January 2014 

NASA/TM–2014-218155 

A Well-Clear Volume Based on Time 

to Entry Point 

Anthony J. Narkawicz, César A. Muñoz, Jason M. Upchurch, James P. Chamberlain, 

and María C. Consiglio 

Langley Research Center, Hampton, Virginia 
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January 2014 

NASA/TM–2014-218155 

A Well-Clear Volume Based on Time 

to Entry Point 

Anthony J. Narkawicz, César A. Muñoz, Jason M. Upchurch, James P. Chamberlain, 

and María 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

A well-clear volume is a key component of NASA’s Separation Assurance concept for

the integration of UAS in the NAS. This paper proposes a mathematical definition of

the well-clear volume that uses, in addition to distance thresholds, a time threshold

based on time to entry point (TEP). The mathematical model that results from

this definition is more conservative than other candidate definitions of the wellclear volume that are based on range over closure rate and time to closest point of

approach.

iii
1 Introduction

The Traffic Alerting and Collision Avoidance System (TCAS) is a family of airborne

devices that are designed to reduce the risk of mid-air collisions between aircraft with

operating transponders [4]. TCAS II, the current generation of TCAS devices, provides resolution advisories (RAs) that direct pilots to maintain or increase vertical

separation when aircraft distance and time parameters cross designed system thresholds. TCAS II RAs are based on a mathematically-derived time tau denoted τ . The

time τ is an approximation to the time of closest point of approach (TCPA, denoted

tcpa) between two aircraft, and an important element of the resolution advisory logic

in TCAS II is to check whether τ falls below a certain threshold value. A formal

definition of a well-clear volume based on the alerting logic of TCAS was presented

in [2].

The present paper further refines the definition of the well-clear volume, which

is a key component of NASA’s Separation Assurance concept for the UAS in the

NAS project [1]. The refinement presented in this paper uses distance and time

thresholds as in [2]. However, it addresses some drawbacks of using τ or the related

quantity modified τ (τmod), which appeared in later versions of TCAS. For instance,

it is well known that for certain scenarios, τ diverges as the aircraft converge upon

the closest point of approach [2]. Thus, in general τ approximates TCPA only for

sufficiently large values.

Instead of τ , τmod, or TCPA, the definition of the well-clear volume presented in

this paper relies on the time to violation of the horizontal threshold. This time is

called time to entry point (TEP, denoted tep). Figure 1 shows a comparison of τ ,

modified τ , TCPA, and TEP for a 90o encounter, where the aircraft separation at the

closest point of approach is zero nautical miles. In this particular encounter scenario,

τ is perfectly coincident with tcpa. It can also be noted that tep ≤ τmod ≤ tcpa. Indeed,

this inequality holds for every scenario. Hence, a well-clear concept based on TEP

is more conservative than a similar concept based on τ , τmod, or tcpa.

2 Preliminaries

Throughout this paper, 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, and the norm of a vector, i.e., s ≡ √s · s, are defined in a

2-D Euclidean geometry. By convention, vectors s and v denote relative positions

and velocities, respectively. Furthermore, the function root, defined by Formula (1),

computes the roots of the quadratic equation ax2 + bx + c = 0. For completeness,

it is defined such that it returns the value 0 when the roots are undefined. In this

paper, the values returned by root are only used in a context where a = 0 and

b2 − 4ac ≥ 0.

root(a, b, c, ) ≡

−b+

√b2−4ac

2a if a = 0 and b2 − 4ac ≥ 0,

0 otherwise. (1)

Assuming constant velocities, the horizontal positions of the ownship and in-
0 10 20 30 40 50 60

0

10

20

30

40

50

60

time (s)

time (s)

p p

||vo|| = 194.3844 kts

track vo

 = 0°

||vi|| = 388.7689 kts

track vi

 = 90°

τ

t

ep

τ

mod

t

cpa

TTHR

Figure 1. Time comparison for 90◦ encounter

truder aircraft at a time t ≥ 0, are given by

so(t) ≡ so + tvo,

si(t) ≡ si + tvi, (2)

respectively. As it simplifies the mathematical development, some definitions in this

paper use a relative coordinate system where the intruder is static at the center of

the system. In this relative system, the ownship is located at s = so − si and moves

at relative velocity v = vo − vi. Therefore, the relative horizontal position of the

ownship with respect to the traffic aircraft at any time t can be defined as follows.

s(t) ≡ s + tv. (3)

Given the relative position s and velocity v, the time of horizontal closest point of

approach, denoted tcpa, is the time t that satisfies  ̇ s(t) = 0. By convention, tcpa is

defined as 0 when the velocities of the ownship and intruder aircraft are identical,

i.e., when v2 = 0. Hence,

tcpa(s, v) ≡



− s·v

v2 if v2 = 0,

0 otherwise. (4)

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. It can be seen from

Formula (4) that when the aircraft are converging, tcpa is always positive.

The time to co-altitude tcoa is the time when the relative altitude and vertical

speed of the aircraft satisfies sz +tcoavz = 0, where sz ≡ soz −siz and vz ≡ voz −viz.

2
Similar to the horizontal case, the product szvz characterizes whether the aircraft

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

paper, time to co-altitude is only relevant when the aircraft are vertically converging.

By convention, when the aircraft are not vertically converging, time to co-altitude

is defined as -1. Formally, time to co-altitude is defined as follows.

tcoa(sz, vz) ≡



−sz

vz if szvz < 0,

−1 otherwise. (5)

The following functions are used to compute the times when the linear relative

trajectory of the ownship with respect to the intruder will intersect a circular area

of radius D around the intruder.

Δ(s, v, D) ≡ D2v2 − (s · v⊥)

2, (6)

Θ(s, v, D, ) ≡ root(v2, 2(s · v), s2 − D2, ). (7)

When Δ(s, v, D) ≥ 0, the function Θ computes the times when the aircraft will lose

separation, if  = −1, or regain separation, if  = 1, with respect to D. In this paper,

time to entry point is only relevant when the aircraft are horizontally converging,

i.e., when s·v < 0. By convention, when the aircraft are not horizontally converging,

time to entry point is defined as -1. Therefore, given a distance D, TEP is formally

defined as follows.

tep(s, v, D) ≡



Θ(s, v, D, −1) if s · v < 0,

−1 otherwise. (8)

3 Well-Clear Volume Based on Time to Entry Point

The definition of the well-clear volume presented in this paper uses a horizontal distance threshold (D THR), a vertical distance threshold (Z THR), and a time threshold

(T THR). Formula (11) provides the formal definition of the well-clear volume, given

absolute state information, in vector form, for the ownship and intruder aircraft.

The volume consists of horizontal and vertical components, whose definitions are

provided by Formula (9) and Formula (10), respectively.

horizontal WCV tep(s, v) ≡ s ≤ D THR or

(s + tcpa(s, v)v ≤ D THR and

0 ≤ tep(s, v, D THR) ≤ T THR).

(9)

vertical WCV(sz, vz) ≡ |sz| ≤ Z THR or (0 ≤ tcoa(sz, vz) ≤ T THR), (10)

WCV tep(so, soz, vo, voz, si, siz, vi, viz) ≡

let s = so − si, v = vo − vi, sz = soz − siz, vz = voz − viz in

horizontal WCV tep(s, v) and vertical WCV(sz, vz).

(11)

3
−8 −6 −4 −2 0 2 4 6 8 −3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

nmi

nmi

p

Area = 17.258 square nmi

Figure 2. Two-dimensional projection of conflict volume for 90◦ encounter

It should be noted that the conflict volume defined by Formula (11) is symmetric

from the point of view of the ownship and traffic aircraft. In fact, the following

proposition can be formally proven.

Proposition 1. For all two-dimensional vectors so, vo, si, vi and real numbers soz,

voz, siz, viz,

WCV tep(so, soz, vo, voz, si, siz, vi, viz) = WCV tep(si, siz, vi, viz, so, soz, vo, voz).

Figure 2 shows the two-dimensional projection of the conflict volume for the

encounter geometry associated with Figure 1. The conflict volume was determined

with D THR = 2037.2 m, Z THR = 213.36 m, and Z THR = 35 s.

4 Conclusion

This paper presents a well-clear definition based on time to entry point (TEP) which

exploits the RA logic in TCAS II. Relevant intermediate definitions and concepts

are developed, and a mathematical definition of this well-clear concept is provided.

The TEP approach has been shown to always provide earlier alerting times than

similar approaches based on τ , τmod, or tcpa. Thus, TEP is a more conservative

approach in terms of alerting time and providing a larger well-clear boundary.

The mathematical development presented in this paper has been mechanically

verified in the Prototype Verification System (PVS) [3]. This level of rigor is justified

by the safety-critical nature of the well-clear volume in the concept of integration

of Unmanned Aerial Vehicles in the National Aerospace System.

4
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. 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.

3. 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.

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

5
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A Well-Clear Volume Based on Time to Entry Point 

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Narkawicz, Anthony J.; Munoz, Cesar A.; Upchurch, Jason M.; 

Chamberlain, James P.; Consiglio, Maria C.

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NASA Langley Research Center

Hampton, VA 23681-2199

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14. ABSTRACT

A well-clear volume is a key component of NASA's Separation Assurance concept for the integration of UAS in the NAS. 

This paper proposes a mathematical definition of the well-clear volume that uses, in addition to distance thresholds, a time 

threshold based on time to entry point (TEP).The mathematical model that results from this definition is more conservative 

than other candidate definitions of the well-clear volume that are based on range over closure rate and time to closest point of

approach.

15. SUBJECT TERMS

Separation Assurance; Time to Entry Point; UAS in the NAS; Well Clear Volume

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