IEEE Robotics & Automation Magazine - December 2013 - 135

Swivel Angle Variability
in Cluster 3,616

1.0

0.35

0.9

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0.25

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1
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180

130

80
10,000
8,000

180

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160
140
ive 120
lA
ng 100
le
80
(°)

4,000

Sw

2,000
0

l

C

er

t
us

In

d

ex

0.6
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0.15
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-50 -40 -30 -20 -10

0

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50

Swivel Angle Prediction Error (°)

Figure 5. The intracluster variability of the swivel angle (clusters are
ordered by increasing value of the average swivel angle).

Figure 6. The intercluster probability distribution of the prediction
error.

identify the relationship between the task variables X and the
swivel angle a, instead of using the entire data set. Moreover,
using clustered quantities, the acquired data are uniformly
distributed in the task space.
Since a nonlinear relationship is expected, a pool of nonlinear regressors, consisting in a mixture of sine/cosine waves
and inverse tangent functions, has been selected. The identification of the nonlinear relationship between a and x has been
turned into the minimization of a linear-in-parameter costfunction. A greedy algorithm has been used to solve the identification problem and to simultaneously reduce the number
of regressors needed to obtain an acceptable approximation

Comparison with Other Methods
In light of the previous discussion, it should be clear that the
trajectories during the experiments were constrained to lie
approximately on the estimated configuration manifold
a - p (X) . It is interesting to compare this conclusion with
previous results obtained in similar studies.

p (X) = 1.39 - 0.88 cos (i + .30)
- 0.49 cos (i + z + 2.98) + 0.21 atan 2 (z, y)
+ 0.64 sin (t) + 0.50 atan 2 (x, y)
+ 0.22 cos (t + i + z) - 0.47 cos (z + 0.60)
(3)
+ 0.40 cos (t + i) .
The computed solution of the optimization problem is
reported in (3), corresponding to a root mean square (RMS)
error of less than 6° (4.59%). Function p ($) in (3) can be used
to predict the value of the swivel angle, a, once the task variables X are known.
The model error has been evaluated on the entire data set
resulting in an RMS value of less than 9° (6.89%), which is a
good compromise between the number of regressors to be
evaluated online and the effectiveness in predicting the
elbow swivel angle. For completeness, the intercluster distribution of the prediction error is shown in Figure 6. As a
graphical comparison, time histories of the acquired and of
the predicted values of the swivel angle are shown together in
Figure 7(a)-(c). As one can see, the identified function p ($)
predicts, with reasonable error, the value of the swivel angle
for a given hand pose for every volunteer.

Methods Based on Synergies/Uncontrolled Manifold
The relationship identified in this article can be interpreted as a
synergy [6], which actually restricts the space of natural configurations of the human arm to a lower-dimensional manifold,
namely an uncontrolled manifold [8]. Notice that the measured
null space variability is confirmed to be small if compared to the
variability allowed by joint redundancy, as reported in [9]. This
article confirms previous assessments on the existence of a configuration manifold where every motion is constrained to lie.
Methods Based on Gaussian Mixtures
In comparison with works that make use of Gaussian Mixture
Model (GMM)/GMR to explain the data distribution [14],
the estimated variability of the swivel angle in each cluster can
be modeled as a Gaussian mixture. As a matter of fact, since
the intracluster variability of the swivel angle is relatively
small, a single-Gaussian model with an expected value of e i
and variance s 2i , namely
P ^a | f ^qh ! X ih ~N ^ei, s 2i h

(4)

is adequate for the purpose of this article. Figure 5 also
reports, for a particular cluster, a comparison between the
variability of the swivel angle and the corresponding singleGaussian model. Following this approach, however, might
result in a excessive number of Gaussians (one for each
cluster). On the other hand, representing the identified correlation with a limited number of Gaussians will result in a
less accurate approximation, if compared with the one
reported in (3).
DECEMBER 2013

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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135
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