Air Traffic Control Quarterly, Vol. 12, No. 1, 2004
A FORMAL APPROACH TO THE ANALYSIS OF AIRCRAFT
PROTECTED ZONE
Rachelle L. Ennis and Yiyuan J. Zhao
The protected zone represents a region around a given aircraft that no other
aircraft should penetrate for the safety of both aircraft, and defines the minimum
separation requirements. In this paper, three major components of the protected
zone and their interplays are identified: a vortex region, a safety buffer region, and
a state-uncertainty region. A systematic procedure is devised for the analysis of the
state-uncertainty region. In particular, models of trajectory controls are developed
that can be used to represent different modes of pilot and/or autopilot controls, such
as path feedback and non-path feedback. Composite protected zones under various
conditions are estimated, and effective ways to reduce sizes of protected zones for
advanced air traffic management are examined.
INTRODUCTION
Commercial aircraft must be sufficiently separated from one another in order to ensure
safety. In fact, there is a region around each aircraft in flight that no other aircraft should
penetrate for the safety of all aircraft. This region is defined in this paper as the protected
zone. The protected zone defines minimum separation standards among commercial
aircraft and constrains the flow of air traffic.
The current separation standards were determined many years ago, but the method
by which separation standards were developed is not well documented. It appears to
have been based upon “radar accuracy, display target size, and controller and pilot
confidence” [Thompson, 1997]. While there have been some significant efforts on
analyzing oceanic separation standards and separation standards for parallel runway
operations, domestic separation standards have been by and large unchanged. Advances
in technology and operational concepts over the past several decades warrant an
investigation into separation requirements.
Significant amount of work has been conducted on conflict alerting and avoidance
that assumes certain minimum separation standards [Kuchar & Yang, 1997, Teo &
Tomlin 2003]. There has also been extensive work on collision risk modeling
[FAA/Eurocontrol 1998, Blom & Bakker 2002]. These models can calculate probabilities
of aircraft collisions under specified flight conditions. Separation requirements may be
analyzed indirectly using these models by evaluating the risk levels that certain
separation standards introduce into the airspace system. To complement the above works,
a formal approach is needed for direct studies of minimum separation standards.
1
Recently, Reynolds & Hansman [2000] identified factors involved in defining
aircraft separation standards and discussed the importance of accurate state information
for controllers in maintaining separation standards. The current paper develops a formal
approach that properly combines these factors in the analysis of the protected zone and
thus minimum separation standards.
Specifically, three distinct components of the protected zone are identified and
their interplay analyzed: the vortex region, the safety buffer region, and the stateuncertainty region. A methodology for systematically estimating the state-uncertainty
region that accounts for different trajectory control modes is developed. Estimates of the
state-uncertainty region are obtained and are then properly combined with those of the
vortex region and the safety buffer region to obtain estimates of the entire protected zone.
Results of this paper may be used for analyses in conflict detection, conflict resolution,
and collision risks.
COMPOSITION OF THE PROTECTED ZONE
In this paper, the protected zone is defined as a region around a given aircraft that, if
violated, the safety of either the given aircraft, or an intruding aircraft, or both, could be
compromised. It represents a fundamental non-intrusion zone around a given aircraft.
The protected zone defines minimum separation standards among aircraft. Its violation is
considered to be a conflict.
Let us first assume that the position of an aircraft can be accurately known
through surveillance. A basic protected zone would consist of just two regions in this
case (Figure 1). The trailing vortices originating from the wing tips of the aircraft create
rolling moments, which could potentially overpower the roll control of a following
aircraft. Additionally, these same trailing vortices can cause vertical turbulence powerful
enough to risk the safety of passengers or even cause damage to a following or a cross
trail aircraft [Rossow & James, 2000]. As a result, the vortex region constitutes a part of
the protected zone. At the same time, a buffer is needed around the aircraft body that no
other aircraft should ever penetrate as a safety protection. This safety buffer region
constitutes the other part of the basic protected zone.
In reality, however, inaccuracies of aircraft surveillance create uncertainties in the
knowledge of aircraft state information for ground ATC and other aircraft. In order for air
traffic controllers and/or other pilots to maintain sufficient inter-aircraft separations, they
need to identify a region in which each aircraft is located over a certain time frame. This
region is called a state-uncertainty region for a given aircraft. Because each point inside
the state-uncertainty region represents a likely aircraft position in actual flight, a basic
protected zone discussed above must be reserved for each point in the state-uncertainty
region; resulting in a composite protected zone as shown in Figure 2.
A related but subtly different concept is called the required action range. In order
to ensure that the protected zone is not violated, involved aircraft must start avoidance
maneuvers well ahead of the protected zone boundary. This is because aircraft cannot
2
make instantaneous position changes due to performance limitations and pilots/controller
reaction times. A required action range represents the smallest relative separation by
which proper corrective actions must be taken in order to maintain the minimum
separation requirements or to avoid a potential conflict in time. It is difficult to
completely separate the definitions of the protected zone and the required action range,
since both depend on pilot action times. For the convenience of systematically analyzing
effects of various factors on separation requirements, the protected zone is defined to be
independent of relative motions and performance limits of aircraft involved in a potential
conflict as well as controller reaction times, with the exceptions that the influence of the
wake vortex depends on the type of aircraft that follows and the estimation of the
protected zone requires the use of an estimated typical pilot maneuver time in avoiding
potential conflicts. In comparison, the concept of required action range is defined to
depend on relative aircraft motions, aircraft performances, as well as pilot/controller
reaction times.
Therefore, there are three related separation requirements. The safety buffer
region represents the absolute minimum separation for safety and must be maintained
during a close encounter. The protected zone defines the separation requirement that
controllers and pilot strive to maintain under normal air traffic control operations. Finally,
the required action range reflects the relative range or time at which controllers and/or
pilots should start to maneuver correctly in order to respect the protected zones.
Once the protected zone is defined, a required action range can be determined. In
Kuo & Zhao [2001] a systematic procedure for theoretically determining least required
action ranges is presented based on optimal control theory. To determine practical
required action ranges for the current ATC system, on the other hand, many factors must
be considered that include relative aircraft geometry, aircraft performance limits, radar
sweep times, controller reaction times, communication times, and pilot response times.
The focus of this paper is on the analysis of the protected zone. The study of required
action ranges will be considered in future works.
MODELING OF COMMERCIAL AIRCRAFT FLIGHT
In order to estimate state-uncertainty regions, models of aircraft dynamics, potential error
sources, and trajectory controls are needed.
Aircraft Equations of Motion
For a first-order analysis of the protected zone, a point-mass aircraft model is adequate.
With the assumption of flat earth and coordinated turns, a set of three-dimensional pointmass equations of motion for a commercial aircraft can be derived as follows.
V = T g − g sin γ − fV
1
Ψ=
g L sin φ − f Ψ
V cos γ
[
3
(1)
]
(2)
1
[g L cos φ − g cos γ + fγ ]
V
x = V cos γ sin Ψ + Wx = Vg sin ΨI
γ =
(3)
(4)
y = V cos γ cos Ψ + W y = V g cos ΨI
(5)
h = V sin γ + Wh
1
T = (Tc − T )
(6)
(7)
τe
where V is the true airspeed, T is the normalized excess thrust, L is the normalized lift,
g is the acceleration of gravity, γ is the air-relative flight path angle, Ψ is the air-relative
heading angle measured clockwise from the North, ΨI is the inertial heading angle, φ is
the aircraft bank angle, ( x, y ) are the aircraft positions in the (East, North) direction, h is
the altitude, ( fV , f Ψ , f γ ) are modeling uncertainties, ( W x , W y , Wh ) are wind components
along (East, North, Up) directions, Tc is the normalized excess thrust control command,
and τ e is the engine thrust response time. From these equations the ground speed can be
determined as
Vg = ( x ) 2 + ( y ) 2 = (V cos γ sin Ψ + W x ) 2 + (V cos γ cos Ψ + W y ) 2
(8)
In these equations, the state variables are [ V , Ψ, γ , x, y , h, T ] , and the control
variables are [ L , φ , Tc ] . In comparison, [ fV , f Ψ , f γ , W x , W y , Wh ] are random disturbances.
Models of Trajectory Control
Flying aircraft are controlled by pilots or autopilots to follow commanded trajectories.
Let us assume that at some time t0 = 0 , an aircraft located at [ x 0 , y 0 , h0 ] is flown to
follow the commands
[hc (t ),Vc (t ), Ψc (t )]
(9)
where the time-dependence allows for full generality in expressing different phases and
maneuvers of aircraft flights. The horizontal part of the commanded trajectory can then
be constructed from
x c = Vc (t ) sin Ψc (t ) + Wˆ x (t )
y = V (t ) cos Ψ (t ) + Wˆ (t )
c
c
c
y
(10)
(11)
with x c (0) = x 0 , y c (0) = y 0 , where Wˆ x (t ) and Wˆ y (t ) are estimated wind components
available to the aircraft.
4
In actual flights, pilots or autopilots may use different modes of trajectory control
to follow flight commands in the presence of navigation errors and disturbances. Actual
control strategies can be very complicated. In this paper, general models of trajectory
controls are developed using the method of feedback linearization [Slotine & Li, 1991].
A vertical trajectory control model is first derived for an altitude command hc (t ) .
The idea of feedback linearization is to derive necessary control functions from a
specification of desired closed-loop response dynamics. For example, the normalized lift
may be used to control the aircraft altitude, and the resulting trajectory control is of
second-order. The desired altitude response dynamics can be stated as
(h − hc ) + KV 1 (h − hc ) + KV 2 (h − hc ) = 0
(12)
where KV 1 > 0 , KV 2 > 0 are feedback control gains. Assuming in consistence with
commercial flights T << L , | γ |<< 1 and for most of the times during a flight | φ |<< 1 ,
Eqs. (1) through (6) lead to
h = V sin γ + γV cos γ + Wh ≈ g ( L − 1) .
Substituting this into Eq. (12), the lift control law can be derived as
L =1−
KV 1 ˆ K
(h − hc ) − V 2 (hˆ − hc ) .
g
g
(13)
In Eq. (13), hˆ and ĥ represent the estimated or measured values from the aircraft
navigation system;
ˆ h = h + nh
hˆ = h + nh
and
(14)
where nh and nh are the corresponding navigation/estimation errors.
The general vertical trajectory control model in Eq. (13) can be used to describe
either a path feedback control or anon-path feedback control. For example if a constant
altitude command hc is used then hc = 0 , and KV 2 > 0 . This control law corresponds to a
path feedback control. On the other hand,
if an altitude rate (climb rate, descent rate,
flight path angle, etc) is specified then hc ≠ 0 but KV 2 = 0 . This corresponds to a non-path
feedback strategy. In addition, the feedback coefficients can be selected as
KV 1 = 2ζ V ωV
KV 2 = ωV2
5
(15)
where the damping ratio ζ V and the natural frequency ωV can be selected to match typical
response characteristics of a human pilot or an autopilot [Warren, 1999]. For a jet
commercial transport, typical values of a human pilot are: ζ V = 0.4 ~ 0.6 and
ωV = 0.05 ~ 0.5 rad/sec. If KV 2 = 0 , KV 1 can be selected to match the typical altitude rate
response time constant, e.g. KV 1 = 0.01 ~ 0.1 sec.
In a similar manner, longitudinal and lateral trajectory control models can be
derived. Let ξ represent the along-track deviation of the actual aircraft position from the
commanded trajectory and η represent the cross-track deviation. Figure 3 leads to
ξ sin Ψc
=
η
or,
cos Ψc
cos Ψc
− sin Ψc
x
− xc
y − yc
η = ( x − x c ) cos Ψc − ( y − y c ) sin Ψc
ξ = ( x − x c ) sin Ψc + ( y − y c ) cos Ψc .
(16)
(17)
Assuming | Ψ − Ψc |<< 1, | γ |<< 1, L ≈ 1, T << L , and neglecting lateral wind
measurement errors, the lateral trajectory control model can be approximated as
sin φ ≈
Vc
K V ˆ KL2
ηˆ
Ψ c − L1 c ∆Ψ
−
g
g
g
(18)
where
ˆ = Ψ − Ψ + n , ηˆ = η + n .
∆Ψ
c
Ψ
η
(19)
When K L 2 = 0 , this lateral trajectory control law describes a non-path feedback control
mode. Otherwise, it describes a path-feedback control mode. The differences between
human pilots and autopilots can again be described by the use of different feedback
control coefficients.
Following the same procedure and approximations, the longitudinal trajectory
control can be modeled by
[
]
(20a)
V
K
K
Tc = sin γ + c − s1 Vˆg − Vc − s 2 ξˆ
g
g
g
[
]
(20b)
Vˆ = V + nV , Vˆg = Vg + nV , ξˆ = ξ + nξ .
(21)
V
K
Tc = sin γ + c − s1 Vˆ − Vc
g
g
where
6
Eq. (20a) describes a non-path feedback airspeed control. In comparison, Eq. (20b)
describes a non-path ground speed control if K s 2 = 0 , or a path-feedback position control
if K s 2 ≠ 0 . The longitudinal position control may be used continuously or periodically.
In the above, the lift control in Eq. (13) corresponds to the “path-on-elevator”
control mode whereas Eq. (20) reflects the “speed-on-throttle” control mode. Other
control modes can be modeled similarly. Details are omitted.
NOMINAL SURVEILLANCE TRAJECTORIES
External observers determine the state information of a given aircraft through
surveillance. A typical surveillance system operates periodically every TS sec. For
example, TS = 12 sec for en route radars. Mathematically, the surveilled aircraft state may
be expressed as
x n ,0 = x0 + ∆x S , y n, 0 = y 0 + ∆y S , hn , 0 = h0 + ∆hS
(22)
Vn , 0 = Vc (0) + ∆VS , Ψn, 0 = Ψc (0) + ∆ΨS
(23)
where ∆( ) S ’s represent surveillance errors and
∆RS = ( ∆x S ) 2 + ( ∆y S ) 2
(24)
is the horizontal position determination error. Short-term intents may be expressed in the
form of constant horizontal acceleration components and vertical rate.
Vn = Vc (0) + ∆VS , Ψ n = Ψ c (0) + ∆Ψ S , hn = hc (0) + ∆hS
(25)
where (∆VS , ∆Ψ S , ∆hS ) represent intent estimation errors.
In general, flight intents may be expressed in a wide variety of forms. Correct
determination of an aircraft’s flight intent is crucial to accurately estimating its future
flight paths. In the current ATC system intent is typically known through pre-filed flight
plans and mandatory compliance with controller commands. In the proposed Free Flight
environment or a failure condition, a situation could arise where the intent is unclear.
However, incorporating intent errors into the construction of the protected zone can
drastically increase its dimensions, since the lack of knowledge of intent magnifies the
state-uncertainty region. For the efficiency of airspace operations, the protected zone
should be defined under most likely conditions. In this paper, the intent errors
(∆VS , ∆Ψ S , ∆hS ) are assumed to be very small, and the protected zone is defined with
known intents. Possible intent errors can be considered in devising strategies for conflict
detection and avoidance [Yang & Kuchar, 1997].
7
Based on surveilled state information of a target aircraft, a nominal surveillance
trajectory can be constructed as an estimate of likely future positions of the target aircraft
over a certain time period [0, t f ] .
Vn (t ) = Vn , 0 + Vn t
(26)
Ψn (t ) = Ψn , 0 + Ψ n t
x n ( t ) = x n , 0 + W x ,n t +
Vn , 0
Ψn
y n (t ) = y n , 0 + W y , n t +
cos Ψn ,0 −
Vn ,0 + Vn t
Ψn
(27)
Vn ,0 + Vn t
Vn
V
2 sin Ψn ,0 −
cos Ψn (t ) + n 2 sin Ψn (t )
( Ψn )
Ψn
( Ψn )
sin Ψn (t ) −
Vn , 0
Ψn
sin Ψn ,0 +
Vn
[cos Ψn (t ) − cos Ψn,0 ]
( Ψn ) 2
hn (t ) = hn , 0 + h!n t
(28)
(29)
(30)
where Wx ,n and W y ,n are wind components estimated by the surveillance system. They are
assumed constant and equal to the estimated
wind components used onboard aircraft:
"
ˆ
ˆ
Wx ,n = Wx , W y ,n = W y . In the case of Ψn = 0 , the horizontal trajectories become
1
xn (t ) = xn ,0 + (Vn,0 sin Ψn ,0 + Wx,n ) t + (V#n sin Ψn ,0 ) t 2
2
1
y n (t ) = y n, 0 + (Vn ,0 cos Ψn ,0 + W y ,n ) t + (V$n cos Ψn ,0 ) t 2 .
2
(31)
(32)
For estimating nominal surveillance trajectories over a long term such as for
oceanic flights, the acceleration intents and the estimated wind components can vary with
time. Numerical integrations can be used to obtain nominal surveillance trajectories.
A PROCEDURE FOR ESTIMATING THE STATE-UNCERTAINTY REGION
The state-uncertainty region is defined as the contour that contains largest likely
deviations of actual aircraft trajectories [ x (t ), y (t ), h (t )] from the nominal surveillance
trajectories [ x n (t ), y n (t ), hn (t )] over a certain time range [0, t f ] .
Expressions of Trajectory Deviations
Consider [a (t ), b(t )] as the along-track and cross-track deviations of actual trajectories
from nominal surveillance trajectories respectively. At a given point in time, these
deviations are defined relative to the current nominal heading Ψn (t ) . Similarly as for Eqs.
(16) and (17), one has
a (t ) = ( x − xn ) sin Ψn + ( y − y n ) cos Ψn
b(t ) = ( x − x n ) cos Ψn − ( y − y n ) sin Ψn .
8
(33)
(34)
In comparison, vertical differences between actual and nominal surveillance trajectories
can be expressed as
∆h = h − hn (t ) = h − hc (t ) + hc − hn = ∆hc (t ) + ∆hS + ∆h%S t
(35)
where ∆hc (t ) reflects deviations of actual aircraft altitudes from commanded altitudes.
With the assumption of negligible intent estimation errors, the differences
between actual aircraft trajectories and nominal surveillance trajectories may be
decomposed into three components (Figure 4): position surveillance errors (∆RS , ∆hS ) ,
propagation of velocity surveillance errors over time (t ∆VS , tVn , 0 ∆ΨS ) , and deviations of
actual trajectories from commanded trajectories (ξ ,η , ∆hc ) . Relating to concepts used for
defining flight paths onboard an aircraft [FAA AC 120-29A, 2002, Kayton & Fried,
1997], (ξ ,η , ∆hc ) represent combinations of flight technical errors and navigation system
errors. For the convenience of discussions in this paper, these combinations will be
referred to as the total onboard system errors.
Monte Carlo Simulations
Eqs. (1) through (21) constitute a complete set of equations. This set can be numerically
integrated forward repeatedly in Monte Carlo simulations [Kuchar 1996] to study
differences between actual aircraft trajectories and nominal surveillance trajectories over
a certain time period.
In the Monte Carlo simulations, modeling errors, wind measurement errors, and
navigation errors are all assumed to be independent, white noise random processes. In
reality, these random processes may not be white. For the purpose of estimating
trajectory deviations, these assumptions are reasonable. In addition, all random processes
are assumed to follow uniform distributions, since they represent the worse case within
specified ranges. In actual flights, random variables may not be of uniform distributions.
The use of different probabilistic distributions affects the size of the estimated stateuncertainty regions somewhat, but does not fundamentally change the basic
characteristics of the region. The uniform probability density function for a generic
random variable z over a specified range is given by
p( z ) =
)
&
1
(
&
'2 B
z ∈ [− B + z , B + z ]
(36)
0
otherwise
where z is the mean value. In the following examples, zero means are assumed.
Finally for good numerical accuracies in simulation studies, all state variables are
normalized as follows:
9
d ( ) Vn , 0 d ( )
V
( x , y, h, ξ , η )
t
, ( x , y , h , ξ ,η ) =
,
τ
=
,
=
.
Vn,0
Vn2,0 g
Vn, 0 g dτ
g dt
Details are omitted.
V =
Choice of the Final Time
An appropriate choice of the final time should be the larger one of the
surveillance interval TS and some typical maneuver time TM needed for conflict
avoidance. Whenever a new surveillance measurement is made on a target, a different
nominal surveillance trajectory for this aircraft can be defined, resulting in a different
state-uncertainty region. On the other hand, another aircraft that is trying to avoid this
aircraft may not be able to change its maneuver plans easily once it initiates the maneuver
and thus would need some stable estimate of the protected zone for the target aircraft. In
other words, the state-uncertainty region for the protected zone definition should contain
the largest potential differences between actual aircraft positions and the nominal
surveillance trajectories during the maneuver time, as illustrated in Figure 5.
According to the FAA [FAA 1983], the pilot response time in case of an
emergency avoidance is on the order of TR = 12.5 sec. This is supported by a human
operator model by Hess [1987]. A typical maneuver time can therefore be estimated as
TM = (2 ~ 3) TR ≈ 40 sec. This time frame is consistent with the alert time used in TCAS
[Williamson & Spencer 1989, Harman 1987]. For oceanic flights, on the other hand,
TS > TM and the surveillance interval TS should be used for estimating the stateuncertainty region.
Estimating the State-Uncertainty Contour
To estimate the contour of the state-uncertainty region from simulation results, a cylinder
with the smallest volume that contains all the simulated data points may be found. For
example, the vertical dimension of the contour can be determined from
H = 2 max ∆hmax, k
k∈[1, 2 ,..., N ]
(37)
where N is the total number of simulated data points, and ∆hmax, k is the largest vertical
deviation during the kth simulation,
∆hmax, k = max | hk (t ) − hn (t ) | .
t∈[ 0 ,t f ]
(38)
Assume the horizontal shape of the state-uncertainty contour to be an ellipse, its
horizontal dimensions may be determined from the following optimization problem
min
A, B
I = AB
(39)
subject to
10
(a max, k ) 2 (bmax, k ) 2
+
≤ 1,
A2
B2
for k = 1,2, *, N
(40)
where a max, k and bmax, k are the largest along-track and cross-track deviation respectively
during the kth simulation
a max, k = max | a k (t ) |, bmax, k = max | bk (t ) | .
t∈[ 0,t f ]
(41)
t∈[ 0 ,t f ]
Assuming B = βA and for a given β , the optimal A can be found as
A=
max
k∈[1, 2 ,..., N ]
0
.
2
/ ( a max, k )
+
(bmax, k ) 2
β2
+
,
.
(42)
Note that when β = 1 , it corresponds to a circle. A few different values of β may then be
taken and the one that offers the smallest AB = βA2 can be selected. In this paper, β = 1
is used.
Alternative to finding the region of smallest volume containing all worse-case
position deviations, one may define contours representing a certain percentile of the
worst-case deviations, such as 95-percentile. The procedure would be very similar.
A Special Solution
In the case of constant heading flights with small heading angle measurement errors, a
simplified solution can be found. Eqs. (16) and (17) suggest
x − x c = ξ sin Ψc + η cos Ψc
y − y c = ξ cos Ψc − η sin Ψc
.
(43)
(44)
Substituting these relations into Eqs. (33) and (34), we obtain
a (t ) = ξ cos( Ψn − Ψc ) + η sin( Ψn − Ψc ) + ( xc − x n ) sin Ψn + ( y c − y n ) cos Ψn
b(t ) = −ξ sin( Ψn − Ψc ) + η cos( Ψn − Ψc ) + ( x c − xn ) cos Ψn − ( y c − y n ) sin Ψ .
(45)
(46)
After some algebra, it can be shown that
a max ≤ ξ max + ∆RS ,max + t f ∆VS ,max
(47)
bmax ≤ η max + ∆RS ,max + t f Vn ,0 ∆ΨS ,max .
(48)
At the same time, Eq. (35) leads to
11
∆hmax ≤ ∆hc ,max + ∆hS ,max .
(49)
In the above, ( ) max represents maximum likely errors or deviations. Using these relations,
Monte Carlo simulations are only needed to determine maximum values of the total
onboard system errors over the time interval [0, t f ] : (ξ max ,η max , ∆hc ,max ) , whereas the
largest likely surveillance errors can just be directly added to the maximum total onboard
system errors to obtain estimates of the state-uncertainty region. This assumption reduces
the number of independent random variables in the Monte Carlo simulations as well as
the number of necessary simulations to obtain accurate estimates of largest likely
deviations. Because Ψc ≈ Ψn is basically true when flight intents are correctly known, this
special approach is used below.
MAXIMUM TOTAL ONBOARD SYSTEM ERRORS (ξ max ,η max , ∆hc ,max )
In the extensive simulations, Simulink is used to generate random numbers at each
integration step, and the fourth-order Runge-Kutta numerical integration method in
Matlab is used to numerically integrate the differential equations forward. To avoid
numerical difficulties of integrating stochastic differential equations, values of the
random processes are sampled once and held constant for each integration step. At the
end of each simulation, the greatest deviations of aircraft positions from commanded
trajectories during the interval [0, t f ] are stored.
In order to determine an appropriate sample size or the number of data points,
simulations were run varying the sample size from 1,000 to 15,000 in 1,000 increments.
The maximum total onboard system errors tend to become stabilized within a certain
range after a sample size of 5,000. In order to be conservative, the sampling size of
15,000 is used for generating the following simulation results.
In the first example, non-path feedback controls in the longitudinal and lateral
direction are assumed whereas in the vertical direction, a path feedback control
maintaining a commanded altitude is used. In particular, the airspeed control of Eq. (20a)
is assumed in the longitudinal direction. A level cruise flight is considered. This example
serves as the reference case for the following studies. Figure 6 plots the 15,000 simulated
maximum aircraft position deviations from the commanded trajectory. The final
time t f = 40 s is used, and velocity commands are assumed to be Vc = 800 ft/s,
1
Ψc = 90 deg, and hc = 0 . Feedback gains in the pilot trajectory control model are
selected as: K S1 = 0.05 , K S 2 = 0 , K L1 = 0.05 , K L 2 = 0 , K V 1 = 0.05 , and K V 2 = (0.05) ,
corresponding to a damping ratio of 0.5 and a response bandwidth of 0.05 rad/sec in both
horizontal and vertical directions. Magnitudes for the various random processes are
selected as: BWx = 20 kn, BW y = 20 kn, BWh = 5 kn, B fV = 0.01g , B fψ = 0.001g ,
2
B fγ = 0.001g , BnV = 5 kn, Bnψ = 0.01 rad, Bnh2 = 5 kn, B nh = 50 ft, Bnξ = 50 ft, and
Bnη = 50 ft. It is interesting to note that the scatters of the total onboard system errors do
12
not depend on the direction of flight (Figure 6). Figure 7 shows a typical single
simulation trajectory with the above random parameters. Note that the maximum
deviation of the aircraft position from the commanded trajectory does not necessarily
occur at the end of the simulation integration.
Effects of feedback control gains on total onboard system errors are now
examined. In this simulation study, the frequency bandwidth of the control loops are
increased to 0.1 rad/sec in both horizontal and vertical directions, and the following
feedback control gains are used: K s1 = 0.1 , K s 2 = 0 , K L1 = 0.1 , K L 2 = 0 , K V 1 = 0.1 ,
and K V 2 = 0.01 . All other random parameters are assumed the same as in the reference
case of Figure 6. The simulation results are summarized in Figure 8, which shows that
tightening the control loops does not affect the horizontal contour of the total onboard
system errors significantly in the case of non-path horizontal trajectory control.
The above simulations are repeated now with the use of path feedback in the
longitudinal and lateral directions. For the nominal case of path feedback control, the
2
following feedback control gains are used: K s1 = 0.05 , K s 2 = (0.05) , K L1 = 0.05 ,
K L 2 = (0.05) , K V 1 = 0.05 , and K V 2 = (0.05) . These correspond to a damping ratio of
0.5 and a response bandwidth of 0.05 rad/sec in both horizontal and vertical directions.
All other random parameters are assumed the same as in the reference case of Figure 6.
A comparison of path and non-path feedback controls is shown in Figure 9. Figure 10
shows changes in total system errors when the response bandwidth is increased to 0.1
rad/sec, or the feedback control gains become: K s1 = 0.1 , K s 2 = 0.01 , K L1 = 0.1 ,
2
2
K L 2 = 0.01 , K V 1 = 0.1 , and K V 2 = 0.01 . Tighter feedback controls lead to smaller total
system errors when the path feedback control mode is used.
The maximum total system errors also depend on the choice of the final time for
the numerical integration. If the non-path feedback control model is used in the
longitudinal and lateral directions, along-track and lateral trajectory deviations of actual
flight from commanded trajectories grow as the final time of integration increases. The
vertical deviations tend to become stabilized within a range, due to the use of path
feedback control in the vertical direction. When the path feedback control model is used
in the horizontal direction, the total system error contour is not very sensitive to the
choice of the final time. Path feedback control modes are clearly better than non-path
feedback modes in terms of maintaining small total onboard system errors. On the other
hand, these modes may introduce additional pilot workload. The use of path feedback
control in the longitudinal direction may require regular throttle motion and thus consume
more fuel.
Simulation studies were also carried out to examine effects of distributions and
magnitudes of various random errors, and the effects of aircraft maneuvers on the total
system error contours. Wind measurement errors seem to play a significant role. The
basic structure of the total system error contour remains the same under the assumption of
zero intent errors. Details are omitted.
13
Using the contour estimation method discussed above, we obtain ξ max = 390 ft,
η max = 390 ft, and ∆hc ,max = 60 ft for the reference case in Figure 6 of horizontal non-path
trajectory feedback control and vertical path feedback control.
ESTIMATION OF THE PROTECTED ZONE
To put together an actual estimate for the protected zone, estimates for the stateuncertainty region, the safety buffer, and the vortex region are needed.
The Safety Buffer Region
In this paper, a cylinder centered at the aircraft center of gravity is defined as the safety
buffer region. The radius of the cylinder needs to be at least larger than half of the aircraft
length plus some safe distance, and the height needs to be larger than the height of the
aircraft plus some distance. For the purpose of obtaining a notion of the protected zone, a
safety buffer region of 0.1 nm (600 ft) in radius and 200 ft in thickness is used [Figure
11]. This would accommodate a B747 that has a length of 232 ft, a wing span of b= 194
ft, and a thickness of 64 ft.
The Vortex Region
In Rossow and James [2000], the vortex region is estimated to be a “200 mile long tube
of the atmosphere that begins at about one wing span in diameter and increases linearly
with distance to about 20 wing spans in diameter.” In actual flight conditions, intensities
of the trailing vortices become sufficiently weak in some distance behind an aircraft. For
the purpose of obtaining some notion of the size of the protected zone in the current
paper, the along-track dimension of the vortex tube is plotted to 3 nm where the diameter
increases 0.1 wingspan per nm, as shown in Figure 11. Strictly speaking, different
separation requirements are needed for different pairs of aircraft. For example, 4 and 5nm
wake separation are required for en route aircraft passing behind large or heavy aircraft.
The trailing vortices drop down in altitude anywhere from 500 to 1,500 feet below
the generating aircraft. A definitive study on the descent distance has not been made, and
adequate measurements are not available either. In the current paper, the vortex tube is
assumed to drop in altitude by a ratio of 1/50 over the backward horizontal distance
during the first 12 nm, which is when the long-wave instability occurs, and then the wake
does not descend any further afterwards1.
The above estimates do not consider the effects of winds and temperature on the
vortex region [Rossow, 2002]. Nor do they consider the dynamics of the vortex region
when the aircraft is turning. Strictly speaking, dimensions of the wake vortex region not
only depend on the characteristics of the generating aircraft, but also on those of the
trailing aircraft as well as ambient atmospheric conditions. It is still a challenge to
1
The authors thank Dr. Rossow for many helpful suggestions.
14
properly incorporate wake vortices into the estimation of the protected zone and
additional work is needed.
Estimation of the State-Uncertainty Region
Consider first the mechanical radar in use by air traffic control. The range
estimation by radar is quite accurate. But, the error in azimuth can be significant. The
resulting error region would then be an ellipse with a major axis perpendicular to the line
of sight to the radar. In practical implementation it is most convenient to assume a
spherically shaped error region with the major axis as the diameter. The worst-case
uncertainty in horizontal position measurement for mechanical radar is assumed to be
∆RS ,max = 0.5 nm [Hochwarth, 2002]. For the altitude measurement air traffic control uses
Secondary Surveillance Radar (SSR). Assuming a Mode C transponder, the altitude
quantization error is about 100 feet. It is assumed here that ∆hS ,max = 150 ft allowing for
possible altimetry errors. Velocity surveillance errors are assumed to be ∆V S ,max = 40 kn,
and ∆ΨS ,max = 0.004 rad. Combining surveillance errors and total onboard system errors,
an estimate of the state-uncertainty region with the use of mechanical radar is given as a
disk shaped region with a radius of 1.0 nm and a depth of 420 ft as shown in Figure 12(a)
GPS receivers are now used onboard aircraft to provide accurate aircraft state
measurements, and these measurements may be broadcast to the ground and other aircraft
in the concept of Automatic Dependent Surveillance-Broadcast (ADS-B) [RTCA, 1998].
Surveillance accuracy would then depend on the onboard GPS system accuracy, and the
frequency as well as effective number of digits in the broadcast. For the purpose of
comparison, the following error ranges are assumed for the use of GPS-based ADS-B
system for position and velocity measurements: ∆RS ,max = 15 m, ∆hS ,max = 22.5 m,
∆VS ,max = 0.12 m/sec, and ∆ΨS ,max = 0.002 rad [Misra & Enge, 2001]. Again combining
various error sources, an estimate of the state-uncertainty region with the use of GPSbased ADS-B concept is given as a disk shaped region with a radius of 0.075 nm and a
depth of 270 ft, as shown in Figure 12(b). This is significantly smaller than the stateuncertainty region assuming conventional ground radar.
Example Estimates of the Protected Zone
Figure 13 shows the top views of the basic protected zone, the protected zone based on
ground ATC radar surveillance, and the protected zone by assuming ADS-B with GPS
navigation, whereas Figure 14 compares the side views of the three types of protected
zones. The demonstrated dependence of the protected zone on surveillance accuracy is
highly consistent with the discussions in Reynolds & Hansman [2000], and with the
current FAA practice, in which the required horizontal separation is 5 nm for controlled
aircraft more than 40 nm from the radar site and 3 nm over the terminal area. In
particular, the estimated protected zone in Figures 13(b) and 14(b) for the use of
conventional radar basically agrees with the current FAA en route separation standards of
5 nm horizontally and 1,000 ft vertically [FAA, 2003].
15
These examples are only used to provide an idea of what the protected zone
would look like. They should not be taken as hard numbers, as different assumptions on
the magnitudes and distributions of the various random processes and pilot response
characteristics will surely change the sizes of the protected zones. On the other hand,
these examples do illustrate the application of the proposed formal procedure for the
analysis of the protected zones. The proposed formal analysis procedure can be used to
study protected zones and thus derive separation requirements for a wide range of traffic
scenarios. Further refinement of the protected zone analysis can be made by using rigidbody aircraft models in conjunction with the analysis procedure developed here.
Reduction of the Protected Zone
In current ATC practice, the protected zone is a standard for all aircraft and aircraft
geometries with the possible exceptions of aircraft takeoffs and landings. Working within
the realm of the current ATC system, the protected zone can most likely be reduced
through improved surveillance accuracy. If the position of an aircraft is known to a
greater degree of certainty, the state-uncertainty region will be smaller. This is the same
conclusion arrived at by Reynolds and Hansman [2000].
If the surveillance errors can be made small enough and become comparable to
the total onboard system errors, reduction of the total onboard system errors can further
reduce the protected zone. This may be achieved by using path feedback control modes
and/or obtaining more accurate onboard measurements of wind components and aircraft
states.
Furthermore, the protected zone for each aircraft and pilot can be unique. As
shown in this paper, the definition of the protected zone depends upon the dynamics of
the aircraft and the characteristics of the pilot. The effective dimensions of the vortex
region would also depend upon the relative characteristics of the aircraft behind a given
aircraft. For example, a heavy aircraft can follow a light aircraft at a closer distance than
a light aircraft can follow a heavy aircraft due to the differing capacities of roll control.
Therefore, further reductions of minimum separation requirements may be achieved
through individually designed and/or dynamically-varying protected zones.
CONCLUSIONS
A formal method for evaluating the protected zone is presented. The protected zone is
defined as a region around a given aircraft that, it violated, would cause danger to either
the own aircraft, or the intruder aircraft, or both. It is related to but different from the
required action range, which is the least relative separation between two aircraft at which
correct avoidance maneuvers must be initiated to prevent the violation of the protected
zone.
Three distinct components of the protected zone are identified as the vortex
region, the safety buffer region, and the state-uncertainty region. The safety buffer region
16
and the vortex region are essentially added to one another. These two regions must then
be considered at all locations in the state-uncertainty region. In this paper, a systematic
procedure is presented for the analysis of the state-uncertainty region, which is defined as
the contour of likely worse-case deviations of actual trajectories from nominal
surveillance trajectories over a certain period of time. These deviations can be
decomposed into position surveillance errors, propagation of velocity surveillance errors,
and total onboard system errors consisting of flight technical errors and navigation
system errors. A Monte Carlo simulation approach is developed to estimate deviations of
actual aircraft positions from either commanded or nominal surveillance trajectories. It is
found that path feedback modes of trajectory control result in smaller total onboard
system errors than non-path feedback modes.
The composite protected zones for various parameter scenarios are presented.
Shrinking surveillance uncertainties is found to be central in reducing the size of the
protected zone. After the surveillance errors become sufficiently small, reduction of total
onboard system errors becomes important for the reduction of the protected zone.
ACKNOWLEDGMENTS
This research is supported by the Terminal Air Traffic Management Branch at NASA
Ames Research Center under NCC2-990. We thank Ron Hess and Vernon Rossow for
many helpful discussions and suggestions. We also thank anonymous reviewers for many
constructive and helpful comments.
ACRONYMS
ADS-B
ATC
FAA
GPS
SSR
TCAS
automatic dependent surveillance- broadcast
air traffic control
Federal Aviation Administration
Global Positioning System
secondary surveillance radar
Traffic Alert and Collision Avoidance System
SYMBOLS
( fV , f Ψ , f γ )
modeling uncertainties
g
h
K( )
acceleration of gravity
altitude
feedback control gains
L
n( )
V
Vg
normalized lift
navigation/estimation errors
true airspeed
ground speed
T
normalized excess thrust
17
TM
TS
( W x , W y , Wh )
( x, y )
η
γ
τe
ω( )
typical time needed for conflict avoidance
surveillance interval
wind components along (East, North, Up) directions
aircraft positions in the (East, North) direction
cross-track trajectory deviation
air-relative flight path angle
aircraft bank angle
air-relative heading angle measured clockwise from the North
inertial heading angle
engine thrust response time
closed-loop response frequency
ζ()
ξ
Xc
Xn
X max
damping ratio
along-track trajectory deviation
commanded value of X
nominal surveilled value of X
maximum value of X
X̂
estimated value of X
φ
Ψ
ΨI
REFERENCES
Blom, H. A. P., and Bakker, G. J. (2002), “Conflict Probability and Incrossing
Probability in Air Traffic Management,” Proceedings of the IEEE Conference on
Decision and Control, December.
Federal Aviation Administration (2002), Criteria for Approval of Category I and
Category II Weather Minima for Approach, AC 120-29A, August.
FAA (2002), Order 7110.65N Air Traffic Control. February. See also:
http://www1.faa.gov/atpubs/ATC/index.htm.
FAA/Eurocontrol (1998), A Concept Paper for Separation Safety Modeling, An
FAA/Eurocontrol Cooperative Effort on Air Traffic Modeling for Separation Standards,
May 20, 1998. http://www1.faa.gov/asd/ia-or/pdf/cpcomplete.pdf.
FAA (1983), Pilot’s Role in Collision Avoidance, AC 90-48C.
Hess, R. A. (1987), “A Qualitative Model of Human Interaction with Complex Dynamic
Systems,” IEEE Transactions on Systems, Man and Cybernetics, 17(1), January-February
1987, pp. 33-51.
Harman, W. H (1987), “TCAS: A System for Preventing Midair Collisions,” The Lincoln
Laboratory Journal, 2(3), pp. 437-457.
18
Hochwarth, J. K. (2002), “A Modularized Approach to Comprehensive System-Wide
Computer Simulation of Air Traffic Systems,” Ph.D. Dissertation, Dept. of Aerospace
Engineering and Mechanics, University of Minnesota, December 2002, Secs. 4.8 & 9.3.
Kayton, M. and Fried, W. R. (1997), Avionics Navigation Systems, Second edition, John
Wiley & Sons, Inc., p. 44, p. 647.
Kuchar, James K. (1996), “Methodology for Alerting-System Performance Evaluation”,
Journal of Guidance Control, and Dynamics, 19(2), March-April, pp 438-444.
Kuchar, James K. and Yang, Lee C. (1997), “Survey of Conflict Detection and
Resolution Modeling Methods”, AIAA, A97-37144, pp. 1388-1397.
Misra, P., and Enge, Per. (2001), Global Positioning System: Signals Measurements and
Performance, Ganga-Jamuna Press, Lincoln, MA, pp. 45, 197.
Kuo, H. V. and Zhao Y. J. (2001), “Required Ranges for Conflict Resolutions in Air
Traffic Management”, Journal of Guidance Control, and Dynamics, 24(7), March-April,
pp. 237-245.
Reynolds, Tom G. and Hansman, R. John (2000), “Analysis of Separation Minima Using
A Surveillance State Vector Approach”, presented at the 3rd USA/Europe Air Traffic
Management R and D Seminar, Napoli, 13-16 June.
Rossow, V. J. (2002), “Reduction of Uncertainties in Locations”, Journal of Aircraft,
39(4).
Rossow, V. J. and James, K. D. (2000), “Overviews of Wake-Vortex Hazards During
Cruise”, Journal of Aircraft, 37(6), pp. 960-962.
RTCA Special Committee-186 (1998), Minimum Aviation System Performance
Standards for Automatic Dependent Surveillance Broadcast (ADS-B), RTCA/DO-242,
February.
Slotine, J. E. and Li, Weiping (1991), Applied Nonlinear Control, Prentice Hall,
Englewood Cliffs, NJ.
Teo, Rodney and Tomlin, Claire (2003), “Computing Danger Zones for Provably Safe
Closely Spaced Parallel Approaches,” Journal of Guidance Control, and Dynamics,
26(3), May-June, pp. 434-442.
Thompson, S. D. (1997), “Terminal Area Separation Standards: Historical Development,
Current Standards and Processes for Change”, MIT Lincoln Laboratory Report ATC-258,
Jan., pp. 16.
19
Warren, A. W. (1999), “Vertical Path Trajectory Prediction for Next Generation Air
Traffic Management”, Presented at the 3rd USA/Europe ATM R&D Seminar.
Williamson, T. and Spencer, N. A. (1989), “Development and Operation of the Traffic
Alert and Collision Avoidance System,” Proceedings of the IEEE, 77(11), pp. 17351744.
Yang, Lee C. and Kuchar, James K. (1997), “Prototype Conflict Alerting System for Free
Flight”, Journal of Guidance, Control, and Dynamics, Vol. 20, No. 4, pp. 768-773.
BIOGRAPHIES
Rachelle Ennis received a bachelor degree in mathematics from Michigan State
University in 1999. She is currently a PhD student at the University of Minnesota
working in conjunction with Dr. Zhao. Her areas of interest are air traffic control and
unmanned aerial vehicles.
Dr. Yiyuan J. Zhao received his Ph.D. in Aeronautics and Astronautics from Stanford
University in 1989 and has been on faculty in the Department of Aerospace Engineering
and Mechanics at the University of Minnesota since. His research interests are
optimization techniques and computer methods with applications to air traffic control
automation, rotorcraft flight operation, and unmanned aerospace systems.
20
FIGURES
Safety Buffer
Region
Vortex Region
Figure 1. A basic protected zone when aircraft positions can be precisely known.
Composite
Protected Zone
State-Uncertainty
Region
Figure 2. A composite protected zone with surveillance errors.
21
∆y
η
ξ
( x, y )
Ψc
∆x
( xc , yc )
Figure 3. Along-track and cross-track errors.
Actual
Total Onboard System Errors
(ξ ,η , ∆hc )
Measured
Commanded
Surveillance Errors
Nominal Surveillance
Figure 4. Error components in trajectory deviations.
t0 = 0
t = tf
t0 = 0
TS
TS
Figure 5. Choice of the final time t f .
22
Figure 6. Scatter plots of non-path feedback control with nominal parameters.
23
Figure 7. Plots of a single simulation aircraft trajectory.
24
Figure 8. Effects of increased feedback gains in non-path feedback mode.
Figure 9. Comparison of path and non-path feedback control.
25
Figure 10. Effects of increased feedback gains in path feedback control.
260 ft
0.1 nm
(a)
0.1 nm
3 nm
495 ft
200 ft
235 ft
260 ft
Safety Buffer
Region
Vortex Region
(b)
Figure 11. A basic protected zone: (a) top view, (b) side view.
26
420 ft
1.0 nm
(a)
270 ft
0.075 nm
(b)
Figure 12. Estimate of state-uncertainty region, (a) assuming mechanical radar, (b)
assuming GPS-based ADS-B.
27
(a)
2.2 nm
2.04 nm
5.1 nm
(b)
0.193 nm
0.35 nm
3.25 nm
(c)
Figure 13. Estimates of the protected zones in the horizontal plane: (a) a basic
protected zone with perfect aircraft state information, (b) assuming mechanical
radar, (c) assuming GPS-based ADS-B.
28
(a)
1015 ft
5.1 nm
(b)
865 ft
3.25 nm
(c)
Figure 14. Side views of estimated protected zones: (a) a basic protected zone with
perfect aircraft state information, (b) assuming mechanical radar, (c) assuming
GPS-based ADS-B.
29