NASA/TM–2010–216857

Time of Closest Approach in

Three-Dimensional Airspace

C ́esar A. Mu ̃noz and Anthony J. Narkawicz

Langley Research Center, Hampton, Virginia

October 2010
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NASA/TM–2010–216857

Time of Closest Approach in

Three-Dimensional Airspace

C ́esar A. Mu ̃noz and Anthony J. Narkawicz

Langley Research Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

October 2010
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Abstract

In air traffic management, the aircraft separation requirement is defined by a minimum horizontal distance and a minimum vertical distance that the aircraft have to

maintain. Since this requirement defines a cylinder around each aircraft rather than

a sphere, the three-dimensional Euclidean distance does not provide an appropriate

basis for the definition of time of closest approach. For instance, conflicting aircraft

are not necessarily in loss of separation at the time of closest three-dimensional Euclidean distance. This paper proposes a definition of time of closest approach that

characterizes conflicts in a three-dimensional airspace. The proposed time is defined

as the time that minimizes a distance metric called cylindrical norm. An algorithm

that computes the time of closest approach between two aircraft is provided and the

formal verification of its main properties is reported.

1
1 Introduction

In a three-dimensional airspace, the separation requirement for two aircraft is specified as a minimum horizontal separation, which is typically 5 nautical miles, and

a minimum vertical separation, which is typically 1000 feet. This requirement is

graphically illustrated by a cylinder, called the protected zone, surrounding each

aircraft. A loss of separation between two aircraft is an overlapping of the aircraft’s

protected zones. A conflict is a predicted loss of separation within a lookahead time

interval, which is typically a few minutes long.

A natural question in the context of state-based separation assurance systems

is, “What is the time of closest approach between two aircraft?” In a 2-dimensional

airspace, the answer to this question is clear: the time of closest horizontal approach

between two aircraft is the time when the Euclidean horizontal distance between the

aircraft is minimal. The time of closest horizontal approach characterizes horizontal

conflicts in the sense that two aircraft are in horizontal conflict if and only if they

are in horizontal loss of separation at this time.

In a 3-dimensional airspace, the protected zone is a cylinder rather than a sphere.

Typically, this cylinder is 30 times wider than high. Thus, the three-dimensional

Euclidean distance does not properly reflect whether the aircraft are separated or

not. For instance, vertical separation of one thousand feet may be safe, but horizontal separation of one thousand feet is not. Thus, in contrast to the 2-dimensional

case, the time that minimizes the 3-dimensional Euclidean distance does not provide

an appropriate definition for the time of closest approach between the aircraft.

The problem addressed by this paper is the mathematical and algorithmic definition of a time of closest approach between two aircraft in a 3-dimensional airspace.

This time is defined as the time that minimizes a distance metric called cylindrical

norm. As indicated by its name, the cylindrical norm accommodates the notion

of aircraft distance to the fact that the protected zone is a cylinder. The rest of

this paper is organized as follows. Section 2 explains the mathematical notation

used in this paper. Section 3 reviews the definition of the time of closest horizontal approach. Section 4 proposes a definition of time of closest approach for a

3-dimensional airspace and states its main property, i.e., that the proposed time of

closest approach correctly characterizes 3-dimensional conflicts. Section 5 summarizes this work and presents concluding remarks.

2 Mathematical Notation

The mathematical development presented in this paper has been fully formalized in

the Prototype Verification System (PVS) [6]. PVS is a mechanical theorem prover

that consists of an expressive specification language based on higher-order logic and a

proof checker for this logic.1 For readability, this paper uses standard mathematical

notation instead of PVS syntax.

1PVS is electronically available from http://pvs.csl.sri.com. The development presented in

this paper is electronically avaialable from http://shemesh.larc.nasa.gov/people/cam/ACCoRD.
The constructions in this paper consider an airspace with two distinguished aircraft, the ownship and the intruder, that are potentially in conflict. The state of

an aircraft consists of its current position and velocity information, and it is represented by vectors in a Euclidean geometry. In a 2-dimensional airspace, the separation requirement is defined by minimum horizontal distance D. In a 3-dimensional

airspace, the separation requirement is defined by a minimum horizontal distance

D and a minimum vertical distance H. The values D and H are considered to be

parametric constants.

The symbols R, R

2

, and R

3

represent the sets of real numbers, the set of 2-

dimensional vectors, and the set of 3-dimensional vectors, respectively. Vector variables are written in boldface and can be denoted by their components. For example,

if w ∈ R

3 and u ∈ R2

, then w = (wx, wy, wz) and u = (ux, uy). The expression

w(x,y) denotes the projection of w in the horizontal plane, i.e.,2

w(x,y) ≡ (wx, wy).

The notation kwk refers to the Euclidean norm of the vector w and the notation

w·w0refers to the dot product of the vectors w and w0. The expression 0 represents

the zero vector, i.e., the vector whose components are 0.

Aircraft trajectories are represented by a point moving at constant linear speed,

i.e., if the current state of an aircraft is given by the position s and velocity vector

v, its predicted position at time t is s+t v. In this paper, the position vector so and

the velocity vector vo represent the current state of the ownship. Similarly, si and

vi represent the current position and velocity vectors of the intruder aircraft. As it

simplifies the mathematical development, the formalization presented here usually

considers a relative view where the intruder is fixed at the origin of the coordinate

system. In this paper, the vectors s and v will commonly be used to denote the

relative position so − si and the relative velocity vo − vi, respectively.

Aircraft predicted trajectories are considered valid for a lookahead time interval,

which is specified by a lower bound B and an upper bound T. The notation [B, T],

where 0 ≤ B < T, represents the interval of real numbers greater than or equal to

B and less than or equal to T, i.e.,

[B, T] ≡ {x ∈ R | B ≤ x ≤ T}.

The values B and T are parametric constants. The upper bound T is usually a

real number. However, the formalism presented here also considers the special case

where T has an infinite value, i.e., T = ∞. In that case, it is assumed that for

all x ∈ R, x < ∞ and min(x, ∞) = x. It is remarked that this abuse of notation

is a syntactical convenience and should not be understood as an extension of real

arithmetic with infinite values. For instance, the value ∞ is not considered to be

in R. Hence, if T = ∞, the expression [B, T] simply represents the interval of real

numbers greater than or equal to B. Furthermore, particular care is taken in this

paper to avoid the use of ∞ in arithmetic expressions other than min(x, ∞) and

relational operations such as x < ∞.

2The symbol ≡ is used in this paper to introduce mathematical definitions.

2
Finally, by convention, names of predicates and functions used in the specification of the problem are written in italics. Functions that represent algorithms to be

implemented in a programming language are written in typewriter font.

3 Time of Closest Horizontal Approach

The horizontal separation requirement can be understood as an imaginary circle of

diameter D around each aircraft and a horizontal conflict between two aircraft as a

future overlapping of these circles. In this paper, an alternative but equivalent view

is considered where the intruder is surrounded by a circle, called protected zone, of

radius D. From this perspective, a horizontal conflict between these two aircraft is

equivalent to the existence of a time within a lookahead time interval at which the

ownship is in the interior of the intruder’s protected zone.

3.1 Horizontal Conflict

Given a lookahead time interval [B, T], where T is possibly infinite, a horizontal

conflict between the ownship and the intruder aircraft occurs when there is a time

t ∈ [B, T] such that the horizontal distance between the aircraft is less than D, i.e.,

k(so + t vo) − (si + t vi)k < D,

where so, vo, si, and vi are all in R

2

. Since (so + t vo) − (si + t vi) = (so − si) +

t(vo − vi), the predicate that characterizes horizontal conflict can be defined in

terms of the relative vectors s = so − si and v = vo − vi, i.e., the relative position

and velocity vectors, respectively, of the ownship with respect to the intruder. The

predicate HorizontalConflict?[B,T], parametric on the lookahead time interval [B, T],

is formally defined as follows.

HorizontalConflict?[B,T](s, v) ≡ ∃ t ∈ [B, T] : ks + t vk < D. (1)

3.2 Time of Closest Horizontal Approach

This section presents a mathematical derivation of the time of closest horizontal

approach between two aircraft for a given lookahead time interval.

Definition 1. The time of closest horizontal approach between the ownship and the

intruder aircraft, for a lookahead time interval [B, T], where T is possibly infinite,

is the minimum time τ in the interval [B, T] that satisfies

∀ t ∈ [B, T] : ks + t vk ≥ ks + τ vk, (2)

where s = so − si and v = vo − vi.

From the definition above, it is not clear that such time τ exists. The rest of

this section provides an algorithm definition of time of closest horizontal approach

and the proof that it satisfies Definition 1.

3
The norm ks+t vk is minimized precisely when its square ks+t vk

2

is minimized.

This squared norm is equal to kvk

2

t

2 + (2 s · v)t + ksk2

, which is a quadratic

polynomial in the variable t. For any quadratic polynomial of the form a t2 + b t + c

where a > 0, the minimum occurs at the point t = −

b

2 a

. Thus, if v 6= 0, then the

minimum of the norm ks + t vk is attained at time −

s·v

kvk

2 . If that time is less than

B, then every t ≥ B satisfies ks + t vk ≥ ks + B vk. Furthermore, if that time is

greater than T, then every time t ≤ T satisfies ks + t vk ≥ ks + T vk. If v = 0,

the distance between the aircraft remains constant. In that case, the minimum time

that satisfies Formula (6) is B.

This motivates the definition of the function tcha[B,T]that computes the time

of closest horizontal approach for the lookahead time interval [B, T].

tcha[B,T](s, v) ≡

(

min T, max B, −

s·v

kvk

2

 if v 6= 0,

B otherwise.

(3)

It has been proved in PVS, using basic algebra, that the time tcha[B,T](s, v) satisfies

Definition 1.

Lemma 1. For any t ∈ [B, T], ks + t vk ≥ ks + τ vk, where τ = tca[B,T](s, v).

The next theorem follows immediately.

Theorem 2. HorizontalConflict?[B,T](s, v) holds if and only if ks+τ vk < D, where

τ = tcha[B,T](s, v).

This theorem states that the ownship and intruder aircraft are in horizontal

conflict within the lookahead time interval [B, T] if and only if they are expected to

be in horizontal loss of separation at the time of closest horizontal approach, which

is given by the function tcha[B,T].

4 Time of Closest Approach in a 3-Dimensional Airspace

In the relative frame of reference, the protected zone surrounding the intruder aircraft is a cylinder of half-height H and radius D. A conflict between the ownship

and intruder aircraft occurs when there is a time within the lookahead time interval

at which the ownship is in the interior of the intruder’s protected zone.

4.1 Conflict

A mathematical definition of conflict is given as follows. The ownship and the

intruder aircraft are in conflict during the time interval [B, T], where T is possibly

infinite, if there exists a time t ∈ [B, T] when vertical separation is lost, i.e,

|(soz + t voz) − (siz + t viz)| < H,

and horizontal separation is lost, i.e.,

k(so + t vo)(x,y) − (si + t vi)(x,y)k < D,

4
s

D

H

Z

X

τ

v

Figure 1. Time (τ ) of closest Euclidean approach in a 3-dimensional airspace

where so, vo, si, and vi are all in R

3

. As in the case of horizontal conflict in Formula (1), the predicate that characterizes conflict is defined on the relative vectors

s = so − si and v = vo − vi.

Conflict?(s, v) ≡ ∃ t ∈ [B, T] : |sz + t vz| < H and

ks(x,y) + t v(x,y)k < D.

(4)

4.2 Euclidean Distance

Identically to the construction in Section 3.2, the time min(T, max(B, −

s·v

kvk

2 )),

when v 6= 0, minimizes the 3-dimensional Euclidean distance between the aircraft

during the time interval [B, T], where s and v are in R

3

. If the protected zone

around an aircraft were a sphere instead of a cylinder, this formula would suffice

as the definition of the time of closest approach. However, since the protected zone

is a cylinder, this definition does not properly characterizes 3-dimensional conflicts.

Indeed, it does not satisfy the property that the aircraft are in conflict during the

time interval [B, T] if and only if they are in loss of separation at this time.

If the dimensions of D and H were comparable, the minimal 3-dimensional

Euclidean distance could be an indicator of the time of closest approach. However,

the typical protected zone, i.e. where D = 5 nautical miles and H = 1000 feet,

is 30 times wider than it is high. Figure 1 provides a side view of the protected

zone on the X, Z-plane where the axes have the same scale. In Figure 1, assume

that vy = 0 and that τ is the time of minimum Euclidean distance for the relative

position s = so − si and relative velocity vector v = vo − vi. This figure illustrates

that although the aircraft are in conflict, they are not in loss of separation at time τ .

Hence, an appropriate definition of the 3-dimensional time of closest approach

between two aircraft, which are surrounded by cylindrical protected zones, cannot

be defined on terms of the minimum Euclidean distance between the aircraft.

5
4.3 Cylindrical Norm

An alternative notion of aircraft distance, which is based on cylindrical norms, is

proposed in [3]. Given a cylinder of radius D and half-height H, the cylindrical

norm of a vector w ∈ R

3

, with respect to D and H, is the quantity

kwkcyl ≡ max 

|wz|

H

,

kw(x,y)k

D



. (5)

With this norm, R

3

is a metric space in the sense of real analysis [7]. That is,

the cylindrical norm satisfies the following properties for any vectors w and u in R

3

.

• Positivity: kwkcyl ≥ 0.

• Nullity: kwkcyl = 0 if and only if w = 0

• Scalability: kk wkcyl = |k| kwkcyl for any k ∈ R

• Triangle Inequality: kw + ukcyl ≤ kwkcyl + kukcyl

The cylindrical distance between the vectors w and u is defined as the cylindrical

norm kw − ukcyl.

The following lemma and theorem, which are proven in PVS, follow directly from

Equation (4) and Equation (5).

Lemma 3. The ownship and the intruder aircraft are in loss of separation if and

only if kso − sikcyl < 1.

Theorem 4. Let s = so − si and v = vo − vi be in R

3

, Conflict?B,T (s, v) holds if

and only if there exists t ∈ [B, T] such that ks + t vkcyl < 1.

4.4 Time of Closest Approach

Using the cylindrical norm, this section generalizes the construction in Section 3.2

to define the time of closest approach for a three dimensional airspace, where the

protected zone is a cylinder.

Definition 2. The time of closest approach between the ownship and the intruder

aircraft, for a lookahead time interval [B, T], where T is possibly infinite, is the

minimum time τ in the interval [B, T] that satisfies

∀ t ∈ [B, T] : ks + t vkcyl ≥ ks + τ vkcyl, (6)

where s = so − si and v = vo − vi.

As in the case of Definition 1, the fact that there exists a time τ that satisfies

Formula (6) is not obvious. The rest of this section provides an algorithmic definition

of the time of closest approach and the proof that it satisfies Definition 2.

The time of closest horizontal approach is constructed by finding the minimum

time in a time interval that minimizes the 2-dimensional Euclidean norm. This

idea can be generalized to define the time of closest approach in a 3-dimensional

6
airspace by finding the minimum time in [B, T] that minimizes the cylindrical norm

ks + t vkcyl. It suffices to minimize the square ks + t vk

2

cyl of this norm. This square

can be written as the maximum of two quadratic polynomials as follows.

ks + t vk

2

cyl = max 

(sz + t vz)

2

H2

,

ks(x,y) + t v(x,y)k

2

D2

!

= max 

1

H2

(v

2

z

t

2 + (2 sz vz)t + s2

z

),

1

D2

(kv(x,y)k

2

t

2 + (2 s(x,y)

· v(x,y))t + ks(x,y)k

2

)



.

(7)

It can be proved that the minimum of the function max(p(t), q(t)), where t ranges

over the interval [B, T] and p and q are two quadratic polynomials of the form

p(t) = a t2 + b t + c and q(t) = e t2 + f t + g, is reached at a point τ that satisfies

one of the following conditions.

• τ = B, or

• τ = T and T 6= ∞, or

• τ is equal to either −

b

2a

or −

f

2e

, or

• τ is such that p(τ ) = q(τ ). If p and q are not equal and a 6= e, then there

are at most two solutions to the equation p(t) = q(t). If this equation has a

solution, then the discriminant of the polynomial (a−e)t

2 + (b−f)t+ (c−g)

must be nonnegative. This discriminant is given by (b − f)

2 − 4 (a − e)(c − g).

Alternatively, if a = e and b 6= f, then there is exactly one solution to the

equation p(t) = q(t), namely −

c−g

b−f

. In both cases, it is easy to see that there

are only finitely many possibilities for τ .

Applying this reasoning to the maximum in Equation (7), there are at most

six possibilities for the time τ . The first four possible times are B, T, −

sz

vz

, and

−

s(x,y)·v(x,y)

kv(x,y)k

2 . The other two possibilites are the solutions to the equation

1

H2

(v

2

z

t

2 + (2 sz vz)t + s2

z

) = 1

D2

(kv(x,y)k

2

t

2 + (2 s(x,y)

· v(x,y))t + ks(x,y)k

2

). (8)

Equation (8) can be rewritten as

A t2 + B t + C = 0, (9)

where

A =

v

2

z

H2

−

kv(x,y)k

2

D2

, (10)

B =

2 sz vz

H2

−

2 s(x,y)· v(x,y)

D2

, (11)

C =

s

2

z

H2

−

ks(x,y)k

2

D2

. (12)

7
Equation (9) has a solutions for t precisely when the discriminant B2 − 4 A C is

nonnegative. In this case, there are two possible solutions, which are equal if this

discriminant is zero, and these solutions are given by the quadratic formula. This

motivates the following definition of the function tca[B,T].

tca[B,T](s, v) ≡

let

A =

v

2

z

H2

−

kv(x,y)k

2

D2

,

B =

2 sz vz

H2

−

2 s(x,y)· v(x,y)

D2

,

C =

s

2

z

H2

−

ks(x,y)k

2

D2

in

T := ∅;

if v(x,y) 6= 0 then

T := T ∪ −

s(x,y)· v(x,y)

kv(x,y)k

2



;

endif;

if vz 6= 0 then

T := T ∪ −

sz

vz



;

endif;

if A 6= 0 and B

2 − 4 A C ≥ 0 then

for ι ∈ {1, −1} do

T := T ∪ (

−B + ι

√

B2 − 4 A C

2 A

)

;

endfor;

elsif A = 0 and B 6= 0 then

T := T ∪ −

C

B



;

endif;

τ := B;

for t ∈ T do

if t > B and ks + t vkcyl < ks + τ vkcyl then

τ := t;

endif;

endfor;

return min(τ, T);

(13)

It has been proved in PVS that the time tca[B,T](s, v) satisfies Definition 2.

8
Lemma 5. For any t ∈ [B, T], ks + t vkcyl ≥ ks + τ vkcyl, where τ = tca[B,T]

(s, v).

The proof of Lemma 5 uses the fact that the quantity ks + t vkcyl is a convex

function of t, which can be proved using basic algebra. From that lemma, the next

theorem follows immediatly.

Theorem 6. Conflict?(s, v) holds if and only if |sz + τ vz| < H and ks(x,y) +

τ v(x,y)k < D, where τ = tca[B,T](s, v).

Theorem 6 states that tca[B,T] properly characterizes 3-dimensional conflicts,

i.e., the ownship and the intruder are in conflict during the time interval [B, T] if

and only if they are expected to be in loss of separation at time tca[B,T](s, v).

5 Conclusion

Mathematical and algorithmic definitions of time of closest approach have been presented for two and three-dimensional airspace geometries. Although the definition

of time of closest horizontal approach is well-known (see for example [4]), the fact

that this definition characterizes horizontal conflicts appears to be a novel approach

to this subject. The major contribution of this paper is the definition of time of closest approach for a three-dimensional airspace and the formal proof of its properties.

The definition of this time is based on the notion of cylindrical distance, originally

proposed in [3].

Since the typical protected zone is very flat, researchers in state-based separation

assurance systems usually define the time of closest approach in the horizontal plane.

This is the approach followed for example by [4], where a 3-dimensional conflict

detection and resolution algorithm is proposed. In the case of conflicting aircraft,

special care has to be taken when the time of closest horizontal approach does not

lie in the interval of conflict.

The cylindrical distance at time of closest separation can be used as a measure

of threat severity in a 3-dimensional airspace. Indeed, this non-dimensional value is

less than 1 in the case of a conflict. The closer this value is to 1, the closer the relative

trajectory is to the envelope of the protected zone. A similar non-dimensional value ε

is proposed in [2] and used in [1] as a measure of conflict risk. This value ε represents

the ellipse distance of an ellipsoid enclosed within the cylindrical separation zone.

Neither the time of closest horizontal approach nor the value ε characterize threedimensional conflicts as tca[B,T] does. It is believed that the proposed time of closest

separation allows for simpler separation assurance algorithms and yields a notion

of distance in a 3-dimensional airspace that is more appropriate for a cylindrical

protected zone. Indeed, this notion is at the base of the formally verified conflict

prevention algorithm presented in [5].

References

1. Richard Barhydt, Todd M. Eischeid, Michael T. Palmer, and David J. Wing. Regaining lost separation in a piloted simulation of autonomous aircraft operations.

9
In Proceedings of the 5th USA/Europe Air Traffic Management R&D Seminar,

Budapest, Hungary, June 2003.

2. Wim den Braven. Analysis of aircraft/air traffic control system performance. In

Proceedings of the AIAA Guidance, Navigation and Control Conference, number

AIAA-95-3363, pages 1721–1733, Baltimore, Maryland, USA, August 1995.

3. Gilles Dowek and C ́esar Mu ̃noz. Conflict detection and resolution for 1,2,. . . N

aircraft. In Proceedings of the 6th AIAA Aviation Technology, Integration and

Operations Conference (ATIO), number ATIO-2007-7737, Belfast, Northern Ireland, September 2007.

4. Jacco Hoekstra. Designing for safety: The free flight air traffic management concept. Technical Report 90-806343-2-8, Technische Universiteir Delft, November

2001.

5. Anthony Narkawicz, C ́esar Mu ̃noz, and Gilles Dowek. Formal verification of air

traffic prevention bands algorithms. Technical Memorandum NASA/TM-2010-

216706, NASA, Langley Research Center, Hampton VA 23681-2199, USA, June

2010.

6. Sam Owre, John Rushby, and Natarajan Shankar. PVS: A prototype verification

system. In Deepak Kapur, editor, Proceeding of the 11th International Conference on Automated Deductioncade, volume 607 of Lecture Notes in Artificial

Intelligence, pages 748–752. Springer, June 1992.

7. Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, third edition,

1976.

10

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5e. TASK NUMBER

5f. WORK UNIT NUMBER

411931-02-51-07-01

6. AUTHOR(S)

C ́esar A. Mu ̃noz and Anthony J. Narkawicz

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, Virginia 23681-2199

8. PERFORMING ORGANIZATION

REPORT NUMBER

L–19933

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

10. SPONSOR/MONITOR’S ACRONYM(S)

NASA

11. SPONSOR/MONITOR’S REPORT

NUMBER(S)

NASA/TM–2010–216857

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category 03

Availability: NASA CASI (443) 757-5802

13. SUPPLEMENTARY NOTES

An electronic version can be found at http://ntrs.nasa.gov.

14. ABSTRACT

In air traffic management, the aircraft separation requirement is defined by a minimum horizontal distance and a minimum

vertical distance that the aircraft have to maintain. Since this requirement defines a cylinder around each aircraft rather than

a sphere, the three-dimensional Euclidean distance does not provide an appropriate basis for the definition of time of closest

approach. For instance, conflicting aircraft are not necessarily in loss of separation at the time of closest three-dimensional

Euclidean distance. This paper proposes a definition of time of closest approach that characterizes conflicts in a

three-dimensional airspace. The proposed time is defined as the time that minimizes a distance metric called cylindrical

norm. An algorithm that computes the time of closest approach between two aircraft is provided and the formal verification

of its main properties is reported.

15. SUBJECT TERMS

Air traffic, conflict detection, time of predicted closest approach, distance at time of predicted closest approach, 3-dimensional

airspace

16. SECURITY CLASSIFICATION OF:

a. REPORT

U

b. ABSTRACT

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c. THIS PAGE

U

17. LIMITATION OF

ABSTRACT

UU

18. NUMBER

OF

PAGES

19

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802