Chamberlain, J. P.

Consiglio, M.C.

Munoz, C. A.

Narkawicz, A. J.

Upchurch, J.M.

NASA Technical Reports Server

A Family of Well-Clear Boundary Models
for the Integration of
Unmanned Aircraft Systems
in the National Airspace System
C. A. Mun˜oz A. J. Narkawicz J. P. Chamberlain
M.C. Consiglio J.M. Upchurch
NASA Langley Research Center
in Support of
the UAS in the NAS Project
May 13, 2014
https://ntrs.nasa.gov/search.jsp?R=20160006946 2019-08-31T02:33:17+00:00Z
See and Avoid vs. Sense and Avoid
2 / 24
A Motivation for a Formal Definition of Well Clear
I The FAA SAA Workshop for UAS defines sense and avoid
as: “the capability of a UAS to remain well clear from and
avoid collisions with other airborne traffic.”
I How will a UAS determine if it is well clear from other
airborne traffic?
I In the absence of an on-board human pilot with the
experience and judgement to determine well clear, a formal
definition is needed to provide guidance to a ground pilot or
possibly an automated algorithm.
I This definition should be more conservative than TCAS, a
system intended to be the last resort in collision avoidance, so
as to be compatible.
I NASA has examined and developed several formal definitions
which considered to be a family of well-clear boundary
models.
3 / 24
The Approach
A key characteristic of NASA’s concept is that the self-separation
threshold is a conservative extension of the collision avoidance threshold
defined by TCAS.1
1000 ft
20
0 
ft
N
M
AC
collision volume
(CAT) collision avoidance threshold
(SSV) self-separation volume
(SST) self-separation threshold
*ATC Separations Services apply as necessary
Volumes and thresholds are shown as cylinders for illustrative purposes only. In
general, these shapes are irregular, with the exception of the collision volume.
1
Consiglio, Chamberlain, Mun˜oz, and Hoﬄer, ICAS, 2012
4 / 24
Interoperability with TCAS RA Logic
I TCAS is a family of airborne devices that are designed to
reduce the risk of mid-air collisions between aircraft equipped
with operating transponders. 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 lbs,
I To ensure compatibility of NASA’s self-separation concept and
TCAS, the mathematical definition of the volume determined
by the SST is considered to be a conservative extension of the
core TCAS II Resolution Advisory logic which checks against
independent horizontal and vertical time and distance
threshold.2
2Mun˜oz, Narkawicz, and Chamberlain, GNC, 2013.
5 / 24
Assumptions
I Two aircraft, the ownship and intruder,
I Accurate aircraft state information is available for both, i.e.,
I Horizontal positions so , si and velocities vo , vi
I Altitudes soz , siz and vertical speeds voz , viz
I Relative position s = so − si and velocity v = vo − vi
I Relative altitude sz = soz − siz and vertical speed vz = voz − viz
I Prediction at a particular time instant of a future well-clear
violation is based on a straight-line trajectory from that time
instant, i.e., constant velocity is assumed.
ownship
intruder
so  
si 
vi
vo
6 / 24
A Family of Well-Clear Boundary Models
Definition of the Well Clear Volume
WCVtvar (s, sz , v, vz) ≡ Horizontal WCVtvar (s, v) and
Vertical WCV(sz , vz),
(1)
Anywhere inside the volume determined by this function, the aircraft are
not well clear.
The function tvar(s, v) is the only change between the models
Horizontal WCVtvar (s, v) ≡ ‖s‖ ≤ DTHR or
(dcpa(s, v) ≤ DTHR and 0 ≤ tvar(s, v) ≤ TTHR),
Vertical WCV(sz , vz) ≡ |sz | ≤ ZTHR or 0 ≤ tcoa(sz , vz) ≤ TCOA.
dcpa(s, v) ≡ r(tcpa(s, v)) = ‖s + tcpa(s, v)v‖,
‖s‖ ≡
√
s2 =
√
s · s
|sz | ≡ soz − siz
7 / 24
A Family of Well-Clear Boundary Models
Definition of the Well Clear Volume
WCVtvar (s, sz , v, vz) ≡ Horizontal WCVtvar (s, v) and
Vertical WCV(sz , vz),
(1)
Anywhere inside the volume determined by this function, the aircraft are
not well clear.
The function tvar(s, v) is the only change between the models
Horizontal WCVtvar (s, v) ≡ ‖s‖ ≤ DTHR or
(dcpa(s, v) ≤ DTHR and 0 ≤ tvar(s, v) ≤ TTHR),
Vertical WCV(sz , vz) ≡ |sz | ≤ ZTHR or 0 ≤ tcoa(sz , vz) ≤ TCOA.
dcpa(s, v) ≡ r(tcpa(s, v)) = ‖s + tcpa(s, v)v‖,
‖s‖ ≡
√
s2 =
√
s · s
|sz | ≡ soz − siz
7 / 24
Parameter: Time Variables and Thresholds
Four choices for tvar(s, v):
τ(s, v) ≡
{
− s2
s·v if s · v < 0,
−1 otherwise,
(2)
tcpa(s, v) ≡
{
− s·v
v2
if v 6= 0,
0 otherwise,
(3)
τmod(s, v) ≡
{
DTHR2−s2
s·v if s · v < 0,
−1 otherwise,
(4)
tep(s, v) ≡
{
Θ(s, v, DTHR,−1) if s · v < 0 and ∆(s, v, DTHR) ≥ 0,
−1 otherwise, (5)
where
Θ(s, v,D, ) ≡ −s · v + 
√
∆(s, v,D)
v2
,
∆(s, v,D) ≡ D2v2 − (s · v⊥)2.
The four well clear volumes are in order of increasing containment
All four models use the same vertical time variable to compare to TCOA:
tcoa(sz , vz ) ≡
{
− sz
vz
if sz vz < 0,
−1 otherwise. (6)
8 / 24
Parameter: Time Variables and Thresholds
Four choices for tvar(s, v):
τ(s, v) ≡
{
− s2
s·v if s · v < 0,
−1 otherwise,
(2)
tcpa(s, v) ≡
{
− s·v
v2
if v 6= 0,
0 otherwise,
(3)
τmod(s, v) ≡
{
DTHR2−s2
s·v if s · v < 0,
−1 otherwise,
(4)
tep(s, v) ≡
{
Θ(s, v, DTHR,−1) if s · v < 0 and ∆(s, v, DTHR) ≥ 0,
−1 otherwise, (5)
where
Θ(s, v,D, ) ≡ −s · v + 
√
∆(s, v,D)
v2
,
∆(s, v,D) ≡ D2v2 − (s · v⊥)2.
The four well clear volumes are in order of increasing containment
All four models use the same vertical time variable to compare to TCOA:
tcoa(sz , vz ) ≡
{
− sz
vz
if sz vz < 0,
−1 otherwise. (6)
8 / 24
Parameter: Time Variables and Thresholds, continued
Horizontal WCVtvar(s, v) ≡ ‖s‖ ≤ DTHR or
(dcpa(s, v) ≤ DTHR and 0 ≤ tvar(s, v) ≤ TTHR)
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
time (s)
tim
e 
(s)
 
 
‖vo‖ =150 kts
track vo =53
◦
‖v i‖ =350 kts
track v i =90
◦
τ
tcp a
τmod
tep
TTHR
Figure : The 4 well clear volumes are in order of increasing containment
9 / 24
Conceptualizing the Well-Clear Boundary
I Sweep the ownship trajectory around 360◦ while holding voz
constant,
I a boundary in three dimensions is determined by calling
WCVtvar along each trajectory,
I project the resulting surface into the horizontal plane
containing so .
Figure : Illustration of a 3-dimensional encounter projected into 2
dimensions
10 / 24
WC TEP
−8 −6 −4 −2 0 2 4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x (nmi)
y 
(nm
i)
 
 
WC_TEP = 6.7 square nmi
Intruder
Ownship
WCVtep (s, sz , v, vz) ≡ Horizontal WCVtep (s, v) and Vertical WCV(sz , vz)
11 / 24
WC TAUMOD
−8 −6 −4 −2 0 2 4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x (nmi)
y 
(nm
i)
 
 
WC_TAUMOD = 6.1 square nmi
Intruder
Ownship
WCVτmod (s, sz , v, vz) ≡ Horizontal WCVτmod (s, v) and Vertical WCV(sz , vz)
12 / 24
WC TCPA
−8 −6 −4 −2 0 2 4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x (nmi)
y 
(nm
i)
 
 
WC_TCPA = 5.9 square nmi
Intruder
Ownship
WCVtcpa (s, sz , v, vz) ≡ Horizontal WCVtcpa (s, v) and Vertical WCV(sz , vz)
13 / 24
WC TAU
−8 −6 −4 −2 0 2 4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x (nmi)
y 
(nm
i)
 
 
WC_TAU = 5.7 square nmi
Intruder
Ownship
WCVτ (s, sz , v, vz) ≡ Horizontal WCVτ (s, v) and Vertical WCV(sz , vz)
14 / 24
Properties of Interest: Symmetry
Definition (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).
The ownship and intruder agree on whether they are well clear.
Theorem (Symmetry)
The well-clear boundary models WC TAU, WC TAUMOD, WC TCPA,
and WC TEP are symmetric for any choice of threshold values DTHR,
TTHR, ZTHR, and TCOA.
15 / 24
Properties of Interest: Inclusion
Theorem (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).
16 / 24
Properties of Interest: Inclusion, continued
−8 −6 −4 −2 0 2 4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x (nmi)
y 
(nm
i)
 
 
WC_TAU
WC_TCPA
WC_TAUMOD
WC_TEP
WCVtvar (s, sz , v, vz) ≡ Horizontal WCVtvar (s, v) and Vertical WCV(sz , vz)
17 / 24
Properties of Interest: 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).
Local Convexity: Along a linear trajectory, the aicraft does not lose well clear,
gain it back, and lose it again.
−8 −6 −4 −2 0 2 4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x (nmi)
y 
(nm
i)
 
 
WC_TEP = 6.7 square nmi
Intruder
Ownship
Figure : WC TEP
−8 −6 −4 −2 0 2 4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
x (nmi)
y 
(nm
i)
 
 
WC_TAU = 5.7 square nmi
Intruder
Ownship
Figure : WC TAU
18 / 24
Properties of Interest: Local Convexity, continued
Theorem
For any choice of threshold values, the well-clear boundary models
WC TCPA, WC TAUMOD, and WC TEP are locally convex.
Theorem
For some choices of threshold values, the well-clear boundary
model WC TAU is not locally convex.
19 / 24
Conclusion
I A formal definition of well clear is motivated by the need for
UAS to operate safely in the presence of other aircraft in the
airspace
I A family of well-clear boundary models is introduced which
are extensions of the TCAS II RA logic
I Characterizing concepts for these models are:
I Symmetry
I Inclusion
I Local convexity
I WC TAU has instances of non-local convexity and is the least
conservative model
I WC TEP is the most conservative model
20 / 24
References
M. Consiglio, J. Chamberlain, C. Mun˜oz, and K. 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.
C. Mun˜oz, A. Narkawicz, and J. 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),
August 2013.
J. Upchurch, C. Mun˜oz, A. Narkawicz, J. Chamberlain, and
M. Consiglio, “Analysis of well-clear boundary models for the
integration of UAS in the NAS,” NASA Technical
Memorandum (submitted), 2014.
21 / 24
The End
Questions?
22 / 24
Encounter Space for Randomly-Generated Trajectories
Intruder
Ownship
h
R
I Ownship position, and horizontal
direction fixed,
I Ownship and intruder horizontal
velocity randomly chosen
849 velocities,
I Intruder horizontal position chosen
from U [pi, 2pi],
I Intruder vertical position chosen
from N (soz , h/6),
I Intruder horizontal velocity direction
chosen from U [0, 2pi],
I Intruder vertical velocity chosen
from N (0, viz,max).
23 / 24
Example Encounters of Interest
−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 
(nm
i)
 
 
‖vo‖ =83.6 kts
track vo =0.0
◦
‖v i‖ =126.8 kts
track v i =45.0
◦
WC_TAU
WC_TCPA
WC_TAUMOD
WC_TEP
Figure : Large difference in tin
−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 
(nm
i)
 
 
‖vo‖ =17.0 kts
track vo =135.0
◦
‖v i‖ =92.7 kts
track v i =90.9
◦
WC_TAU
WC_TCPA
WC_TAUMOD
WC_TEP
Figure : Disagreement in WCVtvar
24 / 24


A Family of Well-Clear Boundary Models for the Integration of Unmanned Aircraft Systems in the National Airspace System

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