Torque Minimization of Dynamically Decoupled 3R Spatial Serial
Manipulators via Optimal Motion Generation
YAODONG LU, VIGEN ARAKELIAN
1ECN, LS2N UMR 6004,
1 rue de la Noë, BP 92101, F-44321 Nantes,
FRANCE
also with
2Mecaproce / INSA Rennes,
20 av. des Buttes de Coësmes, CS 70839, F-35708 Rennes,
FRANCE
Abstract: - This paper proposes an analytically tractable solution for minimizing input torques in decoupled
three-degrees-of-freedom spatial serial manipulators. The solution relies on the generation of motion using a
«bang-bang» profile. The problem is solved in two stages. Firstly, the dynamic decoupling of the manipulator is
accomplished through the redistribution of the moving masses and the relocation of one of the actuators. This
leads to the decoupling of the equations of motion for different degrees of mobility. It is worth mentioning that
this solution represents a symbiosis of two distinct approaches: the redistribution of link masses and the
relocation of one of the manipulator actuators. This innovative approach to dynamic decoupling has not been
previously proposed. At the second stage, the input torques of the actuators are reduced by generating motion
profiles for the manipulator's links using the «bang-bang» law. Thus, thanks to the developed methodology, it
becomes possible to reduce the energy consumption of high-speed manipulators by choosing the optimal
planned motion of their links. To evaluate the effectiveness of this approach, numerical simulations are carried
out using the ADAMS software. A comparative analysis of the trajectories generated by the fifth-order
polynomial profile, widely used in industrial robots, and the «bang-bang» profile has been performed. The
simulation results show act, the use of the «bang-bang» profile allows one to reduce the maximum values of
the input torques. The developed technique allows designers to create high-speed manipulators featuring
decoupled dynamics and diminished energy consumption.
Key-Words: - Manipulator, dynamics, decoupling, torque, «bang-bang» profile, optimal generation of motions
Received: February 24, 2023. Revised: July 21, 2023. Accepted: August 26, 2023. Published: September 14, 2023.
1 Introduction
It is known that the manipulator dynamics are
highly coupled and nonlinear. This complexity
arises from various factors such as changing inertia,
interactions between different joints, and nonlinear
forces like Coriolis and centrifugal forces. The goal
of dynamic decoupling in manipulators is to achieve
conditions that allow for decoupled and linear
dynamic equations. This simplifies optimal control
and energy accumulation in manipulators.
Let us consider the classification the methods for
dynamic decoupling of manipulators. To achieve a
clearer classification, it is convenient to separate
them into two levels. Firstly, let us review two
different approaches that have been developed for
creating dynamically decoupled manipulators: A) by
optimal mechanical design and B) by improved
control, [1]. There are three main methodologies
developed for dynamic decoupling of manipulators
via mechanical transformation:
A1) via mass redistribution;
A2) via actuator relocation;
A3) via addition of auxiliary links.
In order to eliminate the coupling and nonlinear
torques via mass redistribution (subgroup A1), the
inertia matrix must be diagonal and made invariant
for all statically balanced arm configurations. These
properties were described in detail for various
structural solutions in, [2], [3]. Let us explore the
essential properties of mass distribution required to
achieve this objective.
Let and ,, be the joint displacement and
torque of the i-th joint, respectively, then the
equation of motion of the manipulator is given by
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where, is the i-j element of the manipulator
inertia matrix, and is the torque due to gravity.
The first term in the equation represents the inertia
torque resulting from the acceleration of the i-th
joint, while the second term accounts for the
interactive inertia torque arising from the
accelerations of the other joints. The interactive
inertia torque is directly proportional to the
acceleration. The third term captures the nonlinear
velocity torques generated by Coriolis and
centrifugal effects. These nonlinear velocity torques
emerge due to the dependence of the inertia matrix
on the arm configuration. However, if the inertia
matrix becomes diagonal for any arm configuration,
the second term in the aforementioned equation
disappears, eliminating the presence of interactive
torques. In this case, the manipulator's inertia matrix
is referred to as a decoupled inertia matrix. The
significance of the decoupled inertia matrix lies in
the fact that the control system can be treated as a
collection of single-input, single-output subsystems
associated with individual joint motions. This
decoupling of the inertia matrix enables a simplified
control approach, where each joint can be controlled
independently without interference from other
joints. It facilitates the design and implementation of
control strategies for manipula-tors, treating them as
separate entities rather than a complex
interconnected system.
The equation of motion under these conditions
reduces to
where, the second term represents the nonlinear
velocity torques resulting from the spatial
dependency of the diagonal elements of the inertia
matrix. Note that the number of terms involved in
this equation is much smaller than the number of
original nonlinear velocity torques, because all the
off diagonal elements are zero for . This
reduces the computational complexity of the
nonlinear torques.
Another important form of the inertia matrix that
simplifies the dynamics is a configuration-invariant
form. In this case, the inertia matrix remains
constant regardless of the arm configuration. This
means that it is not dependent on joint
displacements, resulting in the elimination of the
third term in the first equation mentioned. As a
result, the equation of motion reduces to a
simplified form
Note that the coefficients and are
constant for all arm configurations. Thus, the
equation is linear except the last term, that is, the
gravity torque. The inertia matrix in this form is
referred to as an invariant inertia matrix. The
significance of this form is that linear control
schemes can be adopted, which are much simpler
and easier to implement.
When the inertia matrix is both decoupled and
configuration invariant, the equation of motion
reduces to:
The system is completely decoupled and
linearized, except the gravity term. Thus, we can
treat the system as single-input, single-output
systems with constant parameters.
The linearization and dynamic decoupling of 3
degrees of freedom serial manipulators via mass
redistribution have been considered, [2]. In this
study, all of the arm constructions that yielded the
decoupled inertia matrices were identified. The
proposed approach is exclusively applied to serial
manipulators with non-parallel joint axes. In the
case of parallel axes, such an approach allows
linearization of the dynamic equations but not their
dynamic decoupling, [3].
It should be noted that in the case of planar serial
manipulators, the mentioned method cannot be used.
Therefore, in serial manipulators with an open
kinematic chain structure, the inertia matrix cannot
be decoupled unless the joint axes are orthogonal to
each other.
Now, let us consider the methods of the sub-
group A2. A commonly used arrangement for
actuating robot manipulators with motorized joints
involves directly attaching motors to the joints. This
design does not involve any transmission elements
between the actuators and the joints. However, in
certain cases, the dynamic decoupling follows from
the kinematic decoupling of motion when the
rotation of any link is due to only one actuator. It is
obvious that it must be accompanied by an optimal
choose the mass properties of certain links. It is
evident that this must be complemented by an
optimal choice of the mass properties of certain
links. The optimal selection of mass properties
involves carefully determining the distribution of
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mass along the manipulator's links. By appropriately
choosing the mass distribution, it is possible to
achieve improved dynamics and reduce the coupling
effects between the joints. This optimization process
aims to minimize unwanted interactions between the
joints, resulting in more efficient and accurate
manipulator performance.
a)
b)
Fig. 1: Dynamic decoupling: via actuator relocation
(a) and via addition of auxiliary links (b)
Figure 1a shows such an example, where the
actuators are installed at the base of the manipulator
and a transmission mechanism is used to rotate the
second link. In this case, ,
with and . In the case of
actuating of both joints: is the joint displacement
of the first link relative to the base; is the joint
displacement of the second link relative to the first
link.
A detailed description of this solution and
different structural architectures can be found in,
[4], [5], [6], [7], [8], [9], [10], [11].
The linearization of the dynamic equations and
their decoupling by adding auxiliary links (subgroup
A3) has also been developed, [12], [13], [14], [15].
In this case, the inclusion of complementary links
enables the optimal redistribution of kinetic and
potential energies, resulting in the linearization and
decoupling of the dynamic equations. The process
of determining the parameters of these additional
links is based on eliminating the coefficients of
nonlinear terms present in the manipulator's kinetic
and potential energy equations. By carefully
selecting and configuring these complementary
links, it becomes possible to effectively redistribute
the energy within the manipulator system. This
redistribution helps to mitigate the nonlinear effects
that arise from the interaction between different
joints and links. Through a systematic approach, the
parameters of the added links can be determined in a
way that cancels out or minimizes the coefficients
associated with the nonlinear terms in the energy
equations. The purpose is to obtain linearized and
decoupled dynamic equations, simplifying control
and improving performance of the manipulator. By
eliminating the nonlinear terms and their
coefficients, the dynamics of the system become
more predictable and manageable, facilitating the
design of efficient control strategies and enhancing
the overall functionality of the manipulator. Figure
1b shows such an example, whereby opposite
rotation of gears 5 and 6, as well as by an optimal
redistribution of mass the dynamic decoupling of
the 2 degrees of freedom serial manipulator is
achieved.
The design concept of manipulators with
adjustable links has been thoroughly investigated in,
[15]. This study introduces a novel approach for
dynamically decoupling serial planar manipulators
by combining mechanical and control solutions. The
key idea is to achieve opposite motions of the
manipulator links (Figure 2) along with an optimal
redistribution of masses. This combination allows
for the cancellation of coefficients associated with
nonlinear terms in the manipulator's kinetic and
potential energy equations. After achieving full
linearization and decoupling of the manipulator
through this methodology, a linear control technique
can be employed. Furthermore, the variation in
payload is accounted for by incorporating forward
compensation into the controller. To ensure stability
of the linearized and decoupled manipulator, a full
state feedback control strategy is implemented.
The proposed design concept and control
approach offer several advantages. By achieving
dynamic decoupling, the manipulator's behavior
becomes more predictable and easier to control. The
linearized model simplifies control algorithms and
allows for the incorporation of compensation
techniques to address changing payloads.
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Stabilization is ensured through the implementation
of a full state feedback loop.
Fig. 2: Dynamic decoupling with adjustable links
This study presents an integrated solution that
combines mechanical design considerations with
control strategies, resulting in a linearized,
decoupled, and stable manipulator system. The
results provide insights into the potential benefits
and feasibility of this approach for enhancing the
performance and control of planar serial
manipulators.
Fig. 3: Two degrees of freedom planar serial
manipulator with added two-link group
In, [16], the dynamic decoupling of the manipu-
lator is achieved by introducing a two-link group,
forming a Scott-Russell mechanism, to the initial
structure. Figure 3 illustrates the robot arm
configuration, which consists of two principal links,
AB and BP, along with a sub-group comprising
links BC and CD. The manipulator's movements are
planar motions perpendicular to the vertical plane
and, therefore, not subject to gravitational forces in
this particular study. The slider D is free to slide
along the link AB and is connected to the link CD
through a revolute joint D. This arrangement forms
a Scott-Russell mechanism when the added sub-
group with links BC, CD, and the slider interacts
with the link BP of the original structure.
The Scott-Russell mechanism, [17], is known for
generating theoretically linear motion by employing
a linkage form with three portions of equal-length
links and a rolling or sliding connection. In this
paper, another property of the mechanism is
utilized: it generates rotations of links with identical
angular accelerations, meaning that the angular
accelerations of links BC and CD are similar. The
combination of opposite motion of links within the
Scott-Russell mechanism and optimal redistribution
of masses enables the cancellation of coefficients
associated with nonlinear terms in the manipulator's
kinetic and potential energy equations.
Subsequently, through optimal control design,
dynamic decoupling is achieved, even in the
presence of changing payload.
This process becomes relatively straightforward
because the modified structure of the manipulator
with the two-link group has effectively canceled out
coupling and nonlinearity. By utilizing the unique
properties of the Scott-Russell mechanism and
integrating it into the manipulator structure, this
study successfully achieves dynamic decoupling and
addresses the impact of changing payloads. The
cancellation of coupling and nonlinearity within the
modified structure paves the way for easier control
and enhanced manipulator performance.
In study, [18], an analysis is presented on the
tolerance capabilities of different decoupled models
of manipulators. Two indices are proposed to
quantify the positioning accuracy of the
manipulator: the angular error of the actuators and
the position error of the end-effector. To analyze the
influence of each variable on the positioning
accuracy, fixed parametric errors are introduced.
The analysis reveals that the variation in length
variables has a higher impact on positioning
accuracy compared to other variables. The mass
parameters are identified as the second most
influential factors, while the inertia parameters have
the least influence on positioning accuracy.
Reducing or canceling coupling and nonlinearity
in manipulators is often essential, as
demonstrated in, [18], [19], [20], [21], [22], [23].
Hence, in this paper, alongside dynamic
decoupling, the minimization of input torques is
considered.
Torque minimization of dynamically
decoupled spatial serial manipulators via
optimal motion generation of the 2R spatial
serial manipulator has been discussed in, [24].
The problem to be solved was relatively simple,
since the dynamic decoupling
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of such a manipulator can be carried out by optimal
redistribution of moving masses.
The central theme of the present study focuses on
torque minimization in dynamically decoupled
spatial serial manipulators through optimal motion
generation, specifically targeting the 3R spatial
serial manipulator. This problem is more complex
compared to the previous case, [24]. The present
research contributes to enhancing fast manipulator
design, allowing for to reduction of the input
torques in manipulators with dynamic decoupling.
The next two sections deal with the dynamics of
the 3R spatial serial manipulator and the decoupling
of its motion equations by mass redistribution and
actuator relocating. The combination of these two
approaches allows for the elimination of dynamic
coupling and non-linearity in 3R spatial
manipulators, which improves their performance
and control.
2 Dynamics of the Manipulator
The manipulator under consideration consists of
three links (Figure 4a): orthogonal links 1 and 2
with rotating angles , and link 3 with rotating
angle, which is parallel to link 2.
a)
b)
Fig. 4: Spatial serial manipulator: a) initial structure;
b) with relocated actuator
We will distinguish the vectors
0
1
θ
,
0
2
θ
and
0
3
θ
relative angular velocities with
00
11
θ = dθ dt
,
22
0
θ = dθ dt
,
33
00
θ = dθ dt
and the vectors
1
θ
,
2
θ
and
3
θ
absolute angular velocities with
0
11
θθ
,
00
2 1 2
θ θ θ
and
0 0 0
3 1 2 3
θ θ θ θ
.
According to Lagrangian dynamics, the
equations of motion can be written as:
where, are the torques; are the generalized
coordinates; is the Lagrangian; is the
kinetic energy and is the potential energy.
The kinetic energy of the manipulator consists
of the sum of the kinetic energies of the links:
1 2 3
K K K K
. Thus,
where, is the axial moment of inertia of link 1;
are the axial moments of inertia of link 2
relative to corresponding coordinate axes of the
system associated with link 2; are the
components of the angular velocity about the
same axes, i.e.
is the
mass of link 2; is the linear velocity of the center
of mass of link 2; are the axial moments
of inertia of link 3 relative to the corresponding
coordinate axes of the system associated with link 3;
are the components of the angular
velocity about the same axes, i.e. ; is the
mass of link 3; is the linear velocity of the center
of mass of link 3. Please note that the center of mass
of link 1 is located on the rotating axis and the
kinetic energy due to the linear velocity is canceled.
With regard to potential energy, we have:
with
10P
;
;
where, is
distance of the center of mass of link 1 from the
joint axis ; is the length of link 2;
is the distance of the center of mass of link 2
from the joint axis ; is the distance of
the center of mass of link 1 from the joint axis.
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Indeed, under the given conditions, it is evident
that the dynamics of the manipulator are coupled
and nonlinear.
3 Dynamics Decoupling of the
Manipulator
Let us carry out the dynamic decoupling of the 3R
spatial manipulator by combining two distinct
approaches: mass redistribution (subgroup A1) and
actuator relocation (subgroup A2).
For this purpose, firstly, let us modify the design
of the manipulator by placing the third actuator on
the axes and coupling it with link 3 by a
transmission. This transmission can be a
parallelogram, a belt transmission, a gear
transmission or other type of motion generation.
Figure 4b shows the modified design of the
manipulator, featuring the delocalization of the third
actuator and the integration of the transmission
mechanism. By combining the mass redistribution
and actuator relocation techniques, it is possible to
achieve dynamic decoupling of the 3R spatial
manipulator.
Under the provided conditions, including the
delocalization of the third actuator, the kinetic
energy of the manipulator can be expressed as
follows:
where (i = 1, 2, and 3) is the generalized angles
defined as (Figure 5):
;
;
.
The components of the angles relative to the
corresponding coordinate axes of the system are
defined as follows: , ;
,
and
;
,
and
; , and are the actuator
velocities.
Fig. 5: Spatial serial manipulator with the absolute
and relative velocities
Now, let us consider that the manipulator is
statically balanced, i.e. . It can be reached
if and
, considering the
location of the center of mass outside .
Then consider that the mass distribution of the
links is such that the following condition is ensured:
We get
Consequently, the derived equations are linear
and decoupled, indicating that the motion of each
link can be treated independently without affecting
the others. Based on these equations, it can be
deduced that by minimizing the maximum values of
the angular accelerations, the torques exerted on the
manipulator can also be minimized. To accomplish
this, the second step involves the generation of
motion profiles for the links. In the context of
motion generation, careful consideration should be
given to the design and implementation of
trajectories for the manipulator's links. By
optimizing the motion profiles, it becomes possible
to achieve smoother and more efficient movements,
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ultimately resulting in reduced torque requirements.
The goal of this second step is to develop an optimal
motion generation strategy that not only ensures
precise control over the manipulator's movements
but also minimizes the torques exerted on the
system. By effectively managing the motion
profiles, the overall performance of the manipulator
can be significantly improved, leading to enhanced
efficiency, accuracy, and longevity of the system.
4 Motion Generation via «Bang-
Bang» Profile
The path generation law, which provides the
minimum of the maximum value of the acceleration
is the «bang-bang» profile (Figure 6):
It is obvious that the maximal values of the
angular accelerations change following the motion
profile, [25]: for quantic polynomial profile
and for «bang-bang»
profile. This simple
comparison shows the difference between the
maximum acceleration values of these two laws.
(a) «Bang-bang» law (b) Trapezoidal law
Fig. 6: «Bang-bang» and trapezoidal motion profiles
However, it is important to note that the «bang-
bang» profile (Figure 6a) presented is based on
theoretical considerations. In practical applications,
actuators are unable to generate discontinuous efforts.
Therefore, it becomes necessary to modify this
motion profile by employing a trapezoidal profile
(Figure 6b). As studies have shown in, [26], it
appears that for given actuator parameters, the
minimizations obtained in the cases of the «bang-
bang» and trapezoidal profiles are very close (less
than 1 %).
Indeed, the application of the «bang-bang» law
in motion generation theoretically leads to a
reduction of approximately 30% in the maximum
value of the angular acceleration. Therefore, to
minimize the maximum values of the angular
accelerations and, consequently, the torques exerted
on the rotating links 1, 2, and 3, it is recommended
to employ the «bang-bang» profile. By reducing the
angular accelerations of the manipulator's links, a
corresponding decrease in the torques can be
achieved.
5 Illustrative Example with CAD
Simulation Results
To create a CAD model and carry out simulations
via ADAMS software, a decoupled 3R spatial serial
manipulator with;;;
;;
; was used. The
initial and final values of the rotating angles ,
and are the following:
, and . Two types
of motions for are simulated: 1) with fifth
order polynomial profile wildly used in industrial
robots; 2) «bang-bang» profile.
According to the simulation results depicted in
Figure 7, Figure 8 and Figure 9, it is apparent that
there is a significant reduction in torques for the two
studied cases.
Fig. 7: Variations of torque (Nm) for two studied
cases
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Fig. 8: Variations of torque (Nm) for two studied
cases
Fig. 9: Variations of torque (Nm) for two studied
cases
In the first case, where the motion is generated
using the fifth order polynomial profile, and in the
second case, where the motion is generated using
the «bang-bang» profile, both cases exhibit a
reduction in torques of up to 30.62%. These findings
confirm the effectiveness of utilizing the «bang-
bang» profile for motion generation in minimizing
the torques exerted on the manipulator's links.
Overall, the simulation results provide
quantitative evidence of the benefits of utilizing the
«bang-bang» profile in terms of torque reduction,
validating the effectiveness of the proposed
approach in improving the manipulator's
performance.
6 Error-Sensitivity Analysis of the
Manipulator
Let us now carry out Sobol sensitivity analysis, [27],
to reveal the sensitivity of errors of the painmeters
of the manipulator. The purpose is to identify
parameters’ errors more affe the input torques.
According to the Sobol method, the first-order
sensitivity index is stated as follows:
and total order index is given as:
where, measures the contribution of the parameter
to the total variance of the response. The total
order index represents the total contribution
(including interactions) of a parameter to the total
variance of the responTor to reduce the
computational burden, the generally used Monte
Carlo estimators are introduced here, [28]:
where, A and B are the matrixes of all the
parameters generated by the Sobol sequence, which
is a uniform sampling method. AB is the combination
of the columns of matrix A and B. The accuracy of
the Monte Carlo estimators depends on the number
of points in every interval of parameters (more
points more accurate). As mentioned above, in this
case, the sensitivity of parameters to the torque 1, 2
and 3 is studied. The Sobol sequence is applied for
the uniform sampling in every interval of
parameters, which has ±5% error, wherein 8000
points are distributed in every interval of parameters
for the sampling. The simulation results are
presented in Table 1.
From the obtained results, it can be seen that
and are the most influencing factors to the input
torques 1 and 3 but for the torque 2, the parameters
2
l
and
3
m
are actually the most influencing factors.
Thus, the most influential parameters
,,,
must be identified more correctly in the control
system as errors in these parameters influence more
on the input torques.
7 Discussion
The application of the «bang-bang» law is widely
known to reduce the maximum value of
acceleration. This property has been successfully
utilized in previous studies, such as, [28], [29], [30]
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to mitigate the impact of inertia forces on the
vibration of the manipulator's base. However,
applying this approach to reduce input
torque presents a more challenging task due to the
dynamic coupling between the degrees of freedom
within the manipulator. The key to solving this
problem lies in the dynamic decoupling of the
manipulator. By achieving dynamic decoupling,
the input torques can be linearized and dependent
on only one input parameter. By utilizing this
approach, a simpler control system can be
implemented, as the input torque is less reliant
on complex interactions between different
degrees of freedom. The current study focuses on
the investigation of the 3R serial spatial
manipulators. However, the proposed
approach can potentially be extended to other
manipulators, provided that their dynamic
decoupling is achievable. Using the outlined
method, a decrease in input torques has been
successfully attained. This was illustrated through
CAD simulations.
By combining dynamic decoupling with the
application of the «bang-bang» law, this study aims
to simplify the manipulator's control and reduce
input torques, thereby enhancing the overall
efficiency of the manipulation system.
8 Conclusion
This paper considers the problem of reducing the
torques in robot manipulators with decoupled
dynamics. The 3R spatial serial manipulator is
considered. The main objectives and findings of the
study are summa-rized as follows: 1) Dynamic
decoupling of the manipulator: the study
successfully achieves dynamic decoupling by
implementing techniques such as mass
redistribution, centre of mass arrangement, and
actuator relocalization. These modifications result in
decoupled and linear dynamic equations for the
manipulator. 2) Proportional relationship between
input torques and angular accelerations: It is
demonstrated that in the dynamically decoupled
manipulator, the input torques are directly
proportional to the input angular accelerations.
This relationship allows for simplified control
and improved manageability of the manipulator. 3)
Application of the «bang-bang» profile for motion
generation: The proposed approach suggests using
the «bang-bang» profile for motion generation,
which effectively reduces the maximal values of
input torques. This motion profile optimization
contributes to enhanced performance and torque
reduction in the manipulator. 4) Efficiency
demonstrated through numerical simulations: the
effectiveness of the proposed solution is illustrated
through numerical simulations conducted using
ADAMS software.
Table 1. Simulation results
These simulations validate the efficiency and
benefits of the suggested approach in reducing
torques and improving manipulator performance. 5)
Sensitivity analysis using the Sobol method: the
study also conducts a sensitivity analysis of design
parameters about the input torques using the Sobol
method. This analysis provides insights into the
impact of different design parameters on torque
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requirements and can inform future optimization
efforts.
Overall, this research makes a new contribution
to the advancement of torque reduction methods in
manipulators with decoupled dynamics. The
combination of dynamic decoupling and motion
generation optimization provides new results for
improving the design and control of manipulators.
The primary accomplishment of this work can be
summarized as follows: when designing 3R spatial
manipulators, it is highly recommended to
incorporate dynamic decoupling, as it greatly
simplifies the control process and justifies the
application of the «bang-bang» law of motion to
reduce input torques.
Acknowledgement:
This study was carried out as part of Yaodong Lu's
Ph.D. dissertation, with the support of the China
Scholarship Council [Grant Number:
202008070129].
References:
[1] Arakelian V. (Editor), Dynamic Decoupling of
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Yaodong Lu, Vigen Arakelian
E-ISSN: 2224-3429
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Yaodong Lu carried out the CAD simulations and
edited the initial version of the manuscript.
- Vigen Arakelian was responsible for conceptuali-
zation and final editing of the manuscript.
Sources of Funding for Research Presented in
a Scientific Article or Scientific Article Itself
The author Yaodong Lu's received a support of
the China Scholarship Council [Grant Number:
202008070129] for the Ph.D. dissertation.
Conflict of Interest
The authors have no conflicts of interest to
declare that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2023.18.15
Yaodong Lu, Vigen Arakelian
E-ISSN: 2224-3429
171
Volume 18, 2023
