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Dynamic programming for trajectory optimization of
engine-out transportation aircraft
Hongying Wu, Nayibe Chio Cho, Hakim Bouadi, Lunlong Zhong, F´elix
Mora-Camino
To cite this version:
Hongying Wu, Nayibe Chio Cho, Hakim Bouadi, Lunlong Zhong, F´elix Mora-Camino. Dynamic
programming for trajectory optimization of engine-out transportation aircraft. CCDC 2012,
24th Chinese Control and Decision Conference, May 2012, Taiyuan, China. pp 98 -103, 2012,
<10.1109/CCDC.2012.6244015>.<hal-00938792>
HAL Id: hal-00938792
https://hal-enac.archives-ouvertes.fr/hal-00938792
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1INTRODUCTION
The failure of engines is a dramatic event for air flight
safety and many incidents and accidents are resulting from
engine failure. Here an undesirable and very special case,
all engines out at a given point of the flight, is considered.
This situation may lead to a crash unless a flyable descent
trajectory towards a safe landing place is performed. There
are many different reasons for engine-out while it appears
that in this situation any wrong decision made by the pilots
may lead to catastrophic consequences.
So it looks quite desirable to develop an emergency
guidance mode for this situation. This new functionality
could be integrated in a Flight Management System which
should be able to select a proper landing site and propose a
feasible trajectory towards this site.
To achieve this purpose there are major steps which should
be performed: establish and analyze the flight dynamics of
an air transportation aircraft with total engine failure
(power off), study the gliding characteristics and flying
qualities of a transportation aircraft, develop a method to
establish safe reachable areas from a given situation and
finally develop a method to optimize a gliding trajectory
towards a possible safe landing place.
In this study, it is supposed that engine out occurs once the
aircraft has already gained some speed and altitude after
take-off.. Only glide of engine-out airplane in the vertical
plane is considered as a start.
2ENGINE OUT FLIGHT DYNAMICS
To establish and analyze the flight dynamics of an air
transportation aircraft with total engine failure (power off),
the classical equations of flight should be slightly adapted
to this particular case.
The aerodynamic forces (drag D, lift L, and side force Y)
are defined in terms of dynamic pressure, reference area
and dimensionless aerodynamic coefficients [1]:
aeD MCSVD ,,2/1 2
GDU
(1-1)
aeL MCSVL ,,2/1 2
GDU
(1-2)
arY MrpCSVY ,,,,2/1 2
GEU
(1-3)
Here V is the airspeed, ȡ is the air density (kg/m3), Į is the
angle of attack, e
G
is the elevator deflection, p is the aircraft
roll rate, r is the yaw rate and M is the current Mach
number, LD CC ,and Y
Care dimensionless aerodynamic
coefficients. CD and CL are supposed related by the polar
model 2
0'L
D
DCKCC where K is a constant.
It is considered that some hydraulic power remains
available to activate the elevators, ailerons and rudders
aerodynamic surfaces, so that dynamic stability as well as
attitude control can still be performed by the flight control
system. Indeed, many transport aircraft are equipped with
an deployable auxiliary turbine (RAM) which allows
insuring in the control channels of the aircraft the
availability of a residual hydraulic power. While the
additional drag generated by the RAM remains minor [2],
the extinction of the aircraft engines results in a noticeable
increase of the drag, while lift and side forces remain quite
the same. The drag coefficient is now given by:
Dynamic Programming for Trajectory Optimization of Engine-out
Transportation Aircraft
Hongying Wu, Nayibe Chio Cho, Hakim Bouadi*, Lunlong Zhong*, Felix Mora-Camino*
*LARA, ENAC, Toulouse 31055,France
E-mail: whyhgh@hotmail.com, hakimbouadi@yahoo.fr, lunlong.zhong@enac.fr, moracamino@hotmail.fr
#Programa de Ingeniería Mecatrónica, UnaB, Bucaramanga, Colombia
nchio@unab.edu.co
Abstract: The purpose of this communication is to contribute to the development of a new trajectory management
capability for an engine-out transportation aircraft. Engine-out is a dramatic situation for flight safety and this study
focuses on the design of a management system for emergency trajectories at this special situation. First the gliding
characteristics and flying qualities of a transport aircraft with total engine failure are analyzed while gliding range
estimation is considered. Then a new representation of the flight dynamics of an engine-out aircraft is proposed where
the space variable is chosen as independent parameter instead of the time variable. This allows to propose a new
formulation of the corresponding trajectory optimization problem and to develop a reverse dynamic programming
solution technique. Simulation results are displayed and new development perspectives are discussed.
Key Words: Flight Safety, Trajectory Optimization, Quasi Steady Glide, Reverse Dynamic Programming
98
978-1-4577-2074-1/12/$26.00 c
2012 IEEE
aDEaeDD MnCMCC ,,,
'
DGD
(2)
where CDE is the additional drag of a shut down engine, and
n is the number of engines of the aircraft.
The flight equations can be written as:
)sin),,((
1
JJTU
gmVD
m
V
(3-1)
)cos),,((
1
JJTUJ
gmVL
Vm
(3-2)
J
cosVx
(3-3)
J
sinVz
(3-4)
where
J
is the path angle,
T
is the pitch angle. Once fuel
dumping has been performed, the mass m of the aircraft is
considered to remain constant. Here x and z are
respectively the current longitudinal and vertical positions
of the aircraft center of gravity. Then, its height above Earth
is given by:
)( xHzh (4)
where )( xH is ground level at position x.
3ESTIMATION OF GLIDING RANGE
A first estimation of gliding range can be obtained by
considering that the aircraft remains in a quasi steady
gliding condition where air speed and path angle change
steadily according to current air density during the whole
descent.
Fig 1. Aircraft forces for quasi steady descent
In this situation the path angle is such as [3]:
)/()/1(arcsin gVf
J
(5)
or according to [4]:
¸
¹
·
¨
©
§
dz
dE
mgf T
//1
J
where 2
2
1VmzmgET (6)
is the aircraft total energy. Here f =L/D is the lift to drag ratio.
According to equation (5), a “minimum” glide angle, max
J
,
is achieved with a maximum lift to drag ratio. Since mgL |,
this corresponds to a minimum drag. Then it can be shown
that:
)'2/1 0
max KC D
J
(7-1)
)'2()2()(
2
1
0
'
0
2
min DD CQCVzD
U
(7-2)
A first approximation to the maximum range to sea level is
then given by:
0
00 '2 D
aCKzR (8)
where z0 is the initial altitude. Now, considering the above
expression of min
D, we get:
0
'/)(/2)( D
CKSzmgzV
U
(9)
Then the air speed decreases during the quasi static glide
descent. For a wide body aircraft, from cruise level, about 8
m/s are lost for a quasi steady initial descent of 1000 m. A
stall constraint can be considered to check the feasibility of
the glide maneuver:
max
)(/2)()( Lstall CSzmgzVzV
U
! (10-1)
or
max0 /1'/ LD CCK ! (10-2)
This condition is a general aerodynamic condition for
gliding feasibility of a given aircraft under a specific
aerodynamic situation. Also, the expression of Dmin shows
that dynamic pressure remains constant during the quasi
static glide . Figure 2. displays the airspeed and stall speed
during steady descent.
Fig 2. Airspeed during steady gliding
Then a relation between the quasi static glide path angle
and the altitude can be introduced and the “minimum” glide
path angle is given by:
)/(
0000
0
max
0
/
1
'2
tan
Rag
D
TzaTP
Q
KC
|
JJ
(11)
where air density in standard atmosphere can be expressed
as z
RT
g
ez
0
)(
UU
with zaTzT 00
)( ,KT q 15.288
0,
mKa /105.6 3
0qu ,KsmR q 22 /287 ,3
0/2250.1 mkg
U
.
Then this angle increases while the altitude is decreasing
during the quasi steady glide, shown in Figure 3. The
maximum flight range Ra is then more accurately
determined by:
¸
¸
¸
¹
·
¨
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
³
11
1
'2
'2
tan
1
1
0
0
0
0
0
00
00
0
0
0
Ra
g
D
D
z
a
z
T
a
Rag
RT
PSC
mg
CKzdzR
J
(12)
This is illustrated in Figure 4. If the aircraft loses engine
power at a higher altitude, it can glide over an increased
2012 24th Chinese Control and Decision Conference (CCDC) 99
range. In the case of the accident occurred on 24/08/2001,
the A330 aircraft glided for 120 km.
With this information, the reachable landing site can be
determined according to some flight planner [5], [6].
Fig 3. ‘– tan Ȗ ’ during quasi steady gliding
Fig. 4 Reachable range for quasi steady glide.
4GLIDE TRAJECTORY OPTIMIZATION
FOR SAFETY
In this section the problem of managing the trajectory of a
transportation aircraft gliding from a given initial flight
situation is considered. Contrarily to the classical max
range gliding problem, by the end of the gliding maneuver,
the aircraft must be in conditions (speed and attitude) to
perform a safe touch down at landing. In this case, the flight
guidance equations written in the aircraft wind axis are
given by equation (3).
Observe the equations that the only independent input
parameter which is available here is the pitch angle,
T
,
which can, even in an engine-out situation, be controlled by
the pilot either through the hydraulic power provided by the
RAT or the auxiliary power unit-APU, or through the trim
control channel.
Here the initial flight conditions are written as:
0000 )0(,)0(,)0(,)0(
JJ
VVhhxx (13)
while the final landing conditions are such as:
11 )(,)()),(()(
J
J
fffGf tVtVtxhth (14)
where V1 and 1
J
should allow a safe landing at altitude
))(( fG txh where function hG is representative of the ground
topography under the considered flight area.
Since final time is unknown and is only characterized by
the satisfaction of the final conditions, the replacement of
independent parameter t by the space variable x allows to
diminish the complexity of the problem since now final xf
is known once the landing site has been chosen. Moreover,
this approach should facilitate the consideration of ground
separation constraints and could make easier the
consideration of the effect of wind over the glide trajectory.
From equations (3) with :
)cos/(1/
J
Vdxdt (15)
we get:
)316()cos),,((
cos
1
)216()sin),,((
cos
1
)116(
2
c
c
c
JJTU
J
J
JJTU
J
J
gmVL
Vm
gmVD
Vm
V
tgz
where “ ’ ” represent the derivative with respect to the
longitudinal position x of the aircraft. The additional instant
constraints are:
],[],,[)()( 00min ff zzzxxxzVxV t (17-1)
^`^ `
)(,min)()(,max maxmax
min
min xxx
JDTTJDT
dd (17-2)
],[)()( 0fG xxxxhxz t (17-3)
Constraints (17-1) and (17-2) prevent from stalling and
constraint (17-3) from some flight into terrain-FIT situation
at an intermediary point of the glide.
Then, different formulations of an optimization problem [7]
can be considered to design a safe glide trajectory. For
example the following criterion could be minimized with
respect to the successive values of
T
along the glide:
2
))()((min fGf xhxh (18)
under final constraints
)1()()1( max1min1 vVxVvV fdd (19-1)
)1()()1( max1min1 gxg fdd
J
J
J
(19-2)
where minmaxmin ,, gvv and max
gare positive margins and
with state equations (16), flight constraints (17) and initial
conditions (13).
The solution of this non linear, strongly constrained
trajectory optimization problem is difficult from the
numerical point of view and a direct on line computation of
its solution does not appear to be feasible. For instance, an
approach based on the minimum principle [8] should result
in a very difficult two point boundary problem since the
resulting Hamiltonian has not an affine structure with
100 2012 24th Chinese Control and Decision Conference (CCDC)
respect to the input parameter. Many other complex
techniques have been developed for trajectory generation
[9], [10], [11] while Dynamic Programming [12] appears to
provide some good perspectives [13],[14]. To apply
effectively a Dynamic Programming solution strategy, a
discretization of this problem appears necessary and the
choice of the space variable x as independent variables for
the flight equations appears most convenient.
5THE PROPOSED SOLUTION STRATEGY
Here dynamic programming is used to generate a feasible
glide trajectory towards a safe landing place. To insure the
satisfaction of the final landing configuration given by the
quality constraints (14), which is a more critical condition,
a reverse approach is adopted. Then the gliding trajectory is
computed backward from these final conditions through the
feasible glide set defined by constraints (17) and the space
discretized state equations (16). With the objective of
getting a smooth flyable trajectory which avoids wasting
unnecessarily the remaining hydraulic energy used to
control the aerodynamic actuators (elevator, THS, flaps and
aero brakes) along the engine-out glide trajectory, a new
optimization criterion is adopted here. This surrogate
criteria allows penalizing large variations on pitch attitude
angle, descent path angle, speed and flight level, so that its
evaluation along a feasible path i
k
Pleading to state iat stage
k is given by a formula such as:
)( Ts
s
Es
s
s
s
Ps
i
kEC T
i
k
''' ¦
OJOTO JT
(20)
Here ss
JT
OO
,and s
ET
O
are positive weights whose values
change with the distance to the landing site.
Dynamic programming, either direct or reverse, considers
at each stage different feasible states and selects for each of
them the best path leading to them from the initial state at
the first stage of the search process. Under a given value of
input parameter i
s
T
at stage s, backwards integration is
used to assess the additional cost involved in going from
state (s,i) to a new feasible state at the next stage of the
search process.
However, whatever the size of the discrete steps adopted to
perform this reverse search process, from one stage to
another, a large number of new states should be generated
to guarantee the accuracy of the resulting solution. This
leads to an explosive number of solutions to be considered
when the stage order increases. So the explosion of the
points must be avoided to insure the computer processes the
problem. After each backward integrating, many points
should be cut by using the dynamic programming principle.
Here, to alleviate this foreseeable computational burden, a
heuristic melting procedure is developed where closer
states to a central state of the current stage in the search
process are deleted while this central state is maintained.
The distance ij
s
' between two states i and j of stage s which
has been adopted to generate these clusters within one stage
is given by:
2
max
2
2
max
2
2
max
2)()()(
J
JJ
OOO J
j
s
i
s
j
s
i
s
z
j
s
i
s
V
ij
sz
zz
V
VV
' (21)
Here a two level weighting has been adopted:
maxmax,zV and max
J
are scaling parameters and zV
OO
,and
J
O
with 1
J
O
O
O
zV are positive relative weightings.
The above approach which has been developed is basically
an open loop approach and requires a very large
computational effort which is unlikely to be performed on
board an aircraft which is already in a critical engine-out
situation. Our proposal here, which should be developed in
the near future is to take profit of the amount of data
generated by the reverse dynamic programming search
process, considering different situations and parameters
such as aircraft initial flight level, altitude and mass, to train
a neural network devise designed to generate pitch angle
directives at each point along the descent so that the glide
trajectory leads safely to the landing situation. Here the
computational burden associated with reverse dynamic
programming is taken into profit to generate the training
data base for the neural network [15].
The generated pitch angle directives can be either sent to
the autopilot when it is still operating or to a flight director.
In that last case this will allow this maneuver to be
performed efficiently in manual mode by the pilot. Observe
that along the glide trajectory, each new solicitation of the
neural network will generate new piloting directives in
accordance with the current situation of the aircraft which
is also the result of external perturbations such as wind.
6SIMULATION RESULTS
A simulation study has been performed using the RCAM
wide body transportation aircraft model [16]. Then
considering the case in which an engine failure occurs
150km away from a possible landing site, different glide
trajectories obtained by reverse dynamic programming are
displayed on Figures 5. and on Figure 6 according to
different initial situations.
It appears that if the aircraft has a large initial total energy,
which means high speed and/or high altitude, the resulting
glide trajectory is not be very smooth: the speed and
altitude are subject to large and rapid changes so that the
aircraft loses energy in excess sufficiently quickly to arrive
to the landing site with acceptable flying parameters. When
initial total energy is not too much excessive, the resulting
glide trajectories result to be smoother.
For example, for initial conditions with an Figure 7. and
Figure 8. display an optimized glide trajectory in the case
in which initial altitude is 10km (FL330) and initial
airspeed is 200m/s (about 400 knot) .
Figure 9. and Figure 10. display the landing range which
can be reached safely by an aircraft whose initial glide
conditions are an altitude of 10km and an airspeed of
2012 24th Chinese Control and Decision Conference (CCDC) 101
180m/s (about 360 knot). The largest obtained glide range
is about 137 km while the shortest obtained glide range is
116 km. Then in that case, the gliding aircraft can reach
safely landing sites located between 116km and 137km
away. Observe on these figures that, the shorter the range,
the rougher is the trajectory. Comparing figures 7. and
Figure 9. , it appears also that with a higher initial airspeed
the gliding aircraft range is also higher.
These numerical results indicate that reverse dynamic
programming can be used to solve the glide trajectory
generation problem and contribute to the design of a glide
trajectory generator either off line or on line.
Fig 5. Optimal glide trajectories with different initial speeds
3-dimensional representation
Fig. 6 Optimal glide trajectories in vertical plane with different
initial altitudes
Fig 7. A smooth optimal glide trajectory in 3-D
Fig 8. A smooth optimal glide trajectory in vertical plane
Fig 9. Trajectories of aircraft in vertical plane with different
landing ranges, 3D view
Fig 10. Trajectories of aircraft in vertical plane with different
landing ranges
102 2012 24th Chinese Control and Decision Conference (CCDC)
7CONCLUSION
The purpose of this communication has been to present the
first results of a study turned towards the design of an
emergency management system able to cope with an
engine-out situation for a transportation aircraft. The main
contributions of this communication are:
- a review of the quasi steady glide range for a
transport aircraft;
-the proposal of a new representation of the flight
dynamics of a gliding aircraft with the
introduction of a spatial dimension as
independent variable;
-the development of a solution strategy based on
backwards integration and reverse dynamic
programming whose feasibility is supported by the
displayed simulation results.
This work should be completed by the integration of lateral
maneuvers and the consideration of the effect of wind over
the glide trajectory. This last point could be tackled by the
development of an adaptive approach based on the online
estimation of wind and the use of a neural machine to
generate control directives on a reactive basis. This remains
for further studies.
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2012 24th Chinese Control and Decision Conference (CCDC) 103
