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A Novel Dynamics Identification Method for Rigid-Body Manipulators

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This work presents a new method of dynamics identification for rigid-body manipulators under the influence of gravity. It is assumed that each link has an actuated joint where the axis of rotation is different from the center of mass, such that the driving torque is influenced by gravitational force. The dynamics model compensation together with the force of gravity are utilized to achieve natural oscillation for each link. The natural oscillation allows the system to be converted into an optimization problem where the frequency of oscillation is minimized. The correct dynamics parameters are already found when the minimum frequency of oscillation is achieved. The proposed method is analyzed and a theorem is presented to support the claims presented in this work, together with implementation results.
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A Novel Dynamics Identification Method for
Rigid-Body Manipulators
Rodrigo S. Jamisola, Jr.and Elmer P. Dadios
Abstract—This work presents a new method of dynamics
identification for rigid-body manipulators under the influence
of gravity. It is assumed that each link has an actuated joint
where the axis of rotation is different from the center of mass,
such that the driving torque is influenced by gravitational force.
The dynamics model compensation together with the force of
gravity are utilized to achieve natural oscillation for each link.
The natural oscillation allows the system to be converted into
an optimization problem where the frequency of oscillation
is minimized. The correct dynamics parameters are already
found when the minimum frequency of oscillation is achieved.
The proposed method is analyzed and a theorem is presented
to support the claims presented in this work, together with
implementation results.
Index Terms—dynamics identification, robot control, natural
oscillation
I. INTRODUCTION
Identifying the dynamics parameters of a given manipulator
remains a challenge when implementing full dynamics control,
especially with higher degrees of freedom (DOFs) manipu-
lators and humanoids. More recent identification procedures
of manipulator arm dynamics include the use of torque data
[1], floating-base motion dynamics [2], iterative learning [3],
neural network aided identification [4], set membership un-
certainty [5], and simultaneous identification and control [6].
Identification procedures that are more robot specific include
[7], [8], [9], [10]. Some examples of identification procedures
for humanoid robot dynamics used neural network [11], and
fuzzy-stochastic functor machine [12]. Recent humanoid robot
modeling and control include fuzzy neural network [13],
global dynamics [14], and ground interaction control [15].
A successful dynamics identification [16] and implemen-
tation on a mobile manipulator that is tasked to perform an
aircraft canopy polishing [17] is one of the few successful
implementations in full dynamics manipulator control. To
the best of our knowledge there is no available study that
quantifies how much deviation from the physical parameter
is acceptable for an identified parameter, such that the full
dynamics control can output the best possible performance.
This statement has two issues to address. First when dynamics
identification is performed, it is normally through a physical
system whose dynamics parameters are unknown. Second,
there is no set standard on qualifying a best performance of a
given manipulator or humanoid. Thus the only way to verify
Dept. of Electronics and Communications Engineering and
Dept. of Manufacturing Engineering and Management, De La
Salle University - Manila, 2401 Taft Ave, 1004 Manila, Philippines
{jamisolar,dadiose}@dlsu.edu.ph. This work is supported by the
University Research Council of De La Salle University - Manila.
A
A
A
mg
l
θ
y
x
1
2
_
(b)
(a)
Fig. 1. Subfigure (a) shows an upright pendulum turning about point Adue
to gravity and an initial displacement from equilibrium position. Subfigure
(b) shows the free body diagram.
the accurateness of the model is through an actual control
performance based on the acceptable limits set by the control
designer.
The method presented in this work will utilize the prin-
ciple of natural oscillation together with gravitational force
to identify the full dynamics of a multi-link rigid body.
The experimental procedure will compensate for Coriolis and
centrifugal forces and gravitational term, then an initial push
is given to allow the system to achieve natural oscillation.
The correct mass, center of mass, and inertial parameters
correspond to the least frequency of oscillation given an
initial perturbation from equilibrium. Analysis of the proposed
method and a theorem are presented to support the principles
behind the experimental procedure in this work, together with
the implementation results.
II. OVERVIEW
A. An Inverted Pendulum
The sum of the kinetic energy Kand potential energy P
of an upright pendulum, as shown in Fig. 1, consisting of a
slender bar with mass mand length lis [18]
K+P=1
2Id
θ
dt 2
1
2mgl cos
θ
(1)
and is constant, where Iis the inertia of the slender bar, gis the
gravitational constant, and
θ
is the angular displacement from
the vertical axis. Taking the time derivativeand assuming small
angular displacements, the above equation can be expressed as
(b)
(a)
_
A
1
2
x
Ay
θ
l
A
mg
τ
A
Fig. 2. Subfigure (a) shows an inverted pendulum. Subfigure (b) shows the
free body diagram such that the gravitational compensation is sent to the
system all the time which results into a system that is floating against gravity.
d2
θ
dt2+mgl
2I
θ
=0 (2)
with constant frequency of oscillation
ω
2=mgl/2I. This rela-
tionship shows the frequency of oscillation being dependent on
the physical characteristic of the pendulum. Thus a different
pendulum with different physical characteristics will give a
different value of the frequency of oscillation. When no torque
given at point A, the only moment acting on the pendulum is
that due to gravity. When an initial displacement is given, it
will cause the system to respond with angular acceleration and
achieve natural oscillation.
Next we consider an inverted pendulum as shown in Fig. 2.
This time, a torque
τ
Ais given at point Aas shown in the
free-body diagram of Subfigure (b). With the torque sent to
the system that is equal to the moment due to gravity, the
inverted pendulum will now “float” against gravity. When
an initial push is given, the inverted pendulum will achieve
angular acceleration such that the effective torque sent to the
system is
τ
A=I¨
θ
+1
2mgl sin
θ
,(3)
that is, the effective torque sent at point Ais the sum of
the given initial push that resulted into an acceleration and
the torque sent to balance off gravity. An oscillation occurs
because the total energy is in the process of being converted
from a purely potential energy to a purely kinetic energy, and
vice versa.
B. The Frequency of Oscillation
For a single degree of freedom, the given initial push
created an angular acceleration necessary for it to achieve
natural oscillation. The value of the gravitationalcompensation
torque that is sent to the system will depend on the values of
the dynamics parameters model. In this dynamics parameter
identification, it is the actual force of gravity that is being
utilized to define the critical boundary for the identification
of the correct values of the dynamics parameters. The main
objective is to minimize the frequency of oscillation such that
the dynamics parameters are not overcompensated which re-
sults into higher frequency of oscillation or under compensated
which results into a system falling under gravitational force.
By doing this, the problem becomes an optimization problem
with an objective function
min
ω
(I,m,r)(4)
where
ω
is the frequency of oscillation that is a function of
inertia I, mass m, and center of mass r.
C. Multi-Linked Rigid Body Dynamics
The general case of the multi-linked rigid body dynamics of
a manipulator will be presented. The torque to be sent to each
joint of the robot is computed by taking the kinetic energy K
and potential energy Pand solving the Lagrange equation of
T=K+P[19]
d
dt
T
˙qi
T
qi=
τ
i(5)
such the joint torque vector is expressed as
τ
=A(q)¨
q+c(q,˙
q)+g(q).(6)
The symbol A(q)is the joint-space inertia matrix, c(q,˙
q)is the
Coriolis and centrifugal forces vector, g(q)is the gravitational
terms vector, and ¨
q,˙
q,qare the joint space acceleration,
velocity, and displacement. The control equation is given as
[20]
u=¨
qdkv(˙
q˙
qd)kp(qqd)(7)
where u=¨
q, the proportional kpand derivative kvgains are
set to zero, and the desired acceleration ¨
qdvelocity ˙
qd, and
displacement qdare set to zero as well. This makes an open-
loop system and subsequently, the first term of Eqn. 6 equals
to zero. The torque sent to the manipulator joints would be
the Coriolis and centrifugal forces and the gravitational terms.
When the system is given an initial push, the total effective
torque now includes that due to acceleration ¨
q.
In [16], it is shown that the lumped inertia terms foundin the
inertia matrix A(q)are the same terms found in Coriolis and
centrifugal forces vector c(q,˙
q). Once all the lumped inertia
terms are identified in A(q), all the inertia terms are already
found. The biggest challenge in implementing the dynamics
identification in [16] is its need for a simplified symbolic
dynamics parameters in order to identify the lumped dynamics
model.
The method presented in this work will attempt to improve
the dynamics identification in [16] by identifying the individ-
ual dynamics parameters and not the lumped models. However,
a limitation of this method is the dominant parameter is the
gravitational term such that the inertias may be expressed as
estimates of masses and centers of mass in order to identify
them. The future direction of this identification procedure is
to treat the inertias as independent parameters.
III. THE PROPOSED DYNAMICS IDENTIFICATION
METHOD
In this section we will show a mathematical proof of
the proposed method to support the claims in this work. In
addition, the corresponding algorithm and its block diagram
will be shown.
A. Analysis of the Proposed Method
Given a physical system such that each link is influenced by
gravitational force, the physical parameters can be identified
by letting the system achieve natural oscillation through the
force of gravity. The following theorem will present support
for the dynamics identification method presented in this work.
Theorem 1. A multi-link rigid manipulator with revolute joints
is under computer control and is influenced by gravitational
force. Its minimum frequency of oscillation
ω
is achieved when
the estimated physical parameter values in the mathematical
model are closest to the values in the physical system.
Proof: Case I. Single Degree-of-Freedom. Equating the
torque between the physical system and the mathematical
model,
τ
=I¨
θ
+g(
θ
)= ˜
Iu+˜g(
θ
),(8)
where the control equation u =¨
θ
dkv(˙
θ
˙
θ
d)kp(
θ
θ
d).
Assuming no friction and setting kv=kp=0and ¨
θ
d=0,
results into u =0such that
¨
θ
+g(
θ
)˜g(
θ
)
I=0.(9)
Because I is constant and assuming small angular displace-
ment, Eqn. 9 can be treated as an undamped linear second-
order system [21] such that,
ω
2=f(g(
θ
)˜g(
θ
)).(10)
By the property of an open-loop computer control [22], g(
θ
)
cannot be strictly equal to ˜g(
θ
)and thus
ω
can only be
minimized. When
ω
is minimum, it is the case when the value
of ˜g(
θ
)is very close to the value of g(
θ
). Furthermore, when
g(
θ
)>˜g(
θ
), the pendulum will fall under gravitational force.
Case II. Multiple Degree-of-Freedom. The torques given at
the joints on both the physical system and the mathematical
model can be expressed as,
τ
=A(
θ
)¨
θ
+c(
θ
,˙
θ
)+g(
θ
)= ˜
A(
θ
)u+˜
c(
θ
,˙
θ
)+˜
g(
θ
).(11)
The control equation u=¨
θ
dkv(˙
θ
˙
θ
d)kp(
θ
θ
d)=0
because kv=kp=0and ¨
θ
d=0. This results into
¨
θ
+A(
θ
)1[(c(
θ
,˙
θ
)˜
c(
θ
,˙
θ
)) + (g(
θ
)˜
g(
θ
))].(12)
With small angular displacements, the inertia matrix A(
θ
)
can be considered constant. The natural oscillation may be
achieved at lower speeds such that model error in the Coriolis
and centrifugal forces is less dominant than the model error
in the gravitational forces. Furthermore, when the physical
system is correctly modeled, the system becomes decoupled
such that each link’s motion can be considered independently.
Thus Eqn. 12 may be treated as linear second-order system
[21] where
ω
2=f(g(
θ
)˜
g(
θ
)).(13)
When
ω
is minimum, this is the case when the values of the
model are very close to their physical parameter values.
B. The Algorithm
The algorithm for the dynamics identification through nat-
ural oscillation (DINO) is presented in the following.
1) Select initially large values for the inertia ˜
I, mass ˜m, and
center of mass ˜rto initialize the dynamics parameters
of the multi-link rigid body. The values of the inertias
may be expressed as masses and centers of mass.
2) For each of the joint i, send the torque
τ
icorresponding
to the Coriolis and centrifugal forces and the gravita-
tional terms.
3) Give the system an initial perturbation to move it from
rest. The system will start to oscillate at a certain
frequency
ω
.
4) Measure the frequency of oscillation
ω
.
5) Adjust the values of the dynamics parameters ˜
Im, and
˜rto minimize the frequency of oscillation
ω
.
6) Repeat the previous step until the minimum frequency
of oscillation
ω
is found such that the corresponding ˜
I,
˜m, and ˜rare the best estimates.
The flowchart of the proposed algorithm is shown in Fig. 3.
The dynamics parameters are initialized at large values to
prepare the system for the experimental procedure. This is to
avoid the system from falling due to insufficient torque sent
at the joints. Because of the initially large joint torque
τ
, the
system will be vertically upright and is at rest when no outside
forces are acting on it, except gravity. Then an outside force,
normally at the end-effector, is applied to perturb the system
from rest that will cause it to start to oscillate. Although the
dominant parameters in the identification procedure are the
gravitational terms, the Coriolis and centrifugalterms still need
to be included in the identification process because the model
helps decouple the system and makes the experimental results
more consistent.
To measure the frequency of oscillation
ω
, normally the
number of periods of oscillation is counted within a predefined
number of update cycles. An incremental step is performed
within the search space of the dynamic parameters to find
the combination of values that correspond to the minimum
ω
.
The dimension of the search space increases as the number of
dynamics parameters to be identified increases. Incrementally
changing the values of the dynamic parameters and measur-
ing the corresponding frequency of oscillation are repeatedly
performed until the minimum
ω
is found. For higher degrees
of freedom, it is highly possible that the identified parameters
will result in a combination of possible range of values. Each
of the possible combinations can be checked later against the
robot response in the actual manipulator control. Admittedly,
the limitation of this experimental method is due to the fact
that the links, whose parameters are to be identified, have to
Min I, m, r
τ
c(θ,θ) + g(θ)
Outside Force
A(θ)θ+c(θ,θ) + g(θ)
.. .
.
τI, m, r
Measure ω
Is
ω
min?
Νο
Yes
Start
Finish
Yes
Init I, m, r
Falls
under
gravity?
Νο
Fig. 3. The flowchart of the algorithm for the dynamics identification through
natural oscillation.
x2
z2
y2
x1
z1
y1
x0
z0
y0
x3
z3
y3
3-DOF
2-DOF
1-DOF
{
{
{
L
L
Lm1
m2
m3
Fig. 4. The frame assignment for 1-DOF, 2-DOF, and 3-DOF manipulators.
be dependent on gravitational force. Other links that are not
under the influence of gravitational force will be addressed
in future work. However, this work shows that for such links,
the force of gravity can be utilized in the identification process
itself.
IV. RESULTS AND ANALYSIS
Simulations are performed to test the proposed algorithm
in identifying the dynamics parameters of one, two, and
three degrees of freedom manipulators. Open dynamics engine
(ODE), an OpenGL program integrated with C code, is used
as the simulation platform. This simulation platform requires
the actual physical model for the system to move in a virtual
world. The dynamics parameters are then provided to create a
model for the simulation. Then the corresponding torques are
sent to the joints using estimated dynamics parameter values.
In this section, the range of estimated values corresponding
to the minimum frequency of oscillation will be presented for
each of the simulated manipulator. The frame assignment is
shown in Fig 4. For all experiments performed, 5,000 update
cycles are considered. The following symbol simplifications
are used Si...n=sin(n
i
θ
i)and Ci...n=cos(n
i
θ
i). In the
simulation experiment, friction is not included and angular
displacements are considered to be small if it is within ±15
degrees.
A. One Degree of Freedom
The simplest case of 1-DOF is shown to test the sensitivity
of the system to numerical errors. Given a mass m1and link
length L, a mass factor fm1is multiplied to the gravitational
term such that the torque sent to the system becomes
g1=fm11
2m1gL S1.(14)
The inertia of the 1-DOF manipulator at the joint is I=1
3m1L2.
Then a initial push is given to start the system to achieve
natural oscillation. The results are shown in Table I.
TABLE I
ONE DEGREE OF FREEDOM RESPONSE
Mass Factor fmNo. of Periods at 5,000 Updates
1.020 8.7
1.010 6.9
1.005 5.8
1.004 5.5
1.003 5.1
1.002 4.8
1.001 4.2
1.000 4.1
0.999 (falls under gravity)
The table shows consistently that as the estimated pa-
rameters start from large values and decrease towards the
correct values, the frequency of oscillation decreases. And
at 0.001 difference of the mass factor, a difference in the
frequency of oscillation is achieved. The minimum frequency
of oscillation is shown to be 4.1 periods per 5,000 update
cycles. A mass factor fm1=0.999 will cause the system to
fall due to gravitational force.
It is noted that when the product of mass m1and length L
results in a multiple of 1.0 (at a tolerance of ±0.001) to the
actual physical model, the resulting frequency of oscillation is
at the minimum. In a sense it is the lumped model of mass
and center of mass that is identified such that any values of
m1and Lwould suffice to model the dynamics parameters as
TABLE II
TWO DEGREES OF FREEDOM RESPONSE
Mass Factor No. of Periods at 5,000 Updates
fm1,fm2,fm3Link 1, Link 2
1.010 5.4
1.005 4.7
1.004 4.5
1.003 4.5
1.002 4.4
1.001 4.2
1.000 4.0
0.999 3.6
0.998 ( falls under gravity )
long as the product of their values corresponding to the actual
physical model and is correct to within 0.001 accuracy. This
precision defines the range of possible values of mass m1and
length L.
B. Two Degrees of Freedom
The symbolic form of the components of inertia A(
θ
),
Coriolis and centrifugal forces c(
θ
,˙
θ
)and gravitational terms
g(
θ
)of the 2-DOFs manipulator is shown in Eq. 15. Although
the inertia matrix is not used in the identification procedure,
the symbolic form here is shown for completeness. Links one
and two have equal link length L, and m1and m2are their
respective masses.
A11 =1
3m1L2+fm1m2L2(4
3+C2)
A12 =A21 =fm1m2L2(1
3+1
2C2)
A22 =fm11
3m2L2
c1=fm11
2m2L2˙q2(q1+˙q2)S2
c2=fm11
2m2L2S2˙q2
1
g1=fm2(1
2m1+m2)LgS1+fm31
2m2LgS12
g2=fm31
2m2LgS12
(15)
In the symbolic model shown, the mathematical model
of the inertias were simplified such that only the principal
moment of inertia about the joint axis is considered. The
other two principal moments of inertia and all the products of
inertia were assumed zero. However, in the simulation model
this is not true, as the ODE simulation mimics the real-world
physical system these values can only be assigned relatively
small values and not completely zeros.
For each of the lumped inertia parameters, mass factors
fm1,fm2, and fm3are used. That is fm1for m2L2,fm2for
(1
2m1+m2)L, and fm3for m2L. Note that the dynamics model
can be identified individually without the lumped symbolic
model. However, the value of the lumped model only served to
show the limits on the products of the identified dynamics pa-
rameters. In the subsequent experiments that were performed,
because the natural oscillations were achieved at low speeds,
all the links approximately moved in unison.
Table II shows the experimental results for 5,000 updates of
a two degrees of freedom robot with three mass factors fm1,
fm2, and fm3. The frequency of oscillation is recorded for the
different values of the mass factors. The robot falls under the
influence of gravity when the mass factors are 0.002 below the
assigned physical mass factor of 1.0. The minimum frequency
of oscillation applies when all mass factors are 0.001 below
1.0. The frequencies of oscillations for both links decreased
accordingly as estimated values decreased towards the actual
physical values.
The inertia parameters in all cases is a function of masses
and centers of mass. The same values affect the gravitational
forces, which are the dominant parameters in the experiment
because the system does not move at a very high speed.
The Coriolis and centrifugal forces are only dominant when
the speed of each joint is high. Thus for this method of
identification procedure, it is advantageous to express the
inertia parameters in terms of masses and centers of mass in
order to identify the parameters of the entire physical system.
C. Three Degrees of Freedom
The symbolic full dynamics model of a 3-DOFs is shown
in Eqn 16. The link masses are m1,m2, and m3for links 1
to 3 and have equal link lengths of L. Five mass factors from
fm1to fm5are used for the identified five lumped parameters.
It is assumed that the principal inertias of each link about the
joint is dominant, such that the other principal inertias and the
products of inertia are assumed zero. The inertias are expressed
as masses and centers of mass. The values of fm1to fm5are
varied to determined the response of the system in finding the
minimum period of oscillation.
A11 =1
3(m1+4m2+7m3)L2+fm4(m2+2m3)L2C2
+fm2m3L2(C3+C23)
A12 =A21 =1
3(m2+4m3)L2+fm41
2(m2+2m3)L2C2
+fm21
2m3L2(2C3+C23)
A13 =A31 =fm21
6m3L2(2+3C3+3C23)
A23 =A32 =fm21
6m3L2(2+3C3)
A22 =1
3(m2+4m3)L2+fm23m3L2C3
A33 =fm21
3m3L2
c1=fm11
2(m2+2m3)L2˙q2(q1+˙q2)S2
fm21
2m3L2(˙q3(q1+q2+˙q3)S3
+(˙q2+˙q3)(q1+˙q2+˙q3)S23)
c2=fm11
2(m2+2m3)L2˙q2
1S2
+fm21
2m3L2(˙q3(q1+q2+˙q3)S3
+˙q2
1S23)
c3=fm21
2m3L2(( ˙q1+˙q2)2S3+˙q2
1S23)
g1=fm31
2(m1+2(m2+m3))LgS1
+fm41
2(m2+2m3)LgS12
+fm51
2m3LgS123
g2=fm41
2(m2+2m3)LgS12
+fm51
2m3LgS123
g3=fm51
2m3LgS123 (16)
Table III shows the experimental response of the 3-DOF ma-
nipulator. In the experiment, the natural oscillation is achieved
TABLE III
THREE DEGREES OF FREEDOM RESPONSE
Mass Factor No. of Periods at 5,000 Updates
fm1,fm2,fm3,fm4,fm5Link 1, Link 2, Link 3
1.010 4.5
1.005 4.0
1.004 3.9
1.003 3.9
1.002 3.7
1.001 3.2
1.000 3.1
0.999 3.0
0.998 (falls under gravity)
0 0.2 0.4 0.6 0.8 1
−20
−10
0
10
θ1
Natural Oscillations at 5,000 Computational Cycles
0 0.2 0.4 0.6 0.8 1
2
3
4
θ2
0 0.2 0.4 0.6 0.8 1
−0.5
0
0.5
θ3
Normalized Steps
Fig. 5. The number of periods for each of the joints for the three degrees
of freedom robot, recorded within 5,000 update cycles. The x-axis showed
the normalized number of cycles. This is the case where fm1=fm2=fm3=
fm4=fm5=1.002 where the number of periods for all links is 3.7.
at slow speed and the entire system moves in unison. This is
helpful but not necessary, in achieving well-defined natural
oscillations. The inertias were also expressed in terms of
masses and centers of mass, and the dominant parameters are
gravitational and not the Coriolis and centrifugal forces. With
the slow speed, the system becomes a typical second-order
system such that the period of oscillation is highly dictated
by gravitational parameters. The 3-DOF response is shown in
Fig. 5.
As in the previous results, there is a distinctive decrease
of the period of oscillation as the mass factors decrease
towards the correct value. At mass factors of 0.998 the system
fell under gravity. The minimum frequency of oscillation is
achieved at a mass factor value of 0.999. The system can
be considered as having a tolerance limit of ±0.001. Such
tolerance value is also demonstrated in mass factor values of
1.004 and 1.003 where the corresponding periods remain the
same. For practical purposes, it can be stated that the tolerance
value is at ±0.002. In most cases, it is hard to quantify the
acceptable tolerance values of an identified parameter because,
to the best of our knowledge, there is no such quantification
study. The correctness of the identified value is only checked
through the accuracy of the robot response. But even with
the quantification of the accuracy of robot response, there
is no set standard on how much is the tolerance limit to be
considered “best performance” given set parameters as speed,
task requirement, and load. Thus the success of most dynamics
identification method normally is based on response of the
manipulator where the identification is being implemented.
And this is dependent on what type of task the manipulator
is required to perform, and is generally not applicable to all
systems.
V. CONCLUSION AND FUTURE WORK
This work showed a new method of identifying the dy-
namics parameters of multi-link rigid bodies that move under
gravitational influence. A mathematical proof is presented to
support the validity of the proposed algorithm. The idea is to
compensate for the Coriolis and centrifugal forces and grav-
itational terms, and apply an outside force for the system to
achieve natural oscillation. Once this is done, the experimental
identification proceeds by minimizing the frequency of oscilla-
tion just before the system falls under the influence of gravity.
The correct values of the dynamics parameters correspond
to the system with the minimum frequency of oscillation.
Results sensitivity and range of the identified experimental
values were shown for one, two, and three degrees of freedom
manipulators.
The future direction of this work is to implement the method
on a physical robot manipulator, where the friction parameters
influence the results. In addition, links that are not directly
under gravitational influence will be considered in future
study. The ultimate goal of this experimental procedure is to
implement it on a humanoid robot moving in a full-dynamics
whole-body control [23].
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