Dynamic parameter identification method for robotic arms with static friction modelling
ACTA IMEKO
ISSN: 2221-870X
September 2021, Volume 10, Number 3, 44 - 50
ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 44
Dynamic parameter identification method for robotic arms
with static friction modelling
Dániel Szabó1, Emese Gincsainé Szádeczky-Kardoss 1
1 Department of Control Engineering and Information Technology, Budapest University of Technology and Economics, Budapest, Hungary
Section: RESEARCH PAPER
Keywords: Identification; dynamics; manipulator
Citation: Dániel Szabó, Emese Gincsainé Szádeczky-Kardoss, Dynamic parameter identification method for robotic arms with static friction modelling, Acta
IMEKO, vol. 10, no. 3, article 8, September 2021, identifier: IMEKO-ACTA-10 (2021)-03-08
Section Editor: Bálint Kiss, Budapest University of Technology and Economics, Hungary
Received January 15, 2021; In final form September 6, 2021; Published September 2021
Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and source are credited.
Corresponding author: Dániel Szabó, e-mail: szabo.daniel@iit.bme.hu
1. INTRODUCTION
Automation is playing an increasingly important role in
industry, where manipulators are used to solve various types of
problems. Precise control algorithms are required to meet the
need for speed and accuracy, and an accurate dynamic model is
needed to use these control algorithms, such as the computed
torque method [1]. Although the robot manufacturer can provide
the necessary values, they are usually either inaccurate or non-
existent; thus, the dynamics of the manipulator have to be
determined through an identification process.
An independently identifiable parameter vector and a
regression matrix are required to create an identifiable model.
The barycentric parameters [2] or the modified recursive
Newton–Euler formula [3] can be used to solve this problem.
In general, the friction effect of the joints is not negligible,
hence the dynamic model of the robot, determined according to
the structure of the manipulator, has to be extended with the
friction model. There are plenty of existing friction models that
can be used in the identification of the dynamic model, but there
has to be a trade-off between model accuracy and computational
complexity.
In the simplest case, only the Coulomb and viscous friction
effects need to be considered in the dynamic model [4]. Although
this can be sufficient in some cases, the stiction effect in the joint
remains unmodelled, and the nonlinearity has to be compensated
for by the controller.
A dynamic friction model can be applied if a high level of
precision is needed in the model, but in this case, the model
cannot be converted into a linear-in-parameters model [5]. This
method can be used only if the dynamic model is known and
only friction compensation is needed.
Discontinuities of the sign function can be eliminated by
using the arctangent function to approximate the step at which
the velocity changes its sign. The continuity of this function is
necessary when using the identified model in the controller for a
smooth control signal when the velocity reaches zero. In [6],
both static and dynamic friction identification were presented
but only for cases where the other dynamic parameters were
known. It is possible to convert the static friction model into a
linear-in-parameters form.
The optimisation of the excitation trajectory is important for
generating a well-conditioned regression matrix. The finite
Fourier series [7] and the polynomial functions [8] are used
frequently as excitation trajectories.
ABSTRACT
This paper presents an identification method for robotic manipulators. It demonstrates how a dynamic model can be constructed with
the help of the modified Newton–Euler formula. To model the friction of the joints, static friction modelling is used, in which the friction
behaviour depends only on the actual velocity of the given joint. With these techniques, the model can be converted into a linear-in-
parameters form, which can make the identification process easier. Two estimators are introduced to solve the identification problem,
the least-squares and the weighted least-squares estimators, and the determination of the independently identifiable parameter vector
to make the regression matrix maximal column rank is presented. The Frobenius norm is used as the condition of the regression matrix
to optimise the excitation trajectories, and the form of the trajectories has been selected from the finite Fourier series. The method is
tested in a simulated environment to achieve a three-degrees-of-freedom manipulator.
mailto:szabo.daniel@iit.bme.hu
ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 45
The proper estimator for a problem can be selected by
investigating the rate of the noise corrupting the measurements
[7]. In this study, the least-squares (LS) and weighted-LS (WLS)
estimation methods were assessed. As was shown in [9], the
maximum likelihood estimation gives better results in
simulations when there are corrupted joint value measurements,
but usually, in real-life scenarios, this noise is negligible compared
with the noise of the torque measurements.
In [10], the dynamics of the payload were determined by an
identification process that measured actuator currents. In this
method, the dynamic parameters of the manipulator were
known.
The present study compares two methods to estimate the
parameters of a robot dynamic model with static friction in each
joint. It demonstrates how the reduced row echelon form of the
regression matrix can be used to find the independently
identifiable variables. The results in [9] are extended in this paper
by presenting a method incorporating the friction model into the
identification process.
The paper is organised as follows. In Section 2, the form of
the dynamic model is introduced, and this section describes how
the modified Newton–Euler formula and the static friction
model are used to obtain its linear-in-parameters form. Section 3
describes the configuration of the measurements, and the
method to determine the independently identifiable parameters
is explained in Section 4.1. The LS and the WLS estimators are
then described. After that, Section 0 introduces the optimisation
of trajectory parameters and the criterion for this optimisation
method. The results of the simulation are in Section 6, and
finally, the conclusions are set out in Section 7.
2. MODEL EVALUATION
2.1. Dynamics of the manipulators
A manipulator can be modelled with an open kinematic chain,
which is composed of links connected by joints. The Denavit–
Hartenberg parameters can be used to define the kinematics of a
given manipulator [11], and the homogeneous transformation
matrices can be defined to calculate the actual position and
orientation of the end effector.
The dynamic model of a manipulator is given by the nonlinear
multiple-input and multiple-output function, which clarifies the
relationship between the joint positions, their first- and second-
order derivatives and the joint torques/forces. The following
equation shows this relationship:
𝝉 = 𝑯(𝒒)�̈� + 𝒉(𝒒,�̇�) + 𝒄(𝒒), (1)
where 𝐪 denotes the 𝑛-by-1 vector of the joint variables (for an
𝑛 degree-of-freedom manipulator), 𝛕 is the vector of joint
torques (or forces), 𝐇 is the 𝑛-by-𝑛 symmetric, positive-definite
joint-space inertia matrix, the Coriolis and centrifugal forces are
modelled by vector 𝐡 and the gravitational effects are given by
vector 𝐜.
Although the model in (1) provides a good physical
interpretation of the system, it cannot be used effectively during
the identification process. To solve this problem, the linear-in-
parameters form of the model can be used:
𝝉 = 𝜱(𝒒,�̇�, �̈�)𝜣, (2)
where 𝛷 denotes the regression matrix and 𝛩 is the parameter
vector.
The values of the inertia matrix, the mass of the link (𝑚𝑗) and
the first moment of the links are contained in 𝛩:
𝛩 = [𝛩1
𝑇
𝛩2
𝑇
… 𝛩𝑛
𝑇]𝑇
𝛩𝑗 = [𝐼𝑥𝑥
𝑗
𝐼𝑦𝑦
𝑗
𝐼𝑧𝑧
𝑗
𝐼𝑥𝑦
𝑗
𝐼𝑥𝑧
𝑗
𝐼𝑦𝑧
𝑗
…
𝑚𝑗 𝑚𝑗𝑠𝑥
𝑗
𝑚𝑗𝑠𝑦
𝑗
𝑚𝑗𝑠𝑧
𝑗
]
𝑇
,
(3)
where 𝐼𝑘𝑙
𝑗
parameters are the entries of the symmetric inertia
matrix and 𝐬𝑗 = [𝑠𝑥
𝑗
𝑠𝑦
𝑗
𝑠𝑧
𝑗
]
𝑇
denotes the position of the
centre of mass of the 𝑗-th link measured in the frame belonging
to the 𝑗-th joint.
To solve the identification problem, it is necessary to calculate
the regression matrix. This can be achieved, for example, with
the Lagrangian formula, which provides a good physical view of
the system; but it is computationally expensive. Hence, a better
solution is the Newton–Euler formula, which has a
computational complexity of 𝒪(𝑛) [3].
First, using the Newton-Euler formula, the kinematics of each
frame of the joints are calculated and transformed into the frame
of the end effector. This is done with forward recursion, and the
velocities, angular velocities, accelerations and angular
accelerations are determined for each frame. Then, the forces
and moments are transformed with backward recursions from
the end-effector frame to the base frame. The applied
torque/force of the given joint is determined by the following
equation in a given step of the backward recursion:
𝛕𝑗 = {
𝐧𝑗
𝑇
(𝐀𝑗
𝑇
𝐳0) rotational
𝐟𝑗
𝑇
(𝐀𝑗
𝑇
𝐳0) translational
, (4)
where 𝐧𝑗 and 𝐟𝑗 are the moment and the force exerted on link 𝑗
by link 𝑗-1 and 𝐀𝑗 is the orthonormal rotation matrix that
describes the rotation between the 𝑗-th and the 𝑗-1-th frame; 𝐳0
is the base vector of the axis of rotation (or translation), which is
selected by the convention [0 0 1]𝑇. This results in linear
terms in the inertias and the masses, while the centre-of-mass
vector will be represented in either linear or quadratic terms.
These quadratic terms can be eliminated by transforming the
inertia tensor: let 𝒞𝑖 denote the coordinate system for the 𝑖-th
link and 𝒞𝑖
∗ denote the frame of the centre of mass of the 𝑖-th
link; let 𝐀𝑖,𝑖∗ be the orthonormal rotation matrix between 𝒞𝑖 and
𝒞𝑖
∗. If 𝐈𝑖 is the inertia tensor around the centre of mass, then the
inertia tensor 𝐈′𝑖 in 𝒞𝑖 is
𝐈′𝑖 = 𝐀𝑖,𝑖∗𝐈𝑖𝐀𝑖,𝑖∗
𝑇 + 𝑚𝑖(𝐬𝑖
𝑇𝐬𝑖𝐈(3𝑥3) − 𝐬𝑖𝐬𝑖
𝑇), (5)
where 𝐈(3𝑥3) is the 3-by-3 identity matrix.
2.2. Friction modelling
The model in (1) must be extended with the torque vector 𝝉𝒇
to model the effect of the friction:
𝝉 − 𝝉𝒇 = 𝑯(𝒒)�̈� + 𝒉(𝒒,�̇�) + 𝒄(𝒒). (6)
There are several methods to model the vector of the friction
torques. In this study, static friction modelling was used,
including stiction, Coulomb and viscous friction effects, and the
Striebeck effect, which was represented using the arctangent
function [5]:
𝝉𝒇 = 𝝉𝒔𝑆0(𝒗) + 𝝉𝒔𝒄
2
π
arctan(𝒗𝛿) + 𝝉𝒗 , (7)
ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 46
where 𝒗 denotes the vector of velocities (thus, the first-order
derivative of the joint variables), 𝝉𝒔 represents stiction, 𝝉𝒔𝒄 is the
difference between Coulomb friction and stiction, 𝛿 represents
the shape of the Striebeck effect, 𝝉𝒗 denotes the coefficient of
the viscous friction and the function 𝑆0 is used as the
approximation of the 𝑠𝑖𝑔𝑛(𝒗) function:
𝑆0(𝒗) =
2
π
arctan(𝒗𝐾𝑣), (8)
where 𝐾𝑣 defines the shape of the function.
In this case, the friction model can be transformed into a
linear-in-parameters form, where the regression matrix and the
parameters belonging to one joint are the following:
𝛷fric
𝑗
= [
2
π
arctan(𝒗𝐾𝑣),
2
π
arctan(𝒗𝛿) ,𝑣]
𝛩fric
𝑗
= [𝜏𝑠
𝑗
, 𝜏𝑠𝑐
𝑗
, 𝜏𝑣
𝑗
]
𝑇
.
(9)
3. MEASUREMENT SETUP
Simulations were used to perform the measurements in a
MATLAB Simulink environment, but the proposed method
could also be used with real measurements. To model the
dynamics and the friction effect, the Simscape Toolbox was
applied.
At this stage, control of the manipulator was performed by
computing the torques automatically using the simulation;
furthermore, the torque measurements were corrupted with
independent zero-mean Gaussian noise. The measurements were
repeated with different variances for the specific joints, and the
joint torques were corrupted with independent zero-mean
Gaussian noise when each joint had different variances. The joint
angles were also corrupted with noise, but this was negligible
compared with the noise of the torque measurements [9].
4. IDENTIFICATION PROCESS
4.1. Determination of the independent variables
According to (2) and (9), the dynamics of the manipulator can
be written in a linear-in-parameters form, but to use it during the
identification process, some changes have to be applied because
of the parameter vector described in (3).
With these parameter vectors, the column-rank of the
regression matrix 𝛷 will not be maximal. This means that
unidentifiable parameters or parameters that are not
independently identifiable also exist.
Three parameter types can be defined [12]:
1. If the 𝑖-th column of 𝛷 is a null vector, then the 𝑖-th
parameter is unidentifiable and the dynamics are
unaffected. In this case, this parameter and the 𝑖-th
column of 𝛷 can be removed.
2. If the 𝑖-th column of 𝛷 is not null and can be expressed
as a linear combination of the other columns, then the
𝑖-th parameter can be identified only in a linear
combination with other parameters.
3. If the 𝑖-th column of 𝛷 is not null and cannot be
expressed as a linear combination of the other columns,
then the 𝑖-th parameter is independently identifiable.
First, all the columns of 𝛷 with only zeros should be
eliminated; the number of independent parameters is then equal
to the rank of 𝛷. These parameters can be expressed in a linear
combination of the original parameters.
To obtain the coefficients for the linear combinations, the
reduced row echelon form of 𝛷 is required:
[𝛷𝑟𝑟𝑒𝑓, 𝐣𝐋] = 𝑟𝑟𝑒𝑓(𝛷), (10)
where 𝐣𝐋 denotes the indexes of the columns with the leading
values 1 in the reduced row echelon form. These indexes will give
a basis in the columns of 𝛷 by indexing out the column of the
regression matrix with 𝐣𝐋, and 𝐣𝐊 is the vector of the indexes that
are not contained by 𝐣𝐋.
Let 𝐋 and 𝐊 be the matrices with columns indexed by 𝐣𝐋 and
𝐣𝐊 from 𝛷, respectively. In this way, a linear combination can be
defined to express each column of 𝐊 with the columns of 𝐋, and
the coefficients can be determined by the columns of 𝛷𝑟𝑟𝑒𝑓
indexed by 𝐣𝐊:
𝐊(: , 𝑖) = 𝛷𝑟𝑟𝑒𝑓(1, 𝐣𝐊(𝑖))𝐋(: ,1) + ⋯+
+𝛷𝑟𝑟𝑒𝑓(𝑚,𝐣𝐊(𝑖))𝐋(: ,𝑚),
(11)
where 𝑚 is the rank of 𝛷, i.e. the number of independent
parameters.
The matrix 𝐊 can therefore be expressed as 𝐊 = 𝐋𝐁, with the
help of 𝐁 defined as:
𝐁 = 𝛷𝑟𝑟𝑒𝑓(1:𝑚,𝐣𝐊). (12)
Two vectors of the parameters are defined by indexing them
with 𝐣𝐋 and 𝐣𝐊:
𝛩𝐿 = 𝛩(𝐣𝐋)
𝛩𝐾 = 𝛩(𝐣𝐊)
. (13)
Thus, a new form of the model can be introduced:
𝛕 = 𝛷𝛩 = [𝐋 𝐊][
𝛩𝐿
𝛩𝐾
] =
= 𝐋[𝐼(𝑚𝑥𝑚) 𝐁][
𝛩𝐿
𝛩𝐾
] = 𝐋𝛩∗,
(14)
where 𝛩∗ = 𝛩𝐿 + 𝐁𝛩𝐾. Consequently, 𝛩𝐿(𝑖) is an
independently identifiable parameter if, and only if, the 𝑖-th row
of 𝐁 contains only zeros.
4.2. LS estimation
If the torques of the actuators are corrupted with zero-mean
Gaussian noise (𝐧𝜏) with the same standard deviation, the
following measurement model can be defined:
𝛕𝑚(𝑘) = 𝛕(𝑘) + 𝐧𝜏(𝑘)
𝑒𝜏𝑖(𝛩
∗,𝑘) = 𝛕𝑚𝑖(𝑘) − 𝛕𝑖(𝛩
∗,𝑘) =
= 𝛕𝑚𝑖(𝑘) − 𝐋(𝑘, :)𝛩
∗,
(15)
where 𝛕𝑚 is the measured value of the torque vector, the error
of the given joint is denoted by 𝑒𝜏𝑖 and 𝑘 denotes the index of
the measurement.
With this measurement model, the standard LS estimator can
be used. The minimisation criterion of the estimator is the
following:
𝑉𝐿𝑆 = ∑∑(
𝑛
𝑖=1
𝑁
𝑘=1
𝑒𝜏𝑖(𝛩
∗,𝑘))2, (16)
where 𝑁 is the number of measurements.
ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 47
The solution to the problem can be given in a closed form
with the help of the Moore–Penrose pseudoinverse of 𝐋:
�̂�𝐿𝑆 = 𝐋
+𝛕𝑚 = (𝐋
𝑇𝐋)−1𝐋𝑇𝛕𝑚. (17)
4.3. WLS estimation
If noises with different standard deviations (𝜎𝜏𝑖) are
corrupting the measurements, the WLS estimator has to be
applied. Here the error of the measurements in the optimisation
criterion are weighted with the reciprocal of the variation of the
noise for a given joint:
𝑉𝑊𝐿𝑆 = ∑∑
(𝑒𝜏𝑖(𝛩
∗,𝑘))2
𝜎𝜏𝑖
2
𝑛
𝑖=1
𝑁
𝑘=1
. (18)
Consequently, the problem can be solved as
�̂�𝑊𝐿𝑆 = (𝐋
𝑇𝚺−1𝐋)−1𝐋𝑇𝚺−1𝛕𝑚, (19)
where 𝚺 is the diagonal covariance matrix of the noise.
5. TRAJECTORY OPTIMISATION
The quality of the identification is dependent on the condition
of the 𝐋 matrix, which can be optimised by using the correct
excitation trajectories. There are several criteria to perform this.
5.1. Optimisation criteria
In several papers, the optimisation criterion to find the
optimal excitation trajectories is the condition number of 𝐋𝑇𝐋
[13], while in [14], the maximisation of the minimum singular
value of 𝐋𝑇𝐋 is used as the optimisation criterion. However,
faster convergence can be achieved if the Frobenius norm is used
as the criterion [8]:
CondF(𝐀) = ||𝐀||F||𝐀
−1||F
||𝐀||F = (∑∑𝐀𝑖𝑗
2
𝑝
𝑗=1
𝑝
𝑖=1
)
1/2
,
(20)
where 𝐀 is an 𝑝-by-𝑝 nonsingular matrix.
Hence, the optimisation problem can be written in the
following form:
min
𝛅
CondF(𝐋
𝑇𝐋)
such that
{
𝐪min < 𝐪 < 𝐪max
�̇�min < �̇� < �̇�max
�̈�min < �̈� < �̈�max
𝐪(0) = 𝐪0 𝐪(𝑡𝑓) = 𝐪𝑓
�̇�(0) = 0 �̇�(𝑡𝑓) = 0
�̈�(0) = 0 �̈�(𝑡𝑓) = 0
, (21)
where 𝛅 vector contains the parameters of the trajectory and the
𝐪𝑚𝑖𝑛,𝐪𝑚𝑎𝑥, etc. vectors are the minimal and maximal joint
values, velocities and accelerations, respectively. By using
equality constraints, the initial (𝐪0) and final (𝐪𝑓) conditions can
be taken into consideration. The initial and final velocities and
accelerations are selected as zero.
The optimisation problem in (21) was solved in MATLAB
using the fmincon function with the interior-point method as the
solving algorithm.
5.2. Trajectory parametrisation
The form of the applied trajectories can be expressed as finite
Fourier series. The definitions of the parametrisation for the joint
variables and their first- and second-order derivatives are the
following [15]:
𝑞𝑖(𝑡) = ∑
𝑎𝑙
𝑖
𝜔𝑓𝑙
𝑀
𝑙=1
sin(𝜔𝑓𝑙𝑡) −
𝑏𝑙
𝑖
𝜔𝑓𝑙
cos(𝜔𝑓𝑙𝑡),
�̇�𝑖(𝑡) = ∑𝑎𝑙
𝑖
𝑀
𝑙=1
cos(𝜔𝑓𝑙𝑡) + 𝑏𝑙
𝑖sin(𝜔𝑓𝑙𝑡),
�̈�𝑖(𝑡) = ∑−
𝑀
𝑙=1
𝑎𝑙
𝑖𝜔𝑓𝑙sin(𝜔𝑓𝑙𝑡) + 𝑏𝑙
𝑖𝜔𝑓𝑙cos(𝜔𝑓𝑙𝑡),
(22)
where 𝜔𝑓 is the fundamental frequency of the Fourier series.
The parameter vector can therefore be defined by an 𝑛 × 𝑀
matrix ([𝑎1
1 … 𝑎𝑀
𝑛 𝑏1
1 … 𝑏𝑀
𝑛]) , where 𝑛 is the number
of joints of the manipulator and 𝑀 denotes the number of
harmonics in the Fourier series.
6. EXPERIMENTAL RESULTS
The implementation of the identification process was
performed in MATLAB, while the dynamics and the friction
modelling were evaluated in MATLAB Simulink (Section 3).
The structure of the three-degrees-of-freedom manipulator
used in this study is depicted in Figure 1, and its Denavit–
Hartenberg parameters are introduced in Table 1. The MATLAB
Symbolic Toolbox was used to determine the dynamics of the
manipulator. The simulated trajectories are shown in Figure 2.
These trajectories were designed by the method described in
Section 0.
The results of the identification and the symbolic expressions
of the final 𝛩∗ parameter vector are shown in Table 2. It can be
seen that 24 identifiable variables were determined.
Figure 1. Model of the three-degrees-of-freedom manipulator, defined by
the Denavit–Hartenberg parameters in Table 1.
Table 1. The Denavit–Hartenberg parameters of the modelled three-degrees-
of-freedom manipulator
𝒊 𝜽𝒊 𝐢𝐧 rad 𝒅𝒊 𝐢𝐧 m 𝒂𝒊 𝐢𝐧 m 𝜶𝒊 𝐢𝐧 rad
1 𝑞1 0.3 0
π
2
2 𝑞2 +
π
2
0 0.5 0
3 𝑞3 0 0.5 0
ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 48
Figure 2. Measurements with optimised excitation trajectories, 𝜎𝜏 = [0.5,0.3,0.7] N·m.
Table 2. Results of the estimations (𝜇: mean, 𝜎2: variance of the estimators)
Symbolic expression Exact values
LS WLS
𝜇 𝜎2 𝜇 𝜎2
𝐼𝑥𝑥
(2)
− 𝐼𝑦𝑦
(2)
−
𝑚(2)𝑠𝑥
(2)
2
0.059 0.059 8.27 ⋅ 10
−04 0.059 6.25 ⋅ 10−04
𝐼𝑥𝑥
(3)
− 𝐼𝑦𝑦
(3)
−
𝑚(3)𝑠𝑥
(3)
2
0.059 0.057 8.09 ⋅ 10
−04 0.058 6.18 ⋅ 10−04
𝐼𝑥𝑦
(2)
0.000 0.002 3.23 ⋅ 10−04 0.001 2.27 ⋅ 10−04
𝐼𝑥𝑦
(3)
0.000 −0.000 2.04 ⋅ 10−04 −0.000 1.47 ⋅ 10−04
𝐼𝑦𝑦
(1)
+ 𝐼𝑦𝑦
(2)
+ 𝐼𝑦𝑦
(3)
+
𝑚(2)𝑠𝑥
(2)
2
+
𝑚(3)𝑠𝑥
(3)
2
−0.117 −0.115 5.25 ⋅ 10
−04 −0.115 4.30 ⋅ 10−04
𝐼𝑥𝑧
(2)
−
𝑚(2)𝑠𝑧
(2)
2
−
𝑚(3)𝑠𝑧
(3)
2
0.000 0.001 2.06 ⋅ 10
−04 0.000 1.50 ⋅ 10−04
𝐼𝑥𝑧
(3)
−
𝑚(3)𝑠𝑧
(3)
2
0.000 −0.001 1.60 ⋅ 10
−04 −0.000 1.14 ⋅ 10−04
𝐼𝑦𝑧
(2)
0.000 −0.001 1.32 ⋅ 10−04 −0.000 1.06 ⋅ 10−04
𝐼𝑦𝑧
(3)
0.000 0.001 7.56 ⋅ 10−05 0.001 5.49 ⋅ 10−05
𝐼𝑧𝑧
(2)
+
𝑚(2)𝑠𝑥
(2)
2
−0.059 −0.060 3.68 ⋅ 10
−04 −0.060 2.68 ⋅ 10−04
𝐼𝑧𝑧
(3)
+
𝑚(3)𝑠𝑥
(3)
2
−0.059 −0.057 3.30 ⋅ 10
−04 −0.058 2.21 ⋅ 10−04
𝑚(2) + 2𝑚(2)𝑠𝑥
(2)
− 2𝑚(3)𝑠𝑥
(3)
1.414 1.415 1.50 ⋅ 10
−04 1.414 6.01 ⋅ 10−05
𝑚(3) + 2𝑚(3)𝑠𝑥
(3)
0.707 0.706 3.82 ⋅ 10
−05 0.707 1.42 ⋅ 10−05
𝜏𝑠
(1)
2.000 1.998 7.97 ⋅ 10
−04 1.998 7.80 ⋅ 10−04
𝜏𝑠
(2)
1.000 0.999 1.42 ⋅ 10
−04 0.999 1.41 ⋅ 10−04
𝜏𝑠
(3)
2.000 1.998 2.36 ⋅ 10
−03 1.998 2.34 ⋅ 10−03
𝜏𝑣
(1)
0.500 0.505 8.26 ⋅ 10
−03 0.506 7.89 ⋅ 10−03
𝜏𝑣
(2)
0.277 0.243 9.26 ⋅ 10
−04 0.244 8.08 ⋅ 10−04
𝜏𝑣
(3)
0.030 0.034 1.60 ⋅ 10
−02 0.035 1.54 ⋅ 10−02
𝜏𝑠𝑐
(1)
−0.300 −0.302 1.45 ⋅ 10
−02 −0.303 1.39 ⋅ 10−02
𝜏𝑠𝑐
(2)
−0.200 −0.176 1.67 ⋅ 10
−03 −0.177 1.65 ⋅ 10−03
𝜏𝑠𝑐
(3)
−0.300 −0.306 3.56 ⋅ 10
−02 −0.307 3.45 ⋅ 10−02
𝑚(2)𝑠𝑦
(2)
0.000 0.000 3.82 ⋅ 10−06 0.000 2.24 ⋅ 10−06
𝑚(3)𝑠𝑦
(3)
0.000 0.000 4.67 ⋅ 10−06 0.000 2.37 ⋅ 10−06
ACTA IMEKO | www.imeko.org September 2021 | Volume 10 | Number 3 | 49
The dynamics have some independently identifiable
parameters (𝐼𝑥𝑦
(2)
, 𝐼𝑥𝑦
(3)
, 𝐼𝑦𝑧
(2)
, 𝐼𝑦𝑧
(3)
, 𝑚(2)𝑠𝑦
(2)
, 𝑚(3)𝑠𝑦
(3)
) and some
unidentifiable variables (e.g. the parameters of the inertia matrix
of the first segment), which do not influence the robot dynamics.
All the friction parameters are independently identifiable, and the
other parameters can be identified in a linear combination with
more parameters.
It can be seen from the results that the LS and the WLS
estimators performed well. With this method, the friction
parameters were also identified with only one measurement
configuration, which simplifies the measurement process. Figure
3 depicts the efficiency of the WLS estimator in the case of the
non-optimised trajectories, using a finite set of Fourier series as
excitation. The difference between the approximated and the
nominal value of the torque vector is negligible.
7. CONCLUSION
This paper presents two methods to solve the identification
problem of the dynamic model of robotic manipulators. By
modelling only the static friction behaviour, the whole dynamic
model can be expressed in linear-in-parameters form. In this
form, both LS and WLS estimators can be used effectively to
approximate the unknown parameters of the robotic arm.
As can be seen in Table 2, all the friction parameters can be
identified independently, hence the method can also be used
when the dynamic parameters are known, but only for friction
compensation.
The advantage of this method is that only one measurement
configuration is needed to obtain all the desired parameters, but
it is not required for moving only one joint at a time. These
results can be used in advanced control algorithms.
8. ACKNOWLEDGEMENT
The research reported in this paper, carried out at the
Budapest University of Technology and Economics, was
supported by the ‘TKP2020, Institutional Excellence
Programme’ of the National Research Development and
Innovation Office in the field of artificial intelligence (BME IE-
MI-SC TKP2020).
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