IEEE Robotics & Automation Magazine - December 2013 - 134

pose X and the swivel angle a such that a = p (X) is likely to
exist. In [13] and [12], such a relationship is conjectured,
assuming, however, that it exists only between the elbow
swivel angle and a partial description of the hand pose. A statistical verification of the existence of this relationship, at least
during the operations considered in the experimental campaign, is thus required.
The first step of this analysis is to cluster the task space.
Each cluster X i is a six-dimensional 5 cm # 5 cm # 10° # 10°
# 10° hypercube. If the correlation between X and a is
believable, at least in one cluster, the corresponding values of
the swivel angle a = h ^qh are, roughly speaking, not so different. Here the standard deviation, s i , has been considered as
an intracluster similarity measure:

z

y

x

a

W

s i = Std 6a i@
Figure 4. The definition of the swivel angle, a , as the angle
between the vertical plane passing through the shoulder and the
wrist and the plane containing the whole arm.

Following the approach proposed in [19], an Extended
Kalman Smoother (EKS) is used to estimate joint variables.
The model consists in a chain of three discrete-time integrators, one for each joint variable. The state-update equation is,
therefore, linear, while the output equations, three for each
marker accounting for the marker kinematics, are nonlinear.
Introducing the process noise, w , and the measurement
noise, v, one obtains:
R
V R
R
V
2
3 V
Sq (k + 1)W SI I e I e /2 I e /6W Sq (k)W
Sqo (k + 1)W S0 I I e I e2 /2 W Sqo (k)W
S
W =S
W + w (k)
WS
Sqp (k + 1)W S0 0 I I e W Sqp (k)W
WS
Sq
W S
W
q (k + 1) T0 0 0 I X qq (k)
T
X
T
X
y (k) = g (q (k)) + v (k),

(1)

where e = 1/200 s is the discrete time-step. The EKS has
been adopted to simultaneously estimate the joint variables
and their time-derivatives.
Redundancy Resolution in the Human Arm
The problem addressed in this section is to check whether
during the experiments a relationship between hand position/
orientation and swivel angle can be identified. Data clustering
was applied to obtain a manageable set of data on which statistical analysis could be performed. Then, an expression of
such a relationship was derived and validated.
Uncorrelation Statistics
Let X = [x y z t i z] T be the vector of hand position
(x, y, z are the Cartesian coordinates of the wrist center) and
orientation (ZYZ Euler angles have been used for t, i, z) . We
want to investigate whether a relationship between the hand
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DECEMBER 2013

q: {X = f ^qh ! X i}.

(2)

Its meaning in this context is straightforward: if in a given
cluster X i this value is small enough, a correlation between
the hand pose X and the swivel angle a exists, at least locally
(i.e., in the considered cluster).
However, in this article, we are interested in proving (or disproving) that a correlation between such variables exists in the
whole set of collected experimental data (see Figure 3). While
for each cluster we were interested in measuring how data were
sparsely distributed around their mean value, to prove the existence of a global relationship, we need to measure how much
the values s i > 0 are different from zero. To this end, the mean
value, the square root of the second moment and the 95th percentile have been taken into account. Their definitions and the
corresponding values computed on the available data are listed
in Table 2. As one can see, the computed values are very small.
In 95% of the clusters, a correlation between the hand pose X
and the swivel angle a is very believable. We thus have clear
experimental evidence that a relationship a = p (X) exists
during the motion, where function p ($) might be identified.
As a further verification of this result, Figure 5 shows the variability of the swivel angle in each cluster.
Least-Squares Identification
Based on the statistical evidence of a static (i.e., time-independent) correlation between the swivel angle a and the hand
pose X during assembly-like movements, an expression for
the function p ($) that describes such a relationship will be
derived. Since in 95% of the clusters, the mean value of the
swivel angle is representative, the clustered values are used to
Table 2. Summary statistics: actual and normalized
with respect to the range of variability a max - a min .
Value
Statistic Definition
m 1 = E 6s i@

Actual

Normalized

2.20°

1.68%

2.94°

2.25%

p 95

5.96°

4.56%

m 2 = E 6(s i) 2@
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