Method for determining maximum conjunction probability of rectangular-shaped objects
View Patent ↗Maximum conjunction probability calculations are refined by addressing rectangular object shapes, even though their orientation may not be known. Object dimensions and relative miss distance, coupled with the aspect ratio of the projected combined covariance, are used to determine the size and orientation of the covariance as well as the object orientation that produces the maximum probability. In treating the objects as rectangles instead of spheres, the probability calculation becomes more realistic by accounting for object shape. In the absence of object attitude information, a footprint is created that completely defines the region where the two objects might touch. This footprint can then be rotated to determine the orientation that produces the largest probability, making it the most conservative estimate for the given conjunction conditions. Sufficiency of the positional data is accessed to meaningfully support the calculations by using the associated standard deviation to demarcate a probability dilution region.
1. A method for determining maximum conjunction probability of rectangular-shaped objects, comprising:
getting primary and secondary object dimensions ht, wd, lt where lt≧wd≧ht;
computing a combined object radius OBJ as
OBJ=r 1 +r 2
where one-half of a diagonal for a rectangular solid encompassing the primary object is denoted as r 1 , where
r
1
=
(
lt
I
)
2
+
(
wd
1
)
2
+
(
ht
1
)
2
2
and a greatest half-width (r 1p ) perpendicular to r 1 is
r
1
p
=
It
1
·
1
-
(
It
1
2
·
r
1
)
2
one-half of a diagonal for a rectangular solid encompassing the secondary object is denoted as r 2 , where
r
2
=
(
It
2
)
2
+
(
wd
2
)
2
+
(
ht
2
)
2
2
and a half-width (r 2p ) perpendicular to r 2 is
r
2
p
=
It
2
·
1
-
(
It
2
2
·
r
2
)
2
;
computing a width factor (w) as
w
=
min
(
r
1
p
+
r
2
,
r
1
+
r
2
p
)
OBJ
;
projecting relative positions of the primary and secondary objects onto an encounter plane to compute a miss distance (dist);
obtaining a combined covariance aspect ratio (AR); and
computing a maximum conjunction probability (P max ) as
P
max
=
OBJ
2
2
·
π
·
AR
·
(
σ
y
)
2
·
∫
-
w
w
∫
-
1
-
y
2
1
-
y
2
exp
[
(
-
1
2
)
·
[
(
dist
+
x
·
OBJ
AR
·
σ
y
)
2
+
(
y
·
OBJ
σ
y
)
2
]
]
ⅆ
x
ⅆ
y
where x and y are variables of integration in the encounter plane such that x lies along a major axis of the combined object radius OBJ, y lies along a minor axis of the combined object radius OBJ, and σ y is a covariance standard deviation along the minor axis in the encounter plane.
2. The method of claim 1 , wherein, when covariance data is available, the method further comprising:
obtaining the combined covariance aspect ratio (AR) by:
getting primary and secondary object covariance;
combining the primary and secondary object covariance;
getting primary and secondary object position vectors;
computing relative position vectors; and
projecting the combined covariance onto an encounter plane;
and computing the miss distance by:
projecting relative positions of the primary and secondary objects on to the encounter plane.
3. The method of claim 2 , wherein, for collision computation at a point of closest approach, the method further comprising:
getting primary and secondary object velocity vectors; and
computing relative velocity vectors in order to make projections onto the encounter plane.
4. The method of claim 3 , further comprising:
determining if P max is unacceptably high; and if so
taking appropriate action;
otherwise ending the method.
5. The method of claim 3 , further comprising:
determining if the combined covariance is in a probability dilution region; and if so
warning an operator;
otherwise ending the method.
6. The method of claim 2 , wherein, for line of sight impingement computation, the method further comprising:
getting an emitter location; and
computing a line of sight vector in order to make projections onto the encounter plane.
7. The method of claim 6 , further comprising:
determining if P max is unacceptably high; and if so
taking appropriate action;
otherwise ending the method.
8. The method of claim 6 , further comprising:
determining if the combined covariance is in a probability dilution region; and if so
warning an operator;
otherwise ending the method.
9. The method of claim 1 , wherein, when covariance data is not available, the method further comprising:
selecting a combined covariance aspect ratio (AR) from the group consisting of AR=1, AR=“best guess” as deduced from historical covariance data, AR=40, and AR=∞.
10. The method of claim 9 , wherein, for collision computation at a point of closest approach, the method further comprising:
getting primary and secondary object velocity vectors; and
computing relative velocity vectors in order to make projections onto the encounter plane.
11. The method of claim 10 , further comprising:
determining if P max is unacceptably high; and if so
taking appropriate action;
otherwise ending the method.
12. The method of claim 9 , wherein, for line of sight impingement computation, the method further comprising:
getting an emitter location; and
computing a line of sight vector in order to make projections onto the encounter plane.
13. The method of claim 10 , further comprising:
determining if P max is unacceptably high; and if so
taking appropriate action;
otherwise ending the method.
14. The method of claim 1 , further comprising:
initializing σ y as:
σ
y
0
=
OBJ
2
·
AR
·
[
(
1
-
3
·
AR
2
)
·
w
·
(
1
-
w
2
)
3
]
3
·
(
w
·
1
-
w
2
+
asin
(
w
)
)
+
[
(
AR
2
+
1
)
2
+
2
·
(
dist
OBJ
)
2
]
and performing iteration on σ y to find the maximum collision probability within bounds of user tolerance.
15. The method of claim 1 , further comprising prescreening by:
setting
r
=
OBJ
dist
and approximating P max as:
P max =0.48394 ·r where r <0.8
P max =0.21329·exp(1.01511 ·r )−0.09025 where 0.8 ≦r< 1.0 and
P max =0.5 where r =1.