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Near-Optimal Guidance Method for Maximizing the Reachable Domain of Gliding Aircraft

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Near-Optimal Guidance Method for Maximizing the Reachable Domain of Gliding Aircraft Trans. Japan Soc. Aero. Space Sci. Vol. 49, No. 165, pp. 137–145, 2006 Near-Optimal Guidance Method for Maximizing the Reachable Domain of Gliding Aircraft By Takeshi TSUCHIYA Department of Aeronautics and Astronautics, The University of Tokyo, Tokyo, Japan (Received November 16th, 2005) This paper proposes a guidance method for gliding aircraft by using onboard computers to calculate a near-optimal trajectory in real-time, and thereby expanding the reachable domain. The results are applicable to advanced aircraft and future space transportation systems that require high safety. The calculation load of the optimal control problem that is used to maximize the reachable domain is too large for current computers to calculate in real-time. Thus the optimal control problem is divided into two problems: a gliding distance maximization problem in which the aircraft motion is limited to a vertical plane, and an optimal turning flight problem in a horizontal direction. First, the former problem is solved using a shooting method. It can be solved easily because its scale is smaller than that of the original problem, and because some of the features of the optimal solution are obtained in the first part of this paper. Next, in the latter problem, the optimal bank angle is computed from the solution of the former; this is an analytical computation, rather than an iterative computation. Finally, the reachable domain obtained from the proposed near-optimal guidance method is compared with that obtained from the original optimal control problem. Key Words: Guidance and Control, Aircraft, Optimal Control Nomenclature Ã: flight path angle of an aircraft gliding at maximum lift-to-drag ratio CD0: zero-lift drag coefficient v: adjoint variable for dimensionless speed CL : slope of lift coefficient : adjoint variable for flight path angle d0: zero-lift drag normalized by weight : air density g: gravity acceleration : bank angle H: Hamiltonian optimal: optimal bank angle h: altitude : dimensionless time h: dimensionless altitude final: dimensionless final time J: performance index present: dimensionless present time K: induced drag coefficient due to angle of attack : heading angle k: ratio of adjoint variables for flight path angle and target: target heading angle dimensionless speed m: mass n: load factor 1. Introduction noptimal: optimal load factor nÃ: load factor of an aircraft gliding at maximum lift- Theoretical, analytical, and numerical methods of optimal to-drag ratio control have been applied to the problem of maximizing the t: time reachable domain of aircraft gliding with no thrust.1) This is tfinal: final time an important problem for the flight of supersonic and hyper- V: speed sonic aircrafts that have been proposed for use in future VÃ: speed of an aircraft gliding at maximum lift-to- space transportation systems. Over the past decade, studies drag ratio of space transportation systems that achieve affordable v: dimensionless speed and reliable access to space have resulted in remarkable ad- x: down range vances in structures, materials, propulsion, and system de- x: dimensionless down range sign technologies. Many researchers have conducted a vari- y: cross range ety of concept studies on space transportation systems. A y: dimensionless cross range two-stage-to-orbit reusable launch vehicle, for example, is : angle of attack a typical concept, in which the first-stage vehicle flies backs Ã: angle of attack for maximum lift-to-drag ratio to the launch site after separation of the second-stage vehi- : flight path angle cle. Although it is usually proposed that the first-stage vehi- cle be equipped with engines for the return flight in addition Ó 2006 The Japan Society for Aeronautical and Space Sciences 138 Trans. Japan Soc. Aero. Space Sci. Vol. 49, No. 165 to those for acceleration flight, an extra increase in the Finally, we show the numerical results and compare them weight of the propulsion system is not preferable if the ve- with the exact solutions. hicle can return to the launch site by gliding.2–4) Therefore, in the areas of guidance and control, it is necessary to accu- 2. Optimal Control for Maximizing the Reachable Do- rately estimate the glide-back performance after separation main and to develop methods for guiding the vehicle to the launch site. In addition, guidance and control play an important role 2.1. Problem formulation in improving the safety and operability of these autonomous Assuming that an aircraft has three degrees-of-freedom vehicles. Under a nominal condition in which there are no and that the Earth is flat, the equations of the motion of a failures or damages, a conventional control system can di- gliding aircraft are given as follows. rect the vehicle along the trajectory calculated in prelaunch, dh but it cannot respond to abnormal conditions occurring dur- ¼ V sin ð1Þ ing flight. For instance, certain emergencies, such as a mis- dt dx sion abortion induced by unexpected engine failures, could ¼ V cos cos ð2Þ occur in the case of a reusable launch vehicle. In order to dt avoid the loss of the vehicle in such an emergency, it is nec- dy ¼ V cos sin ð3Þ essary to reconfigure the guidance method for glide-back dt flight to the launch site with an onboard computer in real- dV 1 m ¼ V2SðC þ K 2Þmg sin ð4Þ time. If the computer judges that returning to the launch site dt 2 D0 exceeds the ability of the vehicle, it needs to find alternative d 1 mV ¼ V2SC cos À mg cos ð5Þ sites that the vehicle can reach and then guide the vehicle to dt 2 L one of them. This paper proposes a guidance method appli- d 1 sin cable to that situation, and of course, to the nominal situa- mV ¼ V2SC ð6Þ dt 2 L cos tion as well; that is, it offers a method for calculating the maximum reachable domain from the present state of the These equations include six state variables and two control aircraft and a real-time optimal guidance by which the air- variables, the angle of attack and bank angle . The aero- craft can fly to a specified point within the domain. dynamic model of the aircraft is simple. Lift and drag are For the real-time solution of the optimal control problem obtained from lift slope, zero-lift drag, and induced drag. for maximizing the reachable domain of the gliding aircraft, The gravity acceleration and air density are kept constant. one of the most frequently applied methods is the singular The simple model easily leads to an optimal guidance meth- perturbation method.5–7) This method simplifies the optimal od, as shown in the following section. Before computing the control problem by classifying equations of motion of an optimal controls and deriving the optimal guidance method, aircraft according to the convergence speed of state vari- the motion equations are rendered dimensionless, because ables on a steady state. It results in a closed-loop law which this decreases the number of parameters in the equations produces approximate optimal solutions. The classification and makes them more suitable for numerical computation. depends on results in literature, and oscillations are ob- We make the variables dimensionless on the basis of the served in flight trajectories. On the other hand, research in- gliding flight condition in which the lift-to-drag ratio is stitutes around the world, including NASA, have planned maximized, because the gliding performance of the aircraft supersonic or hypersonic experimental aircraft for which is decided from that condition. The gliding condition is guidance laws have been proposed. The guidance methods obtained as follows. for aircraft gliding at high speed are based on the terminal rffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi C 2 KC area energy management (TAEM) adopted for the space à ¼ D0 ;à ¼tanÀ1 D0 ; shuttle.8–10) Though this simple method can respond to flight K CL sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ under nominal conditions, it cannot exploit the performance 2mg cos à Và ¼ of an aircraft under abnormal conditions. On the other hand, à SCL recent improvements in computer capability have yielded new methods for real-time optimal control.11) This paper These values give dimensionless variables of t, h, x, y, and V proposes a guidance method quite different from conven- as follows. tional ones. In this method, an onboard computer solves a gt gh gx gy V simplified optimal control problem in real-time to produce ¼ ; h ¼ ; x ¼ ; y ¼ ; v ¼ ð8Þ Ã Ã2 Ã2 Ã2 à near-optimal control values. V V V V V First, this paper defines an optimal control problem for Moreover, two parameters and a dimensionless control maximizing the reachable domain of a gliding aircraft and variable are defined: shows the exact optimal solutions. Second, the optimal con- Ã2 Ã2 à Ã2 V SCD0 V SCL V SCL trol problem is simplified for onboard and real-time compu- d ¼ ; nà ¼ ; n ¼ : 0 2mg 2mg 2mg tation on the basis of the features of the optimal solutions, and then a method for solving the problem is proposed. ð9Þ Nov. 2006 T. TSUCHIYA: Near-Optimal Guidance Method for Maximizing the Reachable Domain of Gliding Aircraft 139 As a result, the following dimensionless equations of motion of the equations of motion, the initial conditions, and severi- are defined. ty of the constraint conditions. This paper takes the policy that optimal control is solved without the constraints, and dh ¼ v sin ð10Þ if the optimal trajectory breaks the constraints, it is modified d so that it satisfies them instead. However, this paper does not dx ¼ v cos cos ð11Þ show these modifications, which remain for future study. d When we performed the optimization under the absence of dy ¼ v cos sin ð12Þ any constraints in flight, the aircraft tried to extend its flight d distance by flying underground.
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