Abstract

AIAA-95-1225-CP

Simultaneous Optimization for Sizing and Flight Trajectory of a Sailplane

Shinji SUZUKI*, and Norihisa KAWAMURAt the University of Tokyo ~9 5 26608

7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan

This paper considers a simultaneous optimization for an aircraft design and a flight trajectory optimization prob- lem. The two optimization problems which have been separated traditionally should be optimized simultane- ously to obtain an extreme high performance. Since si- multaneous design problems must consider a huge num- ber of design variables, this paper presents a block di- agonal S Q P (sequential quadratic programming) ap- proach by introducing the interface variables between the two problems. As numerical examples, the design of a sailplane in a flight distance competition is studied. The need of a simultaneous optimization approach will be demonstrated by comparing a conventional design process, i.e., a lift-drag ratio maximization approach.

1 Introduction

This paper considers the integration of an aircraft siz- ing problem and an optimal trajectory problem. In the aircraft design phase, the configuration and the geomet- rical parameters of an aircraft are optimized for an as- sumed fright path. After an aircraft has been designed, an actual flight path or trajectory is optimized with considering fuel efficiency or flight time. However, an aircraft design and its flight trajectory problem should be formulated simultaneously when an exact flight mis- sion is determined or an extremely high performance level is required.

Numerical optimization techniques have been devel- oped for both aircraft design and trajectory problems. Aircraft design optimization problems have been suc- cessfully solved by using a mathematical programming formulation. vanderplaats(l) optimized a supersonic aircraft design for an prescribed mission profile. Dovi and Wrenn(2) solved a. wing-body transport design for an assumed mission profile. In these studies, design pa- rameters such as wing area, aspect ratio, sweep back

'Associate Professor, Department of Aeronautics and Astro- nautics, Member AIAA

t ~ r a d u a t e S tudent , Department of Aeronautics and Astronautics

Copyright 01995 by the American Institute of Aeronautics and Astronautics, Inc. All right reserved.

angle, and thickness-to-chord ratio were selected as the design variables to be optimized. On the other hand, flight trajectory problems can be formulated as an op- timal control problems(3). While numerous solutions approaches, e.g., calculus of variations, maximum prin- ciple, and dynamic programming have been developed, recent investigations have revealed the advantages of using a mathematical programming approach(4w6). In a mathematical programming formulation, the control variables are discritized for the design variables, and a performance index is minimized or maximized while satisfying boundary conditions and path constraints. I t follows from this that mathematical programming a l g e rithms handle inequality constrains in an efficient man- ner, and do not have as excessive computer-storage re- quirements as dynamic programming. Although both aircraft design problems and flight trajectory problems have been solved by using mathematical programming, it is seldom that these two problems in different disci- plines are solved at the same time.

By increasing computer capabilities and by devel- oping optimization algorithms, we can apply multidis- ciplinary optimization for practical problems. Since multidisciplinary optimization problems must deal with multiple analysis programs in different disciplines and a huge number of design variables, several solution meth- ods have been proposed. Sobieski and o t h e r ~ ( ~ 1 ~ ) have developed decomposition approach based on the sen- sitivity analysis. This paper applies the Sequential Quadratic Programming method(g) as a mathematical programming formulation, and proposes a block diag- onal approach by introducing the interface variables to efficiently solve multidisciplinary optimization prob- lems.

As numerical examples, the design of a sailplane which can be entered in a competition for flight dis- tance is considered. In the competition, it is assumed that a sailplane descends from a height of 1 0 m and the flight distance is measured a t touch down. Such a competition actually exists in ~ a ~ a n ( " ) . The design problem has two interrelated elements: first, how to de- sign the sailplane and secondly, how to control its flight path. Both problems must be solved simultaneously to obtain a maximum flight distance. In the presenting paper, the simultaneous optimization will be compared with a conventional design, i.e. the maximization of a

lift-drag ratio. Note that when a plane descends at the steady speed, the flight distance can be maximized a t the maximum lift-drag ratio.

2 SQP Optimization Problem

A mathematical programming problem is generally for- mulated as

Minimize : d ( x ) (1)

Subject to : ci(z) = 0, i E E [a)

c;(x) > 0, i E K (3)

where x and g5 are the design variables vector and a objective function t o be minimized, and c; denotes an equal constraint (E) or an unequal constraint ( K ) .

A Sequential Quadratic Programming (SQP) method finds the optimal solutions by solving the follow- ing Quadratic Programming problem in a sequential mannedg)

1 T k Minimize : v , 4 ( x k ) d + -d B d 2 (4)

Subject to : c,(xk) + ~ , c , ( z ~ ) d = 0, i E E ( 5 )

c i ( z k ) + V , C ; ( Z ~ ) ~ 2 0, i E I{ (6)

where x k is the vector x a t the k-th iteration step, and d is the update vector (= xk+' - xk) . The matrix B in Eq. (4) denotes the approximation matrix of the Hessian matrix H for the following Lagrange function:

where u is the Lagrange variables vector. To update the matrix B, we employs the BFGS formulation1'.

3 Block SQP for Simultaneous Optimization

The maximum mission performance is obtained through both an aircraft design optimization and a trajectory optimization. Hence, the design variables consist of both aircraft design parameters X D and control param- eters x c . Note that the control variables as a time func- tion are descritized by dividing the time scale into small elements and by approximating the control variables a t each elements (Appendix A).

The objective of the integrated problem is written as

Minimize : 4 ( z D , z c ) (9)

where 4 is related to a mission performance. Con- straints in both problems are given as

Subject to : gD(xD) 2 0 (10)

Subject to : gc(xD, z C ) > 0 (11)

Vehicle Design Phase

Figure 1: Interface variables between two phases

where gD represent design constraints for structural characteristics, flight stability, and aircraft size limi- tations, and gc denote constraints in an optimal con- trol problem, e.g., terminal conditions, path and control constraints (See Appendix A).

In order to solve this problem efficiently, we introduce the interface variables XI which represent aircraft char- acteristic parameters appearing in the optimal control problems. The interface variables such as weight, wing loading, and aspect ratio are contracted with the air- craft design parameters X D (Fig. 1). Consequently, the problem can be formulated as

By introducing the interface variables, we need not cal- culate the derivatives of g5 and gc with respect to X D ,

and we can reconstruct the B matrix in the following block diagonal form:

Since the number of the interface variables XI is usually less than the number of the aircraft design variables X D , the diagonal form of the B matrix is suitable for numerical calculations.

4 Sailplane Design Examples

The design problem of a sailplane for a flight distance competition is considered as numerical examples. In the competition, i t is assumed that a sailplane descends from a height of 10 m and the flight distance is mea- sured at touch down (Fig. 2). The competition has been held every year at Lake Biwa in ~apan(" ) . Except that the runway is limited within 10 m, there are no re- strictions on its size and weight. The design process for this sailplane has two interrelated elements: the aircraft design phase, and the flight control phase.

4.1 Aircraft Design Phase

Table 1: Design variables in aircraft design phase

While several configurations are available, we assume

~i~~~~ 2: ~ l i ~ h ~ trajectory in flight distance competi- a conventional glider as shown in Fig. 3. A sailplane

tion has a straight wing with a single pipe spar('"?). We selected EPPLER748 as its wing section, decided that a

Symbol -

s1, s 2

-41, -42

X I , A2

C n , Cr2 It

yl, y;, dl , d2, d3

n,,, &

Design Variables Mainjrear wings

Areas Aspect ratios

................ Taper ratios

structure frame was made from an aluminum tube, and divided the wing spar into three section so as t o select an optimal tube's diameter. The design variables as shown in Fig. 3 are listed in Table 1 in which the lift coefficient of each wing at the steady glide condition was used to specify the attached angle of each wing. Note that while the maximum load factor and the steady glide speed are not geometrical parameters but ones associate with the flight control phase, these parameters must be given in the aircraft design phase.

X

A

1 O m

The design constraints in the aircraft design phase are summarized in the following way:

Lift coefficients ;y;. at steady glide Distance of two wings

Main wing Spar divide positions Spar diameters v ".- -...

m g Flight conditions --.._ --.___ --. ---..__ Maximum load factor

.-.- --._ .-._ - - Steady glide speed

(a) Geometrical size: Although there are no restric- tions on size and weight in the regulations, we as- sume the following constraints so that a pilot can run up while holding a sailplane:

S2, A2, h2, C 1 2 2. Total length:

L 1 5 (m) ( 18)

3. Spar diameter a t wing tip must be less than wing thickness:

Figure 3: Design variables of sailplane

(b) Structural characteristics: Maximum stress of spar and fuselage pipes must be less than the final stress limit at 1.5 x n,,, load:

Rear wings size is constrained instead of the stress analysis of a rear wing spar:

(c) Flight Condition: To keep the controllability of a sailplane, the steady glide speed V, and lift coef- ficients a t steady glide are restricted as:

Additionally, trim and static stability conditions must be satisfied a t steady glide.

In order to formulate the design constraints above, we employ the following approximations for mathematical modeling:

Aerodynamic characteristics: The lift and the in- duced drag are calculated by using the vortex-line theory(14). T h e wing profile drag is estimated from the two dimensional experimental data of the given airfoil. The parasite drag coefficient of a fuselage is assumed to be 0.06.

Structure characteristics: The stress of each alu- minum tube is calculated by using a beam the- ory in which the load distribution was found in the aerodynamic calculations without consideration of aeroelastic effects. The diameter and the thickness of a tube are assumed as (20 mm, 0.8 mm) for a rear wing, and (60 mm, 1.2 mm) for a fuselage. The diameters of a main wing are selected as the design variables and the thickness of the tube is assumed to be 2 % of the diameter.

Stat,ic stability: We assume that a plane is controlled by shifts of the pilot's weight position. The trim condition and the static stability conditions are given by using the aerodynamic and inertia char- acteristics.

4.2 Flight Control Phase

An optimal control problem is considered for a point mass model t o obtain the optimal control strategy. When the design of a sailplane has been determined, the design variables of this phase can be selected as lift coefficients and the flight time Tf (Appendix A). Note that the time scale was divided into 40 elements. Initial conditions are given as

x(0) = 0 (m) (23)

d o ) = 10 (m) (24) y(0) = -3 (deg) (25)

where we assume the initial speed from the sailplane weight with a pilot weight (60 kg). By integrating the dynamic equations in Appendix B, we formulate the following constrains:

(a) Terminal condition:

Table 2: Interface variables

(b) Path constraint: The flight height is greater than 0 and load factor is less than nmax in each time element:

Interface variables Zero lift drag coeff. Airplane efficiency x aspect ratio Vehicle weight Wing Area Maximum load factor

(c) Control constraint: A lift coefficient is less than stall limit:

CLi 5 cpax (29)

Symbol

CD o eA1 W s1

n m ax

4.3 Optimization

In order to demonstrate the need of simultaneous op- timization, we also consider a maximum lift-drag ratio design. If a sailplane flies downward at a steady glide speed, the flight distance is proportional t o the lift-drag ratio L I D of a plane. Therefore, the two optimization designs are formulated as:

(a) Simultaneous design: The sailplane characteris- tic data used in the dynamic equations are selected as the interface variables z1 which can be defined by using the design variables xo. The interface variables are shown in Table 2. T h e objective func- tion in simultaneous design is the maximization of a flight distance and given as

Minimize : - x f ( z o , z l ) (30)

This objective function is tried to minimized while satisfying the constraints, i.e. the unequal con- straints both in the airplane design phase and in the flight control phase, and the equal constraints which define the interface variables as the functions of the aircraft design variables.

(b) L I D maximum design: In L I D maximization design, the design phase and the flight control phase are separated in the following manner: the objective function in the design phase is the lift- drag ratio of a plane at the steady glide condition, and the objective function in the control phase is the flight distance of the obtained plane. It should be noted that the maximum load factor nYn,, and the steady glide speed & in the aircraft design phase were selected as those values obtained i n the simultaneous design.

Table 3: Characteristics of optimized saiplanes

4.4 Numerical Results

Characteristics Wing area, S1 (m2) Weight, W (kg) Aspect ratio, A1 Wing loading,

H'/& (kg/m2) LID ratio, L I D Flight distance,

.-f (rn)

LID max. 9.42 29.1 40.0

9.46 33.6

330.6

Simultaneous opt. 14.6 35.0 36.4

6.52 29.0

396.7

(a) Initial design The iteration process in SQP refines the initial de-

sign in Fig. 4 (a) to maximize the flight distance in the simultaneous design or to maximize the L I D at the steady glide until all design variables converge. Figures 4 (b) and (c) illustrate the sailplane shapes obtained by the simultaneous design and the maximum lift-drag ra- U I GI

tio design, respectively. The characteristic data of the two planes are compared in Table 3 in which the flight distance for the L I D maximum design was calculated through the control phase optimization for the plane obtained in the design optimization phase.

From Table 3, we can reason the design policies as follows: (1) the L I D maximum design increases wing aspect ratio t o reduce the induced drag, and (2) the si- multaneous design reduces the wing loading (W/S) by increasing wing area within the weight limit. This dif- ference is reflected in the shape of each sailplane. An

(b) Simultaneous Optimization

important point is that flight distance of the simulta- neous design is longer than that of the L I D maximum design.

Figures 5 show the flight paths, the control variables and the flight speeds of the optimized sailplanes. The case of interest here is that the optimal flight trajecto- ries of the two sailplanes are different. The sailplane designed by L I D maximization dives quickly and flies near the ground because the ground effect can reduce the induced drag (see Appendix B). On the other hand, the sailplane designed in simultaneous optimization flies at steady glide in the early flight stage. and then flies near the ground. I t is supposed that the sailplane de- signed in simultaneous optimization can generate the steady flight part because of the lower wing loading W/S. Furthermore, Fig. 5 (c) indicates that the flight speed at the steady glide keeps a lower value. This is hccause the flight distance of the simultaneous design is longer than that of the L I D maximum design.

Finally, we confirmed that the proposed hlock diag- onal SQP algorithm was effective in numerical calcwla- tions. The numerical examples in this study can be solved directly without using the interface variables. The number of iterations and the CPU time for the

(c) L I D maximization

Figure 4: Initial and optimized shapes

simultaneous optimization case are compared in Table

Table 4: Iteration number and CPU time I SOP 1 Block diagonal SQP

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 100 200 300 L

Distance (m)

(a) Flight trajectory

100 200 300 400 Distance (m)

I - > - , "

Number of iterations 1 52 1 43

4. This indicates that the block diagonal approach can save both the iteration numbers and the computational time.

CPU time normalized with SQP case

5 Conclusion

The design process of an airplane and the flight trajec- tory problem were integrated in a simultaneous opti- mization formulation. In order to solve simultaneous optimization problems, we must cope with the diffi- culty in dealing with a huge number of design variables. This paper proposed the introduction of interface vari- ables to obtain the diagonal form of the approximate Hessian matrix in Sequential Quadratic Programming. The sailplane design for a flight distance competition was considered for numerical examples. The need of the simultaneous optimization has been demonstrated by comparing the present design with the conventional maximum lift-drag ratio design. Finally we could con- firm that the introduction of the interface variables in a block SQP algorithm was effective in numerical calcu-

1.0

lations.

0.59

Appendix A: Mathematical Optimization Ap- proach for Optimal Control Problem

(b) Lift coefficient The dynamic equations can be expressed in the follow- ing state equations:

where x( t ) and ~ ( t ) are the vector of states and the vector of controls, respectively. At the initial time, t = 0, and the final or terminal time, Tf , some states may be specified as

100 200 300 400 Distance (m)

(c) Flight speed

Figure 5: Optimal flight solutions

The upper limits or the lower limits of the state or control variables may be determined during, 0 5 t 5 Tf. This type of constraint is called a path constraint and defined as

The control variables u( t ) is sought to minimize or maximize a given performance index

In order t o transform an optimal control problem to a parameter optimization problem, the control vari- ables are approximated by defining discrete parameters. When a set of time functions are introduced, the control variables are represented as

u ( t ) x n l ( t ) u l + n2(t)u2 + ... + n ~ ( t ) u ~ = N ( t ) u p

(35) While any kind of approximation can be utilized, this paper employs a piecewise linear approximation, i.e., the time scale is divided into elements and a control variable in each time element is approximated as a linear function.

When the final time is not specifird, both Tf and up are considered as the design variables to be optimized. By integrating the state equations, we can represent the boundary conditions and the path constraints as the functions of the design variables. As a result, the problem can be formulated as a mathematical program- ming problem.

Appendix B: Dynamic Equations of a Sailplane

The dynamic equations of a sailplane as a point mass m is written as

1 9 j' = -pvSICL--cosy

2m v (37)

j: = v c o s y (38)

7 j = V s i n y (39)

where V, y, and (a, y ) are the flight speed, the path angle, and the position coordinates, and CL, p, and g

are the lift coefficient, the air density, and the gravity acceleration, respectively. The drag coefficient CD is represented as

where Coo, e, and A; are the zerelift drag coefficient, the airplane efficiency, and the main wing aspect ra- tio modified due to the ground effect in the following manner(15) :

where h and b are the wing height from the ground and the wing span. This modification indicates that the in- duced drag decreases when a wing approaches to the ground.

References

(1) Vanderplaats, G.N., 'LAutomated Optimization Techniques for Aircraft Synthesis," AIAA Paper 76-909, Proceedings, AIAA Aircraft Systems and Technology

(2) Dovi, A.R., and Wrenn, G.A., "Aircraft Design for Mission Performance Using Nonlinear Multiobjective Optimization Methods," Journal of Aircraft, Vol. 27, No. 12, Dec., 1990, pp.1043-1049. (3) Bryson, A. E., Jr., and Ho, Y. C., Applied Optimal Control, Blaisdell, Waltham, MA, 1969. (4) Tabak, D., and Kuo, B. C., Optimal Control by Mathematical Programming, Pentice-Hall, Englewood Cliffs, NJ, 1971. (5) Hargraves, C.R., and Paris, S.W., "Direct Trajec- tory Optimization Using Nonlinear Programming and Collocation," Journal of Guidance, Control, and Dy- namics, Vol. 10, No. 4, 1987, pp.338-342. (6) Suzuki, S., and Yoshizawa, T., "Multiobjective Tra- jectory Optimization by Goal Programming with Fuzzy Decisions," Journal of Guidance, Control, and Dynam- ics, Vol. 17, No. 2, 1994, pp. 297-303. (7) Sobieszczanki-Sobieski, J., James, B.B., and Dovi, A.R., "Structural Optimization by Multilevel Decompo- sition," AIAA Journal, Vol. 23, No. 11, 1985, pp.1775- 1782. (8) Sobieszczanki-Sobieski, J., "Sensitivity Analysis and Multidisciplinary Optimization for Aircraft Deign: Re- cent Advances and Results," Journal of Aircraft, Vol. 27, No. 12, 1990, pp. 993-1001. (9) R. Fletcher, Practical Methods of Optimization, Sec- ond Edition, Jhon Wiley, 1987. (10) Azuma, A., "Birdman Rally and Sky Leisure Japan," Journal of the JSASS, Vol. 42, No. 485, 1994, 337-350. (11) Broyden, C.G., "The convergence of a class od double-rank minimization algorithms, Part 1 and 2," Journal of Inst. Math. Appl. Vol. 6, 1970, pp. 76-90, 222-231. (12) Markowski, M.A., The Hang Glider's Bible, Tab Books, 1977. (13) Raymer, D.P., Aircraft Design: A Conceptual Ap- proach, AIAA Education Series, 1989, pp. 689. (14) Bertin J.J., and Smith M.L., Aerodynamics for En- gineers, Second Edition, Prentice-Hall, 1989, pp. 235- 257. (15) Hoerner, S.F., and Borst H.V., Fluid-Dynamic L i ' , Hoerner Fluid Dynamics, NM, 1985.

Meeting, Dallas, T X , September 27-29, 1976. c 7 7