IEEE Robotics & Automation Magazine - December 2018 - 81

RRT* Only

RRT*-PDMPs

First Trajectory

Trajectory at
30,000 Modes

Online Phase
Only

Computational
Time

121.23 s

1,830.53 s

4.21 ms

Performance
Index

21.16

z (m) 2
0
2
0

13.16

2

13.94

y (m)

-2

-2

0
x (m)

Figure 4. A comparison of the efficiency and computational time using sampling-based planning (RRT*) and learning-based planning
(RRT*-PDMPs). The red and blue lines show the body trajectories of aerial manipulators, respectively. The black line indicates the
produced trajectory of the center of the object. Each dashed line and the solid line describe the trajectory of the aerial manipulator
from RRT* and RRT*-PDMPs, respectively.

trajectories, we can generalize the optimal motion by choosing the appropriate style parameters based on the environmental information.
To reveal the relationship between the environment and
the corresponding optimal motion, we built an explicit function that we call the mapping function. The mapping function
is represented by solving a regression problem between environmental parameters and style parameters, in which the
environmental parameters are representative environmental
information. Because of the nonlinear relationship observed
between environmental parameters and style parameters, a
nonlinear model is applied for a zero empirical error, which
indicates that the regression function computes the exact
output style parameters when testing each environmental
parameter in trained demonstrations. For nonlinear modeling approaches, we use Gaussian process regression (GPR),
which is a highly accurate and flexible regression model for
approximating unknown nonlinearities.
The environment can be represented by environmental
parameters, denoted as r ! R k, where k is the number of featured properties representing the environment, e.g., the
dimension and location of obstacles. In the specific setting
reported here, we set k = 8 for the environmental parameters, including the closest x, y, and z positions of obstacle
edges. To obtain a generalized mapping function considering all state variables, we formulate the GPR using the
matrix of stacked sets of style parameters, defined as
S 1:N _ [S <1 , S <2 , f, S 


IEEE Robotics & Automation Magazine - December 2018
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