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Failure Boundary Estimation For Lateral Collision Avoidance
Manoeuvres
James Dunthorne1, Wen-Hua Chen and Sarah Dunnett
Abstract— This paper proposes a method for predicting
the point at which lateral collision avoidance manoeuvres
fail. It starts by defining the kinematic failure boundary
for different encounter geometries and velocities. This relies
on the assumption that the ownship aircraft is able to turn
instantaneously. A simple trajectory model is then developed
to improve the prediction, and with the knowledge of the
kinematic boundary, two optimisation algorithms are used
to estimate the true failure boundary. When the intruder is
travelling quickly when compared to the ownship aircraft,
results have shown that the failure boundary is much larger
due to the growth of the kinematic boundary. Knowledge
of the failure boundary is then used to determine collision
avoidance logic to ensure the correct decision is always made.
Knowledge of the boundary reduces the size and complexity
of the verification problem, and allows sensor resolution
requirements to be formally set.
Keywords: UAVs, Collision Avoidance, Sense & Avoid,
Failure Boundary Estimation, Clearance, Safety, Verification
I. INTRODUCTION
For many years the military has been using Unmanned
Aerial Vehicles (UAVs) in war-zones around the world.
Typically they have been operated within military controlled
airspace, which has allowed the incorporation of greater
functionality, without all of the regulatory hurdles that are
present within the civilian world [1]. It is not beneficial to
constrain commercial use of UAVs to segregated airspace
as this is very costly and time consuming. In order to fly
UAVs in non-segregated airspace, one of the challenges is
to develop and certify a collision avoidance system which is
capable of replicating the ability of a human pilot to “See &
Avoid” other airspace users when other forms of separation
have failed.
Collision avoidance systems are safety critical in the sense
that failure could result in a catastrophic accident with
many lives lost. Because UAVs are operated remotely, by a
human pilot, there is no guarantee that the pilot will be able
to take responsibility for resolving the collision. In some
circumstances full control must be handed to the UAV to
avoid a conflict, and so these systems need to be verified to
a very high degree of confidence.
Recent research on the clearance of flight control and col-
lision avoidance systems has focused on optimisation based
methods [2], [3]. Computational load increases exponentially
1James Dunthorne, Wen-Hua Chen and Sarah Dunnett are from the
Aeronautical and Automotive Engineering Department at Loughborough
University, Loughborough, United Kingdom, LE11 3TU
j.dunthorne@lboro.ac.uk w.chen@lboro.ac.uk
s.j.dunnett@lboro.ac.uk www.lucasresearch.co.uk
with the number of problem dimensions, and so their use is
currently restricted to smaller problems. In order to verify
lateral collision avoidance systems, we first try to understand
the problem using formal verification techniques. One of the
objects of formal techniques are to model and predict when
a system is likely to fail. A stall warning system is a good
example [4].
This paper offers a method of predicting the amount of
time available before a collision avoidance system’s reso-
lution manoeuvre would fail to separate conflicting aircraft
adequately. This is referred to in this paper as Failure
Boundary Estimation (FBE) and has already been identified
as an important tool when in conflict scenarios [5]. In being
able to predict the failure boundary, collision avoidance logic
can be determined, the responsibility of when to manoeuvre
can be managed correctly, the size and complexity of the
verification problem can be reduced, and sensor resolution
requirements can be set.
II. COLLISION AVO IDAN CE ALGORITHM SEL ECT ION
Many collision avoidance resolution algorithms were re-
viewed so that the most promising solutions could be identi-
fied [6]. These included procedural resolutions [7], poten-
tial fields [8], optimisation based methods [9], intelligent
learning algorithms [10] and geometric based methods [11].
The resolution algorithm needed to be an analytical solution
which solved the collision avoidance problem mathemati-
cally. The mathematics could then be analysed and formal
methods could be developed. It was also important that the
algorithm was simple, predictable and was able to fit into
regulatory guidelines. Without these criteria, the algorithm
would struggle to be certified.
The geometric algorithm proposed by Goss [11] satisfied
all of the selection criteria. It calculates the heading com-
mand needed to transform the relative velocity vector VR
outside of the intruders cone of collision, as described in
[12]. By studying the equations used to provide the heading
command, it is possible to predict when the system will fail.
First is a brief explanation of how the collision avoidance
system works.
A. Algorithm Overview
The algorithm in [11] calculates 4 resolution trajectories,
one of which corresponds to a manoeuvre which moves
the relative velocity vector to the right hand edge of the
intruder’s safety bubble (clockwise rotation), as shown in
Figure 1. This shall be referred to as a right hand edge (RHE)
manoeuvre. The other three solutions are not considered, as
they represent either symmetrical or sub-optimal resolutions.
Fig. 1. Geometric algorithm description
The equation in [11] is simplified to two spacial dimen-
sions, xand y, as only lateral resolutions are considered.
ψHC =β+γ+ sin−1VB
VA
sin ψ6AB −β−γ(1)
where,
γ= sin−1R
d(2)
VAand VBare the magnitude of the ownship and intruder
aircraft’s velocities. Ris the horizontal safety bubble radius
as defined by local air regulation (normally 152.4m or 500ft),
dis the distance between the two aircraft, and βis the angle
subtended between the ownship aircraft’s heading, and the
intruders position on its horizon as shown in Figure 2. βis
measured between the limits −πto π.
R
d
VA
VB
βγ
Ownship
Intruder
Cone of
collision
Fig. 2. Definition of encounter variables
ψ6AB is the difference between the intruder’s heading, ψB
and ownship aircraft’s heading, ψA, and is measured between
0and 2π.
ψ6AB =ψB−ψA(3)
III. KIN EMATI C FAILURE BOUNDARY
By inspecting Equation (1) we can find when the algorithm
gives an imaginary solution. These cases define the point in
an encounter where it becomes impossible to resolve the
collision using that manoeuvre even if the ownship aircraft
were able to turn instantaneously. There are two cases when
this happens:
A. Case 1
The first case arises when:
d < R (4)
The implication of this is not very significant, but means
that if the ownship aircraft is already within the intruders
safety bubble, a safe resolution trajectory is not available.
B. Case 2
For the case where d≥R, the end term of Equation (1)
causes a trajectory to give an imaginary solution when:
VB
VA
sin ψ6AB −β−γ>1(5)
The maximum value a sine function can take is +1.
Therefore the kinematic boundary can only exist for cases
where:
VA≤VB(6)
C. Limits
Since we are only considering a RHE manoeuvre, γmust
be positive. Therefore γis constrained as follows:
0≤γ≤π
2(7)
By re-arranging Equation (5), and introducing the limits
above, we obtain the limits of the kinematic boundary.
π−sin−1VA
VB≤ψ6AB −β≤3π
2−sin−1VA
VB
(8)
D. Kinematic Boundary Distance
To calculate the distance, dK IN , at which the kinematic
boundary is breached, we re-arrange Equation (5). When
performing the inverse sine operation, two solutions are
produced, θand π−θ.
θ=ψ6AB −β−sin−1R
d(9)
When within the limits as described in the previous
section, the π−θsolution must be used.
dKI N =R
sin sin−1VA
VB+ψ6AB −β−π(10)
If the lower limit is breached, the θsolution is used.
dKI N =R
sin sin−1VA
VB+ψ6AB −β(11)
If the upper limit is breached, only the boundary as
described in Case 1 applies (i.e. d=R).
E. Kinematic Boundary Heading
When Case 2 is satisfied, the end term of Equation (1)
reduces to π
2. Substituting in the boundary distance, dKI N ,
we are able to find the kinematic boundary heading, ψKI N .
ψKI N =ψ6AB + sin−1VA
VB−π
2(12)
This angle corresponds to a trajectory which maximises
the separation between the two aircraft, and it sits perpen-
dicular to the new relative velocity vector. Turning beyond
this angle reduces separation and so it is used to constrain
the turn.
F. Analysis
Figure 3 shows a plot of the kinematic boundary distance,
dKI N , against the encounter geometry, ψ6AB , for three
different velocity ratios (VA
VB=0.2,0.6and 1.0). It is assumed
that the aircraft are involved in a direct collision so that βcan
be determined from velocity vectors. Limits, as described in
the previous sections, have been applied.
0 1 2 3 4 5 6
0
500
1000
1500
2000
2500
3000
3500
4000
Conflict Geometry, ψ<AB (rad)
Kinematic Boundary Distance, d KIN (m)
VA/ VB = 0.2
VA/ VB = 0.6
VA/ VB = 1.0
ASYMPTOTE
SAFE
REGION
Fig. 3. Kinematic Boundary Distance Variation for Different Velocity
Ratios
The left hand side of the graph corresponds to resolutions
which would be commonly referred to as overtaking ma-
noeuvres. These manoeuvres have a much larger kinematic
boundary distance, compared with the resolutions on the
right hand side of the graph.
When ψKI N = 0, the ownship aircraft is already travelling
in the direction of the kinematic boundary heading, and so
the separation between the aircraft cannot be increased in
that direction. This results in an asymptote at the point:
ψ∞=π
2−sin−1VA
VB(13)
The asymptotic point for each curve corresponds to en-
counter geometries where the relative velocity vector is
perpendicular to that of the ownship aircraft. Those encounter
geometries to the left of the asymptote correspond to conflicts
where the intruder is approaching from behind. In these cases
the RHE manoeuvre causes the ownship aircraft to avoid the
intruder by turning left (i.e. ψKI N <0) For all encounters to
the right of the asymptote, a right hand turn is instigated. For
velocity ratios higher than 1, no kinematic boundary exists,
and therefore the collision is dynamically constrained.
IV. TRA JEC TORY MODELLING
The kinematic boundary distance is based on the assump-
tion that an instantaneous turn can be made. In order to
produce a more realistic estimate of the real failure boundary,
a turn model is required. The ownship aircraft’s position
and heading at time, t= 0 is taken as the reference for
these projection models. The position of the ownship aircraft
through a manoeuvre can be split into three stages:
1) Decision Making
2) Manoeuvring
3) Kinematic Trajectory
VA
VB
ψKIN
Stage 1
Stage 2
Stage 3
tMAN
tKIN
Ownship
Intruder
Fig. 4. Kinematically constrained failure boundary resolution
A. Stage 1 - Decision Making
The first stage models the ownship aircraft’s position
with time up until the manoeuvre is instigated. This can be
estimated using a linear trajectory.
For
0< t ≤tMAN (14)
PA(t) = 0
VAt(15)
where PAis the ownship aircraft’s position with projected
time, t.
B. Stage 2 - Manoeuvring
For the second stage, a simple constant turn rate model is
used. This stage starts at time, t=tMAN , and ends when
the ownship aircraft has finished turning to the kinematic
heading, ψKI N . The time taken to do this can be calculated
as follows:
tT UR N =ψKI N
˙
ψA
(16)
where,
˙
ψA=gtan(φA,MAX )
VA
(17)
φA,MAX is the ownship aircraft’s maximum bank angle.
A nominal value of π
3is chosen for this work. gis the
acceleration due to gravity.
The time at which the manoeuvring phase ends is there-
fore:
tKI N =tM AN +tT U RN (18)
To calculate the ownship aircraft’s position through the
turn, we can use the radius of the turn.
Rturn =V2
A
gtan(φA,MAX )(19)
The ownship aircraft’s position through a turn, as a func-
tion of time, t, can be calculated.
For,
tMAN < t ≤tKI N (20)
PA(t) = Rturn −Rturn cos( ˙
ψA(t−tMAN ))
Rturn sin( ˙
ψA(t−tMAN )) + VAtM AN (21)
C. Stage 3 - Kinematic Trajectory
The final stage requires a linear trajectory to be added to
the position at the end of Stage 2, PA(tKI N ). This linear
trajectory is directed along the kinematic boundary heading.
For,
t>tKI N (22)
PA(t) = PA(tKI N ) + VA(t−tKI N ) sin(ψKIN )
VA(t−tKI N ) cos(ψKI N )(23)
D. Intruder’s Trajectory Model
For the intruder’s trajectory model we make the assump-
tion that the intruder aircraft maintains its current speed and
heading. The intruder’s position can then be modelled by
a simple linear trajectory model similar to Stage 3 of the
ownship model.
The intruder’s initial position, PB,0is calculated using
the intruder’s bearing from the ownship aircraft, β0, and the
distance between the vehicles, d0.
PB,0=d0sin(β0)
d0cos(β0)(24)
The future position of the intruder is therefore:
PB(t) = PB,0+VBtsin(ψ6AB )
VBtcos(ψ6AB )(25)
V. FAILURE BOU NDARY ESTIMATION
The FBE method involves two key stages:
1) Minimum Distance Estimation - Finding the minimum
distance between the two aircraft given that the own-
ship aircraft manoeuvres at a specific time.
2) Failure Boundary Search - Finding the manoeuvring
time which causes the two aircraft to miss each other
by a distance equal to the safety bubble radius, R.
Both of these searches are done using simple optimisation
algorithms as shown in Figure 5.
Fig. 5. Failure Boundary Estimation Process
A. Minimum Distance Estimation
In order to search for the minimum distance, dmin, be-
tween the two aircraft if the ownship manoeuvres at time
tMAN , we can use an optimisation function in Matlab such
as FMINCON. This allows you to find a local minimum of
a constrained non-linear multi-variable function.
min f(t)such that
A.t ≤b(26)
Our objective function f(t)in this case is the projected
distance between the two aircraft, dproj . This can be calcu-
lated using trigonometry.
dproj (t) = q(PBy (t)−PAy (t))2+ (PB x(t)−PAx (t))2
(27)
subject to the following time constraint:
t≥0(28)
The SQP algorithm was chosen as the solver as it con-
verges quickly and reliably. It is started at the point t= 0.
The optimisation function outputs 2 values, the time at the
point of closest approach, tmin, and the distance at the point
of closest approach, dmin, which is used to search for the
failure boundary.
B. Failure Boundary Search
The second component searches for the failure boundary
by finding the manoeuvring time, tM AN , which gives a
minimum distance, dmin, equal to the safety bubble radius,
R. We can also set this up as an optimisation problem, with
the objective function:
derror (tM AN ) = |R−dmin(tM AN )|(29)
subject to the following constraints:
0≤tMAN ≤tT C (30)
where tT C is the time to collision and can be estimated from
the relative velocity, VR, and distance, d.
tT C =d
VR
(31)
The SQP algorithm is once again chosen as the solver, and
the function is started at the point tMAN = 0.
This function outputs 2 values, the estimation of the failure
boundary, tfail, and the value of derr or at that point.
VI. SIMULATION RESULTS
A higher fidelity simulation model is used to assess the
effectiveness of the FBE method. An encounter model gener-
ates collisions by adjusting ψ6AB at 10 degree increments.
At each increment, the aircraft are placed at a fixed time
to collision, tT C . The resolution is simulated, and if the
minimum distance between the two aircraft is greater or
equal to the safety bubble radius, R= 152.4m(500ft),
tT C is reduced. A local boundary search algorithm is used
to adjust the time to collision so that it converges onto the
boundary. An F2B Bristol Fighter [14] dynamic model is
used to model the roll dynamics. The turn dynamics are
modelled by integrating the rate of change of heading with
time.
ψA=Zt
0
gtan(φA)
VA
δt (32)
Change in the ownship aircraft’s position is found by
integrating its velocity vector at each time step. At this stage
no uncertainty is added in the simulation environment.
A. High Speed Scenario
For the high speed scenario, the ownship aircraft’s velocity
is set quite high compared with the intruder’s, VA= 125m/s
and VB= 25m/s. The kinematic boundary, and FBE meth-
ods are compared to the boundary found from simulation.
See figure 6.
0 1 2 3 4 5 6
0
1
2
3
4
5
6
7
8
Conflict Geometry, ψ<AB (rad)
Failure Boundary Time (s)
Simulated Failure Boundary
Kinematic Failure Boundary
FBE Method
SAFE REGION
Fig. 6. Constant turn rate model, high speed scenario results
In these scenarios the boundary distance tends to a con-
stant across all conflict geometries. This is because the
ownship aircraft’s dynamics dominate the problem. It can be
seen that the FBE method provides a much better prediction
of the boundary found from simulation. In a high speed
scenario, the intruder is always in front of the ownship
aircraft, and since the boundary is fairly flat, a left hand turn
would be just as effective as a right hand turn. Therefore
rules of the air should be followed (i.e. turn to the right).
B. Low Speed Scenario
For the low speed scenario we set VA= 25m/s and VB=
125m/s. The results are presented in figure 7.
0 1 2 3 4 5 6 7
0
5
10
15
20
25
30
35
40
45
50
Conflict Geometry, ψ<AB (rad)
Failure Boundary Time (s)
FBE Method
Kinematic Failure Boundary
Simulated Failure Boundary
SAFE
REGION
Fig. 7. Constant turn rate model, low speed scenario results
For the low speed scenario, the failure boundary is much
larger due to the growth of the kinematic boundary. The
asymptote is predicted very well by the kinematic boundary
and the FBE method provides an accurate prediction of the
boundary found from simulation. When the intruder is on
the right hand side of the ownship aircraft (ψ6AB > π) the
RHE manoeuvre is much more effective.
C. Same Speed Scenario
For this scenario, both aircraft are set to a speed of 75
m/s. The results are presented in Figure 8.
01234567
0
5
10
15
Conflict Geometry, ψ<AB (rad)
Failure Boundary Time (s)
Kinematic Failure Boundary
Simulated Failure Boundary
FBE Method
PARALLEL
RESOLUTION
TRAJECTORIES
SAFE
REGION
Fig. 8. Constant turn rate model, same speed scenario results
For small values of ψ6AB the aircraft get stuck and end
up flying along parallel trajectories. When VA
VB= 1 the
kinematic boundary heading, ψKI N =ψ6AB . The relative
velocity for these types of conflict resolutions is very small,
and so a boundary would need to be developed which would
ensure that the relative velocity after a manoeuvre did not
fall below a particular threshold.
D. Collision Avoidance Logic
The UAV can use the knowledge of the failure boundaries
to determine the correct course of action during an encounter.
The boundary of the manoeuvre which moves the relative
velocity vector to the left hand edge of the safety bubble
(a LHE manoeuvre) is found easily, by reflecting the graph
through the point ψ6AB =π. Both of the boundary times
are converted to distances using the relative velocity, and are
plotted for the Low Speed Scenario. See Figure 9.
Fig. 9. Application of Failure Boundary Estimation to Determine Collision
Avoidance Logic
Each area can be divided up into different logic states
depending upon the boundary predictions.
If a collision were to be detected at Point 1 (d= 7km),
both manoeuvres would be available. The remote pilot would
be advised of the conflict and the UAV should maintain its
heading according to rules of the air (ROA). At Point 2
(d= 1.2km) the RHE manoeuvre would no longer available,
and it would no longer be safe to follow ROA, so the UAV
automatically performs a LHE manoeuvre before the full
failure region is entered.
VII. CONCLUSIONS & FUTURE WO RK
A method has been presented which successfully predicts
the failure boundary for lateral collision avoidance manoeu-
vres. Encounters where the intruder is travelling at high
speeds compared with the ownship aircraft were found to
have large failure boundaries. Some conflicts where both
aircraft were travelling at similar speed were shown to result
in parallel trajectories. Knowledge of the failure boundaries
is used to formulate collision avoidance logic, ensuring that
the correct decision is always made. FBE is a crucial step
which is needed to reduce the size and complexity of this
verification problem, and can also be used to generate sensor
requirements.
Future work will involve the modification of the turn
model to include simple roll dynamics. This should drasti-
cally improve the accuracy of the FBE method. A boundary
will be introduced based on a relative velocity limit, and used
to prevent aircraft from getting stuck on parallel trajectories.
The FBE method will be validated in X-Plane and realistic
uncertainty shall be introduced. Finally it will be validated
with real flight tests performed on a small fixed wing UAV.
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