Aerospace and Electronic Systems Magazine September 2016 - 35


Liu and Foina

Figure 3.

The 2D optimal safety control is applied to a 3D quadrotor model.

= 50 m/s, by assuming a maximum horizontal thrust of
translation drag of ctvmax [41].

and

Dtdelay: In all the feasible collaborative SAA technologies discussed previously, ADS-B gives the largest bounded communication delay, 2.0 s. We use this conservative value throughout the
simulations, to approximate the worst case behavior [25], [42].
rmin: The minimum separation distance rmin is tunable to satisfy
future minimum separation standards. It is set at 20 m, or twice the
size of a typical helicopter, throughout the simulations.
The simulation is expected to be realistic, although a hardware
implementation is necessary to confirm the effectiveness of the hybrid controller. It will be left as future work.

SAFETY CONTROLLER
In this section, two sets of simulation results are presented. First,
we examine how the 2D safety controller behaves when applied
to a full 3D quadrotor model. Then, we present the WCMMD
and WCERT results after running algorithm 1, covering all practical accelerations and relative velocities for our specific setup.
All safety sets are produced by gridding 120 points evenly for
.

Safety Controller Behavior
Figure 3 shows the behavior of the safety controller. Two factors,
namely the magnitude of the maximum acceleration and drag
force, are investigated. The dashed red lines are the reference
signals generated from the high-level safety controller, while the
black solid lines are the tracking signals from the low-level sliding
mode angle controllers [37]. The quadrotor is originally at rest.
The parameters are

In Figure 3(a), when amax is low, say 2 m/s2, the constant acceleration assumption holds very well, but in sacrifice it takes a long
SEPTEMBER 2016

time (4.9 s) to finish the avoidance maneuver. At the beginning of
the avoidance maneuver, the tracking delay between the dashed
red line and the solid black line is 0.28 s, which results in an earlier
reaction time (ERT) of 0.1 s.
The avoidance duration is shortened to half as much (2.6 s)
when amax = 10 m/s2, but the desired pitch qd has some variation
at the beginning and the end of the avoidance. The variation exists because the quadrotor could not achieve the desired acceleration instantaneously due to RI. Nonetheless, the assumption
of applying constant maximum acceleration remains valid. The
tracking delay is 0.56 s, which gives an ERT of 0.2 s. When
amax = 5 m/s2, the result is somewhere in between. As amax increases, the tracking delay increases, and ERT increases almost
proportional to the tracking delay. Therefore, without drag, ERT
is solely determined by the tracking delay at the beginning of the
avoidance maneuver.
In the normal case with drag force (Figure 3(b)), the variation
in desired pitch (qd) is worse, although the desired roll (fd) stays
almost constant. qd is generally increasing during the avoidance
maneuver, especially at amax = 10 m/s2. It makes sense because as
the velocity goes up, the drag force goes up, and to generate an
equivalent amax, a larger pitch angle is needed. The avoidance duration is longer, ranging from 7.9 s (when amax = 2 m/s2) to 3.1 s
(when amax = 10 m/s2).
However, with drag, ERT is not a strong function of tracking
delay anymore. With similar tracking delays, the ERT is 1.54 s
when amax = 2 m/s2 and 0.42 s when amax = 10 m/s2. This result is the
complete opposite to the no-drag case. As amax increases, the drag
force increases faster, and to counteract the drag, a larger ERT is
required.

Worst Case Scenarios
In this section, we discuss the simulation results of the
safety controller. The WCMMD and WCERT are observed for a
range of maximum horizontal relative velocity v and maximum
horizontal accelerations amax. In these worst case scenarios, the

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