IEEE Robotics & Automation Magazine - September 2021 - 131

with m ,c LBi
and m ,c UBi
as the lower and the upper bounds,
respectively, and where the inequalities are intended to be
componentwise.
Alternatively, by following the approach in [13], it is possible
to require the center of mass of link i to lie inside an ellipsoid
Ei
, with center x Rci ! described by
3
E = ! --R xx Qx x
ic
i
" x
where Q Rsi !
ii i
s
c
i
3 |( )( ) # 1,,
T
(6)
33 is a positive definite matrix defining the
#
shape and orientation of the ellipsoid in the link frame. Finally,
as further constraints considered in [13], the overall mass
of a link might be required to lie inside a region Si
(which
could be an ellipsoid as well; see the " CLS-3 " section). This
constraint, together with the physical consistency, is
addressed as S-density realizability in [13]. As an example,
Figure 1(a) reports the ellipsoid S4
fourth link and the ellipsoid E4
(in orange) containing the
(in red) in which the center
of mass mc
4
4 constraint is to lie.
Estimate of the Dynamic Parameters
In this section, we review the main identification methods
listed in the " Literature Analysis " section and which will be
the objects of our comparison. More mathematical details,
together with instructions to run the code, can be found in
the supplemental material available in [26].
Here, it is worth considering that, apart from the CAD
method, the other four identification approaches require either
that data be acquired or that data be reconstructed from the
robot's onboard sensors, consisting of N joint configurations
(( ), (),( ))qqq
tt t
...
! 12 f
ii i
, together with the corresponding torque
vector ()tix , with ti being the generic time instant and
{, ,, }.
iN These data, together with (1) or (2), are generally
used to build an overdetermined system of linear equations
in the unknown vector of dynamic parameters br in the case
of the ULS method and fullr in the cases of the CLS-1, CLS-2,
and CLS-3 methods in the form of
R
r
x
== rr=
$
T
S
S
S
S
S
where Y()$
x
x
()
()
t
t
1
22
1
x tNN
()
and ()r $
hh
()$
X
W
V
W
W
W
W
T
S
R
S
S
S
S
Y
Y
t
t
Y ()() t
()$
()
()
X
W
V
W
W
W
W
d4
r
()
$$ $()
Y()
,
(7)
y3
d3
y1
are either the base or full regressor computed
in the ith configuration and the base or full vector of
the dynamic parameters, depending on the identification
method adopted.
CAD Estimate
The simplest way to obtain the vector of the dynamic parameters
fullr is to retrieve it from CAD data, which many robot
manufacturers make available to the community. The link
parameters can be automatically extracted by CAD software
once, for instance, properties like the density of each
(a)
d1
z1
x1
z0
z2
x0
y0
(b)
Figure 1. The Kinova Jaco2 robot considered here as test case.
(a) In orange, the external ellipsoid bounding, for example,
the fourth link and, in red, the internal ellipsoid containing the
center of mass. (b) The reference frames of the Kinova Jaco2
robot according to the Denavit-Hartenberg convention.
SEPTEMBER 2021 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
131
x2
y2
z4
z3
x3
y4
x4
x7
d7
x5
d5
y5
z5
x6 y6
component are specified. However, due to the complexity of
the robot structure, the many components involved, and the
uniform density assumption, which is usually made, the
extracted data might be significantly different than the real
data. Moreover, parameters related to friction, when relevant,
are unavailable with a CAD approach. On the other hand, the
physical constraints in the " Constraints on the Robot Model "
section are met by construction.
ULS
The ULS method [5] is a popular method for robots and, in
general, for systems identification that, in the case considered
and in virtue of (7), consists of solving the following minimization
problem in the unknown vector br :
min ()().bbYY
r
r --r
r
xr xr
T
bb
b
r
(8)
In particular, identification requires proper design of the
experiments, and it is important to design exciting trajectories
to provide accurate and fast parameter estimation, even
in the presence of measurement noise, unmodeled dynamics,
and external disturbances. Most works in the field of
robot identification relate the condition number of regressor
Yb
r
to the reliability of the data [5] and design the identification
trajectory so as to minimize this condition number,
as in [4].
However, despite its simplicity, the ULS approach leads
to an estimate of br for which the physical properties
z7
y7
z6
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