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



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).



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



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

197

211

225

239

253

267

281

295

309

323

337

351

365

379

393

407

421

435

449

463

477

491

505

519

533

547

561

575

589

603

617

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



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)



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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.

experimental identification of the dynamics model for cartesian...

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