Table 2. Different rigid body models for the dynamic parameter identification process of the
3-DOF RPS parallel manipulator.
6. Application to a 3-RPS robot
In this section, the results of the identification process, implemented by the authors over a 3-
RPS parallel manipulator, are presented. In the first part, the approach based on considering
the physical feasibility in the identification process is presented. In the second part of the
section, the identifiability of the dynamic parameter for the 3-RPS robot is evaluated. In both
sections, the identification process is validated using a simulated manipulator built by
making use of the ADAMS dynamic simulation program. After that, it is applied over a real
one constructed at the Polytechnic University of Valencia, see Fig. (1).
6.1 Identification considering the physical feasibility
Because of the noise in the input data and/or the discrepancies between the actual parallel
robot and the dynamic model used in the identification process, some of the inertial
parameters obtained using LSM methods result physically unfeasible. Thus, the necessity
for a constrained optimization process to ensure physical feasibility appears clearly. In this
subsection, the results are shown as a comparison between the original actuator forces and
those calculated using the identified dynamic parameters in the case of; a) linear friction
models and. b) nonlinear friction models.
The dynamic model of this manipulator, trajectory optimization and identification process
were built in FORTRAN programming language with the aid of the NAG library and the
NLPQL Sequential Quadratic Programming subroutine (Schittkowski, 2000).
Simulated Robot
In the simulated robot, nonlinear friction model is considered at all the joints of the robot. It
can be represented by the following relation,
δ
S
v vS
F(v) = F + (F − F )e
+ F v
(32)
C
S
C
v
Where FC, FS and Fv are the Coulomb, static, and viscous friction coefficients, respectively,
vS is the Stribeck velocity and δs is the stiction transition velocity. This model consists of five
parameters and captures the Coulomb, static, viscous and Stribeck friction forces (Olsson et
al., 1998). After calculating the external original forces, errors are introduced assuming a
normal distribution producing the perturbed forces. Now, based on these perturbed forces,
36
Parallel Manipulators, Towards New Applications
identification is carried out considering asymmetric linear friction models for all joints,
using LSM and optimization, and symmetric nonlinear friction models using optimization.
The corresponding identification errors are shown in Table (3). where ε is the Relative
RA
Absolute Error and is defined as,
*
∑ τ − τ
idnti
i
i
ε
=
(33)
RA
*
*
∑ τ − τ
i
i
where, *
τ and τ
are the actual applied force and those calculated using the dynamic
idnt
model applying identified dynamic parameters, and *
τ is the average of *
τ .
ε (%)
RA
Perturbed Original
Linear (LSM)
11.96
8.26
Linear (Opt.)
12.44
8.85
Nonlinear (Opt.)
11.02
7.28
Table 3. Identification errors based on simulated manipulator.
As can be seen in Table (3), considering the case where linear friction models were used in
the identification process, when the physical feasibility had been ensured, i.e. identification
by optimization, the error increased. On the contrary, when this was accompanied by the
nonlinear friction models, the results were improved considering both the perturbed and the
original forces. Note that in this step, identification was carried out simulating both types of
the mentioned identification process error sources.
Actual robot
Now, after verifying the identification process over the simulated manipulator, the results
are shown in detail considering the identification of the dynamic parameters of the actual
manipulator. Starting with the optimization process for the exciting trajectory and changing
the initial estimations, different optimized trajectories were obtained. An example of such an
optimized trajectory is that one presented previously in Fig. (2). Hereafter, the identified
dynamic parameters were obtained basing on anther optimized trajectory with a
corresponding condition number of 638.
A PID controller is used in order to determine the control actions. The control actions were
applied with a frequency of 100Hz, at which measurement were also taken. The total
duration of the optimized trajectory is 7.5s. Trajectories were repeated several times, the
applied control actions were averaged and then a second order lowpass digital Butterworth
filter was applied. For the identification process, 75 configuration points are extracted every
0.1s.
When the LSM was used in the identification process, a non physically feasible base
parameters were found. Hence, the identification process was held using the nonlinear
constrained optimization process where the physical feasibility of the obtained inertial
parameters was ensured. Fig. (3) shows a graphical comparison between the actual forces
and those calculated using the dynamic parameters identified by LSM and optimization
considering asymmetric linear friction models at the prismatic joints.
Dynamic Parameter Identification for Parallel Manipulators
37
In order to justify the use of nonlinear friction models in the identification process rather
than those which are linear, as the friction phenomenon in the considered joints has this
tendency, a thorough error comparison was made. This is established considering three
different sets of dynamic parameters identified by: LSM, optimization in the linear case if
linear friction models are considered and optimization in the nonlinear case. In order to
make an overall judgment, error comparison takes place over the same trajectory used in the
identification process (that one of a condition number of 638) and others which are excited,
including the low velocity one. The resulted calculated error in each case is shown in Table
(4).
As can be observed from Table (4), considering identification by the optimization case and
excluding the trajectory used in the identification process, the errors in the predicted applied
forces considering the identified nonlinear friction models are lower than those
corresponding to the linear friction ones for all trajectories. This shows that the dynamic
model that includes nonlinear friction models has a better overall response. On the other
hand, the dynamic parameters obtained by the LSM give the lowest error for all of the test
trajectories. However, the calculated errors using the identified parameters found by means
of the test trajectories became bigger, and almost doubled, contrary to those calculated by
optimization, which kept the same order. Furthermore, the identified dynamic parameters
using the optimization process are physically feasible.
6.2 Identifiability of the base parameters
The identification process, as has been pointed out in the previous section, has the ability to
obtain a physical feasibility solution; however, the constrained optimization problem is
cumbersome. This occurs because constraint equations are functions of the terms of the
inertia tensor calculated with respect to the center of gravity of the corresponding body, and
the linear relation between the base parameter vector and the physical parameters is not just
one.
In addition, the solution of the nonlinear problem does not guarantee that the set of physical
parameters found has been identified accurately. Another approach that can be used for
parameter identification is to evaluate the physical feasibility after the identification process
has been carried out (Yoshida et al., 1996) along with the statistical analysis of variances in
the resulting parameters. Hence, two aspects are verified: Base Parameters with σ < 15%
pi
and physical feasibility. If the parameter accomplishes these criteria, the parameter is
considered properly identified.
For example, here, the identifiably of the dynamic parameters of a 3-RPS parallel robot is
addressed considering a simulated manipulator whose inertial parameters have been
obtained from the CAD models and the friction parameters has been obtained from an
indirect parameter identification process performed by the authors (Farhat et al., 2006).
Noise was added to the generalized forces as well as the independent generalized
coordinates and their time derivates.
The three models previously introduced in Table (2) were used in the identification process.
Friction was identified using symmetrical linear models that include Coulomb and viscous
frictions. When the parameters identified by using Model 1 were analyzed, only 4 of the 34
parameters, including friction and rotor and screw inertias, had σ lower than 15%, and
pi
some of the identified parameters were physically unfeasible. This can be demonstrated in
Table (5) (marked by *), where it can be seen the values of parameters of the simulated and
the identified models, respectively.
38
Parallel Manipulators, Towards New Applications
Forces (N)
τ
org
τLSM
τopt
600
400
200
0
-200
-400
0
1
2
3
4
5
6
7
1000
500
0
-500
0
1
2
3
4
5
6
7
1000
500
0
-500
0
1
2
3
4
5
6
7
Time (s)
Fig. 3. Results from the LSM (τLSM) and the optimization process (τopt).
ε (%)
RA
Trajectory condition number
Friction
model
563 638 718 601 492 LowVel
LSM
12.8 9.48 14.6 17.9 17.9
15.5
Linear
Opt 26.1 20.5 24.4 23.9 23.9
19.2
Nonlin
Op 24.7 21.2 23.5 23.4 23.4
18.0
Table 4. Error comparison considering linear and nonlinear friction models.
Results of the number of parameters properly identified, when model 1-3 was used in the
identification process, are listed in Table (6). The table includes also the average relative
error of the identified parameter relative to the exact parameter ( ε ),
AV
Dynamic Parameter Identification for Parallel Manipulators
39
1
Φ − Φ
i
i
ε
=
∑
(34)
AV
n
Φ
p
i
where Φ the exact values of the parameters and Φ is the vector containing the identified
i
i
parameters. An interesting fact is that despite that Model 1 achieving the lowest ε , the
RA
corresponding ε value was the highest. In addition, only 4 parameters were properly
AV
identified. The difference between models 2 and 3 in ε
was about 1.5%. As 12 parameters
RA
are identifiable, identification was performed using only these parameters (Model 4). This
model includes 3 inertial parameters of the links related to the platform and marked by (†)
in Table (1).
Identified
Parameter Exact
Values
σ %
Values
pi
Izz(4)+Iyy(5) 0.1555
-86.2016*
80.1596
Izz(6)+Iyy(7) 0.1555
-66.6355*
62.0087
Fv(1) 3272.0 3296.73 2.5677
Fc+(1) 227.96 1659.0181 53.3048
Fc-(1) 228.04 -1210.55* 73.1859
Jr(1)+JS(1)
483.10 505.03 13.8778
Table 5. Some of the base parameters identified using Model 1
Model
ε
%
ε
RA
AV %
Number of Parameters
1 4.57
5.37
37/4
2 4.58
2.94
24/12
3 4.63
3.06
18/12
4 4.73
3.17
12/12
Table 6. Results of ε
and ε
from different models.
RA
AV
This result could indicate that, because of the topology of the parallel manipulator and in
the presence of measurement noise, 12 parameters from which 3 are of the links inertial
parameters can be used for modeling and simulating the 3-RPS parallel manipulator
behavior.
Following the same procedure, a dynamic parameters identification process was applied
over an actual parallel 3-RPS manipulator. The resulted ε
values and the number of
RA
parameters properly identified are shown in Table (7). Comparing Table (6) and Table (7),
the level of ε
in the actual manipulator was doubled, but the numbers of parameter
RA
properly identified was found similar (12 for Model 4). The identified links base parameters
of Model 1 and 4 are presented in Table (8) along with those of the simulated manipulator.
As can be observed, the identified parameters of the actual manipulator using model 4 and
the original CAD values of the simulated manipulator are comparable. Contrary to those
identified using Model 1 where a significant difference appears.
The fact that 12 parameters can be properly identified is reasonable. On the one hand the
topology itself of the parallel manipulator, does not allow finding well-excited trajectories.
Additionally, some base parameters have a little contribution to the dynamic behavior of the
40
Parallel Manipulators, Towards New Applications
model; for example, during the movement the accelerations of the limbs are smaller than the
platform. On the other hand, the friction of the linear actuator of the real manipulator was
found to be high, this difficult even more the identifiability of the base parameters of the
links.
Finally, Model 4 was validated. Parameters obtained from one trajectory were used to
compute the forces for another one that had not been used for identification. Fig. (4) depicts
this comparison. As can be seen, the estimated and measurements forces are very close.
Model
ε
%
RA
Number of Parameter
1 8.40
37/2
2 8.43
24/9
3 8.53
18/12
4 8.62
12/12
Table 7. ε
from actual 3-RPS Manipulator.
RA
Real
Real
Base Parameter
CAD
Manipulator
Manipulator
Model 4
Model 1
mx(3)+0.5774my(3) -
-2.47 -2.59
1.16
0.4563(m(3)+m(2))
m(5)-
10.83 13.72
-3.29
2.531my(3)+m(3)+m(2)
m(7)+2.531my(3) 5.42 6.95
-0.557
Table 8. Rigid Body Base Parameters Model 1 vs Model 4.
7. Conclusions and further research
In this chapter, the problem of the identification of inertia and friction parameters for
parallel manipulators was addressed. In the first part of the chapter an overview of the
identification process applied to parallel manipulators was presented. First, the dynamic
model was obtained in a systematic way starting from the Gibbs-Appell equations of
motion. This dynamic model was reduced to a subset of parameters called base parameters
by means of SVD. After that and to ensure the minimal input/output error transmissibility,
approaches to obtaining optimized trajectories that have to be used in the identification
process were presented. In the second part of the chapter, a direct identification approach
was implemented on a 3-DOF RPS parallel manipulator considering the physical feasibility
of the identified inertial parameters. To this end, a procedure based on a nonlinear
constrained optimization problem has been reviewed. In addition, nonlinear friction models
were included in the dynamic formulation subjacent to the identification process. In the last
part of the chapter, a study of the identifiability of the base parameters was presented. It
based on both analyzing the relative standard deviation of each parameter and considering
its physical feasibility. For this approach a simulated manipulator was necessary for
studying and evaluating models used in the identification process. For future research, a
systematical approach is expected to be found, based on statistical frameworks and physical
feasibility, for studying the identifiability of the dynamic parameter without the necessity of
a simulated manipulator. Concepts presented here for parameter identification of parallel
Dynamic Parameter Identification for Parallel Manipulators
41
manipulators can be extended to other areas. For instance, vehicle components (Butz et al.,
2000; Serban & Freeman, 2001; Chen & Beale, 2003; Sujan & Dubowsky, 2003) and ultimately
the novel humanoid systems (Gordon & Hopkins, 1997; Silva et al., 1997; Kraus et al., 2005).
Forces (N)
500
0
-500
0
1
2
3
4
5
6
7
1000
500
0
-500
0
1
2
3
4
5
6
7
1000
500
0
-500
0
1
2
3
4
5
6
7
Time (s)
Fig. 4. Measurement Forces (-red) and Forces from Identified Parameter (–o blue)
8. Acknowledgements
This research has been supported by the Spanish Government grants project DPI2005-08732-
C02-01 and DPI2006-14722-C02-01, cofinanced by EU FEDER funds. The third author thanks
the University of Los Andes Scholarship Programs for helping to finance the junior doctoral
studies.
42
Parallel Manipulators, Towards New Applications
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