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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.

REFERENCES

[1] CAA, CAP722 - Unmanned Aircraft System Operations in UK

Airspace, Guidance

[2] A. Varga and A. Hansson, Optimization Based Clearance Of Flight

Control Laws. Lecture Notes In Control And Information Sciences,

Springer 2012

[3] S. Srikanthakumar, C. Liu and W. H. Chen, Optimization-Based

Safety Analysis of Obstacle Avoidance Systems for Unmanned Aerial

Vehicles. Journal of Intelligent & Robotic Systems, January 2012,

Volume 65, Issue 1-4, p. 219-231

[4] E. L. Wiener, Beyond the Sterile Cockpit. Human Factors: The Journal

of the Human Factors and Ergonomics Society, February 1985, vol.

27, no. 1 p. 75-90

[5] M. Perez-Batlle, E. Pastor, X. Prats, P. Royo, and R. Cuadrado,

Maintaining separation between airliners and RPAS in non-segregated

airspace. ICARUS Research Group, Technical University of Catalonia

[6] J. R. Dunthorne, W. H. Chen and S. Dunnett, Collision Avoidance

Failure Boundary Identiﬁcation For The Clearance of Civil Unmanned

Aircraft. 2nd Year Internal Report, Loughborough University 2013

[7] S Degen, Reactive image-Based Collision Avoidance System for

Unmanned Aircraft Systems. Queensland University of Technology,

May 2011

[8] K. Sigurd and J. How, UAV trajectory Design Using Total Field

Collision Avoidance. MIT 2003

[9] S. L. Waslander, G. Inalhan and C. J. Tomlin, Decentralized Optimi-

sation Via Nash Bargaining. 2004 Stanford

[10] N. Durand, J. Alliot and J. Noailles, Collision avoidance using neural

networks learned by genetic algorithms. IEA-AEI 96

[11] J. Goss and R. Rajvanshi and K. Subbarao, Aircraft Conﬂict De-

tection and Resolution using Mixed Geometric and Collision Cone

Approaches. American Institute of Aeronautics and Astronautics

[12] A. Chakravarthy and D. Ghose, Obstacle Avoidance in a Dynamic

Environment: A Collision Cone Approach. Transactions on Systems,

Man, and Cyberntics - Part A: Systems and Humans, Sep. 1998, V.28,

Num 5, p. 562-574

[13] D. E. Grilley, Resolution requirements for passive sense and avoid,

Alion Science and Technology Jan 2005

[14] T. I. Fossen, Mathematical models for control of aircraft and satellites.

Department of Engineering Cybernetics Norwegian University of

Science and Technology, 2011.

[15] R Oliveira and G Puyou, On the use of optimization for ﬂight control

laws clearance: a practical approach. IFAC September 2011 p9881-

9886