IEEE Robotics & Automation Magazine - June 2020 - 107

Trajectory Planning
Obstacle-avoidance trajectories are computed using a simple
geometric approach. Specifically, if an obstruction is present
along the way to the goal position, a waypoint is added to the
path at a specified avoidance distance ra away from the obstacle center. A course is finally generated to visit all waypoints
on the path by using a minimum jerk trajectory generation
[17]. It should be noted that we use this path-planning method due to its simplicity in implementation in simulations and
experiments; however, the overall proposed framework is
independent from the choice of path-planning approach.

d m = max
max min < p d (t) - p x (p) <,
d ! D t ! [0, T ] ! [0, T ]

Simulation-Based Reachability
In this case study, we use a simulationbased RA. During the offline stage,
we run each training trajectory under the worst-case scenario which, in
our example, is the largest possible

(2)

p

where p d is the position of the vehicle under disturbance d.
Here, d m is used as an upper bound for the actual deviation
from the trajectory, and it is conservative since it is the maximum deviation measured through the entire trajectory. The
position-reachable sets are then generated as follows:
R ( p x, t) = " p (t) : < p (t) - p x (t) < # d m , .

(3)

After generating the reachable sets, the trajectory is labeled
safe or unsafe according to (1). In Figure 3, we show the
reachable sets of two sample trajectories.
Offline Training
The environment has a designated rectangular pickup area
that is limited between [0.0, 1.3] m in the x-axis and [−1.0,
1.0]  m in the y-axis. The drop-off point is located at
p g = [4.0, 0.0] m. There are two obstacles between the pickup
area and drop-off location, positioned at p o1 = [2.0, 0.1] m
and p o2 = [3.0, -0.1] m. For training, 294 points are uniformly distributed in the pickup area and used as the initial
and final positions. For trajectory generation, seven different
avoid distances are considered: ra ! " 0.3, 0.35, 0.4, 0.45, 0.5,
0.6, 0.7 , m.
Two NNs were trained: one for the drop-off operation
and the other for the pickup task. To implement the NNs,
we chose Keras (https://keras.io), a deep-learning library
capable of running on top of Tensorflow (https://tensorf low.org) through a set of application programming
interfaces written in Python. For all layers (input, hidden, and output), we use a sigmoid activation function.
The NN is composed of three input nodes (the x-y initial
position and avoidance distance pair), one hidden layer
of 40 nodes, and one output that determines whether the
label is safe or unsafe. We trained two different NNs, one

Safe

Unsafe

1
0.5

y (m)

Pickup/Drop-Off Task
The first case study that we present in this article is a pickup/drop-off task, an operation that is commonly used in
factory applications where a vehicle moves back and forth
between a warehouse and a workstation. The environment
has a designated pickup area (warehouse) and drop-off position (workstation), with obstacles at known locations in
between. The vehicle is tasked to move from a point inside
the pickup area to the drop-off location. Once it reaches the
drop-off site, it can move back to a new point in the pickup
area. To complete its mission safely, the UAV needs to
decide whether the planned trajectory, parametrized by the
initial position p 0, final goal p g , and avoidance distance ra,
is safe and if not, replanning is required. In this case, replanning is executed by adapting ra .
To train an NN to make safety decisions in this scenario,
two sets of trajectories with different avoidance distances are
generated and labeled using RA: one set links a rich set of initial positions in the pickup area to the
drop-off position, and the other connects the drop-off position to a rich set
1
of final positions in the pickup area.
0.5
The NN queries the initial and final
positions and the avoidance distances;
0
if they are unsafe, it checks a larger
-0.5
avoidance distance until it outputs a
-1
safe decision.
-1
0

disturbance attainable in the environment. Under this condition, for a given trajectory p x, the maximum deviation d m is
calculated as follows:

y (m)

Disturbance Model
The external disturbance considered in this article is bounded
in magnitude: < d < # D max , 6d ! D, where D max is the
upper bound to the disturbance magnitude and D is the set
of all possible disturbances. Here, we assume that the online
disturbance is unknown but constant through time. This is a
reasonable assumption, as wind disturbance generally follows
a Brownian motion and does not change erratically during
short periods of time [15], [16].

1
x (m)
(a)

2

Desired Trajectory
Goal Position

3

0
-0.5
-1
-1

0

Initial Position
Reachable Sets

1
x (m)
(b)

2

3

Obstacle

Figure 3. The reachable sets for two sample trajectories. (a) A safe trajectory. (b) An
unsafe trajectory.

JUNE 2020

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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107
https://keras.io/ https://tensorflow.org https://tensorflow.org
IEEE Robotics & Automation Magazine - June 2020

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