Tuning the Dynamic Response of a Redundant Robotic System Using Its

Dominant Natural Frequencies

Carlos Saldarriaga

, Marcelo Fajardo-Pruna

, Carlos G. Helguero

and Jonathan Leon-Torres

Facultad de Ingenier

ıa en Mec

anica y Ciencias de la Producci

on, Escuela Superior Polit

ecnica del Litoral, ESPOL,

Campus Gustavo Galindo Km 30.5 V

ıa Perimetral, P.O. Box 09-01-5863, Guayaquil, Ecuador

Keywords:

Redundant Robotic System, Dynamic Response, Dominant Natural Frequencies, Fast Fourier Transform,

Robotic Manipulator.

Abstract:

Robotic systems often encounter challenges in achieving desired dynamic responses, especially when they

possess redundant degrees of freedom. This paper proposes a methodology to identify a redundant robotic

system’s dominant natural frequencies and tune its dynamic response through appropriate damping. The sys-

tem’s natural frequencies are accurately identiﬁed by analyzing displacement data and leveraging the power

of fast Fourier transform tools. These frequencies serve as critical parameters for modifying the response

behavior, enabling enhanced control and stability. To validate the effectiveness of the proposed methodol-

ogy, simulations are conducted on a 7-degree-of-freedom redundant Panda robotic manipulator. The results

demonstrate the methodology’s potential to optimize the dynamic performance of complex robotic systems,

opening avenues for improved efﬁciency, safety, and overall system performance.

1 INTRODUCTION

Robotic systems are pivotal in various industries,

from manufacturing and automation to healthcare and

space exploration (Cen and Melkote, 2017). Achiev-

ing precise and controlled dynamic responses is es-

sential for ensuring these systems’ optimal perfor-

mance, safety, and efﬁciency. However, this task be-

comes more challenging in the presence of redundant

degrees of freedom, which offer increased ﬂexibility

but also introduce complexities in controlling and tun-

ing the system’s response (Urrea and Pascal, 2017). In

this paper, we propose a methodology that utilizes the

fast Fourier transform (FFT) to identify the dominant

natural frequencies of a redundant robotic system and

subsequently tunes its dynamic response through ap-

propriate damping.

Identifying modal parameters, such as natural fre-

quencies, is crucial for understanding and character-

izing the dynamic behaviour of a robotic system. By

accurately identifying these frequencies, we can gain

insights into the system’s vibrational modes, allow-

https://orcid.org/0000-0001-9014-681X

https://orcid.org/0000-0002-5348-4032

https://orcid.org/0000-0002-6992-0572

https://orcid.org/0009-0003-5857-279X

ing us to predict and manipulate its dynamic response

(Gonul et al., 2019; Garnier and Subrin, 2022). Tra-

ditionally, identifying natural frequencies involved

experimental modal analysis techniques, which re-

quired physical measurements on the robotic system.

While these methods provide valuable information,

they can be time-consuming, expensive, and may not

always be practical for complex systems. The ad-

vent of computational tools and numerical simula-

tions has opened up new possibilities for modal pa-

rameter identiﬁcation, offering faster and more cost-

effective alternatives (Chen et al., 2014).

The proposed methodology offers several advan-

tages. Firstly, it eliminates the need for extensive ex-

perimental modal analysis, saving time and resources

(Quqa et al., 2020). Secondly, computational tools

provide a more ﬂexible and versatile approach to

modal parameter identiﬁcation. Thirdly, tuning the

system’s response through damping modiﬁcation en-

ables enhanced control, stability, and performance of

redundant robotic systems (

Ilman et al., 2022). It of-

fers a more efﬁcient and cost-effective approach to

modal parameter identiﬁcation by leveraging compu-

tational tools and numerical simulations. The fast

Fourier transform algorithm allows us to convert dis-

placement data from the time domain to the frequency

domain, enabling the identiﬁcation of dominant peaks

166

Saldarriaga, C., Fajardo-Pruna, M., Helguero, C. and Leon-Torres, J.

Tuning the Dynamic Response of a Redundant Robotic System Using Its Dominant Natural Frequencies.

DOI: 10.5220/0012191000003543

In Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2023) - Volume 2, pages 166-172

ISBN: 978-989-758-670-5; ISSN: 2184-2809

that correspond to the natural frequencies of the sys-

tem. Finally, our methodology offers ﬂexibility and

adaptability, allowing it to be applied to various re-

dundant robotic systems. The methodology presented

in this paper is particularly useful for systems that

have redundant degrees of freedom, but it is not a lim-

itation. In case of non-redundant systems the process

becomes somewhat simpler but the method still ap-

plies. In general mechanical systems (not necessarily

a robotic manipulator) the redundancy property would

be analogous to having an unconstrained system, or a

system that is not attached to inertial frames, which

cause the appearance of rigid body modes of motion

with natural frequency ω

= 0.

The Panda robotic manipulator is a suitable case

study for validating our methodology. With its 7 de-

grees of freedom and redundant kinematic structure,

the Panda manipulator exhibits complex dynamic be-

haviour, making it an ideal candidate for testing the

effectiveness of our approach. Through forward kine-

matics and dynamics simulations, we obtain displace-

ment data that accurately represents the manipulator’s

response under speciﬁc operating conditions (Fig-

ure 1).

Figure 1: Franka Emika Panda robot (Saldarriaga et al.,

2022).

Applying the fast Fourier transform to the ob-

tained displacement data. We analyze the result-

ing frequency spectrum to identify the dominant nat-

ural frequencies of the Panda robotic manipulator.

These frequencies represent the system’s inherent vi-

brational modes and provide valuable insights into its

dynamic behavior. Determining the natural frequen-

cies lays the foundation for optimizing the system’s

response through damping modiﬁcation.

Once the dominant natural frequencies are identi-

ﬁed, we proceed to tune the dynamic response of the

redundant robotic system through appropriate damp-

ing. Damping is crucial in controlling oscillations

and attenuating unwanted vibrations within a system.

By strategically introducing damping at speciﬁc fre-

quencies, we can effectively modify the system’s re-

sponse behaviour and enhance its stability and perfor-

mance (Kao and Saldarriaga, 2021).

The damping modiﬁcation technique proposed in

our methodology considers the relative magnitudes

of the peaks in the frequency spectrum. By ana-

lyzing the spectral content of the system, we iden-

tify the frequencies that require additional damping to

achieve the desired response. The damping modiﬁca-

tion can be implemented in an actual physical system

using various techniques, such as introducing passive

dampers or adjusting the control parameters of the

robotic system, largely limited by the sampling of the

system and the accuracy of the dynamic model.

Through simulations on a 7-degree-of-freedom re-

dundant Panda robotic manipulator, we validate the

effectiveness of our methodology. The results demon-

strate our approach’s potential to improve the dy-

namic performance of complex robotic systems.

2 METHODOLOGY

In the proposed methodology, the FFT algorithm con-

verts the time-domain displacement data obtained

through simulations or physical measurements into

the frequency domain, clearly representing the sys-

tem’s spectral content. We can identify the dominant

peaks that correspond to the system’s natural frequen-

cies by analyzing the resulting frequency spectrum.

To illustrate the efﬁcacy of our methodology, we

conduct simulations on a 7-degree-of-freedom redun-

dant Franka Emika Panda robotic manipulator.

With the obtained joint displacement data, we

perform the FFT analysis to extract the natural fre-

quencies of the Panda manipulator. The FFT algo-

rithm decomposes the displacement data into its con-

stituent frequency components, providing a spectrum

that highlights the dominant frequencies present in the

system. By identifying the peaks in the frequency

spectrum, we can precisely determine which natural

Tuning the Dynamic Response of a Redundant Robotic System Using Its Dominant Natural Frequencies

167

frequencies characterize the dynamic behaviour of the

Panda manipulator the most.

Our methodology proposes a damping modiﬁca-

tion technique based on the identiﬁed natural frequen-

cies that contribute the most to the response.

Once the dominant natural frequencies are iden-

tiﬁed, we modulate the system’s dynamic response

through appropriate damping selection. Damping is a

critical parameter that affects the decay rate of oscil-

lations in a system. By strategically updating certain

damping elements that affect speciﬁc frequencies, we

can effectively control and tune the response behavior

of the robot.

The dynamic equation of motion of a robotic sys-

tem is governed by

M(q)

q(t) + G(q,

q(t) + v(q) = τ

+ τ

ext

(1)

where q contains the n joint angles, M is the mass

matrix, G the matrix that contains the gyroscopic

non-linear terms, v is the vector that compensates

for gravity, and τ

ext

is the external torques vector.

In order to impose and establish a compliant behav-

ior (Villani and De Schutter, 2008) to the system, an

impedance controller is established by setting τ

[−Kq(t) − C

q(t) + v(q) + G(q,

q(t)], so that the

system becomes

M(q)

q(t) + C

q(t) + Kq(t) = τ

ext

(2)

and a multidimensional mass M, damping C and stiff-

ness K relationship is obtained for the robot, where

the stiffness and damping matrices can be mapped

from the Cartesian task space through the Jacobian

matrix J

K = J

J + K

+ K

(3)

C = J

J (4)

where K

= J

J and



∂J

∂q

 

∂J

∂q



...



∂J

∂q



An appropriate selection of the damping parame-

ters is not a very straight forward job, especially for

redundant robots. By generating the parameter study

of the elements of the damping matrix C after re-

moving or handling the redundant degree(s) of free-

dom (Saldarriaga et al., 2022), we can modulate the

contribution of each mode of vibration λ

and cor-

responding natural frequency ω

of the system, for

example, those previously identiﬁed by FFT tools re-

sults, in a sound, analytical manner.

In summary, and as illustrated in Figure 2, we in-

tend to modulate the response of the mechanical sys-

tem by the choice of the damping parameters, after the

dominating frequencies are identiﬁed from the FFT

results plots, in a systematic manner by the use of an

analytical methodology that allows us to consider the

case of unconstrained mechanical systems, or redun-

dant robots without losing generality, which would

not be possible without the analytical tool described.

Start

Experimental

data q

(t)

FFT

Improvement of

dominant λ’s

through C

Control

criteria

satisfied?

End

For a given robotic task:

parameters and q

(t)

Figure 2: Flowchart of the methodology carried out in the

experiments.

3 SIMULATIONS

In order to collect robot joint displacements data, the

Franka Panda was modeled in Wolfram Mathematica,

where the dynamic response of the system was ob-

tained by numerically solving the differential equa-

tions. The NDSolve function was used to simulate

and obtain an impedance behavior as the one in Equa-

tion 2 imposed through stiffness and damping param-

eters that were deﬁned in the Cartesian space, and

then mapped into the joint space through Equations 3

and 4; while the inertia mass matrix of the robot was

obtained according to the robot structure and conﬁg-

uration as in (Murray et al., 1994).

The system was handled and solved in a very sim-

ilar manner as a general case of free vibration of dis-

crete systems (Meirovitch, 2001) with respect to an

equilibrium joint conﬁguration. An initial condition

(displacement) of approximately 10cm was imposed

to the end-effector in the Y direction for every simu-

lation, with zero initial velocity, or from rest. Only

the damping matrix C

was updated according to the

intended dynamic response of the system and the ana-

lytical tool. The simulation data consisted of joint dis-

placements with transients and steady state responses,

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

168

after imposing an initial displacement to the end-

effector (with corresponding joint initial conditions

through inverse kinematics) and releasing from rest.

The initial robot conﬁgura-

tion for every simulation was q

[0.072;−0.379;0.167;−2.548;0.0748;2.17;0.185]

rad, the chosen Cartesian stiffness matrix was

=diag(2000, 2000, 2000, 100, 100, 100) in SI

units for both translation and rotation, respectively.

An initial arbitrarily low damping matrix C

was

chosen as 0.1I

to illustrate the methodology, after

that, the procedure described in the previous Section

was carried out to modulate the dynamic response

of the system through the FFT tool and the damping

parameters selection. The plot results of the most

signiﬁcant joints are shown in Figure 3, we can see

how the response is underdamped with large settling

time.

Figure 3: Joint Displacements for the arbitrarily low initial

damping matrix C

. Plots of the most signiﬁcant joints.

4 RESULTS AND DISCUSSION

After obtaining the joint displacements of free vibra-

tion response for the ﬁrst set of parameters, the FFT

tool showed that the dominant modes λ’s were those

corresponding to the natural frequencies 2, 3.2, and

3.6 Hz as shown in Figure 4, which after generating

the parameter study for the starting conﬁguration and

the system, correspond to the lowest frequencies in

the modal space, as shown in Figures 5 and 6, and that

gives us a very good understanding on how to select a

new damping matrix to improve the response.

Thanks to the information provided by both the

FFT results and the parameter study, a new damp-

ing matrix that improves the response of the system

was chosen as C

=diag(70.3;125.4;43;0.6;1;0.1),

which was used to generate and obtain a new simu-

lation and FFT plots, as shown in Figure 8. Now the

(a) FFT Joint 1.

(b) FFT Joint 2.

Figure 4: FFT results for the arbitrarily low initial damping

matrix C

. Plots of the most signiﬁcant joints.

dominant natural frequencies have changed, are not as

prominent as before and depend on the intended joint

to be analyzed. This new damping matrix C

gen-

erates the following damping ratios and natural fre-

Tuning the Dynamic Response of a Redundant Robotic System Using Its Dominant Natural Frequencies

169

Figure 5: Parameter study of element (3,3) of the damping

matrix showing the effect on each mode.

Figure 6: Parameter study of element (2,2) of the damping

matrix showing the effect on each mode.

quencies on the system in the modal space

ζ =







0.2

0.21

0.70

0.57

0.31







; ω







2.01

3.31

3.91

12.67

17.13

58.7







Note that none of the modes are overdamped, and that

the lowest frequencies have damping ratios greater or

equal to 0.2.

Through the parameter study on the element (1,1)

of the damping matrix C

shown in Figure 9, gener-

ated after a value for element (3,3) was chosen to be

126 following the guidelines of Figure 5, and main-

taining the value for element (2,2), we were able to

modulate or damp out the response even further. Af-

ter the second parameter study a new damping matrix

was chosen as C

=diag(219;125.4;126;0.6;1; 0.1)

Figure 10 shows us the joint displacements of the

system when using the C

damping matrix. As it

Figure 7: Joint Displacements for the C

damping matrix

after a parameter study. Plots of the most signiﬁcant joints.

can be seen, most of the transients have disappeared

as expected from theory.

It is also worth pointing out that due to the redun-

dancy of the system, there are zero-potential-energy

(ZP) motions that belong to the null space of the joint

stiffness matrix K, and move or make the robot go

into a new equilibrium conﬁguration, different from

the starting one, as it can also be seen in Figures 3

and 7, and cannot be removed unless the redundancy

is taken care of for the task.

Once the responses have been damped out, the re-

sults from the FFT tool are not as visible or notori-

ous as for the case for underdamped systems, which

makes sense according to theory.

In a more practical or experimental sense, if we

compare the numerical damping values of C

and

, they are signiﬁcantly different, but if we com-

pare the responses, the differences are small. This

may incur in an unnecessary larger control effort

when using C

that would probably generate out of

range torques, strangely slow motions, or instabili-

ties. Our theoretically sound methodology can help

us avoid all these situations.

5 CONCLUSIONS

Through simulations on a 7-degree-of-freedom re-

dundant Panda robotic manipulator, we have demon-

strated the effectiveness of our approach and achieved

the desired outcomes fulﬁlling the objectives of accu-

rately identifying the natural frequencies and optimiz-

ing the system’s response behavior through appropri-

ate damping modiﬁcation.

By strategically introducing damping at speciﬁc

frequencies, we were able to modify the system’s re-

sponse behavior and enhance its stability and perfor-

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

170

(a) FFT Joint 1.

(b) FFT Joint 2.

Figure 8: FFT results for the C

damping matrix after a

parameter study. Plots of the most signiﬁcant joints.

mance. Our methodology considers the relative mag-

nitudes of the peaks in the frequency spectrum to

identify the frequencies that require additional damp-

ing.

0 50 100 150 200 250 300 350 400 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Damping ratio

(Cc(1,1))

58.704

17.1693

12.3702

3.923

3.3188

2.0537

Figure 9: Second parameter study of element (2,2) of the

damping matrix showing the effect on each mode.

Figure 10: Joint Displacements for the C

damping matrix

after a second parameter study. Plots of the most signiﬁcant

joints.

The signiﬁcance of our methodology lies in its

practicality, efﬁciency, and adaptability. Using com-

putational tools and numerical simulations eliminates

the need for extensive experimental modal analysis,

saving time and resources. Moreover, the approach

can be applied to a wide range of redundant robotic

systems, offering ﬂexibility in its implementation.

This approach can signiﬁcantly contribute to the

ﬁeld of robotics by enabling the optimization of dy-

namic performance in complex systems. Further

research and experimentation can build upon this

methodology to address more sophisticated robotic

applications and propel advancements in the ﬁeld.

Tuning the Dynamic Response of a Redundant Robotic System Using Its Dominant Natural Frequencies

171

(a) FFT Joint 1.

(b) FFT Joint 2.

Figure 11: FFT results for the C

damping matrix after a

second parameter study. Plots of the most signiﬁcant joints.

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