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then m1(x) = 0.333, m2(x) = 0.333, and m3(x) = 0.333. This is the basis of hypertrapezoidal fuzzy membership functions and has proven to be quite useful in inferring operational flight segments of an aircraft. One additional parameter is needed for defining an N-dimensional fuzzy partitioning. The crispnessfactor determines how much overlap exists between the sets of two adjacent prototype points. We chose to define the
range of the crispness factor, s, to be [0, 1]. For s= 1, no overlap exists between the sets, and the partitioning reduces to a minimum distance classifier. Figures 14b and 14c show the resulting partitions of the above example for the two extremes s= 0 and s= 1.
Given a sensor measurement, x, the HFMFs can now be calculated using standard trigonometry.  First, a distance measure, r|j, is calculated for each pair of prototype points, as shown Eq. 3. Here, d(x,y) is the Euclidean distance
i
between xand y.
22
(,li)-dx(
dx ,lj)
r(x)= (3)
ij d2 (ll)
,
ij
Then the pair-wise membership functions are calculated for each pair of prototype points, as shown in Eq. 4.
vv vv
Here, vji is a vector from ljto l, vjx is a vector from ljto x, and vviji jxis the dot product of the two vectors.
 
0; r
x .-s
() 1
 ij  
    
 
mij(x)= 1; rij x s 1
() ￡- (4) 
  vv s 
i - 2 lli 
 vjvjx d(j, )
2
 ; otherwise 
( - d(ll, ) 
 1 s)2 ji
 
 
Finally, the degree of membership, mi(x), of measured input, x, can be determined based on product inference as shown in Eq. 5. Here Mis the number of fuzzy sets in the partition. 
M
Hm
ij (x) j=.i
m (x) MM  (5)
i =g1
. Hmkj (x))
 
)
k=1 .j=.1 k .
Notice that Eqs. 3, 4, and 5 are general for N dimensions, including N=1. These three equations allow for the design of N-dimensional membership functions using only N+1 parameters. Additionally, the desirable property of Eq. 1 is enforced.
Figure 15 shows an example of the fuzzy sets that were defined for light twin aircraft using hypertrapezoidal fuzzy membership functions. In Fig. 15 all but two variables (rate of climb and indicated airspeed) are fixed so that a 3D plot could be drawn using just those two variables. As can be seen, the relationships between variables in fuzzy sets can be defined more richly than they could be defined with one-dimensional sets. Correlation is modeled and with the small number of parameters, the technique scales to many dimensions. 
Using HFMF models of flight segment, we were able to accurately infer flight segments directly from aircraft state variables. Figure 16 compares the inferred flight segments to the flight segment that the pilot stated he was in during the data collection process.
m(x)
climbout
1
takeoff finalapp
landing
0
2000
1000

160
0 140
120 -1000
ROC [ft/min] 100
-2000 80 IAS [knots]
Figure 15. Plots of four fuzzy sets for an altitude of 500 feet above ground.
Figure 16. Experimental results for Hypertrapezoidal Flight Segment Identifier.

V. Conclusion and Future Work
Over a decade of research has shown the value of explicitly modeling flight segments and evaluating those models in real-time. Flight segment identification (FSI) is an enabler for context-aware pilot advising, procedure guidance, and automated Highways-In-The-Sky. We explain the difference between State-based FSI, which identifies the flight segment that the pilot is currently flying, and Procedural FSI, which identifies the flight segment that the pilot should be flying. Pilot advisories and intent-based conflict detection were both successfully implemented and flight tested on the NASA SATS project. Our SATS demonstration showed that flight segment identification enables “smarter” avionics which can support new, efficient flight procedures for the NAS.
　
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