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Jason M. Upchurch, César A. Muñoz, Anthony J. Narkawicz, 

James P. Chamberlain, and Maria C. Consiglio 

Langley Research Center, Hampton, Virginia 








Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and Space Administration. 
Abstract

The FAA-sponsored Sense and Avoid Workshop for Unmanned Aircraft Systems

(UAS) defines the concept of sense and avoid for remote pilots as “the capability

of a UAS to remain well clear from and avoid collisions with other airborne traffic.”

Hence, a rigorous definition of well clear is fundamental to any separation assurance

concept for the integration of UAS into civil airspace. This paper presents a family

of well-clear boundary models based on the TCAS II Resolution Advisory logic.

Analytical techniques are used to study the properties and relationships satisfied

by the models. Some of these properties are numerically quantified using statistical

methods.

iii
Acronyms

CAT Collision Avoidance Threshold

CDF Cumulative Distribution Function

NAS National Airspace System

NMAC Near Mid-Air Collision

RA Resolution Advisory

SAA Sense and Avoid

SST Self-Separation Threshold

SSV Self-Separation Volume

TCAS Traffic Alerting and Collision Avoidance System

TCPA Time to Closest Point of Approach

UAS Unmanned Aircraft Systems

1 Introduction

One of the major challenges of integrating Unmanned Aircraft Systems (UAS) into

the airspace system is the lack of an on-board pilot to comply with the legal requirement that pilots see and avoid other aircraft in their vicinity. To address this

challenge, the final report of the FAA-sponsored Sense and Avoid (SAA) Workshop

for Unmanned Aircraft Systems [2] defines the concept of sense and avoid for remote

UAS pilots as “the capability of a UAS to remain well clear from and avoid collisions

with other airborne traffic.” Under this definition, a rigorous definition of well clear

becomes fundamental to any sense and avoid concept that involves UAS.

NASA’s Unmanned Aircraft Systems Integration in the National Airspace System (UAS in the NAS) project aims at conducting research towards the integration

of civil UAS into non-segregated airspace operations. As part of this project, NASA

has developed a sense and avoid concept for UAS that extends the concept outlined by the SAA Workshop [1]. The NASA concept includes a volume, namely the

Self Separation Volume (SSV), located between the Collision Avoidance Threshold

(CAT), defined by collision avoidance systems, and the Self-Separation Threshold

(SST), defined by self-separation systems [2]. The SSV represents a well-clear boundary where aircraft inside the SSV are considered to be in well-clear violation. This

volume is intended to be large enough to avoid safety concerns for controllers and

see-and-avoid pilots, but small enough to avoid disruptions to traffic flow. A key

characteristic of NASA’s concept is that the SSV is a conservative extension of the

CAT defined by the Traffic Alerting and Collision Avoidance System (TCAS).

TCAS is a family of airborne devices that are designed to reduce the risk of midair collisions between aircraft equipped with operating transponders [10]. TCAS II,

the current generation of TCAS devices, is mandated in the US for aircraft with

greater than 30 seats or a maximum takeoff weight greater than 33,000 pounds.

Although it is not required, TCAS II is also installed on many turbine-powered

general aviation aircraft. Version 7.0 is the current operationally-mandated version

of TCAS II, and Version 7.1 has been standardized [8]. In contrast to TCAS I, the
first generation of TCAS devices, TCAS II provides resolution advisories (RAs).

RAs are visual and vocalized alerts that direct pilots to maintain or increase vertical

separation with intruders that are considered collision threats. TCAS II resolution

advisories can be corrective or preventive depending on whether the pilot is expected

to change or maintain the aircraft’s current vertical speed. Corrective RAs are

particularly disruptive to the air traffic system since they may cause drastic evasive

maneuvers. For this reason, they are intended as a last resort maneuver when all

other means of separation have failed.

The core of the TCAS II RA logic is a test that checks distance and time variables

for the horizontal and vertical dimensions against a set of pre-defined threshold

values. To ensure interoperability between NASA’s SAA concept and TCAS, the

mathematical definition of the volume SSV is based on the TCAS II Resolution

Advisory Logic [5]. The definition of SSV follows the same logic, but uses different

thresholds that conservatively extends the collision avoidance threshold provided

by TCAS. This paper further generalizes the definition of the well-clear violation

volume presented in [5] and presents a family of mathematical well-clear boundary

models that are all based on the TCAS II RA logic. Formal and statistical techniques

are used to study properties of this family of models.

The formal development presented in this paper is part of the NASA’s Airborne Coordinated Resolution and Detection (ACCoRD) mathematical framework,

which is electronically available from http://shemesh.larc.nasa.gov/people/

cam/ACCoRD. All theorems in this paper have been formally verified in the Prototype Verification System (PVS) [7], an automated theorem prover.

2 Distance and Time Variables

Distance and time variables are important elements of any separation assurance

concept. These variables are functions over the aircraft current states which are

compared against distance and time thresholds. Many conflict detection and resolution systems rely on the time of closest point of approach and the distance at

that time as their main time and distance variables [4]. This section describes some

additional distance and time variables that are particularly relevant to the definition

of a well-clear boundary model.

This paper assumes that accurate aircraft surveillance information is available

as horizontal and vertical components in a three-dimensional (3-D) airspace. Letters in bold-face denote two-dimensional (2-D) vectors. Vector operations such as

addition, subtraction, scalar multiplication, dot product, i.e., s · v ≡ sxvx + syvy,

the square of a vector, i.e., s

2 ≡ s · s, and the norm of a vector, i.e., ksk ≡ √

s

2,

are defined in a 2-D Euclidean geometry. Furthermore, the expression v

⊥ denotes a

particular 2-D right-perpendicular vector of v, i.e., v

⊥ ≡ (vy, −vx), and 0 denotes

the 2-D vector whose components are 0, i.e., 0 ≡ (0, 0).

The mathematical models presented in this paper consider two aircraft referred

to as the ownship and the intruder aircraft. For the ownship, the current horizontal

position and velocity are denoted so and vo, respectively. Its altitude and vertical

speed are denoted soz and voz, respectively. Similarly, the horizontal position and

2
velocity of the intruder aircraft are denoted si and vi

, respectively, and its vertical altitude and speed are denoted siz and viz, respectively. As it simplifies the

mathematical development, this paper uses a relative coordinate system where the

intruder is static at the center of the coordinate system. In this relative system,

s = so −si and v = vo −vi represent the horizontal relative position and velocity of

the aircraft, respectively. Furthermore, sz = soz − siz and vz = voz − viz represent

the vertical relative position and speed of the aircraft, respectively.

Assuming constant relative horizontal velocity v, the horizontal range between

the aircraft at any time t is given by

r(t) ≡ ks + tvk =

p

s

2 + 2t(s · v) + t2v2. (1)

The time of horizontal closest point of approach, denoted tcpa, is the time t that

satisfies  ̇r(t) = 0, i.e., t = −

s·v

v2 . The dot product s · v characterizes whether the

aircraft are horizontally diverging, i.e., s · v > 0, or horizontally converging, i.e.,

s · v < 0. By convention, tcpa is defined as 0 when v = 0. Hence, tcpa is formally

defined as

tcpa(s, v) ≡

(

−

s·v

v2 if v 6= 0,

0 otherwise.

(2)

It is noted that tcpa(s, v) > 0 when the aircraft are horizontally converging, tcpa(s, v) <

0 when the aircraft are horizontally diverging, and tcpa(s, v) = 0 when the aircraft

are neither converging or diverging. The distance at time of closest point of approach

is defined as

dcpa(s, v) ≡ r(tcpa(s, v)) = ks + tcpa(s, v)vk. (3)

In the vertical dimension, assuming constant relative vertical speed, the relative

altitude between the aircraft at any time t is given by

rz(t) ≡ |sz + tvz|. (4)

The time to co-altitude tcoa is the time t that satisfies rz(t) = 0, i.e, t = −

sz

vz

.

Similar to the horizontal case, the product szvz characterizes whether the aircraft

are vertically diverging, i.e., szvz > 0, or vertically converging, i.e., szvz < 0.

This paper defines time to co-altitude as −1 when the aircraft are not vertically

converging. Therefore,

tcoa(sz, vz) ≡

(

−

sz

vz

if szvz < 0,

−1 otherwise.

(5)

Formula (5) is well defined since szvz < 0 implies that vz 6= 0.

2.1 Horizontal Time Variables

A (horizontal) time variable is a function that maps a relative horizontal position

and velocity into a real number. This real number is negative when the aircraft

are horizontally diverging. When the real number is non-negative, this number

represents a time that, in a separation assurance logic, is intended to be compared

3
against a time threshold. In this paper, the time threshold is called TTHR. An

example of a time variable that is used in conflict detection logics is tcpa [4].

The time variable used in earlier versions of the TCAS detection logic is called

tau, denoted τ [8]. Tau estimates tcpa, but is less demanding on sensor and surveillance technology than tcpa. Indeed, τ is simply defined as range over closure rate,

where closure rate is the negative of the range rate, i.e., τ = −

r(0)

r ̇(0) = −

ksk

s·v

ksk

= −

s

2

s·v

.

This paper defines τ as −1 when the aircraft are not horizontally converging. Formally,

τ (s, v) ≡

(

−

s

2

s·v

if s · v < 0,

−1 otherwise.

(6)

For a limited number of scenarios, the values of τ and tcpa coincide. However, in

most scenarios, the value of τ tends toward infinity as the aircraft approach the

closest point of approach. In general, τ is a good approximation of tcpa, but only for

large values. For that reason, TCAS II uses a modified variant of τ called modified

tau, denoted τmod [8]. Modified tau provides a better estimation of tcpa and has a

more desirable behavior than τ in the proximity of the closest point of approach.

In [3], modified tau is defined such that τmod = −

r(0)2−DTHR2

r ̇(0) =

DTHR2−s

2

s·v

. Similar to

τ , τmod is defined as -1 when the aircraft are not horizontally converging, i.e.,

τmod(s, v) ≡

(

DTHR2−s

2

s·v

if s · v < 0,

−1 otherwise.

(7)

The definition of τmod in Formula (7) depends on DTHR, which is a horizontal distance

threshold. This threshold is called DMOD in the TCAS II RA logic, and its actual

value depends on a sensitivity level based on the ownship’s altitude [8].

In [6], a time variable called time to entry point, denoted tep, is proposed. Time

to entry point is defined as the time to loss of horizontal separation with respect

to DTHR assuming straight-line aircraft trajectories. Similar to tcpa, tep decreases

linearly over time. Time to entry point is formally defined as

tep(s, v) ≡

(

Θ(s, v, DTHR, −1) if s · v < 0 and ∆(s, v, DTHR) ≥ 0,

−1 otherwise,

(8)

where

Θ(s, v, D, ) ≡

−s · v + 

p

∆(s, v, D)

v2

, (9)

∆(s, v, D) ≡ D2v

2 − (s · v⊥)2

. (10)

The function Θ is only defined when v 6= 0 and ∆(s, v, D) ≥ 0. In this case, it

computes the times when the aircraft will lose separation, if  = −1, or regain

separation, if  = 1, with respect to D. When the aircraft are not horizontally

converging or ∆(s, v, D) < 0, time to entry point is defined as -1. Formula (8) is

well defined since the condition s · v < 0 guarantees that v 6= 0.

4
2.2 Properties of Horizontal Time Variables

A useful property of a time variable is symmetry. A time variable tvar is said to be

symmetric if and only for all s, v,

tvar(s, v) = tvar(−s, −v). (11)

Symmetry guarantees that in a pairwise scenario both the ownship and intruder

aircraft compute the same value for the time variable. Hence, checking a symmetric

time variable against a given time threshold returns the same Boolean value for both

aircraft.

Theorem 1. The time variables τ , tcpa, τmod, and tep are symmetric.

It is possible to define time variables that are not symmetric. For instance,

a time variable that computes the first time when the intruder aircraft enters an

elliptical area aligned to the ownship trajectory is not symmetric for every scenario.

However, any time variable can be transformed into a symmetric one by using min

and max operators. For instance, the time variables min(tvar(s, v), tvar(−s, −v)) and

max(tvar(s, v), tvar(−s, −v)) are symmetric for any time variable tvar.

Figure 1 shows a graph of τ , tep, τmod, and tcpa versus time for an initial scenario

where the ownship and intruder aircraft are located at (0 nmi, −3.25 nmi) and

(−6.25 nmi, 0.25 nmi), respectively, flying at co-altitude. Furthermore, the ownship

ground speed is 150 kn, heading 53o, and the intruder ground speed is 350 kn,

heading 90o.

1

In this scenario, the distance threshold DTHR used in the definition of

τmod and tep is 1 nmi. This scenario illustrates that while tep, τmod, and tcpa decrease

over time, the time variable τ decreases up to some point, but then it abruptly

increases in the vicinity of the closest point of approach. Moreover, when these time

variables are checked against a time threshold TTHR, represented by the horizontal

line at 30 seconds, the time variable tep crosses the time threshold first, followed

by τmod, then tcpa, and finally τ . Interestingly, this ordering property holds for any

converging scenario and any choice of common threshold values.

Theorem 2. Let s, v be such that s · v < 0, ksk > DTHR, and dcpa(s, v) ≤ DTHR,

i.e., the aircraft are horizontally converging, are outside the distance threshold DTHR,

and their distance at time of closest point of approach is less than or equal to DTHR.

Then the following inequalities hold

tep(s, v) ≤ τmod(s, v) ≤ tcpa(s, v) ≤ τ (s, v). (12)

3 A Family of Well-Clear Boundary Models

A well-clear boundary specifies the set of aircraft states that are considered to be

in well-clear violation. Following the TCAS detection logic, the well-clear boundary

models in this paper are specified by a logical condition that simultaneously checks

1Aircraft headings are measured in true north clockwise convention, i.e., 0o points to the north

and degrees are positive in clockwise direction.

5
0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

time (s)

time (s)

kvok =150 kn

t r ack vo =53◦

kvik =350 kn

t r ack vi =90◦

τ

tcp a

τm o d

tep

TTHR

Figure 1: Time vs. τ , tcpa, τmod, tep

horizontal and vertical violations. A horizontal violation occurs if the current range

is less than a given horizontal distance threshold DTHR. A horizontal violation also

occurs if distance at time of closest point of approach is less than DTHR and a given

time variable tvar is less than a given time threshold TTHR. In the vertical dimension,

a similar comparison is made. Vertical well clear is violated if the relative altitude

is less than a given altitude threshold ZTHR or if the time to co-altitude is less

than a given vertical time threshold TCOA. The distance and altitude thresholds are

considered to be positive numbers, i.e., DTHR > 0 and ZTHR > 0. The time thresholds

are considered to be non-negative, i.e., TTHR ≥ 0 and TCOA ≥ 0. Formally, this wellclear violation condition can be denoted as follows.

WCVtvar (s, sz, v, vz) ≡ Horizontal WCVtvar (s, v) and

Vertical WCV(sz, vz),

(13)

where

Horizontal WCVtvar (s, v) ≡ ksk ≤ DTHR or

(dcpa(s, v) ≤ DTHR and 0 ≤ tvar(s, v) ≤ TTHR),

Vertical WCV(sz, vz) ≡ |sz| ≤ ZTHR or 0 ≤ tcoa(sz, vz) ≤ TCOA.

The logical condition WCVtvar defines a family of well-clear boundary models

where tvar can be instantiated with any time variable, and DTHR, TTHR, ZTHR, and

TCOA are set to threshold values of interest. The fact that the time thresholds TTHR

and TCOA can be zero allows for the definition of well-clear boundary models that

do not depend on time thresholds. For instance, when TTHR = 0 and TCOA = 0,

WCVtcpa specifies the loss of separation condition for a cylindrical volume of radius

6
DTHR and half-height ZTHR around one of the aircraft. Indeed, in this case, WCVtcpa

is logically equivalent to the logical condition ksk ≤ DTHR and |sz| ≤ ZTHR.

The TCAS II RA core logic provided in [5] is used by WCVτmod, where DTHR,

TTHR, ZTHR, and TCOA are set to the TCAS II thresholds DMOD, TAU, ZTHR, and

TAU, respectively. The actual values of these thresholds are given in a table indexed

by sensitivity levels based on the ownship’s altitude [8]. In the TCAS II RA logic,

the logical condition dcpa(s, v) ≤ DTHR in the horizontal check is called horizontal

miss-distance filter and, in that condition, DTHR is set to the miss-horizontal distance

threshold HMD, which is equal to DMOD. The well-clear boundary model defined

in [6] is obtained by WCVtep, where TCOA = TTHR.

Henceforth, the well-clear models specified by WCVτ , WCVtcpa , WCVτmod, and

WCVtep will be referred to as WC TAU, WC TCPA, WC TAUMOD, and WC TEP,

respectively. The rest of this section studies properties and relations satisfied by

these models.

3.1 Symmetry

A well-clear boundary model specified by WCVtvar , for a given time variable tvar, is

symmetric if and only if

WCVtvar (s, sz, v, vz) = WCVtvar (−s, −sz, −v, −vz). (14)

In other words, in a symmetric well-clear boundary model, both the ownship and

intruder aircraft have the same perception of being well clear or not.

Theorem 3 (Symmetry). If tvar is symmetric, the well-clear boundary model specified by WCVtvar is symmetric. Hence, by Theorem 1, the well-clear boundary models

WC TAU, WC TCPA, WC TAUMOD, and WC TEP are symmetric for any choice

of threshold values DTHR, TTHR, ZTHR, and TCOA.

3.2 Inclusion

Figure 2 illustrates the violation areas for the well-clear boundary models WC TAU,

WC TAUMOD, WC TCPA, and WC TEP for the scenario of Figure 1. The threshold values used in this scenario are DTHR = 1 nmi, TTHR = TCOA = 30 s, and

ZTHR = 475 ft. The violation areas in these figures are similar to the conflict contours proposed in [9]. The points in these areas represent future locations of the

ownship where a well-clear violation will occur assuming that the intruder aircraft

continues its current trajectory and the ownship either continues its current trajectory or instantaneously changes its direction but keeps its ground speed.

Figure 3 overlays the violation areas for the four boundary models. This figure

illustrates that for a common set of threshold values, the violation area of WC TAU

is included in the violation area of WC TCPA, which is included in the violation area

of WC TAUMOD, which is included in the violation area of WC TEP. Theorem 4

below states that this inclusion property always holds for any encounter geometry

and choice of common threshold values. Theorem 4 is a consequence of Theorem 2.

7
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4

−6

−5

−4

−3

−2

−1

0

1

s

o

 = (0.00 nmi, −3.25 nmi)

v

o

 = (119.8 kn, 90.3 kn)

s

i

 = (−6.25 nmi, 0.25 nmi)

v

i

 = (350.0 kn, 0.0 kn)

x (nmi)

y (nmi)

WC_TAU

Area = 4.8 nmi2

Time to Violation = 84 s

Violation Duration = 10 s

Ownship

Intruder

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4

−6

−5

−4

−3

−2

−1

0

1

s

o

 = (0.00 nmi, −3.25 nmi)

v

o

 = (119.8 kn, 90.3 kn)

s

i

 = (−6.25 nmi, 0.25 nmi)

v

i

 = (350.0 kn, 0.0 kn)

x (nmi)

y (nmi)

WC_TCPA

Area = 5.1 nmi2

Time to Violation = 73 s

Violation Duration = 32 s

Ownship

Intruder

(a) WC TAU (b) WC TCPA

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4

−6

−5

−4

−3

−2

−1

0

1

s

o

 = (0.00 nmi, −3.25 nmi)

v

o

 = (119.8 kn, 90.3 kn)

s

i

 = (−6.25 nmi, 0.25 nmi)

v

i

 = (350.0 kn, 0.0 kn)

x (nmi)

y (nmi)

WC_TAUMOD

Area = 5.3 nmi2

Time to Violation = 73 s

Violation Duration = 32 s

Ownship

Intruder

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4

−6

−5

−4

−3

−2

−1

0

1

s

o

 = (0.00 nmi, −3.25 nmi)

v

o

 = (119.8 kn, 90.3 kn)

s

i

 = (−6.25 nmi, 0.25 nmi)

v

i

 = (350.0 kn, 0.0 kn)

x (nmi)

y (nmi)

WC_TEP

Area = 5.8 nmi2

Time to Violation = 71 s

Violation Duration = 34 s

Ownship

Intruder

(c) WC TAUMOD (d) WC TEP

Figure 2: Violation areas for scenario of Figure 1

−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4

−6

−5

−4

−3

−2

−1

0

1

s

o

 = (0.00 nmi, −3.25 nmi)

v

o

 = (119.8 kn, 90.3 kn)

s

i

 = (−6.25 nmi, 0.25 nmi)

v

i

 = (350.0 kn, 0.0 kn)

x (nmi)

y (nmi)

 WC_TAU

 WC_TCPA

 WC_TAUMOD

 WC_TEP

Figure 3: Overlay of violation areas for scenario of Figure 1

8
Theorem 4 (Inclusion). For all s, sz, v, vz and choice of threshold values DTHR,

TTHR, ZTHR, and TCOA, the following implications hold

(i) WCVτ (s, sz, v, vz) =⇒ WCVtcpa (s, sz, v, vz),

(ii) WCVtcpa (s, sz, v, vz) =⇒ WCVτmod (s, sz, v, vz), and

(iii) WCVτmod (s, sz, v, vz) =⇒ WCVtep (s, sz, v, vz).

A key consequence of Theorem 4 is that of the four well-clear boundary models,

WC TEP provides the most conservative safety margins in terms of having the

largest violation area and the earliest time whereby a well-clear violation is defined

to occur. The remaining models can be ordered from most conservative to least

conservative as WC TAUMOD,WC TCPA, and WC TAU.

3.3 Local Convexity

As illustrated by Figure 2, the violation areas are not geometrically convex. However, Figures 2(b)-(d) show that from the point of view of the ownship, any ray that

points towards the violation area has only one intersecting segment. This property

is referred to as local convexity. It can be verified by inspection of Figure 2(a) that

this property does not always hold in the case of WC TAU. A formal definition of

local convexity follows.

Definition 1 (Local convexity). A well-clear boundary model specified by WCVtvar ,

for a given time variable tvar, is locally convex if and only if there are no times

0 ≤ t1 ≤ t2 ≤ t3 ≤ T such that

1. the aircraft are not well clear at time t1, i.e., WCVtvar (s+t1v, sz +t1vz, v, vz),

2. the aircraft are well clear at time t2, i.e., ¬WCVtvar (s + t2v, sz + t2vz, v, vz),

and

3. the aircraft not well clear at time t3, i.e., WCVtvar (s + t3v, sz + t3vz, v, vz).

Thus, a well-clear boundary model is locally convex if for any ownship straightline trajectory there is at most one time interval where the aircraft are not well

clear.

Theorem 5. For any choice of threshold values, the well-clear boundary models

WC TCPA, WC TAUMOD, and WC TEP are locally convex.

As illustrated by Figure 2(a), the well-clear boundary model WC TAU is not

locally convex for all choices of threshold values. In particular, it can be seen in

Figure 1, assuming straight-line trajectories, that for the same encounter scenario

the aircraft will have a well-clear violation at 91 s, 7 seconds later they will be

well clear, and 7 seconds after being well clear, they will have another well-clear

violation.

Theorem 6. For some choices of threshold values, the well-clear boundary model

WC TAU is not locally convex.

9
Figure 4: Depiction of a randomly-generated encounter (top view)

4 Preliminary Statistical Analysis of Well-Clear Boundary Models

This section presents a preliminary statistical analysis of the well-clear models defined in Section 3, with the goal of characterizing the models in terms of relevant

metrics. In particular, the metrics used for comparison are: (1) the violation areas

associated with well-clear violations, and (2) the times when a well-clear violation

first occurs. These metrics serve to validate the inclusion relation given by Theorem 4.

4.1 Random Encounter Generation

The encounter space used in the statistical analysis presented in this section consists

of a half cylinder of radius, R, and height, h. The top view of this situation is shown

in Figure 4 for an arbitrary encounter, and the three-dimensional view is shown in

Figure 5 for a different arbitrary encounter.

The ownship initial position is chosen to be constant as so = (0, −

R

2

) and soz =

h

2

.

The ownship horizontal velocity component vo is randomly chosen from a Burr

distribution with parameters α = 37.0896, c = 2.6351, k = 1.00604 and is intended

to be representative of a fixed-wing UAS, based on the distribution of the velocity

characteristics of 849 fixed-wing UAS [11]. The ownship vertical velocity component

voz is chosen to be zero. The motivation for this particular encounter space is to

create stress scenarios whereby encounters are biased to result in violations, where

the violations cover a broad range of encounter geometries. The details of the

encounter space parameters follow.

10
Intruder

Ownship

h

R

Figure 5: Depiction of a randomly-generated encounter (3-dimensional view)

The intruder initial position in the horizontal plane, si, is randomly chosen from

a uniform distribution to be on the left half cylinder circumference as shown in

Figure 4. The intruder initial position in the vertical plane, siz, is chosen from

a normal distribution having a mean of zero and a standard deviation of h

(2)(2.99) ,

where siz is set to be either h

2

or −

h

2

if the random variable falls in the upper or

lower tail of the distribution, respectively. The intruder initial horizontal velocity

magnitude is chosen from the same distribution as the ownship. Furthermore, the

intruder’s horizontal velocity vector direction is chosen from a uniform distribution

to terminate on the small circle shown in Figures 4 and 5. The intruder vertical

velocity magnitude is randomly chosen from a normal distribution having a mean

of zero and a standard deviation of viz,max

(2)(2.99) , where viz is set to be either viz,max or

−viz,max if the random variable falls in the upper or lower tail of the distribution,

respectively.

4.2 Computation of Violation Area

Given a set of initial position and velocity states for the ownship and intruder,

a well-clear violation area is generated by first indexing the ownship trajectory

through 360 degrees over N steps while holding kvok and voz constant, that is, the

ownship trajectory is swept around a cone of constant height, where each of the

N trajectories is assumed to remain constant. Next, for each of the N ownship

trajectories generated, the time interval for any well-clear violation, [tin, tout], is

computed for the given trajectory. Then, the associated line segment in threedimensional space is projected onto a two-dimensional plane containing the ownship

initial position. As discussed in Section 3.3, the WC TAU model is not locally

convex. Hence, there may be aircraft states that yield multiple instances of tin and

tout for a given trajectory. In such cases, each segment is considered separately. The

resulting geometry for an example encounter are illustrated in Figure 6.

In the analysis presented in this paper, a violation area associated with a wellclear violation is considered to be such a projection into two dimensions. It can be

verified that for the special case of voz = 0, the height of the cone collapses to zero,

11
Figure 6: Projection of three-dimensional violation volume into two dimensions

and the original and projected volumes coincide.

The area of the two-dimensional violation area is computed as follows. First,

consider a differential area in the polar coordinate system given by

dA =

1

2

(r

2 − r2

)dθ, (15)

where r corresponds to the distance from the ownship initial position to the position

at time tin, r corresponds to the distance from the ownship initial position to the

position at time tout, and dθ corresponds to the differential angle between adjacent

ownship trajectories, that is,

dθ =

2π

N

.

Thus, the analytical violation area is determined as

lim

N→∞

X

N

k=1

X

M

i=1

π

N

(r

2

i,k − r

2

i,k), (16)

where M represents the number of violation regions on the kth trajectory, and if

a well-clear violation does not occur for the kth trajectory then ri,k and ri,k are

defined to be zero for that trajectory.

The algorithm implemented to compute the numerical approximation of the

violation area is given by

X

N∗

k=1

X

i

π

N∗

(r

2

i,k − r

2

j,k), (17)

12
0 20 40 60 80 100 120 140

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Numb e r of par titions (N )

P

n

N|

|A i + 1−

A i|

|A i + 1| ≤

0.01

o

Figure 7: Estimate of probability density function of N for 1000 trials

where N∗ denotes a particular choice for N. It can be verified that this estimate

converges to the actual area as N∗ approaches infinity.

The particular choice for N∗ used for the analysis in the remainder of this paper

was arrived at through a Monte Carlo experiment which was run until 1000 random

encounter trajectories resulting in well-clear violations were accumulated. For each

well-clear violation, N was initialized with N = 2 and the corresponding violation

area was calculated using Formula (17). The value of N was then incremented by

one and the violation area recomputed. This process continued until the relative

difference between the computed area and the previously-computed area was below

1%. Thus, the value of N for any randomly-generated encounter was chosen as

the number of partitions that first satisfied the 1% condition. An estimate of the

distribution of N for the 1000 well-clear violations is shown in Figure 7. This figure

was used as a basis for choosing N∗ = 360, which is assumed for the remainder

of the analysis in this paper. Thus, the differential angle for the velocity sweep is

necessarily 1◦.

4.3 Analysis

In the subsequent analysis, the choice is made to present comparisons of WC TAU,

WC TCPA, and WC TAUMOD relative to WC TEP. That is, the analysis and

metrics presented are with respect to WC TEP. In particular, the two metrics

used for the following discussion are: (1) the relative difference in violation areas,

13
determined as

∆A(tvar) = A(WCVtep) − A(WCVτ )

A(WCVtep)

, (18)

∆A(tvar) = A(WCVtep) − A(WCVtcpa )

A(WCVtep)

, (19)

∆A(tvar) = A(WCVtep) − A(WCVτmod)

A(WCVtep)

, (20)

and (2) the difference in time when a well-clear violation will first occur, determined

as

∆tin(tvar) = tin(WCVtep) − tin(WCVτ ), (21)

∆tin(tvar) = tin(WCVtep) − tin(WCVtcpa ), (22)

∆tin(tvar) = tin(WCVtep) − tin(WCVτmod). (23)

The statistical analysis of violation areas is obtained from a Monte Carlo simulation with 10,000 well-clear violations (i.e., greater than 10,000 trials). Each

trial consisted of a random encounter scenario having the geometry discussed in

Section 4.1. For each trial, if the random encounter resulted in a joint well-clear

violation for all models, the corresponding areas were computed, followed by the

relative area differences with respect to WC TEP. This process was repeated until

10,000 joint well-clear violations were accumulated, and the cumulative distribution

function (CDF) for WC TAUMOD and WC TCPA with respect to WC TEP was

then computed (see Formulas 18-20). Figure 8 shows the results of the Monte Carlo

experiment, where the threshold values used for the simulation were TTHR = 30 s,

DTHR = 1 nmi, and ZTHR = 475 ft. The preliminary analysis of the CDFs in Figure 8

reveal that: (1) the areas for WC TAUMOD and WC TEP differ by less than 25%

in approximately 95% of the well-clear violations, (2) the areas for WC TCPA and

WC TEP differ by as much as 55% in approximately 95% of the well-clear violations, and (3) the areas for WC TAU and WC TEP differ by as much as 70% in

95% of the well-clear violations. The Monte-Carlo results provide an experimental

validation of the inclusion property discussed in Section 3.2.

The second metric used to analyze the well-clear models is tin, the time when a

well-clear violation first occurs (see Formulas 21-23). During the same Monte Carlo

experiment previously discussed, if a joint well-clear violation for an initial set of

randomly-generated ownship and intruder positions and velocities occurs, then the

time when the well-clear violation occurs for each model is computed using only

the initial positions and velocities. For each encounter resulting in a well-clear

violation, the time difference with respect to WC TEP is computed for WC TAU,

WC TCPA, and WC TAUMOD. Upon accumulation of 10,000 well-clear violations,

the CDFs of the time difference in time were generated. The results of the Monte

Carlo experiment are shown in Figure 9.

The CDFs in Figure 9 show that: (1) the difference between tin for WC TAUMOD

and WC TEP is limited to approximately 15 s, (2) the difference between tin for

14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δA(tva r) (relative difference)

P

{

δ

A(tva r) ≤

∆

A(tva r)|W C Vt va r}

∆A(τ )

∆A(t cp a)

∆A(τm o d)

Figure 8: CDF of relative difference in area with respect to WC TEP

0 5 10 15 20 25 30 35

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δti n(tva r) (s )

P

{

δ

ti n(tva r) ≤

∆

ti n(tva r)|W C Vt va r}

∆ti n(τ )

∆ti n(t cp a)

∆ti n(τm o d)

Figure 9: CFD of absolute difference first time to violation with respect to WC TEP

15
−2.5 −2 −1.5 −1 −0.5 0 0.5 1

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

x (nmi)

y (nmi)

kvok =83.6 k t s

t r ack vo =0.0◦

kvik =126.8 k t s

t r ack vi =45.0◦

WC_TAU

WC_TCPA

WC_TAUMOD

WC_TEP

Figure 10: Encounter geometry of interest: large difference in tin

WC TCPA and WC TEP is limited to exactly 30 s, which is TTHR for the experiment, and (3) the difference between tin for WC TAU and WC TEP may slightly

exceed the 30-second TTHR.

While Figures 8 and 9 provide a visual validation of the inclusion property,

additional insight can be gained by considering some examples designed to illustrate

the implications of each particular well-clear model. Figures 10 and 11 show two

such examples.

Figure 10 shows an encounter geometry designed to illustrate a case when all of

the well-clear models indicate a well-clear violation will occur at some time in the

future given the initial ownship and intruder trajectories, yet a significant difference

in tin exists for each approach. In particular, the value for tin for each model is:

1. WC TAU 42 s,

2. WC TCPA 41.7 s,

3. WC TAUMOD 23.9 s,

4. WC TEP 11.7 s.

Thus, the maximum ∆tin is WC TEP − WC TAU = 30.3 s. This scenario depicts a

situation where every model will eventually determine a well-clear violation exists

for the current, constant-velocity trajectory, however there is a wide range in tin,

the time when such a violation first occurs. The figure also demonstrates another

case of violation of the local convexity property for the WC TAU model.

16
−2.5 −2 −1.5 −1 −0.5 0 0.5

−1.6

−1.5

−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

x (nmi)

y (nmi)

kvok =17.0 k t s

t r ack vo =135.0◦

kvik =92.7 k t s

t r ack vi =90.9◦

WC_TAU

WC_TCPA

WC_TAUMOD

WC_TEP

Figure 11: Encounter geometry of interest: non-agreement over WCVtvar

Figure 11 shows a case when all but one well-clear model results in a well-clear

violation for the initial trajectory. In particular, WC TCPA, WC TAUMOD, and

WC TEP produce well-clear boundaries in any horizontal direction the ownship may

travel, however, the WC TAU model produces a region on the ownship’s current

trajectory in which the ownship may pass without incurring a well-clear violation.

This example was selected to illustrate that there are other important, characterizing

properties of the models appropriate for investigation beyond area and tin. This

paper does not present an extensive analysis of such considerations.

5 Conclusion

A family of well-clear boundary models is presented. This family generalizes the

TCAS II Resolution Logic with different possible definitions of horizontal time variables including tau, time to closest point of approach, modified tau, and time to

entry point. Analytical techniques are used to study the properties of this model.

For instance, it has been formally proved that the well-clear model based on time

to entry point is more conservative than tau, time to closest point of approach, and

modified tau for any scenario and any common choice of threshold values. Furthermore, it is shown that all the models in this family are symmetric, i.e., the ownship

and intruder aircraft have the same perception of being well-clear or not at any

moment in time. Except for the model based on tau, all the models are locally

convex meaning that there is at most one interval of time when the aircraft are not

well-clear, assuming straight-line trajectories.

17
Some of these properties are validated through numerical quantification using

statistical methods. In particular, random encounters are generated in Monte Carlo

fashion, and distributions for area and tin are determined for 10,000 data points.

This analysis represents a preliminary look at some characterizing properties of the

family of well-clear boundary models.

The mathematical development presented in this paper has been mechanically

verified in the Prototype Verification System (PVS) [7]. This level of rigor is justified

by the safety-critical nature of the well-clear concept to the integration of Unmanned

Aircraft Systems in the the National Aerospace System.

References

1. Mar ́ıa Consiglio, James Chamberlain, C ́esar Mu ̃noz, and Keith Hoffler. Concept

of integration for UAS operations in the NAS. In Proceedings of 28th International Congress of the Aeronautical Sciences, ICAS 2012, Brisbane, Australia,

2012.

2. FAA Sponsored Sense and Avoid Workshop. Sense and avoid (SAA) for Unmanned Aircraft Systems (UAS), October 2009.

3. Jonathan Hammer. Horizontal miss distance filter system for suppressing false

resolution alerts, October 1996. U.S. Patent 5,566,074.

4. James Kuchar and Lee Yang. A review of conflict detection and resolution

modeling methods. IEEE Transactions on Intelligent Transportation Systems,

1(4):179–189, December 2000.

5. C ́esar Mu ̃noz, Anthony Narkawicz, and James Chamberlain. A TCAS-II resolution advisory detection algorithm. In Proceedings of the AIAA Guidance Navigation, and Control Conference and Exhibit 2013, AIAA-2013-4622, Boston,

Massachusetts, 2013.

6. Anthony J. Narkawicz, C ́esar A. Mu ̃noz, Jason M. Upchurch, James P. Chamberlain, and Mar ́ıa C. Consiglio. A well-clear volume based on time to entry

point. Technical Memorandum NASA/TM-2014-218155, NASA, Langley Research Center, Hampton VA 23681-2199, USA, January 2014.

7. S. Owre, J. Rushby, and N. Shankar. PVS: A prototype verification system. In

Deepak Kapur, editor, Proc. 11th Int. Conf. on Automated Deduction, volume

607 of Lecture Notes in Artificial Intelligence, pages 748–752. Springer-Verlag,

June 1992.

8. RTCA SC-147. RTCA-DO-185B, Minimum operational performance standards

for traffic alert and collision avoidance system II (TCAS II), July 2009.

9. J. Tadema, E. Theunissen, and K.M. Kirk. Self separation support for UAS. In

AIAA Infotech@Aerospace 2010, number AIAA-2010-3460, Atlanta, GA, USA,

April 2010.

18
10. U.S. Department of Transportation Federal Aviation Adminstration. Introduction to TCAS II Version 7.1, February 2011.

11. Unmanned Vehicle Systems International (UVSI). RPAS: Remoted Piloted Aircraft Systems - The Global Perspective. Blyenburgh & Co., Paris, France, 10th

edition, June 2012. uvs-info.com.

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Analysis of Well-Clear Boundary Models for the Integration of UAS 

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The FAA-sponsored Sense and Avoid Workshop for Unmanned Aircraft Systems (UAS) defines the concept of sense and 

avoid for remote pilots as “the capability of a UAS to remain well clear from and avoid collisions with other airborne traffic.” 

Hence, a rigorous definition of well clear is fundamental to any separation assurance concept for the integration of UAS into 

civil airspace. This paper presents a family of well-clear boundary models based on the TCAS II Resolution Advisory logic. 

Analytical techniques are used to study the properties and relationships satisfied by the models. Some of these properties are 

numerically quantified using statistical methods. 

15. SUBJECT TERMS

National airspace; Safety; Separation assurance; Unmanned aircraft systems; Well-Clear boundary; Well-Clear violation

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