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 : 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 n : 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 V : 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 : 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 2Þ mg 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 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 ¼ D0 ; ¼ tan 1 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 V ¼ of an aircraft under abnormal conditions. On the other hand, 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 2 2 2 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- 2 2 2 V SCD0 V SCL V SCL trol problem is simplified for onboard and real-time compu- d ¼ ; n 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. To get rid of such solutions, dv 1 n2 we present a single inequality constraint, ¼ d v2 þ sin ð13Þ d 0 v2 n 2 hðtÞ 0 ð0 < t < tfinalÞ: ð20Þ d n cos cos ¼ ð14Þ With a direct collocation method, the optimal control prob- d v lem is converted into a nonlinear programming problem, d n sin ¼ ð15Þ and we solve the problem using a sequential quadratic pro- d v cos gramming method.12) To discretize the state and control var- These equations include only two parameters, d0 and n , that iables, they are divided into 100 nodes from the initial time must be determined. The control variables of these equa- to the final time. As described later, the optimal trajectory tions are the bank angle and a load factor n. consists of a transitional trajectory, in which the variables As an optimal control problem for maximizing the do- change greatly, and a steady flight trajectory. Allocating main in which the aircraft can glide, a performance index many nodes in the transitional trajectory improves calcula- is defined. The heading angle at the initial time when the air- tion accuracy. craft starts gliding is fixed at 0. We specify a target heading 2.2. Optimal control solutions angle target at the final time tfinal when the aircraft reaches Figures 1–4 show optimal solutions resulting from some the ground. To compute the optimal control by which the values of the target heading angle target. The gliding condi- aircraft flies in the direction of the heading angle target as tion for the maximum lift-to-drag ratio of the aircraft is far as possible, the following maximized performance index ¼ 3:684 deg; ¼ 9:755 deg; V ¼ 127:8 m/sec: is defined. ð21Þ J ¼ x ð Þ cos þ y ð Þ sin ð16Þ final target final target After the angles of attack and bank angles drastically change The maximum reachable domain results from substituting for the first approximately 100 seconds, they converge on the target heading angle target into various values. the state of Eq. (21); the aircraft finally flies at the maximum For numerical simulations, we set the parameters g ¼ lift-to-drag ratio. These are reasonable flights for maximiz- 9:80665 m/sec2 and ¼ 1:225 kg/m3 for Eqs. (1)–(6). ing the flight distance. Just before getting to the ground, the The specifications of the aircraft are m ¼ 19 Mg and S ¼ aircraft increases its angle of attack to extend the flight dis- 2 83 m , and its aerodynamic coefficients are CL ¼ 6:0 tance. It flies at a low altitude, and falls when its velocity is 2 2 3 10 , CD0 ¼ 1:9 10 , and K ¼ 1:4 10 . The initial too small to provide the lift necessary to support its weight. conditions at t ¼ 0 are This motion is not realistic, and so the following section xð0Þ¼yð0Þ¼0; hð0Þ¼3000 m ð17Þ does not consider the controls required to produce it. It is important to bear in mind that, within the context of this Vð0Þ¼500 m/sec; ð0Þ¼5 deg; ð0Þ¼0 deg: ð18Þ study, the optimal flight consists of steady gliding flight at The terminal condition at a final time tfinal, which is also an a maximum lift-to-drag ratio, preceded by a transitional optimized variable, is flight. The transitional flight is important for extending the flight distance, because a reachable point is automatically hðt Þ¼0: ð19Þ final determined in the steady gliding flight. In other words, it These parameters are used to compute the maximum gliding is important to determine the extent to which the aircraft range of the aircraft with an altitude of 3000 m, a velocity of can gain altitude and fly forward in the transitional flight. 500 m/sec, and a flight path angle of 5 degrees. The flight speed and path angle of the initial condition are An aircraft in actual flight should satisfy certain con- larger than those of the steady gliding flight. To change straint conditions: the angle of attack limit, dynamic pres- the initial large kinetic energy into potential energy, the sure limit, load factor limit, etc. The constraints may be ac- aircraft increases the path angle. The maximum value of tive in some intervals of optimal trajectories; the aircraft the path angle depends on the target heading angles. For may fly along the constraint conditions. This paper, howev- target ¼ 0, the aircraft increases the path angle up to 49.7 er, does not add the constraints to the optimization problem. degrees, and then rises by 5676 m. The flight speed decreas- This is because the constraints make it hard to understand es during the ascent flight. Then the aircraft begins descend- the essence of the optimal solutions. The intervals where ing. Note that the flight speed and path angle become small- the constraints are active change greatly by the parameters er than those of the gliding flight just before the conver- 140 Trans. Japan Soc. Aero. Space Sci. Vol. 49, No. 165
20 80 18 Target Heading Angle 70 ψ = target 0 deg 60 16 ψ = target 60 deg ψ = 50 14 target 120 deg [deg] ψ = 180 deg 40 α 12 target [deg] γ 30 10 20 8 10 Target Heading Angle
Path Angle ψ = 6 0 target 0 deg
Angle of Attack ψ = target 60 deg 4 -10 ψ = target 120 deg ψ = 2 -20 target 180 deg 0 -30 0 50 100 150 200 250 300 350 400 450 500 0 100 200 300 400 500 Time t [sec] Speed V [m/sec]
Fig. 1. Angle of attack history for the optimal control solution. Fig. 3. Path angle vs. speed for the optimal control solution.
120 Target Heading Angle ψ = 100 target 0 deg ψ = 60 deg target 9 ψ = 120 deg 80 target 8 [deg] ψ = target 180 deg σ 7 6 60 [km] h 5 4 40 3 60 Bank Angle
Altitude 2 50 40 1 [km] 30 y 20 0 -60 20 -40 -20 10 0 20 0 Downrange 40 60 0 0 50 100 150 200 250 300 350 400 450 500 x [km] 80 Crossrange Time t [sec] Fig. 4. Flight trajectories for the optimal control solution. Fig. 2. Bank angle history for the optimal control solution. only a few minutes with a standard desktop computer. How- gence. For all the target heading angles, the flight speed and ever, this is still too much time for real-time computation path angle of the aircraft converge on V and , respective- with an onboard computer. The following section explains ly, while oscillating. Although we expected that uniform a method for producing near-optimal control in real-time convergence, rather than the oscillating convergence, would with an onboard computer. be the global optimal, the results showed that uniform con- vergence was the local optimal, and that the global optimal 3. Real-Time Optimal Guidance Method was an oscillating flight trajectory. If the target heading angles are set to be values other than 3.1. Near-optimal control method 0, the histories of the bank angles change drastically. The The preceding section discretized the state and control peak of the bank angle occurs during the transitional flight; variables of the optimal control problem, and solved the the bank angle increases gradually and exceeds 90 degrees nonlinear programming problem. It was impossible to ob- as the target heading angle increases. Not only the bank an- tain the optimal solution in real-time, though the optimiza- gle but also the angle of attack and the load factor are large. tion problem was comparatively easy. Another typical solu- Though some constraints are necessary in actual flight, this tion method for the optimal control problem is based on cal- paper does not consider them, as previously mentioned. The culus of variations. This method regards the optimal control bank angle decreases after it has reached the maximum, and problem as a variational problem and solves a two-point or then the angle reaches 0 degrees by the beginning of the multipoint boundary value problem that is a necessary con- steady gliding flight. dition satisfied by an optimal solution, using a shooting This problem was not difficult to solve. In fact, the solu- method, etc. Because this method requires repetition of inte- tion of each optimal control problem could be obtained in gration from the initial time to the final time to obtain the Nov. 2006 T. TSUCHIYA: Near-Optimal Guidance Method for Maximizing the Reachable Domain of Gliding Aircraft 141 solution that satisfies the boundary conditions, the computa- dv 1 n2 tion time of a problem with a long integration interval is ¼ d v2 þ sin ð22Þ d 0 v2 n 2 large. Moreover, it is necessary to choose an initial solution d n cos near the optimal solution, because the robustness of this ¼ ð23Þ method is inferior to that of the method based on the nonlin- d v ear programming method. This paper overcomes these faults Here, the only control variable is the load factor n. The by the following ideas that adopt a solution method based on state variables at the present time present are vð presentÞ and the variational problem for real-time near-optimal control. ð presentÞ, and the state variables at the final time final are The first idea was to divide the three-dimensional optimal vð finalÞ¼1 and ð finalÞ¼ of the steady gliding flight. control problem into two optimization problems: a gliding The performance index of the optimal control problem is distance maximization problem in which the motion of the then defined as Z aircraft is limited to a two-dimensional vertical plane, and final an optimal turning flight problem in a horizontal direction. J ¼ v sinð Þd ; ð24Þ Then, we first solve the optimal control for the former prob- present lem, the gliding distance maximization of an aircraft flying which corresponds to the glide distance. The speed in the in a straight path without turns. Second, if the aircraft needs direction perpendicular to the velocity vector of the steady to turn to the target heading angle, the bank angle is deter- gliding flight is v sinð Þ. The increase in the value mined by minimizing the reduction in the performance in- obtained from integrating v sinð Þ in the transitional dex that was obtained by solving the former problem. This flight extends the glide distance of the aircraft. use of two problems is simpler than the original three- The Hamiltonian of the optimal control problem is dimensional flight range maximization problem. Though 1 n2 the former flight distance maximization problem requires H ¼ v sinð Þþ d v2 þ sin v 0 v2 n 2 a shooting method, it can be easily solved because the fea- n cos tures of the global optimal solution have already been found þ ; ð25Þ in the preceding section. When we guess the value of a con- v trol variable at the initial time and integrate the equations to where v and are adjoint variables. @H=@n ¼ 0 gives the obtain a trajectory, the trajectory indicates whether this es- optimal control timated control variable is smaller or larger than the optimal n 2v one. This paper also proposes a solution method by which n ¼ k; k ¼ : ð26Þ d the latter optimal turning problem is analytically and easily 2 0 v solved using the results of the former flight distance maxi- The differential equations of the adjoint variables are mization problem. 2 @H 1 n The second idea was as follows. As a performance index _v ¼ ¼ sinð Þþ2 vd0 v @v v3 n 2 of the flight distance maximization problem in the vertical n cos plane, we determine not the flight distance between the þ ð27Þ 2 initial point and the terminal point where the aircraft reaches v the ground, but a function of the downrange distance and @H sin _ ¼ ¼ v cosð Þþ v cos : ð28Þ altitude difference in the transitional flight. The optimal @ v flight is composed of a transitional flight and a steady glid- It should be noted that the Hamiltonian H is a constant value ing flight. It is not necessary to include the steady gliding 0 because the Hamiltonian does not depend on the time ex- flight in the flight distance maximization problem because plicitly and final is unspecified. Substituting Eq. (26) into the flight distance of the steady gliding flight is automatical- Eq. (25) gives ly decided. We have only to optimize a performance index n 2 cos 1 defined from the terminal point in the transitional flight k2 k d v2 sin þ v sinð Þ¼0: 4d v 0 that corresponds to the flight distance in the total flight. 0 v This shortens the interval of the integration to be executed ð29Þ repeatedly. The following explains how to solve the optimal control Details of the algorithm are described below. nð presentÞ at the present time present. First, an initial guess 3.2. Gliding distance maximization problem of an optimal load factor nð presentÞ to state variables The aircraft glides in the vertical plane from the present vð presentÞ and ð presentÞ is chosen and adjoint variables speed and path angle. This subsection provides a method vð presentÞ and ð presentÞ are computed from Eqs. (26) for obtaining the optimal control value of a load factor, and (29). Equations (22), (23), (27) and (28), whose initial which the aircraft should take at the present so as to values are vð presentÞ, ð presentÞ, vð presentÞ and ð presentÞ, maximize its reachable glide distance. We define two are integrated through Eq. (26) from which the load factor state variables, a dimensionless speed v and a path angle is calculated. The computed trajectory provides the informa- , and extract the following equations of motion from tion as to whether the guessed nð presentÞ is larger or smaller Eqs. (10)–(15). than the optimal value. Figure 5 shows some examples 142 Trans. Japan Soc. Aero. Space Sci. Vol. 49, No. 165
near to those of the steady gliding flight, the optimal solution 40 can be obtained in a shorter amount of time. 0.61 τ = τ 30 present 3.3. Optimal turning flight problem 20 The preceding subsection gives the optimal control 0.57 nð presentÞ at the present time present when the aircraft glides [deg] 10 γ τ in the vertical plane. The performance index J in the preced- n( present ) 0.10 ing subsection indicates the maximum reachable distance of 0 = 0.5894 the aircraft. If the aircraft flies in the vertical plane without -10 turning, it should take the load factor obtained in the preced- Path Angle -20 ing subsection that maximizes J. A difference between the target heading angle and the present one, however, requires v(τ )= 1.0 -30 final γγτ = * turning flight. The turning flight reduces the maximum ( final ) -40 reachable distance of the aircraft flying in the vertical plane; 0.8 1.0 1.2 1.4 1.6 1.8 2.0 it decreases the value of the optimal performance index J. Dimensionless Speed v Therefore, the optimal bank angle maximizes a negative value dJ=d , a variation of J due to the turning angle. Fig. 5. Initial guess of nð presentÞ and trajectories for ð presentÞ > . The solution of nð presentÞ is 0.5894. Since the optimal performance index J is a function of the state variables v and , a derivative dJ=d is represented as dJ @J dv @J d d 5 ¼ þ : ð30Þ τ d @v d @ d d n( present ) 2.0 0 = 1.7740 Here, it should be noted that, in optimal control theory, the adjoint variables vð presentÞ and ð presentÞ obtained in the preceding subsection have the following characteristics.13) -5 [deg] 1.7685 @J @J γ ¼ vð presentÞ; ¼ ð presentÞð31Þ -10 τ = @v ¼ present @ ¼ present v( final ) 1.0 γγτ = * ( final ) Substituting Eqs. (13), (14), (15), and (31) into Eq. (30), we -15 1.7741 Path Angle τ =τ compute the optimal load factor noptimal and bank angle present optimal that maximize the derivative dJ=d at the present -20 1.78 1.5 time present. The results are sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi -25 1 k cos n ¼ n v v2 þ sin þ ð32Þ 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 optimal d v Dimensionless Speed v 0 kn 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 6. Initial guess of nð presentÞ and trajectories for ð presentÞ < . The optimal ¼ cos ; ð33Þ 1 k cos solution of nð presentÞ is 1.7740. 2 2d0 v þ sin þ d0 v where the guessed nð presentÞ is larger or smaller for where k is the ratio of adjoint variables = v at the present ð presentÞ > . Figure 6 also indicates some examples for time present, which was already computed in the preceding ð presentÞ < . Whereas solid curves that converge on subsection. vð finalÞ¼1 and ð finalÞ¼ are the optimal solution tra- In summary, we propose the following method for solving jectories of nð presentÞ, dashed curves and dash-dotted curves the maximum range in the target direction of target: if there are the trajectories for which nð presentÞ is smaller and larger, is a difference between the target heading angle target and respectively. It is possible to judge whether the guessed the present angle , the aircraft makes the turning flight with nð presentÞ is too large or small by the intersections of the tra- a load factor of Eq. (32) and a bank angle of Eq. (33). The jectories and ¼ . This judgment makes it possible to algorithm is shown in Fig. 7. If target ¼ , the aircraft takes calculate the optimal load factor with a bisection algorithm the load factor obtained in the preceding subsection without which uses larger and smaller values of load factors. In the banking. The method for examining whether the aircraft bisection algorithm, we can stop the computation according can reach a specified point and the method for guiding the to the computing speed and necessary solution accuracy. aircraft to a point inside the maximum reachable domain Though the algorithm needs to iterate the numerical integra- are the same: the aircraft makes the turning flight with tion, the time required for the solution can be reduced due to Eqs. (32) and (33) until the heading angle corresponds to the following advantages of this method: the definition of the direction of the point. After that, the aircraft continues the problem is simpler than that of the original three-dimen- to fly with the result described in the preceding subsection. sional optimal control problem and the integral interval is This method allows the aircraft to fly to the desired point only a period of the transitional flight. If the states become with a sufficient altitude margin. Nov. 2006 T. TSUCHIYA: Near-Optimal Guidance Method for Maximizing the Reachable Domain of Gliding Aircraft 143
τ Time present
Guidance Distance Maximization τ ( Compute n( present ) , k )
τ Guess n( present )
λ λ τ Compute k , v , γ at present γ λ λ Integrate v , , v , γ
No τ τ n( present ) proper? Correct n( present )
Yes
Optimal Turning Flight σ ( Compute noptimal , optimal )
Yes ψ τ =ψ No ( present ) target ?
= τ σ noptimaln( present ) Compute noptimal , optimal σ = optimal 0 from Eqs. (32) and (33)
τ Update present
Fig. 7. Flowchart of the near-optimal guidance method.
10 90
Target Heading Angle 80 Target Heading Angle ψ = ψ = target 0 deg target 0 deg 8 ψ = 70 ψ = target 60 deg target 60 deg ψ = ψ = 120 deg target 120 deg [deg] target 60 ψ = [deg] ψ = 180 deg α target 180 deg target 6 σ 50
40 4 30 Bank Angle
Angle of Attack 20 2 10
0 0 0 50 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 Time t [sec] Time t [sec]
Fig. 8. Angle of attack history for the near-optimal control solution. Fig. 9. Bank angle history from 0 to 50 sec for the near-optimal control solution.
3.4. Numerical simulation To evaluate the near-optimal guidance method in the pre- with the control obtained from the two-dimensional flight ceding subsection, this subsection conducts numerical sim- distance maximization problem. A comparison of the trajec- ulations utilizing the same parameter values and initial con- tories by the proposed guidance method with the optimal ditions stated in the previous section. Figures 8–11 show the trajectories in the previous section reveals that the flight flight trajectories. For target 6¼ 0 deg, flight histories up to control which pulls the aircraft up immediately before hit- the time when the heading angle reaches the target angle ting the ground does not appear in the proposed guidance target are all the same. The aircraft stops turning when the method. Apart from that control, the flight histories for the heading angle corresponds to the target angle, and it flies target heading angle target ¼ 0 deg are exactly the same. 144 Trans. Japan Soc. Aero. Space Sci. Vol. 49, No. 165
60 500
50 400 40
[deg] 30 300 γ 20 Target Heading Angle 200 10 ψ = target 0 deg Path Angle ψ = 60 deg 0 target Computation Time [sec] ψ = target 120 deg 100 ψ = -10 target 180 deg
-20 0 100 200 300 400 500 Exact Optimal Control Near-Optimal Control Speed V [m/sec] Fig. 12. Comparison of computation times. Fig. 10. Path angle vs. speed for the near-optimal control solution. on the optimal solutions varies according to initial guesses. We determined the most suitable parameter values by trial and error, and the initial guesses were chosen on the assumption that the appearance of the optimal solutions was clear. On the other hand, for near-optimal control, we employed the fourth-order Runge-Kutta method with a time 9 step of 0.01 sec as a numerical integration, and the control 8 values were computed at intervals of 0.1 sec in the flight 7 6 time. Figure 12 shows that the near-optimal control can be [km]
h 5 computed at high speed, with a computation time that is 4 about one-tenth of that required for the exact optimal con- 3 60 trol. Moreover, it is impossible to compute the exact optimal
Altitude 2 50 40 control values in real-time because the entire trajectory 1 [km] 30 y between the initial and final states is optimized in the exact 0 20 -60 -40 optimization process. In contrast, the near-optimal control -20 0 10 Downrange 20 40 0 method is capable of real-time computation. x [km] 60 80 Crossrange
Fig. 11. Flight trajectories for the near-optimal control solution. 4. Conclusions
This paper proposes a guidance method for computing the For target 6¼ 0 deg, however, the reachable range resulting near-maximum reachable domain from the present state of from the guidance method is smaller than that in the optimal an aircraft and the real-time optimal guidance by which solutions. The difference increases with the turning angle. the aircraft can fly to a specified point within the domain. The flight range is decreased by about 22% for a target head- For real-time optimal control, we divide the reachable do- ing angle target ¼ 180 deg. This is because the guidance main maximization problem into two optimization prob- method proposed in this paper divides the optimal control lems: a gliding distance maximization problem in the verti- problem into two problems: a flight distance maximization cal plane and an optimal turning problem. First, the former problem and an optimal turning flight problem. The latter problem is solved by iterating the numerical integration of problem produces the control values without information the flight trajectory. Since the features of the optimal solu- on the difference between the target heading angle and the tions computed are shown in the first half of this paper, present angle. we can easily obtain the optimal solution. The optimal bank Figure 12 compares computation times of the proposed angle is analytically computed from the latter problem based near-optimal control method and the exact optimal compu- on the optimal solution of the former, in the event that the tation in the previous section. The computation time implies aircraft needs to make a turn in order to reach a specified elapsed time to obtain the optimal solutions for all the cases target heading angle or destination. Comparison of the flight of target. As the computation time depends largely on the trajectories and controls between the guidance method performance of a computer and parameters of numerical based on the above algorithm and the exact optimal computations, the result shown in the figure is only an solutions shows a good agreement of the qualitative features example. For instance, not only are many parameters in and a difference in the reachable range of less than 20–30%. the exact optimal computation, the elapsed time to converge Moreover, the proposed method, which can be used for Nov. 2006 T. TSUCHIYA: Near-Optimal Guidance Method for Maximizing the Reachable Domain of Gliding Aircraft 145 real-time computation, can obtain the approximate optimal Glideback and Flyback Boosters, J. Spacecraft Rockets, 38 (2001), control values faster than the exact optimization method. pp. 752–758. 5) Naidu, D. S. and Calise, A. J.: Singular Perturbations and Time Scales If an aircraft is forced to make a gliding return in an in Guidance and Control of Aerospace Systems: A survey, J. Guid. emergency situation, the proposed guidance method has Control Dynam., 24 (2001), pp. 1057–1078. the ability to locate more candidate airports than the conven- 6) Shapira, I. and Ben-Asher, J. Z.: Singular Perturbation Analysis of tional guidance methods. Moreover, this method can even Optimal Glide, J. Guid. Control Dynam., 27 (2004), pp. 915–918. 7) Sheu, D. L., Chen, Y. M. and Chern, J. S.: Optimal Three-Dimensional deal with the situation in which the aircraft must change Glide for Maximum Reachable Domain, AIAA Atmospheric Flight the target heading angle during flight, such as in response Mechanics Conference, AIAA, Washington, D.C., 1999, AIAA to the sudden appearance of a no-fly zone. It is necessary 99-4245. to consider computational methods where the control values 8) Kraemer, J. W. and Ehlers, H. L.: Shuttle Orbiter Guidance System for the Terminal Flight Phase, Automatica, 13 (1977), pp. 11–21. can be obtained applying stricter models of motion and 9) Youssef, H. and Lee, H.: Development of X-33 Guidance, Navigation various constraints. This research is now underway. and Control System, AIAA Guidance, Navigation, and Control Conference, AIAA, Washington, D.C., 1999, AIAA 99-4126. 10) Tragesser, S. G. and Barton, G. H.: Autonomous Intact Abort System References for the X-34, AIAA Guidance, Navigation, and Control Conference, AIAA, Washington, D.C., 1999, AIAA 99-4253. 1) Vinh, N. X.: Optimal Trajectories in Atmospheric Flight, Elsevier, 11) Mayne, D. Q., Rawlings, J. B., Rao, C. V. and Scokaert, P. O. M.: New York, NY, 1981, Chap. 7. Constrained Model Predictive Control: Stability and Optimality, Auto- 2) Tsuchiya, T. and Mori, M.: Optimal Conceptual Design of Two-Stage matica, 36 (2000), pp. 789–814. 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