IEEE Robotics & Automation Magazine - June 2020 - 25

In [15], we assumed that the parameter w + N (n, R) follows a Gaussian distribution and that the maximum likelihood estimation of n is the empirical mean of all ws, each of
which corresponds to one demonstration and is obtained by
solving a least-square problem.
Here, we extend the Gaussian distribution to a GMM and
consider task parameter queries q as inputs:
K

w = ~ (q) + % N ( n k (q), R k (q))z k,
k=1

(3)

where z k is an element of a K-dimensional binary variable
z = (z 1, z 2, f, z K )T , with only one particular element being
equal to one and all other elements being zero. The probability of the kth element of z being equal to one is

MDN suggested for VMP generalization also works for
ProMP generalization.
In general, for MP generalization, M human demonstrations for different task parameter queries are collected.
The purpose is to learn an MDN [~($)] mapping from the
task parameter query q to the parameter distribution of the
MP parameter w. For MDN, we have an assumption that
the number of mixture components K of the output GMM
is known.
The MDN
Using neural networks to learn the functions in (4) results in
an MDN (see [16]). The mixing coefficients r k ($), mean
n ($), and covariance R ($) are represented by the network
branches, as shown in Figure 3. These network branches

p (z k = 1) = r k (q).
The probability of the result w given the task parameter q is
p (w | q) =

K

/ r k (q) N ( n k(q), R k (q)).

k=1

(4)

MP Parameter

Task Parameters

We assume that the number of the mixture components K
of the GMM is known and that " r k ($), n k ($), R k ($) ,Kk = 1 are
(a)
(b)
the functions to be learned.
The advantage of the VMP over DMP is its ability to Figure 2. A comparison of (a) a ProMP and (b) a VMP. The
dashed curves are demonstrations, and the solid curves are
adapt to intermediate via points by modifying the elementa- generated trajectories for different goals. The red circles indicate
ry trajectory h(x) (see [15]). Apart from the via-points the starts and goals missed by the ProMP.
adaptation, extending the force term
in a DMP to a GMM makes VMPs
and DMPs exchangeable. Since the
ProMP lacks the elementary trajectory and has no hyperparameters y 0
and g, the ProMP parameter funcMP
Generalization
tion w = ~ (q) determines both the
motion trajectory shape and its start
and goal. In many tasks, the start and
(a)
(c)
goal are a part of the task parameter
queries. With a VMP, we reduce the
MDN
K
learning complexity, because one
K
requirement of the task, i.e., reaching
{µk}k = 1
a new goal, is directly satisfied by the
hyperparameter g.
K
For example, in Figure 2, in contrast to a VMP, not all of the trajectoK
{Σk}k = 1
ries generated by the ProMP reach the
goal if we use the entropy MDN and
select the most probable parameter w
from the output distribution. In some
π
tasks, however, the goal is not a part of
the task parameters and is necessary
for the task execution. For these tasks,
(b)
the VMP requires learning an additional mapping from the task parameters to the goals, whereas the ProMP Figure 3. The MDN proposed for MP generalization. As an example, (a) with the target
as the task parameter indicated by the red circle, the system (b) generates an MP
provides a more compact solution. parameter corresponding to (c) the motion of throwing the ball (see the red curves) on
As shown in Figure 2, the entropy the target.
JUNE 2020

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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IEEE Robotics & Automation Magazine - June 2020

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