zotero-db/storage/NZNVMXSS/.zotero-ft-cache

Source of Acquisition NASA Ames Research Center

Nhan Nguyen* NASA Ames Research Center;Moffett Field, CA 94035

This paper presents a study of a minimum time-to-climb trajectory analysis for aircraft flying in a two-

dimensional altitude dependent wind field. The t i e optimal control problem possesses a singular control

structure when the lift coefficient is taken as a control variable. A singular a r c analysis is performed to obtain

an optimal control solution on the singular a r c Using a time-scale separation with the flight path angle treated

3f 2 f& f*a+th,e gw-ez&$Edifyof the

cent-.! ,&&e. is re&c& by pEy&&bp +he lift cQe%-&Et

control. A further singular a r c analysis is used to decompose the ori,@nal optimal control solution into the flight

path angle solution and a trajectory solution as a function of the airspeed a n d altitude. The optimal control

solutions for the initial and final climb segments are computed using a shooting method with known starting

values on the singular arc T h e numerical results of the shooting method show that the optimal flight path

angle on the initial and final climb segments are constant. The analytical approach provides a rapid means for

analyzing a time optimal trajectory for aircraft performance.

I. Introduction

The climb performance of an aircraft is an important design requirement for establishing trajectories to reach a

specified altitude and airspeed after takeoff in some optimal manner. For transport aircraft, a climb segment may

follow a trajectory designed to achieve an optimal fuel consumption or a minimum time. Trajectory optimization

problems to minimize aircraft fuel consumption or time of climb had been studied by various contributors in the 70’s

and ~O’S.’-~ In recent years, minimum time problems have also been examined for many space systems.8-10 Many

approaches are found in literature for analyzing optimal flight trajectories for a minimum time-to-climb of an aircraft.

One such method is based on the singular perturbation method that has been investigated for the minimum time-to-

climb p r ~ b l e r n . ~T?he~ singular perturbation method performs a time-scale separation of the fast and slow states in

fight dynamic equations so that the dimension of the problem is reduced by the order of the,fast states. If the aircraft

is modeled as a point mass with three state variables: altitude, speed, and flight path angle, then the flight path angle is

considered as a fast state and therefore its differentialequation can be treated in a quasi-steady state approximation.“?

This allows the ffight path angle be treated as a control variable for the &mum time optimal control

Another popular method is based on the energy state approximation method;1,4dMch

es

for computation by combining the altitude and speed variables into the energy state variable that represents the sum

of the kinetic energy and potential energy of the aircraft during climb. Because of the order reduction in the state

equations, the energy state approach enjoyed a popularity in minimum time optimal control problems. The total

energy leire: curves &us c m be viewed as clirves of sub-optimal climb path.” Climbs OF &ves along the eEergy leve!

curves theoretically is supposed to take virtually little time. In some climb maneuvers, an aircraft often has to execute

an energy dive to trade altitude for speed. Once a desired speed is zttained, the aircraft climbs out of the current energy

level curve and transitions into a curve that takes the aircraft to the next energy level curve. This transition curve cuts

across the energy level curves in an optimal manner such that the altitude can be gained in a minimum time. This

curve is h o w n as the energy climb path. Fig. 1illustrates an energy climb maneuver.

The energy climb path turns out to be a singular arc control problem that can be analyzed by the Pontryagin’s

minimum principle.’2 In brief, the minimum principle introduces the concept of Hamiltonian functions in analytical

dynamics that must be minimized (or maximized) during a trajectory optimization. The optimization probiems are

formulated using calculus of variations to determine a set of optimality conditions for a set of adjoint vatiables &&’

*ResearchScientist, Intelligent Systems Division, M a l Stop 269-1

1 of 16 Amencan In&~~toef Aeronauacs and Astronautics

provide the sensitivity for the control problems.'; This is known as an indirect optimization method which typically involves a high degree of analytical complexity due to the introduction of the adjoint variables that effectively doubles the number of state variables. Furthermore, the solution method frequently involves solving a two-point boundary value problem. Nonetheless, the adjoint method provides a great degree of mathematical elegance that can reveal the structureof a problem. In some cases, exact analytical optimal solutions can be obtained. Except for simple problems, many trajectory optimization problems require numerical methods that can solve two-point boundary value problems such as gradient-based methods, shooting methods, dynamic programming, etc.

Fig. 1 - Time-Optimal Energy Climb Path
Within the framework of the Pontryagin's minimum principle, the singular-arc optimal control method is an intermediate method for the trajectory optimization. The existence of a singular arc in the time optimalcontrol can simplify the trajectory optimization significantly. Briefly, the singular arc is described by a switching function that minimizes a Hamiltonian function when the Hamiltonian function is linear with respect to a control variable.
In this study, we will examine an aspect of the minimum time-to-climb problem for an aircraft flying in the
presence of a two-dimensional atmospheric wind field. An analytical solution for the singular arc is obtained. Wind
patterns at a local airport can affect the climb performance of aircraft. While the time-optimal climb problems have been thoroughly studied in flight mechanics, the effect of winds are usually not included in these studies. A solution method of a minimum-time to climb will then be presented for computing a minimum time-to-climb flight trajectory.

In our minimum time-to-climb problem, the aircraft is modeled as a point mass and the flight trajectory is strictly confined in a vertical plane on a non-rotating, flat earth. The change in mass of the aircraftis neglected and the en,@.ne
thrust vector is assumed to point in the direction of the aircraft velocity vector. Ln addition, the aircraft is assumed to
fly in an atmospheric wind field comprising of both horizontal and vertical components that are altitude-dependent. The horizontal wind componentnormally comprises a longitudinaland lateral component. We assume that the aircraft motion is symmetric so that the lateral wind component is not included. Thus, the peainent equations of motion for the problem are defined in its the state variable form as

+ h = vsin y W h

(1)

T-D- W~iny

?j=

-w,cos*/-whsiny

m

+ + = L - Wcosy mu

wzsiny-whcosy v

(3)

2of !6 Amencan Institute of Aeronauhcs and .4stronautics

where h is the altitude, v is the speed, y is the flight path angle, T is the thrust force, D = CDqS is the drag force,
L = CLqS is the lift force, VV is the aircraft weight, rn is the aircraft mass, and w, = w, (h)and wh = wh (h)are
the respective temporal average horizontal and vertical wind field components as functions of the altitude. Thus, the time rate of change of the wind field can be computed as

I.

tiJ,=wzh=wj.(vsiny+wh)

(4)

I.

W h = W h h = W 6 ('U SinA{f W h )

(5)

where the prime denotes the derivativewith respect to the altitude h.The gradient of the wind velocity with respect to
the altitude is also called a vertical wind shear.I6 The problem is now posed as a minimization of the time-to-climb from an initial speed and an initial altitude to
a h a 1 speed and a final altitude at a final time t f . To formulate the time-optimal control problem, we consider the following cost function

J=lfdt

(6)

subject to state constraints by Eqs. (1)-(3). The 'boundary conditions for the problem are the iritial and final altitude md aiispeed of the aircraft 2s

where M is the Mach number as a function of the altitude and airspeed. The flight path angle y (0) or y ( t f )may be
free or fixed. To solve for the time-optimalcontrol problem of Eq. (6), we apply the Pontryagin's minimum principle which can
be stated as follows:
Let z( t ) : (0,t ) -+ Rn be state variables and u(t) : (0,t f ) -+ Rm be in a set of admissible control U that
- guides a dynamical system described by 2 = f (x(t),u( t ) )from an initial state 2 (0) to a final state x ( t f )where
f (x,u): R" x Rm --i Rnis some function, there exist a set of variables X ( t ): (0,t ) Rn,called adjoint variables,
such that we have the following necessary conditions for optimality casted in a Hamiltonian canonical structure

where H (z( t ),u (t),X (t))is the Hamiltonian function of the dynamical system which, for a minimum time-to-climb
optimal control problem, is defined as

H .( (9> 21 (t)1 (9)= 1+ AT (4 f .( (t),u

(9)

The optimal control is given by the following necessary condition

u*(t)= argminH (x*(t),u(t),X (t))
UEU

If the dynamical system is autonomous, that is J is not an explicit function of time, the Hamiltonian function is required to be zero throughout the optimal solution6 I

.( ( t ), (t>I ( 9 )= 0

(113

A singular control exists if the control variable u appears linearly in the Hamiltonian function H co.rres.pond$g

-a2-H - 0

(12)

dU2

3 of 16

American Institute of Aeronautics and Asronautics

Let S(z,A) : Rn x Rn ---f Rmbe a switching function defined by
S=- dH du
Since the control u does not appear explicitlyin S,it cannot be determined explicitly. However, by differentiating S repeatedly until the control u appears explicitly,then the control u is so determined. For the control u to be optimal, the Kelley's condition must be satisfied
(14)
For this probIem, the Hamiltonian function is defined as

+ ) x,
+

( L - W cosy

W, sin y - wh c o s y

mv

(15)
V

w h i z Ah, A, and A, a e :he adjoizt uzr,ab!es. We define the specific excess thrust F and the Wing loading factor n as functions of the altitude, Mach number &I,
and Reynolds number Re as

where CD,Ois the profile drag coefficientand K is the induced drag parameter. Then, the necessary conditions for optimalityresult in the following adjoint differential equations

Ah

dH
= --

= -A hwk-Av

[ E9--

(" 1 i ' ) 'I w,cosy+whsiny (vsiny+wh)- w,cosy+wAsiny wh "

+ -) + siny - w i cosy) (siny wh

w i (wi siny - w i cosy

V

V

dH
v -- --

= -Ahsiny-X,

dV

siny

A ---=d-HA 7 - dr

+ gsiny

1
w,

sin 2y

I
- wh cos 2 7 i-

V

In order to determine the extremalcontrol to achieve a fastest climb path, we take the lift coefficient CLas a control

variable. Then the optimal control is one that renders the Hamiltonian stationary

. . .

I

'I

-d=H dCL

-A,

2KC;qS m

+A,--=OqS mv

4 cf 16

S C Z = - A, 2KvAV

American Institute of Aeronautics and Asacnautics

subject to the inequality constraint C L ,5~Cz~5 ~C L , ~ ~ , .
The optimal lift coefficient in Eq. (21) may be solved by a two-point boundary value problem using a numerical method such as a shooting method or a gradient descent method. Such a numerical solution often does not reveal a structure of the optimal control analytically. In most cases, it is found that this problem can be approximated as a singular optimal control problem by making an assumption that the induced drag parameter K is usually small and therefore can be neglected. ln fact, the ideal induced drag parameter for an elliptical wing loading is given by
K=- 1
TAR
where AR is the aspect ratio. For a typical transport aircraft, the wing aspect ratio is about 7 so that the ideal K
parameter is about 0.045. Thus, this assumption is reasonable. Under this assumption, we see that d2H/dCZ = 0 and so there exists a singular c011tr01 with a switching function
S=X,- qs mv
Since qS/mv > 0, we then obtain a bang-bang extremal control as

The bang-bag control law is thus viewed as a sub-optimal solution to the minimum time optimal control problem.
Equation (24) states that the minimum-time-to-climbtrajectory is approximated by three sub-optimal ai-cs. The first arc is a trajectory on which the aircraft flies at some initial altitude and airspeed at a maximum lift coefficient until
it intercepts with a singular arc trajectory defined by the second switching condition. The s i n ~ l aarrc is an optimal flight trajectory for the fastest climb' and is also c d e d an energy climb path (EQ) since it is a path that crosses a set
+ of level curyes of constant energy heights E = h v2/2gas illustrated in Fig. 1. At some point on this trajectory,the
aircraft climbs out of the singular arc and flies along a final arc with a minimum lift coefficient to arrive at the final
altitude and airspeed. To find a singular control, we use the fact that S = 0 and S = 0 on the singular arc to establish that A, = 0 and
A, = 0. Hence, this allows us to eliminate A, in Eq. (20) by solving for Ah

L

J

The remaining adjoint equations (18) and (19) now become

A, DifferentiatingEq. (25), which is equivalentto computing S = and then substitutingEq. (27) into the resulting

5 of 15 American Institute of Aeronauhcs and Astronautics

expression yield

+ [w; cos 27 w i sin 2y - wh (W; tan

+

cos y

V

By ecpati3-g Eq. (28) to Eq. (261, the singular arc for a minimum time-to-climb is now obtained as

+ + f (h,v,7 ,n) = (1 - UI)F

(1

d F v2 dF

~

2 V)-dv

- --
g dh

+a3n+aa

= 0

with the coefficients a],a2, a3, and a4 described by the following functions of the wind field parameters

Wh (w;
a1 =

tan y - wh’ >

9

+ sin y (3 tan2 y) - 2wk cosy + v cos2 y

Equation (29) is a partial differential equation in terms of the specific excess thrust F that describes an optimaI

climb path for a minimum time-to-climb solution for an aircraft flying in the presence of an altitude dependent at-

mospheric wind field. Examining Eq. (29) reveals that there is a high degree of cross coupling between the horizontal

wind field and the vertical wind field up to the second derivatives of the wind fields. Thus, not only the

gradients affect the optimal climb path, but the wind field curvatures also play a role as well. Equation (29) results

in a parabolic equation in terms of CL,which can be solved to give a feedback control for the lift coefficient on the

singular arc as a function of the three state variables h,v, and y.

A certaiE simplificat;,o?1 of Eq. (29) can be Dade by comidering the concept of fast and slow states in flight

dynamics. Ardema had shown that in optimal trajectory analysis, the three-state model of a point mass aircraft exhibits

a time-scale separation behavior whereby the state variables h and v possess slower dynamics than the state variable

y.3 This time scale separation is normally treated by a singular perturbation analysis by replacing the fast state equation

with the following equation

L - W c o s y w,sin y - w h cosy

€A/ E

mu

-?-

V

(34)

where E is a small parameter.

Equation (34) has an inner solution and an outer solution, which is the steady state solution obt-ned by sstfing

E = 0. The inner solution can usually be solved by the method of matched asymptotic expansion. The inner solution

is also called a boundary-layer solution in reference to the historical origin of the singular perturbation method in

the fluid boundary layer theory. The overall solution is domiiiated by the outer solution with the inner solution only

6of !5

.4merican Insnmte of Aeronauncs and Astronauhcs

affecting a small initial time period. As a first order approximation, the inner solution can sometimes be ignored. In
this case, Eq. (34) results in the following load factor

' ) n = cosy - (v sin y +- wh) (w: sin y - wh cosy

(35)

9

Equation (35)thus dethrones the number of state equations from three to two by converting the state variable y into a control variable in place of the lift coefficient CL. Substituting Eq. (35) into Eq. (29) now yields the optimal climb path function

v2dF v 9 ah 9
+ -u 9

We have thus reduced the optimal climb path by removing the dependency on the lift coefficient.We now examine some special cases of the general optimal climb path function f (h,v, 7):

1. Steady Wind Field:

For a steady wind field, the gradients and curvatures vanish, thereby reducing the optimal climb path function

to the following

f ( h , v ) = F + vd-F dv

v2 g

dF
dh

=

O

(37)

Equation (37) is the well-known result for the ECP without an atmospheric wind field. Thus, the optimal climb path in the presence of a steady wind field is effectively the same as that without the wind field effect. This result is not surprising and can be explained by the fact that since the inertial reference kame is attached to the air mass, whether the air mass is moving at a constant speed or remains stationary, the speed of the aircraft relative to the air mass is the same, thus resulting in the same optimal climb path.
2. Horizontal Wind Field: In the presence of a horizontal wind field only, the optimal climb path function becomes

+ f (h,V, 7 ) = F V-

+ 2vwj. sin2y tan 7 v3wj: sin2ytan y

9

g2

v2 (w.)"

-

sin y tan2ycos2y = 0 (38)

g2

Specializing Eq. (38) for small flight path angles by invoking sin y M y and cosy M 1yields

" v2 (wi)2]

v 2 d F /2uw; v3w, +

y3 =

(39)

L

J

3. Vertical Wind Field:

7 of 16 American Institute of Aeronautics and Astronautics

In the presence of a vertical wind field only, Eq. (36) becomes

For small flight path angles, Eq. (40) reduces to

(F) 2v (W.)'
-

= 0 (41)

9

-v.e ~ c a wl ind geld is especially important for microburst problems. k microburst is a wind shea distuibznce

characterized by an unusually strong downdraft near the ground surface that presents an extreme hazard to

aircraft during take-off or landing.

In. Optimal Solution
During a climb, the aircrah flies under a maximum continuousthrust from take-off to a point in the flight trajectory envelope where it intersects with the optimal climb path. The aircraft then maintains its course along the optimal climb path until it reaches some point on the optimal climb path where it climbs out to the final altitude and airspeed. Thus, there exist three climb segments during a climb as illustrated in Fig. 1 labeled as A, B, and C where B is the singular arc optimal climb path. The suboptimal solutions for the climb segments A and C may be defined by lines of constant
+ energy heights E = h v2/2g.'l
From the optimal control perspective, the initial and final climb segments are to be determined by requiring the
Hamiltonian function as defined in Eq. (15) to be zero. In addition, the adjoint variable A, is no longer restricted
to zero according to the first and last switching conditions in Eq. (24). Thus, in general, ,the optimal solutions for the initial and final flight segments are considerably more complex than the optimal climb path solution and usually involve solving a two-point boundary value problem.
First, we shall consider the singular arc optimal solution. Along this singular arc, allthe state and adjoint variables are functionsof the altitude h, airspeed v, and flight path angle n/. However, it can easily be shown that the flight path angle y on the singular arc optimal energy climb path is in turn a function of the altitude h and airspeed v. Therefore, the two variables h and v uniqueIy determine the optimal climb path. In particular, for the case of small flight path angles for which Eqs. (39) and (41) apply. The optimal climb function f (h:v, y) can be written as

where f2 are explicit functions of v and h.
The time derivative of f (h,v, 7)is evaluated as

taking into account that the assumption $ = 0 is built in10 the climb path function.

Substituting Eqs. (1)-(3) into Eq. (43) then yields the following polynomial equation that can be .mlyed for,&he

fight path angle -/

+ + c5Y5 c4y4

3
c,(h, yi = 0

(44)

2=0

8 of 16

American InsatUte of Aeronautics and Astronautics

where the coefficients c, are (45)

c 4 ( h , v ) = - v3-f-3 ah

3f3 ( V W' x f w h W 6 + g
du

(46)

Equation (44) gives the flight path angle as a function of the altitude and speed. Thus the optimal climb surface
function f (h,ZI, y) can be replaced by a climb path function f (h, u,-/ (h,v)) upon embedding the solution of the
flight path angle from Eq. (44). The adjoint variables along the optimal climb path are now determined from enforcing the conditions H = 0 and
A, = 0 on the singular arc as

The foregoing analysis has shown that the along the singular arc, the time optimal trajectory is known in the 21 - h plane as well as the flight path angle and the adjoint variables. This information greatly facilitates the optimal solutions for the climb segments on the initial and final arcs. Since the initial conditions on the initial climb segment are known according to the problem statement and the fact that its solution must terminate on the optimal climb path function for which all the state and adjoint variables are known, then the optimal solution for the initial arc can be computed using a shooting technique to integrate backward from some point B on the optimal climb path to the initial point A as shown in Fig. 1. Likewise, to compute the optimal solution for the final arc, a shooting method may be used to integrate forward from some point C on the optimal climb path to the final point D that terminates at the desired altitude and airspeed. A shooting method may be established as follows:
The flight path angle y not on the optimal climb path function can be solved by setting the Hamiltonian in Eq. (15) to zero with the usual small angle assumption, thus resulting in the following quadratic equation

Equation (50) is then used to eliminate the flight path angle expression in the adjoint equation in the adjoint differential equations (18) and (19), which can then be parameterized in terms of the altitude as the independent variable instead of time using the following transformation with the small angle assumption

[gg ( w l -dX-h - _Ah -- -xhw;, - x u + dh h

-

f W i Y ) ( V y f w h ) - ( W k -!-W 6 y ) W b ]

vy wh

(51)

'V -k wh
In addition, the airspeed can also be parameterized as a function the altitude as

Equations (50)-(53) are thed integrated either forward or backward using the known state and adjoint variables on the optimal climb path as the starting values at points B and C. The shooting method then iterates on these known values on the optimal climb path until the conditions at points A and D are met.

9 of 15 hencan Inshrute of Aeronauim and Asnanauncs

We compute the time optimal climb trajectory of a transport aircraft m a horizontal wind field. The aircraft has a
maximum thrust T,, which varies as a function of the altitude as
where p is the density, po is the density at the sea level, and a is a constant. Using Eq. (35) with the small flight path angle approximation, the specific excess rhrust i s computed as
The optimal climb path function, Eq. (39), is equal to

The aircraft has the EoIlowing values: T,, =40000 Ib, W =200000 lb, S =1591 ,'tf CD= 0.0343, and a = 0.7. We will compute the singular arc for two typical horizontal wind field problems. The first wind field problem is a low-altitude wind shear disturbance described by the following model

w,

=

a

l

e

_ h_ hi

-

a

2

e

-

k
q

(57)

where, for the problem, a1 = 250 Wsec, a2 = 200 Wsec, hl = 6000 ft, and h2 = 1000 ft. The second wind field probIem is a high-altitude wind field problem described by the followingpower law model

where, for the problem, a3 = 250 ft/sec, h3 = -1000 Wsec, h4 = 30000 ft, and (Y = f.
The two types of wind field are plotted in Fig. 2.

Fig. 2 - Wind Shear and Power Law Models
1Oof 16 American Institute of Aeronautics and Astronautics

As can seen, the wind shear profile is characterized by a strong wind field gradient near the ground which can be hazardous to aircraft during take-off and landing.
To find the singular arc, we decompose the optimal climb path function, Eq. (56), in the individual functions f z (h, u ) according to Eq. (42). For a horizontal wind field problem, Eq. (44) is then formulated as a 4th degree polynomial function in terms of the flight path angle y. A physically correct root is obtained that yields the flight
path angle on the singular arc. The singular arc function f (h,v) then becomes a nonlinear function in terms of h
and v. To compute the singular arc trajectory in the v - h plane, we find the zero solution of this function using a Newton-Raphson method as follows:

[ ] df (hV,i) -1

= - dv

f (hz,vi)

(59)

The computed singular arc time-optimal climb paths for both wind field profiles are plotted in Fig. 3. For reference,

we also compute the singular arc for zero wind disturbance. As can be seen, the singular arc climb paths are steep

paths that rapidly increase the altitude with a relatively smaller change in the air speed for the case of no wind. The

power law profile follows a similar pattern as the no-wind case, although the ground speed intercept is less. This

would mean that with the power law profile, the aircraft has to enter the singular arc climb path at lower speed than

if *,ere

~0 wind. T& sir@&- &-c fGr

wind she= pregle is

pLost&eres+&qg

iz

that

thp

sn~ert i s Y " '

xran'a tinn '-I----- A

significantover a wide range at a very low altitude. This is due to the strong wind field gradient over a short altitude.

At high altitude, the three singular arcs are converging, so the effect of wind field is less pronounced at high altitude.

Fig. 3 - Singular Arc Time-Optimal CIimb Paths

Once the singular arc has been determined, the values of h.and v along the singular arc are plugged into the 4th

degree polynomial to solve for the flight path angle along the singular arc. The lift coefficients are also computed.

Figs. 4 and 5 are the plots ~f the Eight path angle md Et ccefficient on the singdar arc c h ? ?paths. The flight path

angle for the no-wind case generally decreases with altitude. At high altitude, the flight path angle for the two wind

profiles converge to that of the no-wind case. The wind shear case as usual shows a drastic change in the flight path

angle along the singular arc at low altitude. The lift coefficientgenerally increases with altitude for the no-wind case.

The lift coefficient for the wind shear case varies gready at very low altitude and is quite large at ground level due to

the low ground speed intercept required to enter the climb path.

.

We next compute a complete trajectory from take-off to some final altitude and air speed. We only consider the

wind shear case. The initial ground speed at take-off is about Mach 0.2 or 224 ftlsec and the desired air speed at

15000ft is Mach 0.5 and 0.6. The solutions not on singular arc require solving a two-point boundary value problem.

However, since the adjoint variables are completely determined on the singulararc, we can solve the trajeeio$ off G'

singular arc quite easily using a shooting method by integrating Eqs. (51) to (53) either forward or backward stasting

from the singular arc. The adjoint solution on the singular arc is plotted in Fig. 6.

11 of 16

Ainencan Institute of Aeronautm and Astronautics

Fig. 4 - Flight Path A-fgie on Singular Arc

Fig. 5 - Lift Coefficienton SingularArc

-0 6

05

1

15

2

25

3

h. R

x 10'

h, R
Fig. 6 - Adjoint Solution on SinguIar Arc
12 of 16 American Institute of Aeronautics and Astronautics

The complete trajectory is plotted in Fig. 7. To climb to 15000ft and Mach 0.5, the climb trajectory is comprised of three segments. The first segment, segment AB, is the take-off segment on which the aircraft Bies from take-off ground speed to intercept the singular arc at point B. This occurs at a very low altitude. The second segment, or singular arc se,sment BC, is the minimum-time to climb path that takes the aircraft to a higher altitude in a fastest time. At some point on this singular arc, the aircraft begins to depart at point C and flies on the final segment, segment CD, to the final altitude and airspeed. We note that the three segmentsjoin together and are tangent at points B and C. On
this path, the departure slope is to the left of the singular arc. Inthe region to the left of the singular arc, because the
altitude is continuously increasing, the thrust is maintained at the maximum value. The situation for Mach 0.6 corresponding to the final segment EF is different. Since the aircraft must arrive at
Mach 0.6 which is to the right of the singular arc. Because the departure slope must be tangent to the singular arc and curve to the left, there is no point below the final altitude where this tangency exists. As a result, the aircraft must fly past the final altitude and then reduce the engine thrust to let the potential energy be convert to b e t i c energy once the aircraft begins to slow down. For the problem, we use a first-ordermodel to describe the engine thrust reduction from the maximum value to an idle value at roughly 20%of the maximum thrust. The segment EF is sub-optimalinthat the engine thrust varies so that it becomes a control variable in addition to the flight path angle. Nonetheless, it is a very good approximation since the segment EF approximateIy follows a total energy curve.
0 Y. n/sec
Fig. 7 - Climb Trajectory
Fig. 8 - Lift Coefficienton C h b Trajectory
13 of 16 American Institute of Aeronautics and Astronautics

The lift coefficient variation on the climb trajectory is shown in Fig. 8. The lift coefficient vanes greatly from

a maximum take-off value to about 0.4 at a very low altitude. Depending on the wind shear profile, this may not

be feasible due to the slow retraction of flaps and slats deployed during take-off. The flight path angIe on the climb

trajectory is plotted in Fig. 9. The flight path angle for the initial segment AB and the finalse-sment CD are computed

using Eq. (50) as a function of the adjoint solution computed from the shooting method. It turns out that the computed

flight path angle is very nearly constant as shown in Fig. 10. Thus, an approximate trajectory can easily be computed

by only integrating Eq. (53) using a constant flight path angle at a point of tangency to the singular arc. The final

segment EF is computed using this approach.

To examine the optimality of the computed solution, we compute the values of the Hamiltonian function along

various climb segments. The segm

o the values of the Hamiltonian function are

equal to zero. The climb segment

arly illustrated by the non-zero value of the

Hamiltonian function. Nonetheless, it is a reasonable approximate solution, given that the problem would have been

formulated with the engine thrust as an additional control variable, which would result in a more complex problem.

Fig. 9 - Flight Path Angle on Climb Trajectory

2 31 2 305
8 23
0
2295 2 29 2 za6
5 3098
- 5 3096
% 53094 s
5 3092

280

400

600

800

iuoo

1200

1400

hn

5 309

08

09

11

12

13

14

15

h. fi

x 10'

Fig. 10 - Flight Path Angle Computation

14 of 16 American InstiNte of Aeronauhcs and Asironautics

-7
0

0” *“-lb

0.5

1

1.5

2

25

3

= “rI

0.8

09

1

1.1

12

1.3

14

1.5

Fig. i i - Viiues of Xamiitoniati Function
V. Conclusions
A solution to a minimum time-to climb problem in an altitude dependent two-dimensional wind field has been presented. This problem possesses a singular control structure when the lift coefficient is taken as a control variable with the induced drag effect neglected. The optimal climb path on the singular arc is obtained as a function of the three flight state variables and the lift coefficient as a control. A time-scale separation is used to reduce the dimensionality of the optimal climb path function by converting the lift coefficient control into the flight path angle control. Using the singular arc analysis, this three-dimensional function can be further decomposed into a solution for the optimal flight path angle and a two-dimensional climb trajectory as a function of the airspeed and altitude. The known adjoint variables on this optimal trajectory significantly simpiifiesthe general optimal control solution for the initial and final climb segments. A shooting method is formulated to solve numerically the trajectories of the initial and final segments using the known solutions on the optimal climb trajectory. A numerical example for a wind shear profile is computed to demonstrate the analytical method. It is found that the computed flight path angle off the singular arc is nexiy a constant and therefore can be used as a very good approximation. The analytical approach provides a rapid means for estimating climb trajectory of aircraft flying in the presence of atmospheric wind.

lParsons, M.G., Bryson, A.E., Jr., and Hoffman, W. C., “Long-Range Energy-State Maneuvers for Minrmum Tme to S p e d e d T e m a l

Condihons”, 11th AIM Aerospace Sciences Meeting, Washington, D.C., AIAA-1973-229, January 1973.

2Bre&well, J.V., “Ophmd Fbght-Path-Angle Transihons 111 Mmrrmm Time Anplane Chmbs”, American Insntute of Aeronauhcs and As-

uonauhcs and American Astronauhcal Society, Astrodynamm Conference, San &ego, CA, AIAA-1976-795, Angust 1976.

3Ardema, M.D., “Solution of the &mum Time-to-Climb Problem by Matched Asymptohc Expansions”, AIAA Journal of Guidance,

Control, and Dynam~csV, ol. 14, No. 7, 1976, pp. 843-850.

4Kelley, H.J., Cliff, EM., and \.her, X.G., “Energy Mmagen?e% of Three-D?mensionalMlAmm+Tbe Intercept”, Journal of Guidance,

Control, and Dynamics, V01.10, No. 6, 1987, pp. 574-580.

5Ardema,M.D., “Lmneanmtion of the Boundary-Layer Equations of the Mhmum Time-to-Climb Problem”,Journal of Guidance, Control, and Dynamics, Vol. 2, No 5, 1979, pp. 434.436.

6Hedndc,J.K. and Bryson, A.E., Jr., ‘Three-DimensionalMinunum-T1meTurns for a Supersonic Aircraft”, Journal of Aircraft Vol. 9, No.

2, 1972, pp 115-121.

7Bryson, A.E., Jr. and Sla.@ry, R.A., ‘‘Minunurn-TimeMotionsof 2 Two-Arm Robot”, AIpA Guidance,Navigahon and Control Conference,

Portland, OR, August 1990.

8Subchanand Zblkowsln, R., Cranfield Umveisity,“Mmimum-TimeOphmal TraJectonesfor Terxllnal Bunt Manoeuvre”. AIAA Guidance,

Navigation, and Control Conference,AIAA 2005-5968, August 2005.

- gTitus,H., Cooper, C.R, and Fdlon, M.P., “Mbmum Tme and Fuel
Programsand TechnologiesConference, HuntsvdIe, AL,AIM-19944589,

and Sliding Mode September 1994.

Controllers

for

Spacecraft

Amtude

Co.nuo.l”,.SpacIe-

loh.lbassam, B.A., “Optunal Near-Mimmum-Tme Control Design for Flexible Smctures“, Journal of Guidance,Control, and Dynmcs, Vol. 25, NO. 4,2002, pp. 618-625).

15 of 16 Amencan Institute a€Aeronautics and Astronautlcs

I1Vuh,N.X., Flight Mechanics of High-Pei$ormance A‘rcrafi, Cambndge Uruversity Press, 1993
”Pontryagn, L ,Boltyansku, U ,Gamkrelidze, R ,and Mishchenko, E., The Marhemaacal Theory of Opnmal Processes, Interscience, 1962
13Leitmann, G ,The Calculus of Vananons and Opnmal Control, Spnnger,2006. l4Nadu, D S and C&se, A.J,“Smgular Perturbahons and TimeScales in Guidance and Control of Aerospace Systems A Survey”,Journal of Guidance,Control, and Dynarmcs, Vol 24, No 6,2001, pp 1057-1078 15Bryson, A., and Ho, Y,Applied Optimal Control: Ophmizatlon, Estrmanon, and Control*Hemsphere Pubhshing Co , 1975 16Nelson,R C ,Flight Stabilizy and Automanc Control,McGraw-Hll1, 1997
16 of 16 American hshtute of Aeronauncs and Astronaudcs