IEEE Robotics & Automation Magazine - June 2020 - 31

demonstration, we speed up the motion to 1 s and set a fixed
joint goal. The robot hand always opens at 0.55 s. Then, we
record the location of the ball when it drops to the ground. By
fixing the goals and speed of the motions, these hit-ground
locations are dependent only on the shapes of the joint trajectories. In the experiment, we let the robot face the wall, and it
can bounce the ball off the wall to the target.
We let the robot throw 50 times with different human
demonstrations and randomly split the collected data into
30 for training and 20 for testing. We train an MDN (K = 2)
on 30 demonstrations. For the testing, we use only the hitground locations of the other 20 demonstrations as task
parameter queries, which guarantees that all of the hit-ground
locations are reachable. During the testing, we place a plate on
the ground to indicate the current query. Successful task execution is to throw the ball onto the plate either directly or by
bouncing it off the wall. In the experiment, with 10 samples,
the robot missed only two out of 20 target hit-ground locations. In Figure 9, for one specific task parameter query, we
show how two of 10 MP parameters, which correspond to
two different modes, result in different paths of the ball.

cost to improve the performance of the MDN further. The
comparison of different MDNs shows that the new cost functions perform better than the original one, especially when
the set of demonstrations is relatively small.
What we did not consider here is the extrapolation of the
method to areas outside the demonstration range. Since the
MDN is learned fully from demonstrations, its extrapolation
capability is limited. Current methods dealing with the
extrapolation problem focus only on a specific set of task
parameters, such as the TP-GMM and via-points adaptation
of the VMP, described in [5] or our previous work [15]. The
extrapolation of MP generalization to arbitrary task parameter queries is still unsolved.
Recent approaches, such as those in [13] and [14], also
take task-relevant sensory inputs into account and learn them
together with the robot motions. For human-robot

Success Rate

Discussion and Conclusions
This work addressed the problem of MP generalization to different tasks and was concerned with two aspects. First, to take
the multiple modes and models of human demonstrations
into account, we propose using an MDN for the mapping
from the task parameter query to the MP parameter distribution. The experiments show that the MDN-based approach
outperforms techniques used in previous works. Second, to
further reduce the occurrence of mode and model collapse
during training of the MDN, we propose the entropy cost
function. Moreover, for some tasks, we introduce the failure

0.8
0.6
0.4
0.2
0

Baseline
SVR ± Σ

(2)

GPR ± Σ
MDN

(3)

Mode 1

(a)

Mode 1
(b)

(1)

Mode 2

Figure 8. The results of the hit-the-ball experiment show that the
MDN (red) outperforms the baseline (gray), GPR ! R (blue), and
SVR ! R (green).

(1)

Mode 1

30
50
Number of Samples

(2)

(3)

Mode 2

(4)

Mode 2

(c)
Figure 9. (a) There are 4 DoF used for throwing the ball. (b) The robot throws the ball directly to the target. (c) The robot bounces the
ball to the target off the wall.

JUNE 2020

IEEE ROBOTICS & AUTOMATION MAGAZINE

IEEE Robotics & Automation Magazine - June 2020

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