Experimental Identification of the Dynamics Model for Cartesian...
Transcript of Experimental Identification of the Dynamics Model for Cartesian...
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 52
190804-5252-IJMME-IJENS © August 2019 IJENS I J E N S
Abstract— This paper proposes an experimental-based approach
to estimate the kinematic parameters and develop a cascade
controller of a Cartesian robot. The aim is to satisfy the position
accuracy in the trajectory execution, guaranteeing a high
dynamics. The proposed procedure consists of a set of
experimental tests executed on a reference trajectory, varying the
velocity and acceleration in a specific range. The model is based on
the Least-Square-Estimation and the Genetic Algorithms. Once
the kinematic parameters have been calculated and evaluated, the
controller has been built using an iterative procedure to estimate
the PID gains of the position, velocity and current loops. Finally,
the overall system has been validated through a set of reference
trajectories, comparing the observed results with the predicted
values. The RMSE index of torque has shown a congruence
between the obtained results with a maximum value lower than 8.0
Nm.
Index Term— Cartesian robot, Dynamics, Cascade control,
Parameters-based model, Genetic algorithm.
I. INTRODUCTION
IN robotics applications, position accuracy and high
dynamics have to be satisfied, even if these features may be
conflicting [1-6]. One of the main challenge is to implement a
flexible, robust and reliable control system. The development
of an effective dynamic model is the successful factor to
achieve a robust controller. Robot dynamics concerns with the
relationship between the forces acting on a robot and the
accelerations of its parts. Dynamic models may be classified in
two main approaches [5]: the direct model - starting from the
joint loads and knowing the joint positions-velocities, it allows
to obtain the joint accelerations, and the inverse model - given
the joint accelerations-velocities-positions, it defines the
corresponding resultant loads acting on the joints.
The model definition is based on the use of Lagrange or
Newton-Euler formulations. The advanced model-based or
forces-torques control algorithms have been derived from an
appropriate model selection [6]. For an optimal regulation, it is
required to correctly detect and recognize the dynamic factors
[3]. Nevertheless, the parameters of a robot are not always
known in advance, so it is needed to apply effective procedures
able of tuning the dynamic model. Usually, a conventional
robot implementation procedure is composed by a set of
sequential stages starting from modelling, experimental
campaign and kinematic parameter optimization to fine-model
tuning phase [7]. Data collection and signal processing are the
critical activities to verify if the developed model reflects the
real behavior of the robot. In order to estimate the dynamic
parameters, a number of techniques may be used [8, 9]. Least-
Squares-Estimation (LSE) and Maximum Likelihood
Estimation (MLE) approaches are most common and applied
methodologies [10]. Other algorithms (e.g. Genetic-Algorithms
(GAs), adaptive deep-learning techniques, data-driven
methodologies) may be preferred with the intensification of
data-driven methods in 4.0 era [7-11]. The mentioned methods
are used to identify the optimum parameter levels of the
unknown factors in the white-box configuration setting.
In this work, Authors have developed an experimental-based
approach to define the appropriate kinematic model parameters
of a Cartesian robot with 3 degrees of freedom (DOFs). Two
techniques have been selected and evaluated: Least-Square and
Genetic algorithms. In particular, the study presents the
problem formulation and working principles, defining the
selected algorithms used to estimate the kinematic parameters.
The proposed procedure consists of executing a set of
experimental tests on X-axis and Y-axis, varying the velocity
and acceleration in a specific range. Then, a cascade control is
presented and validated through simulations and tests,
executing a number of reference trajectories.
II. PROBLEM STATEMENT AND DYNAMIC MODEL
This study focuses on a Cartesian robot made of modular
aluminum profiles. The robot configuration presents three
linear translations along the X-Y-Z axes. Figure 1 shows the
reference system of the Cartesian scheme (red color). X-axis is
independent while Y and Z-axes are coupled. The structure has
a vertical cross configuration. The motors M1, M2, M3 provide
the linear movement through endless screws, directly connected
to compliant joints (Y-axis, Z-axis) or toothed belt (X-axis).
Table I summarizes the main features of each individual axis:
stroke, maximum speed and acceleration.
Experimental Identification of the Dynamics
Model for Cartesian Robot
F. Aggogeri, N. Pellegrini*, F. Piaggesi and R. Adamini,
Università degli Studi di Brescia, Italy
TABLE I
MAX STROKE, VELOCITY AND ACCELERATION OF THE CARTESIAN DEVICE
Symbol X-Axis Y-Axis Z-Axis
Stroke
[mm] 250.0 240.0 240.0
Velocity
[𝑚𝑚
s]
80.0 230.0 230.0
Acceleration
[𝑚𝑚
s2]
5000.0 5000.0 5000.0
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 53
190804-5252-IJMME-IJENS © August 2019 IJENS I J E N S
A. Dynamic model definition
The definition of the Cartesian dynamic model may be
obtained using the Euler-Lagrange or the Newton-Euler
equations. The mathematical formulation in joint-space,
expressed from Lagrange equation [12], is stated by eq. (1):
𝑀(𝑞)�̈� + 𝐶(𝑞, �̇�)�̇� + 𝜏𝑓 = 𝜏 (1)
Where 𝑞(𝑡) = [𝑞1(𝑡), 𝑞2(𝑡), … , 𝑞𝑛(𝑡)]𝑇 ∈ ℝ𝑛 is a vector of
joint (1,…,n) position. Joint velocity vector is expressed by
�̇�(𝑡) ∈ ℝ𝑛, and joint acceleration vector is stated by �̈�(𝑡) ∈ ℝ𝑛.
The other terms correspond to the inertia matrix: 𝑀(𝑞) ∈ℝ𝑛×𝑛 , while 𝐶(𝑞, �̇�) regards Coriolis, Centrifugal and
Gravitational terms, while 𝜏𝑓(𝑡) ∈ ℝ𝑛 and 𝜏(𝑡) ∈ ℝ𝑛 indicate
the friction forces and the joint torque vector, respectively.
Figure 2 describes the input-output relation of each elements
regarding X-axis and Y-axis.
B. Kinematic parameter estimation techniques
Authors selected two techniques to estimate the kinematic
parameters of the dynamic model: “Least Squares Estimation”
and “Genetic Algorithm”, processed in Matlab software. The
LSE method is an approach that uses the regression analysis to
approximate the solution of overdetermined system [13]. It is a
well-known method in robotics [14], and it allows the
assessment of the inertial parameters obtained from the joint
torques and position. The main limitation is the noise sensitivity
[15]. This drawback affects the accuracy degradation in
determining the kinematic parameters. To overcome this
constraint, it is essential to use an identification trajectory that
avoids the excitation of the robot’s dynamics. An alternative
solution is the introduction of the noise filters to the sampled
signals. The Genetic Algorithms (GAs) are stochastic global
search formulations. GAs apply the evolutionary principle to
general optimization formulations, allocating multi-search-
points to working spaces and associating each search-point with
appropriateness indicator according to the error of constraints-
and-objective functions [16]. GAs are used in several
applications from robotics to industrial and manufacturing
environment [15, 17].
C. Identification of the reference trajectory
A reference trajectory has been identified and designed to
evaluate the selected techniques in dynamic model
development. Each axis performed a forward and backward
movement with a speed equal to 20% of the maximum speed,
and with an acceleration equal to 20% of the maximum
permitted. Then, the same movement was executed with the
same speed and acceleration increased from 40% to 80%. The
procedure has been reiterated, increasing the speed to 40% and
80% of the permitted range. The scope was to excite the device
dynamics, covering the maximum ranges of the possible
scenarios during the robot usage. In particular, the experimental
test duration (X-axis, Y-axis) was lower than 300.0 s or at least
equal to 25 complete forward and reverse cycles. The torque,
velocity and acceleration values have been collected and used
by the algorithms in kinematic parameter estimation.
D. Kinematic parameters estimation
A preliminary analysis has been performed to evaluate the
LSE algorithm applied to X-axis. By solving equations 2 and 3,
the following parameters were estimated: J the equivalent
inertia, c1, the coefficient of dynamic friction and c0, the
coefficient of static friction.
𝑥 = Φ†𝐶 (2)
𝑥 = [
𝐽𝑒𝑞
𝐶1
𝐶0
] , Φ = [�̈�1 �̇�1 𝑠𝑖𝑔𝑛(�̇�1)⋮ ⋮ ⋮
�̈�𝑡 �̇�𝑡 𝑠𝑖𝑔𝑛(�̇�𝑡)
] , 𝐶 = [𝐶1
⋮𝐶𝑡
] (3)
where 𝒙 is the unknown-parameters-vector and 𝚽 is the
known-values matrix (acceleration, velocity and sign of
velocity). Using the column with the velocity sign, the formula
has been linearized. 𝑪 is the vector of the acquired torques at each sampling time.
The GA technique has been used to identify the parameters
of Y-axis. The parameters identified were: 𝑱 that is equivalent
inertia, 𝒄𝟎 and 𝒄𝟏, that are static and dynamic frictions, 𝒄𝒑 is a
parameter that associates the torque to the position of the axis,
Fig. 1. The Cartesian robot and X-Y Axes subsystems
Bellow
Coupling
Z - axis
Y – axis subsystem
X – axis
assembly
Ball screw
Pulley
Chassis Motor X-axis
Motor
Z-axis
Motor
Y-axis
Y
Z
X
X – axis subsystem
a)
b)
Fig. 2. Dynamic model scheme: X-axis (a) and Y-axis (b).
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 54
190804-5252-IJMME-IJENS © August 2019 IJENS I J E N S
𝑫 is a term that includes all residues. The results are shown in
Table 2.
The RMSE index has been adopted and calculated to
quantify the robustness of the proposed procedure. It quantifies
the error between the predicted and observed values of the
torque observed on 𝑇 times. It is equal to the square root of the mean of the squares of the deviations on period T [17-21], as
stated by equation (4):
𝑅𝑀𝑆𝐸 = √(∑ (�̂�𝑡−𝑦𝑡)2𝑇
𝑡=1
𝑇) (4)
III. THE CASCADE CONTROL LOOP
The device control has to determinate the force and torque
resultants that the actuators need to provide in executing of the
required trajectory. The use of a feedback loop control permits
to minimize the difference between the expected values and the
actual observations collected by the robot sensors.
In this way, a cascade control may be applied when the
dynamics is divided in a slow part, concerning with the external
loops, and a fast part, assigned to the internal loops. Each loop
has the corresponding PID controller. The selected configuration is based on the result of the fast
dynamics of the inner loop that provides the fastest attenuation
of the disturbance, minimizing potential effects on the primary
output [22-25]. Three-nested PIDs have been used for position,
speed and current, as shown in Figure 4. The position loop was
composed by the proportional gain, P, while the speed and
current loops had the proportional and integral gains, PI.
The closure of the loops was executed by the actuator, with
a frequency higher than the commercial RP-1 controller. The
position and speed loops had a frequency of 2.0k Hz, the current
loop had a frequency of 8.0k Hz, while the frequency of the RP-
1 controller was 500 Hz. The current loop was the most inner and fastest loop, with a sampling rate of 8.0 kHz.
Ad-hoc tuning phase was performed to establish the current
loop gains, by setting the position and velocity values gains as
described in Table 3. Three main scenarios has been considered
and compared. This approach is modular and valid for
additional cases.
The current loop gains have been defined when the obtained
torque value differed from the expected torque parameter of 5%
- 10%. At first, a proportional gain was set equal to 0.02 while
the integral gain was equal to zero. Then, the proportional gain
has been increased by 50% every time, until the current loop was unstable. The PID gains were progressively reduced by
10%, until the oscillation disappeared. This operation has been
repeated for the other PID gains. Figure 5(a) describes the
scheme of the current loop.
The velocity loop on the drive was controlled by a PI scheme
with proportional and integral actions. It was the second inner
loop, with a sampling rate of 2.0 kHz. The velocity loop has
been tuned executing the proposed procedure with the motor
free to rotate. Then, the motor has been connected to the robot,
the proportional and integral gains have been refined with an
iterative procedure. In this phase, a velocity square wave with
the reference value changing from 0% to 10% of the maximum value of the speed parameter was used. Figure 5(b) shows the
scheme of the velocity loop.
The position loop on the drive was managed by a P
controller, the proportional gain. It was the third loop, with a
sampling rate of 2.0 kHz. The motor has been connected to the
robot and the proportional gain “KpP” was increased up to
unstable condition of “Epos” position loop feedback [26].
Figure 5(c) shows the control scheme of the position loop.
TABLE II
KINEMATIC PARAMETER ESTIMATION
Symbol Description X-Axis Y-Axis
ET Estimation
Technique Least Square
Genetic
Algorithm
J [kgm2] Equivalent
Inertia 0.000089 0.000100
c0 [Nm] Coefficient of static
friction 0.2198 0.3230
c1 [kgm2
s]
Coefficient of
dynamic friction 0.000312 0.000500
cp [Nm] Parameter that
correlate the Torque
and Axis position
- 0.000500
D [Nm] Residuals term - 0.006200
RMSE [Nm] Root Mean Square
Error 7.993 3.799
Fig. 4. The cascade control loops for a robot joint
TABLE III
PID GAINS ESTIMATION FOR CASCADE CONTROL
Scenario Loop Gain Value
1 Position FwF 1.00
1 KpP 150.00
1 Velocity KpS 0.12
1 KiS 20.00
2 Position FwF 0.50
2 KpP 170.00
2 Velocity KpS 0.17
2 KiS 15.00
3 Position FwF 1.00
3 KpP 200.00
3 Velocity KpS 0.30
3 KiS 17.00
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 55
190804-5252-IJMME-IJENS © August 2019 IJENS I J E N S
A validation campaign has been executed to verify if the
estimated parameters were correctly aligned with the
parameters obtained by the experimental tests. Figure 6 shows
a comparison between the PID gain scenarios for X-axis
defined in Table 3.
Figure 6(a) states the reference trajectory used to validate the
gain parameters while Figure 6(b) highlights the deviations
between Scenario 1 and Scenario 3 related to velocity [rad/s]. The oscillating case is the configuration with the highest value
of integral gain (blue curve). The optimal parameter setting is
Scenario 3 (black color), that shows the fastest response without
perturbations at 10.0 s and 27.0 s – 30.0 s (critical periods).
Figure 6(c) presents the torque analysis. The optimal parameter
configuration is shown by Scenario 3, that minimizes the
perturbations in critical periods.
IV. EXPERIMENTAL CAMPAIGN VALIDATION
The validation of the overall system has been verified
executing a set of trajectories and comparing the measured
torques with the estimated values. Figure 7 describes an
example related to a reference trajectory executed on X-Y-axes,
respectively. The aim was to reproduce a standard robot task in
working space and conditions.
The results show that the RMSE error between the predicted
torques and the measured torques is lower than 8.0 Nm,
confirming the robustness of the proposed approach.
In particular, Figure 8 (a-b) presents a comparison between
the observed and estimated values of acceleration and torque
related to Y-axis using the GA techniques. The RMSE index,
calculated on the validation model, has a value equal to 4.78
Nm. In the same way, Figure 8 (c-d) shows the comparison
between the accelerations and torques applying the LSE
estimated parameters.
The value of RMSE is equal to 7.99 Nm. This deviation is
due to the torque signal noise.
a)
b)
c)
Fig. 5. The current loop (a), velocity loop (b) and position loop (c).
a)
b)
c)
Fig. 6. The predefined reference trajectory (a), X-axis velocity comparison
between scenarios 1-3 (b) and X-axis torque comparison between scenarios
1-3 (c)
-1.0
-2.0
-3.0
-4.0
-5.0
-6.0
-7.0
-8.0
-9.0
Cu
rren
t p
osi
tion
[rad
]
Time [s]
0.0 10.0 20.0 30.0 40.0 50.0
-70
-60
-50
-40
-30
-20
-10
0
1
15 29 43 57 71 85 99
113
127
141
155
169
183
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211
225
239
253
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295
309
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337
351
365
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393
407
421
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449
463
477
491
505
519
533
547
561
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589
603
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631
645
659
673
687
Position
Serie1
80
60
40
20
0
-20
-40
-60
-80
Cu
rren
t velo
cit
y [
ra
d/s
]
Time [s]
0.0 10.0 20.0 30.0 40.0 50.0
-60
-40
-20
0
20
40
60
Differenza Vel
Serie1 Serie2 Serie3
Scenario 1
Scenario 2
Scenario 3
4.0
3.0
2.0
1.0
0.0
-1.0
-2.0
-3.0
-4.0
To
rq
ue [
Nm
]
Time [s]
0.0 10.0 20.0 30.0 40.0 50.0
-3
-2
-1
0
1
2
3
4
Differenza coppie PID
Serie1 Serie2 Serie3
Scenario 1
Scenario 2
Scenario 3
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 56
190804-5252-IJMME-IJENS © August 2019 IJENS I J E N S
This result may be considered acceptable since the noise
generated by the static friction has not been evaluated in the
preliminary test.
V. CONCLUSION
One of the most critical task of a robot is to satisfy the required trajectory accuracy guaranteeing a high dynamics. In
this study, an experimental-based approach to estimate the
kinematic parameters of a Cartesian robot is presented. The
proposed procedure consisted on a set of experimental tests
executed on a defined trajectory. The aim is to excite the device
dynamics, covering the maximum ranges of the possible
scenarios during the robot usage. The dynamic model focuses
on the Least-Square-Estimation and the Genetic Algorithms.
Based on the calculated kinematic parameters, the cascade
controller has been developed by a sequential procedure to
estimate the PID gains of the position, velocity and current loops. Finally, the controlled Cartesian robot performance has
been confirmed through a set of predefined reference
trajectories, comparing the observed results with the expected
values. For a practical implementation, the RMSE index has
shown a congruence of the obtained results with a maximum
value lower than 8.0 Nm, equal to 4.78 Nm and 7.99 Nm for
GAs and LSE, respectively. Further works could investigate the
torque signal noise monitoring, the extension of kinematic
parameter identification techniques (in addition to LSE and
GAs) and the inclusion of vertical axis in the robot dynamic
model.
a)
b)
c)
d)
Fig. 8. The comparison between the observed and estimated accelerations
(a) and torque (b) (Y-axis – Genetic Algorithm) – accelerations (c) and torque (d) (X-axis – LSE technique)
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.0
Accele
ra
tio
n [ra
d/s
2]
Time [s]
0.0 1.0 2.0 3.0 4.0 5.0-2000
-1500
-1000
-500
0
500
1000
1500
2000
Acc. Estimated
Acc. Measured
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.0
Torq
ue [
Nm
]
Time [s]
0.0 1.0 2.0 3.0 4.0 5.0
Torque Estimated
Torque Measured
-1.5
-1
-0.5
0
0.5
1
1.5
4.0
3.0
2.0
1.0
0
-1.0
-2.0
-3.0
-4.0
Accele
ra
tio
n [ra
d/s
2]
Time [s]
0.0 1.0 2.0 3.0 4.0 5.0
Acc. Estimated
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000Acc. Measured
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.0
Torq
ue [
Nm
]
Time [s]
0.0 1.0 2.0 3.0 4.0 5.0
-1.5
-1
-0.5
0
0.5
1
1.5
Torque Estimated
Torque Measured
a)
b)
Fig. 7. The reference position for the trajectory on Y-axis (a), and X-axis (b).
80
60
40
20
0
-20
-40
-60
-80
Refe
ren
ce p
osi
tio
n [
mra
d]
Time [s]
0.0 1.0 2.0 3.0 4.0 5.0
-60
-40
-20
0
20
40
60
AXE:IV(2)
-10
-20
-30
-40
-50
-60
-70
-80
-00
Refe
ren
ce p
osi
tion
[m
ra
d]
Time [s]
0.0 1.0 2.0 3.0 4.0 5.0
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
AXE:IV(2)
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:04 57
190804-5252-IJMME-IJENS © August 2019 IJENS I J E N S
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Francesco Aggogeri is an Associate Professor of
Applied Mechanics at University of Brescia. His
main research interests include Applied Mechanics,
Robotics, Mechatronic Devices, Rehabilitation
Engineering and Vibration control. He is the author
of a number of papers (ISI-Scopus) and peer-review
conference proceedings papers. Francesco Aggogeri
is responsible of national and international research
projects. He is the unit coordinator of PROGRAMS
(H2020 project) and he has served as responsible of
IntegMicro (EU-FP7 project) and Copernico (EU-
FP7 project).
Nicola Pellegrini is an Assistant Professor of Applied
Mechanics at University of Brescia. He has been
involved in projects related to integrated mechatronic
systems, Product development, Motion planning and
Soft computing simulation. He is author of a number
of papers published in international journals and
conferences. He has collaborated to National and
European research programs and projects funded by
private companies.
Filippo Piaggesi received the B.S. degree in
Automation engineering from University of Brescia,
Italy, in 2016 and the M.S. degree in automation
engineering from University of Brescia, Italy, in 2018.
He is currently collaborating with Mechanical and
Industrial department of University of Brescia. His
research interest focuses on the development of
Cartesian Robot dynamic models and the application
of algorithms for the estimation of dynamic
parameters.
Riccardo Adamini is currently a Full Professor of
Applied Mechanics at University of Brescia.
He is author of a number of publications and his
research interest includes mechanical engineering,
mechanisms and their applications, kinematics and
dynamics of industrial systems.