IEEE Robotics & Automation Magazine - June 2020 - 113

Since the NN is not completely verified due to the undecided regions, points from these regions are added to the
existing training set, and the NN is retrained as explained in
the "NN Retraining" section. These points are shown by yellow dots in Figure 10(b). After retraining with the addition of
these points, the NN performed with 0% FPs and roughly
1.9% FNs, which was expected since the NN was trained to be
more conservative. Figure 10(b) gives the verification results
with the retrained NN, and, as can be noted, the entire goal
region was verified by Verisig.

any collision. In Figure 11(b), we repeat the same case without
using the NN decisions; here, the UAV moves to the intermediate goal positions even if the NN decision is unsafe. As
expected, there are instances where the UAV crashes or gets
very close to the obstacles. These results confirm that NNs
can be used to monitor safety properties of motion planning
operations using the composition of smaller verified regions into
larger, more complex, and untrained environments.
Experimental Results
The same case study was also performed in experiments following a similar setup as the one presented in
the previous example. NN training was done on a
smaller primitive environment with one obstacle by
performing 100 flights with our aerial testbed under a
wind disturbance blowing in the +y direction. An initial-final position pair was labeled unsafe if the reachable sets generated using the approach explained in the
"Simulation-Based Reachability" section collided with

y (m)

Simulation Results
In this simulation, the trained NN is used to make decisions
about the safety of a trajectory in the cluttered environment presented in Figure 11. First, the primitive box space
used for training is superimposed around each obstacle
(the square areas inside Figure 11) in such a way that there
is only one obstacle in each primitive space; otherwise,
the results of the NN may not be reliable due to the difference from the training conditions.
Once the mission is started, the
quadrotor picks the closest point to the
final goal inside the target region of the
3
first primitive as an intermediate loca2
tion. The NN makes a decision about
the safety of this intermediate goal
1
from the current position of the UAV. If
0
the decision is deemed safe, the UAV
moves to this intermediate position. If
-1
the NN decision is unsafe, it searches
-2
for a safe goal location in the target
Wind
region of the current primitive. This
-3
search is performed by randomly que14
0
2
4
6
8
10
12
rying points in the target area of the
x (m)
primitive, starting from a closer prox(a)
imity of the initially selected goal and
radially enlarging the search area if no
3
safe goals are obtained immediately.
2
Note that, to deploy such an approach,
the NN must contain at least one safe
1
point in the goal area of the primitive.
0
This process continues until the UAV
reaches its final destination.
-1
Figure 11(a) shows the trajectory fol-2
lowed by the quadrotor in this environWind
ment. The queried intermediate goal
-3
positions found by the NN to be unsafe
14
0
2
4
6
8
10
12
are shown by red dots in the goal regions
x (m)
in the primitives areas, while the safe
(b)
intermediate goals traveled to by the
Initial Position
Actual Path
Obstacles
UAV are shown by cyan dots. Wind disFinal Goal Position
Desired Trajectory
turbance is present throughout the
Candidate Goal Positions
Intermediate Goal Positions
entire mission, blowing in the northeast direction, as shown by the orange
arrow inside the figures. The UAV is Figure 11. The navigation simulation in a cluttered environment. (a) The safe replanning
able to complete the mission without using NN decisions. (b) The planning without using NN decisions.
JUNE 2020

IEEE ROBOTICS & AUTOMATION MAGAZINE

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IEEE Robotics & Automation Magazine - June 2020

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