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

Chamberlain, James P.

Consiglio, Maria C.

Munoz, Cesar A.

Narkawicz, Anthony J.

Upchurch, Jason M.

NASA Technical Reports Server

   
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 
 
 
 
 
 
 
 
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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.
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, pro-
vides resolution advisories (RAs) that direct pilots to maintain or increase vertical
separation when aircraft distance and time parameters cross designed system thresh-
olds. 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)
tim
e 
(s
)
p p
||vo|| = 194.3844 kts
track vo = 0
°
||vi|| = 388.7689 kts
track vi = 90
°
τ
tep
τmod
tcpa
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 ‖v‖2 = 0. Hence,
tcpa(s,v) ≡
{
− s·v‖v‖2 if ‖v‖2 = 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) ≡
{
− szvz 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) ≡ D2‖v‖2 − (s · v⊥)2, (6)
Θ(s,v, D, ) ≡ root(‖v‖2, 2(s · v), ‖s‖2 −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 dis-
tance 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
nm
i
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, Ce´sar Mun˜oz, 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. Ce´sar Mun˜oz, Anthony Narkawicz, and James Chamberlain. A TCAS-II resolu-
tion advisory detection algorithm. In Proceedings of the AIAA Guidance Nav-
igation, 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. Introduc-
tion to TCAS II Version 7.1, February 2011.
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Narkawicz, Anthony J.; Munoz, Cesar A.; Upchurch, Jason M.; 
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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.
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Separation Assurance; Time to Entry Point; UAS in the NAS; Well Clear Volume
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