector ([ 0 0 0] ) ( V 0 ), FPA ( γ 0 ), heading angle ( ξ 0 ), AoA ( α 0 ), and AoB ( σ 0 ) ( u, u ) α σ that maximizes the horizontal downrange distance ( ), ind the control history x under various constraints. T deg x = x y z V u = u u U = u u u u s deg s 1 [,,,, γ, ξ, α, σ ] [ α, σ ] { α, σ R : 40 α, σ 40 } Min J[ x( t), u( t), t, t ] = x st.. x = V cosγ cosξ y = V cosγ sin ξ z = V sin γ D{ V, z} V = g sin γ m L{ V, z} g cos γ γ = mv V L{ V, z}sin σ ξ = mv cos γ α = u, ( t, x, y, z σ = z, u α σ, V, γ, ξ, α, σ ) = (0,0,0,125000t,5714t/s, 1.3deg,0,15deg,0) ( z, V ) = (500 t,335t/s) 25 z 8.33 (t/s) (0,,0,0, 90deg, 90deg, 10deg, 90deg) x (,,,,90deg,90deg,50deg,90deg) 2 2 ( 2.5g,0,0) ( n q, Q ) (2.5g,700lb/t,60BTU/t -s) The overall goal is to maximize the downrange distance, so this is implemented by minimizing the negative o the objective variable, x. Also, this particular ormulation has an inequality control constraint limiting the AoA and AoB rates. Limits on the states are represented by the inequality box constraints. The ininity signs were not actually modeled as such, but made very large numbers in order to shrink the searchable region. The end-point conditions were deined or inal altitude z, velocity V, and sink rate ż. The path constraints o normal load n z, dynamic pressure q, and heating rate Q, although not modeled in this proo-o-concept, are represented as inequality box constraints. 7

11 2.4 Solving the Optimal Control Problem To demonstrate optimality, irst a theoretical analysis is perormed based on optimal control theory. To demonstrate the necessary conditions needed or optimality the irst step requires the ormulation o the Hamiltonian: T H( λ, xut,, ) = Fxut (,, ) + λ (,,) xut (12) where F() is the Lagrange cost and () is the vector ield or the right hand side o the dierential equations o motion. Thereore, λx λ y λ z z T λv 8 3 λ = ; x V λ R = R (13) γ z λ ξ λ α λ σ (14) z 1 z0 2 H() = λx[ V cosγ cosξ] + λy [ V cosγ sinξ] + λz[ Vsin γ] + λv ρ0e SreV CD{ α} gsin γ 2m z 1 z z { }sin 0 2 cos re L z γ ρ e S V C α σ 2 γ 0e SreV CL{ } g ψ αuα σuσ + λ ρ α + λ + λ + λ 2m V mv cosγ dh x = RHS o dynamicequations d λ (15) The Hamiltonian Minimization Condition (HMC) is based on Pontryagin s Minimum Principle such that the optimal control must minimize the Hamiltonian with respect to control. For this problem the control is subject to an inequality constraint so the Karush-Kuhn-Tucker (KKT) Theorem is applied by taking the Lagrangian o the Hamiltonian: T (16) H(..., u, µ ) = H() + µ h where µ is a KKT multiplier and h is the control constraint vector. The appropriate necessary condition is: H H h = + µ = 0 u u u T (17) 8

12 and substituting (14) and the controls into (17), the HMC becomes OMIT FOR NOW! (18) Also, the multiplier-constraint pair must satisy the ollowing KKT Complimentarity Conditions (CC): µ 0 i uασ, = 40 = 0 i 40 < u < 40 0 i u 40 ασ, = ασ, ασ, 40 i µ 0 λασ, 0 u = 40 i µ 0 λασ, 0 (19) To determine the inal value o the Hamiltonian, the Endpoint Lagrangian, given as, T E( υ, x, t ): = Ex (, t ) + υ ex (, t ) x + υ ( z z ) + υ ( V V ) z V (20) is substituted into the Hamiltonian Value Condition (HVC): E Ht ( ) + = 0 Ht ( ) = 0 t This indicates that the inal value or the Hamiltonian should be approximately zero or this problem. Also, applying the Terminal Transversality Conditions (TTC), gives some indication o the inal value o the dual variables that can be used later to conirm numerical results. 1 0 υ z (22) E υv λ( t ) = x Likewise, the Hamiltonian Evolution Equation (HEE) is used to indicate the nature o the Hamiltonian with respect to time such that: (23) H dh = 0 = 0 t dt Here, the Hamiltonian is constant with respect to time. Combining HEE (23) with HVC (21), the Hamiltonian should be 0 or all time. (21) 9

13 The above analysis is used throughout this study to veriy that the numerical results satisy the necessary KKT conditions or optimality. For example, the co-state values at the nodes, provided by the Covector Mapping Theorem in the Legendre Pseudospectral Method, can be substituted into the necessary conditions above to evaluate optimality. 3. Sampled-Data Feedback Control The architecture overview or the o-line and on-line trajectory generation is shown in Fig.1 with the basic premise o the sampled-data closed-loop RV guidance and control portrayed in Figures 2 & 3. Mission Objective (cost & constraints) Disturbances DIDO Trajectory Generation (o-line, open-loop) u (), t x () t * * 0 0 DIDO Trajectory Generation (on-line, closed-loop) * u i 0 ( t t ) Vehicle (nonlinear / uncertain) i x Mission Objective (cost & constraints) i x = ( x,... x ), t i DIDO t + t t i * u i 0 ( t t ) Propagation (RK4) i x = ( x... x... x ) i i DIDO t t + t t Measurement Noise Fig. 1: O-Line and On-Line Trajectory Generation Control Architecture The purpose o the architecture shown in Fig.1 is to illustrate the overall systems use o an oline optimal trajectory solution, successive on-line optimal trajectory solutions, and a simple Runge-Kutta propagation scheme to integrate the equations o motion in order to determine the current vehicle state vector. The lowchart or the actual control algorithm implemented in this work is shown in Fig.2. Start t = t ; x = x ; x = x t i i 0 0 i = cost ; m = m * 0 O-line open-loop o DIDO with.: t 0,x,x 0 End Yes x = x No RK 4 i t t i m=m * * * DIDo ( m i.. m i.. m ) + t Dido t t t i t t i * * m = ( m... m ) i x x ( x.. x.. x ) i i + t Dido t t t i Dido t + t i t = t + cost t x i = ( x,... x ) t = t + t i Dido t + t i i Dido t Fig 2: Closed-loop Control Algorithm [12] 10

14 The initial guidance command (i.e. the reerence trajectory), is generated o-line by computing the open-loop optimal control using DIDO sotware to provide the initial control signal, ( α 0, φ 0) and an estimate o the inal downrange distance, x. The closed-loop optimal control is then solved on-line by using the o-line solution as the start-up values to a new open-loop optimal control problem with sampled data ed back rom the previous DIDO run. As the next DIDO run is in progress, the equations o motion are propagated in real-time rom the previously measured state vector as its initial condition using the last interpolated control vector, or i not available, the appropriate portion o the nominal control. Upon DIDO completion, the vehicle s current state is provided rom the propagation routine and a new optimal trajectory is computed using the same constraints as the o-line problem. The successive re-optimization is illustrated in Fig.3 to help clariy what is happening. Not shown are the new complete optimal open-loop trajectories beginning rom each successive initial condition (IC). NOTE: Propagated (RK4) segments are exaggerated or illustration purposes only. IC 2 (t 1= t 0+t DIDO) IC 1 (t 0) RK4 1 RK4 2 IC 3 (t 2= t 1+t DIDO) e-error-balls o-line, open-loop re. trajectory Fig. 3: Successive Re-Optimization Concept To model external disturbances in the orm o wind, wind velocity components W, W, W were augmented to the kinematical and dynamical equations o motion or the ( x y z) closed-loop implementation to give the ollowing equations x = Vcosγ cosξ + W y = Vcosγ sinξ + W z = Vsin γ + W DV {, z} V = gsinγ W cosγ + W sin γ m LV {, z} gcosγ 1 γ = + mv V V γ γ LVz {, }sin σ ξ = mv cosγ z x y ( x z ) ( W xsin W zcos ) (24) 11

15 At lower altitudes, these wind velocity components typically orm a wind shear that is commonly represented by a parabolic gradient as a unction o altitude. For this proo-o-concept study, the external disturbances are kept arbitrary; hence, details o the wind gradient are not included in the model, but rather uniorm, constant components are used in the equations above such that the W terms are zero. 4. Results and Discussions The maximum downrange trajectory o a generic RV is studied to illustrate the perormance o the above on-line closed-loop control scheme. 4.1 Open-Loop Control Solution For the reentry trajectory, the open-loop maximum-downrange solution is shown in Fig.4. It is seen that the optimal control given by the αφ-rate, modulation drives the RV to a maximum downrange o approximately 1,129,593 t (185 nm or 344 km) over a total light time o 567 seconds and within the allowable tolerance o 6.0% or the desired end-point conditions. 12

16 Fig. 4: Optimal Open-Loop States Fig. 5: Optimal Open-Loop Virtual Controls For numerical optimization methods, it is always advisable to veriy solution easibility and that the necessary optimality conditions are indeed satisied Demonstration o Computational FEASIBILITY via Propagation Comparison The easibility o the computational solution can be validated by comparing the DIDO results to the propagated states via a separate ODE Runge-Kutta propagator. By interpolating the values o the control unction, u(t i ), at the LGL points and then integrating the dierential dynamical equations, x = ( x, u( t), t), via MATLAB s ode45 solver, a comparison o error norms can be made with the DIDO trajectory results. Results showed that the open-loop system response does satisy the end-point conditions within an acceptable error range. There is some norm error 13

17 between the DIDO states and the ode45-propagated states; however, this doesn t necessarily invalidate the easibility o the computational solution. There is an obvious tradeo between accuracy and computational execution time. According to Re. [24], solutions using greater than 60 nodes (or a similar problem) will provide acceptable results, but at the cost o signiicantly more execution time. One way to improve the accuracy is to increase the number o nodes or to relax the endpoint constraint. Although this model only ixes inal altitude, inal velocity, and inal sink rate, a real vehicle must arrive to the landing site in a controllable attitude and reasonable approach speed. As such, constraining the vehicle s inal attitude could be included in the inal conditions. Since the ocus o this paper is to demonstrate successul implementation o the optimal closed-loop eedback, the astest execution time possible is sought within acceptable error tolerances. For more on DIDO perormance or this type o problem, reer to Re.[24] Demonstration o Computational OPTIMALITY via Bellman s Principle Comparing the numerical results to the above theoretical analysis o the HVC, HEE, HMC, and TTC conditions validates the optimality o the computational solution. The HVC stated in the theoretical analysis indicates that the Hamiltonian should be 0 at the inal time (i.e. H(t )=0). From the HEE, it can be shown that the Hamiltonian is constant with respect to time. Combining these two conditions, the Hamiltonian should be 0 or all time, clearly evident in Fig.6. Likewise, the Lagrange multipliers or the dynamics at the inal time match the theoretical TTC results (not shown). Fig. 6: Hamiltonian rom Open-Loop Solution Another test to conirm computational optimality is to apply Bellman s Principle o Optimality. This principle essentially states that by using any point on the original optimal trajectory as an initial condition to a new problem (with all other problem ormulation parameters the same) should result in the same optimal trajectory with the same or better cost. Although it is not shown here, this method was used to validate optimality. 14

18 Although the previous open-loop solution results in a easible trajectory and desired end-point conditions that satisy the necessary conditions or optimality, it does not account or any external disturbances as would be the case in real applications. Well known is that without some orm o eedback, disturbances, in the orm o parameter uncertainty, sensor measurement errors (e.g. noise), or unknown/unpredictable external orces/moments, can result in signiicantly degraded system perormance. To illustrate the eects o external disturbances or this reentry problem, a simulated wind gust was applied over a period o 20 sec beginning at 200 sec into the light. Sensor measurement errors and parameter uncertainty can be simulated by assuming the role o the errors rom the numerical propagation. The eects o the wind on the open-loop (OL) trajectory are seen in Fig.7. Fig.8 shows the closed-loop (CL) results or the variables o interest. Fig. 7: Open-Loop (OL) System Response with Nominal Wind Gust 15

19 Fig. 8: System Response with Nominal Wind Gust To urther illustrate how the closed-loop system can recover rom the eects o an external disturbance, Fig.10 compares the results o an exaggerated 600 t/sec (~355 knot) downburst o wind, oten reerred to as a microburst, applied over a 60 second period. It is evident rom igures 11 and 12 that the open-loop system cannot recover rom the microburst. The vehicle is slammed into the ground or the open-loop trajectory (-5207 t), where as the closed-loop trajectory corrects or the microburst and is able to achieve a inal altitude o 490 t, within 10 t o the desired inal altitude and within the allowable end-point tolerance. 16

20 Fig. 9: System Controls or Nominal Wind Gust Fig. 10: System Response to Exaggerated Wind Gust 17

21 Wind Region o applied external disturbance Fig. 11: Eects on Altitude with Exaggerated Wind Gust Fig. 12: Eects on Velocity with Exaggerated Wind Gust 18

22 Success o this eedback method depends on relatively ast re-computation time. Computation o the irst o-line, open-loop optimal control trajectory takes 11 seconds and the subsequent openloop optimal control updates are computed within 0.60 to 11.5 seconds. The eedback computation times are shown over the entire trajectory in Fig.13. These trajectories were generated on a Dell Optiplex Desktop with a Pentium M, 3.40 Ghz processor, and 1.0 GB o RAM. Fig. 13: Feedback Computation Times or System with Nominal Disturbance For this problem, there were no error-balls ( ε ) around the initial conditions or each successive update problem as Pooya used in the time-optimal satellite slew maneuver problem [12]. This means that it is more likely that the re-optimized closed-loop trajectories will not ollow the original, o-line open-loop response. According to Bellman s Principle, without the presence o disturbances, the re-optimized closed-loop trajectory should ollow the original optimal openloop trajectory. By allowing an epsilon-error ball around the initial conditions, there should be less distinction between the old and new trajectories; hence, the new trajectory is literally a neighboring extremal. When disturbances are present, the optimal trajectory changes with each re-optimization step and are eedback to the system. This results in a dierent control trajectory as seen in Fig.14. This unique control history applied to the system with external disturbances is capable o countering their eects and driving the vehicle to the desired end point conditions, primarily altitude and velocity in this problem. A potential drawback or this approach has to do with the act that the optimal controls, at each successive re-optimization step, can essentially be reset to some drastically dierent value within the control constraint bounds as opposed to the states that must start at the previous position vector. This jump in the controls is illustrated in Fig.14 and can be resolved by placing a similar tolerance or ε -error-ball on the initial control vector or each re-optimization problem, such as u(0) k + 1 u k < ε, where ε is some predeined tolerance. 19

23 Fig. 14: Example o Successive Re-Optimization with Control Jumps Another potential problem is that o convergence. I or some reason, one o the re-optimization steps does not converge, it can crash the entire closed-loop process; however, additional logic can be built into the eedback algorithm to prevent the convergence issue rom crashing the system. Such logic, as previously mentioned, may include using the previous good solution or some inite time and then attempting a new re-optimization. In the event o requent or repetitive non-convergence issues, the only option may be to revert back to using the open-loop response or the remaining trajectory. O course this would only be practical i the remaining time-to-go is relatively short and there are minimum uncertainties and/or disturbances during this time. I not, at least a hard crash is prevented and the possibility o a easible trajectory still exists. 5. Conclusions In this paper, a sampled-data eedback control law, constructed rom successive optimal openloop solutions, was used to generate online, optimal trajectories or the reentry o a generic reusable launch vehicle. It was shown that rapid re-computation o the open-loop optimal control can eectively be used or online trajectory generation in the presence o various disturbances and uncertainties. A signiicant advantage o this control scheme is that it does not require any advance knowledge o eedback computation times; hence, no prediction method is 20

24 required. As demonstrated, solutions to diicult problems that may require eedback such as online trajectory generation in the ace o disturbances are now possible. Acknowledgment The authors would like to extend their appreciation to Dr. Pooya Sekhavat, visiting post-doc research assistant at the Guidance, Navigation, and Control (GNC) Laboratory, Naval Postgraduate School, or his help with implementing and debugging the sampled-data eedback algorithm. His eorts are greatly appreciated. Reerences 1. Harpold, J C., and Graves, C.A., Shuttle Entry Guidance, The Journal o Astronuatical Sciences, Vol. 27, No. 3, pp , Jul-Sep Oppenheimer, M.W., and Doman, D.B., Reconigurable Control Design or the X-40A with In-Flight Simulation Results, Proceedings o the 2005 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Oppenheimer, M.W., and Doman, D.B., Reconigurable Inner Loop Control o a Space Maneuvering Vehicle, Proceedings o the 2003 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Bolender, M.A., and Doman, D.B., Nonlinear Control Allocation Using Piecewise Linear Functions, Journal o Guidance, Control, and Dynamics, Vol.27, No. 6, pp , Doman, D.B., and Oppenheimer, M.W., Integrated Adaptive Guidance and Control or Space Access Vehicles, Volume 1: Reconigurable Control Law or X-40A Approach and Landing, AFRL IAG&C Technical Report, Wright-Patterson AFB, OH, Schierman, J.D., Hull, J.R., and Ward, D.G., On-Line Trajectory Command Reshaping or Reusable Launch Vehicles, Proceedings o the 2003 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Schierman, J.D., Hull, J.R., and Ward, D.G., Adaptive Guidance with Trajectory Reshaping or Reusable Launch Vehicles, Proceedings o the 2002 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Shaer, P.J., Ross, I.M., Oppenheimer, M.W., Doman, D.B., Optimal Trajectory Reconiguration and Retargeting or a Reusable Launch Vehicle, Proceedings o the 2005 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Schierman, J.D., Hull, J.R., In-Flight Entry Trajectory Optimization or Reusable Launch Vehicles, Proceedings o the 2005 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Carson, J.M., Epstein, M.S., MacMynowski, D.G., and Murray, R.M., Optimal Nonlinear Guidance With Inner-Loop Feedback or Hypersonic Re -Entry, Paper submitted to American Control Conerence, Milam, M.B., Mushambi, K., and Murray, R.M., A New Computational Approach to Real-Time Trajectory Generation or Constrained Mechanical Systems, Conerence on Decision and Control, Pooya, S., Fleming, A., and Ross, I.M., Time -Optimal Nonlinear Feedback Control or the NPSAT 1 Spacecrat, Proceedings o the 2005 IEEE/ASME, Jul Schierman, J.D., Ward, D.G., Monaco, J.F., and Hull, J.R., Reconigurable Guidance Approach or Reusable Launch Vehicles, Proceedings o the 2001 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Pesch, H.J., O-Line and On-Line Computation o Optimal Trajectories in the Aerospace Field, Prepared or the 12 th Course in Applied Mathematics in the Aerospace Field o the International School o Mathematics, Ettore Majorana Centre or Scientiic Culture, Sicily, Sept Ross, I.M., and Fahroo, F., Legendre Pseudospectral Approximations o Optimal Control Problems, Lecture Notes in Control and Inormation Sciences, Vol. 295, Springer-Verlag, new York, pp ,

25 16. Ross, I.M., and Fahroo, F., Issues in the Real-Time Computation o Optimal Control, Mathematical and Computer Modeling, An International Journal, Vol. 40, Pergamon Publication, to appear. 17. Ross, I.M., and Fahroo, F., Pseudospectral Knotting Methods or Solving Optimal Control Problems, Journal o Guidance, Control, and Dynamics, Vol. 27, No. 3, pp , Ross, I.M., Fahroo, F., and Gong, Q., A Spectral Algorithm or Pseudospectral Methods in Optimal Control, Proceedings o the 6 th IASTED International Conerence on Intelligent Systems and Control, Honolulu, HI, Ross, I.M., and Fahroo, F., User s Manual or DIDO 2002: A MATLAB Application Package or Dynamic Optimization, NPS Technical Report AA , Department o Aeronautics and Astronautics, Naval Postgraduate School, Monterey, CA, June Josselyn, S. and Ross, I.M., Rapid Veriication Method or the Trajectory Optimization o Reentry Vehicles, Journal o Guidance, Control, and Dynamics, Vol.26, No. 3, pp , Strizzi, J., Ross, I.M., and Fahroo, F., Towards Real-Time Computation o Optimal Controls or Nonlinear Systems, Proceedings o the 2002 AIAA Guidance, Navigation, and Control Conerence, AIAA Paper No , Aug Regan, F.J., and Anandakrishnan, S.M., Dynamics o Atmospheric Re-Entry, AIAA Education Series, Washington D.C., Betts, J.T., Practical Methods or Optimal Control Using Nonlinear Programming, Society or Industrial and and Applied Mechanics (SIAM), Philadelphia, PA Shaer, Patrick J., Optimal Trajectory Reconiguration and Retargeting or the X-33 Reusable Launch Vehicle, Thesis, Naval Postgraduate School, Monterey, CA,