American Institute of Aeronautics and Astronautics

1

Pseudo-Spectral-Method Based Optimal Glide in the

Event of Engine Cut-off

Maya Dekel K.1 and Joseph Z. Ben-Asher

2

Faculty of Aerospace Engineering

Technion - Israel Institute of Technology, Haifa, Israel, 32000

Maximum range optimal glide is of great interest in different flight situations. A

common example is the event of engine cut-off during manned or unmanned flight.

If the engine cut-off occurs at low altitude flight conditions, a controller for

maximum range is needed in order to obtain safe landing; this problem may be

approached as an optimal range glide problem which can be solved using Pseudo

Spectral methods. The paper is focused on the computation of optimal trajectories,

based on the minimum principle guidelines using pseudospectral methods. The

problem is solved using a three dimensional point-mass modeling, and the final

location and heading are tested in order to get the maximum range envelope as a

function of the initial conditions, the atmospheric conditions, the vehicle

aerodynamic characteristics and the landing zone angle. The structure of the

optimal trajectories and the structure of the resulting landing area are studied and

the features of the optimal policies are identified. It is shown that the trajectory

cannot be adequatly described by a combination of simple cirular arcs and straight

lines. The landing areas are all convex. The presence of wind affects the landing

area; an approximation to this effect is proposed.

Nomenclature

α = Angle of attack

β = Angle of side slip

LC = Lift coefficient

*

LC = Optimal lift coefficient

,L maxC = Lift coefficient upper limit

,L minC = Lift coefficient lower limit

DC = Drag coefficient

0DC = Zero lift drag coefficient

k = Induced drag parameter

1 Graduate Student

2 Professor, AIAA Associate Fellow

AIAA Guidance, Navigation, and Control Conference08 - 11 August 2011, Portland, Oregon

AIAA 2011-6596

Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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L = Lift force

D = Drag force

ψ

= Heading angle

φ = Roll angle

maxφ

= Roll angle upper limiter

t = Time

h = Altitude

v = Velocity

γ = Flight path angle

q = Dynamic pressure

slρ = Air density at sea level

ρ = Air density

M = mass

W = weight

g = Gravity constant

S = wing surface

H = Hamiltonian

J = Cost function

x = Range in x direction

y = Range in y direction

xW = Wind in the x-axis direction

yW = Wind in the y-axis direction

ןλ

= Lagrange multipliers

θ = Landing zone angle

iw

= PSM weight in node i.

I. Introduction

Since the early years of flight, optimization of trajectories has been a challenging issue. The range

optimization problem cannot be described well enough by a linear model and the only method, which

considers all dynamic effects, is optimal control theory which casts the problem in the form of a two point

boundary value problem (TPBVP).

Approximate methods have emerged to overcome the mathematical difficulties of TPBVP. One such

method is Neighboring Extremals 1-2

. This method assumes a numerical optimal solution to a reference

problem and regards different initial values as perturbations to the reference problem. By linearizing the

problem around the reference solution, a set of linear equations is obtained and an analytical solution can

be obtained. Shapira, (Ref. 3) presented an approximated analytical solution for the non-linear optimal

range glide problem in two dimensional space (2D), based on state variable timescale separation and the

singular perturbation theory. Ref. 3 shows convergence of the approximated problem to the exact optimal

trajectories. Adler, (Ref. 4) obtained approximate solutions to the problem in three dimensiona (3D) based

on primitive motion elements and graph theory. Three types of basic flight primitives were analyzed:

straight flight, gliding flight and turning at a constant rate.

Enormous effort has been spent on the development of computational methods for generating accurate

solutions of optimal control problems. Pseudo Spectral (PS) optimal control is a direct computational

method for solving complex nonlinear optimal control problems. PS methods were originally developed for

the solution of partial differential equations (Ref. 3). However, over the last 15 years, PS techniques have

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emerged as powerful computational methods for solving optimal control problems 5-9

. In recent years, PS

optimal controllers have been used to solve problems such as UAV trajectory generation, missile guidance

and similar problems. The success of PS methods is a result of recent advances in theory, algorithms, and

computational power. These advances in algorithms and technologies make it possible to solve highly

complicated nonlinear optimal control problems in real-life applications. Software tools have increased

significantly the opportunity to explore complex optimal control problems both for academic research and

industrial-strength problems.

The paper is focused on the computation of accurate optimal glide trajectories using pseudo spectral

methods. The problem is successfully solved in the 3D domain, where various final location and heading

are tested in order to get the maximum range envelope as a function of the initial conditions, the

atmospheric conditions, the vehicle aerodynamic characteristics and the landing zone angle. The structure

of the optimal trajectories and the structure of the resulting landing area are studied and the features of the

optimal policies are identified.

The paper is organized as follows. The next section describes the mathematical modeling of the aircraft in

glide and presents the optimal problem formulation. Sections III presents and discusses the computational

results as obtained by the PS method. Section IV verifies the optimality of the results by checking the

maximum principle. It is presented after the results to emphasize the fact that this is a-posteriori

verification. Finally, the last section summarizes the main results.

II. Problem Formulation

A. Model Assumptions

The aerial vehicle in this research work is represented by a point mass in a non-rotating flat earth.

The drag force (D) is assumed to be quadratic as follows:

2

0

=

=

D

D D L

D C qS

C C kC+ (1)

The lift force (L) is:

= LL C qS (2)

where q in the above expressions is the dynamic pressure:

21

=2

q vρ (3)

The air density ( )ρ is assumed, for simplicity, to be exponential in altitude:

( ) = expsl

ref

hh

hρ ρ

−

(4)

refh is a reference altitude and sl

ρ is the air density at sea level.

The air-vehicle numerical characteristics are given in Table 1.

0Dc 0.01

K 0.0216 2[ ]S ft 151

[ ]m lb 800

Table 1: Air Vehicle Characteristics

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For this air-vehicle we obtain the following results for maximum range glide according to classical gliding

theory (the velocity is at sea level or the equivalent air speed):

,= 0.68

= 0.0294[ ] = 1.68[ ]

= 81[ / ]

L glide

glide

glide

C

rad deg

V ft sec

γ − − (5)

B. The 3D Dynamic Model

The 3D dynamic model is described by six equations of motion for six state variables: the range x and y,

the altitude h, the velocity v, the glide angle γ and the heading angle ψ .

The equations of motion are as follows:

( )

= cos cos

= cos sin

= sin

= sin

1= cos cos

sin=

cos

x

y

dxv W

dt

dyv W

dt

dhv

dt

dv Dg

dt m

dL w

dt mv

dL

dt mv

γ ψ

γ ψ

γ

γ

γφ γ

ψ φγ

+

+

− −

−

(6)

Where x

W and yW are wind values in the x-axis and y-axis directions, respectively.

C. Problem Formulation

The optimal controllers * ( )LC t and

*( )tφ are defined as the controllers ( )L

C t and ( )tφ that steer the

aerial vehicle to its maximal range, given some boundary conditions. Therefore, the cost function is defined

as:

( )0.52 2=

f fJ x y− + (7)

The controllers in our case are bounded as follows:

, ,= 1.5, = 0.5L max L minC C − (8)

and

= 50deg

maxφ (9)

III. Computational Results

The computations have been performed by the software package GPOP (Ref. 10) and have been verified, in

part, by the software package DIDO (Ref. 11). We have tested various cases as described below.

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Case 1 - Final altitude is fixed

In the sequel we will use the following boundary conditions: 0

= 0x , 0

= 0y , 0

= 1500h ft ,

0= 200 /v ft sec ,

0= 0γ . In Case 1, only the final altitude condition is fixed at sea level ; all other final

conditions are free. Fig. 1 presents the time history for the state variables. The 2D results in the vertical

plane agree with the results of Ref. 3: a straight flight without changing direction in steady state and initial

and final boundary layers.

Figure 2 presents the optimal control time history and shows a good compatibility between the output from

GPOPS and the MP (to be explained below).

Figure 1. 2D Optimal Trajectory

Figure 2. 2D Optimal control, Maximum Principle Vs. Numerical output, = 0fh

0 200 400 600 8000

2

4

6

x 104 Range Vs. Time

time [sec]

x [

ft]

0 200 400 600 8000

500

1000

1500

Altitude Vs. Time

time [sec]

h [

ft]

0 200 400 600 800

50

100

150

200Velocity Vs. Time

time [sec]

V [

ft/s

ec]

0 200 400 600 800-20

0

20

40Glide angle Vs. Time

time [sec]

γ [d

eg]

0 200 400 600 800-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time [sec]

CL, GPOPS output

CL, Maximum principle

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Case 2 - The final altitude and y are fixed

In this case the final altitude and the final range in the y-axis direction are fixed, while the final heading

condition is still free.

We will use the landing zone angle θ as defined by Fig. 3. It can easily be inferred that an envelope point

,( ,max( ))f given fy x is also an envelope point ,( , max( ))f given frangeθ .

Figure 3. Definition of θ

To deal with the ambiguity this case is divided to two sub-cases:

i. For the first two quadrants of the envelope, there is no imposed limit on the x final.

ii. For the other two quadrants of the envelope, x final is limited to negative values.

Various conditions offy have been tested. The nature of the solution in the vertical plane has not been

changed. We still have a steady state solution and two boundary layers. The case with = 0fy is identical

to case 1. In all other cases, the solution in the horizontal plane comprises an initial turn, follows by a long

straight flight. There is no turn at the final time. Figures 4-5 present the case with = 30000[ ]fy ft .

Notice that *

LC in steady state flight complies with the classical gliding theory.

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Figure 4. 3D Optimal trajectory, = 30fy kft

Figure 5.

3D Optimal controls, Maximum Principle Vs. Numerical output, = 30fy kft

A typical solution in the vertical plane comprises:

• A short initial boundary layer, where the vehicle climbs and the velocity decreases

• A steady state flight, as in the classical gliding.

• Final boundary layer, where the vehicle losses all its energy and the velocity decreases to

its minimal value (Clmax).

In the horizontal plane, we have an initial layer and a steady state flight, and there is no final layer. The

horizontal typical solution in this part comprises:

• A short initial boundary layer, where the vehicle direction is changed to its final direction. The

roll angle increases and then decreases back to zero. Notice that the roll rate is not constant and

therefore, the initial part of the trajectory is not a circular arc.

• A steady state flight, the roll angle is zero, as in classical gliding.

0 200 400 600 8000

0.5

1

1.5

time [sec]

CL

Optimal CL Command

GPOPS output

MP

0 200 400 600 800-5

0

5

10

15

20

time [sec]

φ [

deg]

Optimal Roll Command

GPOPS output

MP

0 200 400 600 8000

2

4

6

x 104 x Vs. Time

time [sec]

x [

ft]

0 200 400 600 8000

2

4

6

x 104 y Vs. Time

time [sec]

y [

ft]

0 200 400 600 8000

500

1000

1500

Altitude Vs. Time

time [sec]

h [

ft]

0 200 400 600 800

50

100

150

200Velocity Vs. Time

time [sec]

V [

ft/s

ec]

0 200 400 600 800-20

0

20

40

60Glide angle Vs. Time

time [sec]

γ [d

eg]

0 200 400 600 8000

10

20

30heading Vs. Time

time [sec]

ψ [

deg]

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Table 2 summarizes the final results for various values of yf.

[ ]fx kft [ ]fy kft Range[kft] [ ]degΘ Time [sec]

0 68.995 68.995 90 845

69.421 0 69.421 0 847

-67.612 10 68.347 172 844

62.562 30 69.383 26 846

-60.688 30 67.698 154 840

52.738 45 69.327 40 846

47.984 50 69.300 46 846

-46.951 50 68.589 133 844

Table 2. Summary of Results for various fy ,

The optimal range glide envelope is presented in Figure 6.

Figure 6. Maximum range glide envelope

The envelope is clearly convex and is nearly a circle. In particular, it can be seen that the maximum range

when > 0(| |< 90)fx Θ and the maximum range when < 0(| |> 90)fx Θ are close. It may be

concluded that because of the relatively small turn radius, the initial turn does not cause a great loss of

energy.

Case 3 - The final altitude, heading and y are fixed

This test case has similar conditions as the previous case. Additionally the final heading is also fixed to

zero ( )= 0fψ . For example with = 30[ ], 0f fy kft x < we obtain the results in Fig. 7-9

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Figure 7. 3D Optimal trajectory, = 30fy kft , = 0, 0f fxψ <

Figure 8. 3D Optimal control, Maximum Principle Vs. Numerical output

= 30fy kft , = 0fψ

The typical solution in the vertical plane is similar to the previous case. In the horizontal plane, however,

since a final direction is required, we get three phases. The horizontal typical solution becomes:

• A short initial boundary layer, where the vehicle direction is changed to a value very close to

the θ direction. The roll angle increases and then decreases back to zero.

• A steady state flight, the roll angle is zero, as in the classical gliding.

• A short final boundary layer, where the vehicle direction is changed to the required final

heading. The roll angle in this phase gets to its maximal level.

Table 3 and Figure summarize the final results for various values of yf. (same terminal heading).

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[ ]fx kft [ ]fy kft Range[kft] [ ]degΘ Time [sec] flightψ

,com maxφ

0 87.56 68.756 90 843.51 90.371 50

68.691 10. 69415 8.2829 846.54 8.3425 13.184

-66.843 10. 67587 171.49 839.88 170.66 50

66.453 20 69398 16.75 846.44 16.87 26.161

-60.688 30 67698 153.7 840.26 154.24 50

56.579 40 69291 35.259 845.51 35.464 50

47.891 50 69235 46.234 845.6 46.502 50

34.299 60 69112 60.246 845.05 60.588 50

-32.612 60 68290 118.53 841.2 119.15 50

Table 3. Summary of Results for various fy , = 0fψ fixed

Figure 9. Maximum range glide envelope , = 0fψ

It can be seen again that the maximum range when > 0(| |< 90)fx Θ and the maximum range when

< 0(| |> 90)fx Θ are close. It can also be seen that the envelope in this case (Figure ) and the envelope

of the case before (Figure 6) are also close. Due to the relatively small turn radius, the initial turn and the

final turn do not cause a great loss of energy (and consequently range).

Case 4 - The final altitude and y are fixed with wind effect

In this case, we added 25m

sec wind in y direction. Figures 10-11 present the results under the wind effect

with = 40[ ]fy kft (final heading is free).

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Figure 10. 3D Optimal trajectory, = 40fy kft , = 25 /yW m sec

Figure 11.

Optimal controls, Maximum Principle Vs. Numerical output,

= 40fy kft , = 25 /yW m sec

Table 4 and Figure summarize the results.

0 200 400 6000

0.5

1

1.5

time [sec]

CL

Optimal CL Command

GPOPS output

MP

0 200 400 600-20

-15

-10

-5

0

5

10

time [sec]

φ [

deg]

Optimal Roll Command

GPOPS output

MP

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[ ]fx kft [ ]fy kft Range[kft] [ ]degΘ Time [sec] [ ]f degψ ,com maxφ

38.625 0 38.625 0 514.67 -47.758 39.73

48.034 10 49064 11.76 591.9 -38.921 32.14

55.090 20 58.608 19.953 654.16 -31.56 25.57

60.465 30 67.498 26.389 707.27 -24.966 19.945

64.472 40 75.873 31.816 752.38 -18.691 14.794

0 142.94 142.94 90 931.91 90.498 50

63.465 100 118.44 57.6 899.5 22.585 16.525

67.451 60 90.276 41.654 731.52 0 0

66.339 80 103.93 50.333 858.71 8.226 47.246

51.008 120 130.39 66.971 918.74 41.45 28.463

40.055 130 136.03 72.875 925.48 53.811 35.282

20.177 140 141.45 81.8 930.76 72.842 44.416

Table 4: Summary of Results for various fy , with wind effect

Figure12. Maximum range glide envelope with wind effect

It be seen that the maximum range when > 0(0 < < 180)fy Θ and the maximum range when

< 0(180 < < 360)fy Θ are significantly different. The envelope is shifted in the positive y-axis

direction. This expected result shows that the vehicle can not reach some relatively close landing areas, but

it can reach some other far landing areas if it uses the wind direction. It can also be seen that the envelope

of the landing area is also convex and the solution qualitative behavior of initial boundary layer, steady

state flight and final boundary layer is still valid. Notice that in the horizontal plane, there is no final layer

because, in this case, no final heading is required.

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Remark: The terminal speed for the no-wind case always corresponds to ,L maxC . This is not the case with

the presence of wind where the terminal air speed is the maximum between two values: the ,L maxC speed

and the wind component in the landing direction (to avoid negative ground speed.

Figure 13 presents a comparison between the envelope without wind and the envelope with wind of 25 m/s.

One can observe that the wind shifted the envelope by approximately the product of the flight time and the

wind velocity. Thus for any wind velocity, it is becomes possible to approximate the maximum range

envelope without actually calculate it.

Figure 6. Wind effect, the envelope is shifted by

f yt W⋅

IV. Minimum Principle Verification

The Hamiltonian part that depends on L

C and φ is the following:

1 sin

( , ) = coscos

L v

D LH C L

m mv mvγ ψ

φφ λ λ φ λ

γ− + + (10)

By the Minimum Principle, the Hamiltonian has to be minimized by the control,

thus:

*

*

=

=L

CL

argmin H

C argmin H

φφ

(11)

The first partial derivatives:

( )

21 1= 2 cos sin 0

cos 2

= sin cos cos 02 cos

v L

L

L

H v SkC

C v v m

v SCH

m

γ ψ

γ ψ

ρλ λ φ λ φ

γ

ρλ φ γ λ φ

φ γ

∂− + + = ∂

∂− − =

∂

(12)

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The solutions to (12) yield the stationary values for the controls. The optimal controllers according to the

minimum principle become:

, ,

*, ,

, ,

*

1 1 sincos

2 cos

>=

<

1arctan

cos

=

stationary stationary

stationary

stationary

stationary stationary

L L min L L max

v

L max L L maxL

L min L L min

min max

C if C C Ckv

C if C CC

C if C C

if

γ ψ

ψ

γ

φλ φ λ

λ γ

λφ φ φ φ

λ γ

φφ

≡ + ≤ ≤

≡ ≤ ≤

>

<

stationary

stationary

max max

min min

if

if

φ φ

φ φ φ

(13)

In all cases, one can see a fairly good compatibility between the solution from GPOPS (and DIDO) and the

analytical results, see Figures 2,5 8 and 11. Some numerical inaccuracy exists at the boundary layers.

Remark: The analytical graph is not bounded in the figures; one should use the correct bounds for the

comparison.

V. Summary and Conclusions

The problem of optimal glide was solved in three dimensions for cases with and without wind. The

solution maintains a typical behavior that can be characterized by three segments:

• First layer: The vehicle climbs (when there is sufficient energy); the velocity decreases;

fast roll rate for a very short time. The turn trajectory is not a circular arc.

• A steady state layer: gliding at a constant equivalent airspeed and a small negative glide

angle as in classical theory; zero roll angle (constant heading).

• Final layer: The vehicle loses most of its energy; the velocity decreases to minimum (see

the last remark of Section III); maximum roll (if necessary) for a very short time.

One possible and useful application of this work is the event of an engine cut-off when the pilot must

know the maximum gliding range in each possible flight direction, in order to evaluate his chances of a safe

landing in a specific landing zone. The research supplies such envelopes for two possible scenarios. The

first is for the case the pilot only needs to reach the landing zone and the final heading is not relevant for

the safe landing (e.g. sea landing). The second is for the case the pilot not only needs to get to the safe

landing zone, but also needs to land in a specific heading, for example in a small airport with only one

landing strip. In the second case, the maximum range that can be reached is smaller than for the first case.

A constant wind effect was also tested. The wind shifted the envelope by approximatelyy flight

W t⋅ . Thust the

pilot can evaluate the maximum range envelope with constant wind without actually calculates it.

References .

1. Eisler, G. R., Hull, D. G. 1993 , "Guidance Law for Planar Hypersonic Descent to a point", AIAA

Journal of Guidance Control and Dynamics, 16, pp. 400-402 2. Ben-Asher, J. Z., Optimal Control Theory with Aerospace Applications, AIAA, 2010. pp. 139-

159

American Institute of Aeronautics and Astronautics

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3. Shapira I., "Approximated Analytical Solution for Optimal Range Glide". PhD Dissertation , The

Technion - Israel Institute of Technology, November 2004. 4. Adler A. , Optimal flight paths for emergency landing, The Technion - Israel Institude of

Technology 5. Elnagar G., Kazemi M. and Razzaghi M. "The Pseudo-spectral Legendre Method for discretizing

Optimal Control Problems", IEEE Transactions on Automatic Control , Vol. 40, No. 10, October

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Control Problems" ,IEEE Transactions on Automatic Control, Vol. 33, No. 4, 1988, pp. 333-340 7. Elnagar G. and Kazemi M, "Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear

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Journal of Guidance, Control and Dynamics, Vol. 24, No. 2, 2001, pp. 270-277. 9. Benson D. A., Huntington G. T. , Thorvaldsen T. P. and . Rao A. V, "Direct Trajectory

Optimization and Costate Estimation via an Orthogonal Collocation Method" AIAA Guidance,

Navigation and Control Conference and Exhibit, August 2006, Keystone, Colorado 10. User's manual for GPOPS: A MATLAB Application Package for Solving Optimal Control

Problems. 11. I. M. Ross, User's manual for DIDO: A MATLAB Application Package for Solving Optimal

Control Problems, Elissar LLC., www.elissar.biz, 2007