IEEE Robotics & Automation Magazine - September 2021 - 133

consequence, that the joint-space inertia matrix is strictly
positive definite ( ()
Bq 0( ).
The first requirement is usually verified by resorting to a
proper metric, typically the reconstruction error, and, eventually,
relative information such as the percentage error. In this
article, the following errors are considered:
w== ,
|
N r
,
w
is defined as
r
r
r
22
x
|
r
(9)
where N is the number of samples of the data set at hand and
| 2
| = )
2
()()
()
xr xr
xr
--
rbb
t
YY
Y
T
r
By
defining v| o=
rbb
t
full fullt ()Yfull full
t
T xr
r
r-for
ULS
for CAD,CLS1,2,3
.
22 / , with ν as the difference between
the number of samples and the number of dynamic parameters
identified, an estimate of the covariance matrix Mt
estimated parameters is [5]
t
MY Yb b
= v21T
-
rr
()
(10)
for all of the techniques. For CLS-1, CLS-2, and CLS-3, which
identify the full vector of dynamic parameters, the covariance
is actually computed on the base representation of the
obtained estimation.
The requirement on the positive definiteness of ()
Bq
is actually a binary constraint and is met via construction
by the CAD estimates and algorithms CLS-2 and
CLS-3; however, the ULS and CLS-1 methods may or
may not meet it. Since the method CLS-1 outputs the
full vector of dynamic parameters, it is possible to check
on their value for the fulfillment of the constraints,
although this is not possible in the ULS case; in addition,
since symbolic computation of the positive definiteness
of ()
Bq is computationally intractable,
numerical methods must be implemented to check this
property. In this case, the numerical validation of the
condition ()
Bq 0( is valid for a specific joint configuration
q, and a sample-based approach is needed to
check it against the joint space. Obviously, one single
counterexample is sufficient to invalidate the property,
while the opposite is true only in a probabilistic fashion
for a large number of samples.
Identification and Validation Trajectories
Numerical conditioning on the various optimization
problems to be solved needs to be properly guaranteed
[5], [21], [22]. Considering the stacked regressor matrices
Yb
r
in (7), it is required that its condition number be
r
in (7), it is, instead,
kept as small as possible while increasing the minimum
singular value. In the case of Yfull
required that the smallest nonzero singular value be as
large as possible.
Table 3. The dynamic parameters
in linear combinations.
Parameters
ll
=+
yy
cy
b11,,zz2
b =- +++ +mmmm mcz
l ()lld m mmm
dm mmm
23 4567md()
b =- ++ +++
ll ()
32 23 3
2
xx
,, ,zz
zz
b =+ + +++
md()
42,,zz3
yy
cy
3
2
b =+ ++ +
b =- ++ ++
ll ()
63 34 4
2
xx
,, ,zz
zz
b =+ ++ +
ld ()
73,,zz4
yy
yz
4
2
cy
xx
54
,,
mm m
b =- ++ +
ll ()
10 44 5 zz
,, ,
zz
yy
5
2
b =+ ++
ld ()
mmcz
11 45 5
2
yz
b13 65
cy
xx
zz
b14 55 6
=- +
b15 56zz
yy
yz
,,
=+
ll l
ll
,,
,, ,zz
=+
ld m,,
mmcz
cz
,,
xx
,, ,
yy
yy
,,
xx
b19 67yy
b20 77
==+
ll
,,yy
=+
-
ll
b16 55 6
=b17
67
,,zz dm m
67
b12 44 56 74 5
=- +,,
dm
md mcy
567
b =+ ++md()95 cz67
l lld mm
54 675 cz
43
,,
mm mm
l ()lld mm m
dm mm
567
b83 34 56 73 4
=- ++ -
,,
dm mm dmcy
()
67
456 7
32
,,
456 7
of the
Table 2. The identifiability of the parameters.
Parameters
1
m
m ,cx
m ,cy
m ,cz
lxx
lyy
lzz
lyz
lxy
lxz
Red cells: not identifiable; blue cells: identifiable in linear
combination; white cells: identifiable alone.
2
3
4
5
6
7
cy
b18 67 6 zz
=+
ll l
SEPTEMBER 2021 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
133
IEEE Robotics & Automation Magazine - September 2021

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2021
