IEEE Robotics & Automation Magazine - June 2020 - 30

(a)

(b)
Figure 7. In the hit-the-ball experiment, the desired final ball location is the input of the MDN, which is denoted by a transparent box. (a)
The robot hits the ball from its right side, and the ball bounces off the border and stops at the target. (b) The robot hits the ball directly
toward the target.

successful task execution. As shown in Figure 8, increasing the
number of samples improves the success rate.
In this specific task, the number of samples helps for two
reasons. One trivial reason is the setup of the task, which
allows successful task executions by chance: the VMP guarantees that the robot hits the ball, and the table borders limit the
final ball locations. The other reason is that the MDN learns
the correct distribution, which gives a high probability to the
correct MP parameter, which is, unfortunately, not precisely
the mode. Sampling from the correct distribution has a greater chance of finding the correct solution than directly selecting the most probable mode.
To prove that the latter exists with the MDN for this task,
we consider a uniform distribution of the MP parameters as
the baseline, whose interval is determined by the minimal
and maximal components of the MP parameters, which correspond to the 50 demonstrations. In addition to the baseline,
we construct Gaussian distributions by considering the GPR
and SVR outputs as mean vectors and with a fixed-variance
matrix (R = 0.01I ).
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JUNE 2020

As shown in Figure 8, the MDN outperforms the others
for all sample numbers. For the baseline, it coincides with the
intuition that its success rate is almost proportional to the
sample number because it does not learn from the demonstrations. GPR and SVR have better performances than the
baseline because they draw the samples close to their output
MP parameters. However, the samples drawn around their
outputs are totally by chance because of the fixed-variance
matrix. In contrast, the MDN learns a relatively correct distribution output. Hence, it already achieves a high success rate
with a smaller sample number.
The Throw-the-Ball Experiment
To further evaluate our methods, we let the humanoid robot
ARMAR-6 throw a ball at a specific target. The arms of
ARMAR-6 have eight degrees of freedom (DoF) each. To
simplify the task, we used only four of them without loss of
generality (see the 4 DoF in Figure 9). The demonstrations
were conducted by a human using kinesthetic teaching. After
learning the corresponding MP parameter for each



IEEE Robotics & Automation Magazine - June 2020

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