A Numerical Integration Scheme For The Dynamic Motion Of Rigid Bodies Using T...IJRES Journal
The dynamics of rigid bodies have been studied extensively. However, a certain class of time-integration schemes were not consistent since they added vectors not belonging to the same tangent space (so3), of the Lie group (SO3) of the Special Orthogonal transformations in E3. The work of Cardona[1,2], and later Makinen[3,4], highlighted this fact using the rotation vector as the main parameter in their derivations. Some other programs in multibody dynamics, such as the work of Haug[5], rely on the Euler parameters, instead of the rotation vector, as the main variable in their formulations. For this class of programs, different time-integration schemes could be used .This paper discusses one such a scheme. As an example of application, the spinning top was used in this paper. For such a problem, the approximate change of the potential energy was found to be an upper bound to the change in the actual total energy during a time step.
Kinematics Analysis of Parallel Mechanism Based on Force Feedback DeviceIJRES Journal
Kinematic analysis of mechanism is the fundamental work of force feedback device research.The
composition of Delta mechanism based on Omega.7 force feedback device was illustrated in this paper.The
kinematic loop equations of Delta mechanism was established according to its geometric relationship,also the
inverse kinematics solution of Delta mechanism were obtained. And the numerical forward kinematics were
calculated by Newton iteration algorithm.Finally,The analysis of velocity and acceleration was carried out
through matrix operations.Kinematic analysis of Delta mechanism provides a theoretical basis for following
study.
Kane’s Method for Robotic Arm Dynamics: a Novel ApproachIOSR Journals
Abstract: This paper is the result of Analytical Research work in multi-body dynamics and desire to apply
Kane’s Method on the Robotic Dynamics. The Paper applies Kane’s method (originally called Lagrange form
of d’Alembert’s principle) for developing dynamical equations of motion and then prepare a solution scheme for
space Robotics arms. The implementation of this method on 2R Space Robotic Arm with Mat Lab Code is
presented in this research paper. It is realized that the limitations and difficulties that are aroused in arm
dynamics are eliminated with this novel Approach.
Key Words: Dynamics, Equation of Motion, Lagrangian, , Robotic arm, Space Robot,
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
A Numerical Integration Scheme For The Dynamic Motion Of Rigid Bodies Using T...IJRES Journal
The dynamics of rigid bodies have been studied extensively. However, a certain class of time-integration schemes were not consistent since they added vectors not belonging to the same tangent space (so3), of the Lie group (SO3) of the Special Orthogonal transformations in E3. The work of Cardona[1,2], and later Makinen[3,4], highlighted this fact using the rotation vector as the main parameter in their derivations. Some other programs in multibody dynamics, such as the work of Haug[5], rely on the Euler parameters, instead of the rotation vector, as the main variable in their formulations. For this class of programs, different time-integration schemes could be used .This paper discusses one such a scheme. As an example of application, the spinning top was used in this paper. For such a problem, the approximate change of the potential energy was found to be an upper bound to the change in the actual total energy during a time step.
Kinematics Analysis of Parallel Mechanism Based on Force Feedback DeviceIJRES Journal
Kinematic analysis of mechanism is the fundamental work of force feedback device research.The
composition of Delta mechanism based on Omega.7 force feedback device was illustrated in this paper.The
kinematic loop equations of Delta mechanism was established according to its geometric relationship,also the
inverse kinematics solution of Delta mechanism were obtained. And the numerical forward kinematics were
calculated by Newton iteration algorithm.Finally,The analysis of velocity and acceleration was carried out
through matrix operations.Kinematic analysis of Delta mechanism provides a theoretical basis for following
study.
Kane’s Method for Robotic Arm Dynamics: a Novel ApproachIOSR Journals
Abstract: This paper is the result of Analytical Research work in multi-body dynamics and desire to apply
Kane’s Method on the Robotic Dynamics. The Paper applies Kane’s method (originally called Lagrange form
of d’Alembert’s principle) for developing dynamical equations of motion and then prepare a solution scheme for
space Robotics arms. The implementation of this method on 2R Space Robotic Arm with Mat Lab Code is
presented in this research paper. It is realized that the limitations and difficulties that are aroused in arm
dynamics are eliminated with this novel Approach.
Key Words: Dynamics, Equation of Motion, Lagrangian, , Robotic arm, Space Robot,
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Comparison of a triple inverted pendulum stabilization using optimal control ...Mustefa Jibril
In this paper, modelling design and analysis of a triple inverted pendulum have been done using
Matlab/Script toolbox. Since a triple inverted pendulum is highly nonlinear, strongly unstable
without using feedback control system. In this paper an optimal control method means a linear
quadratic regulator and pole placement controllers are used to stabilize the triple inverted
pendulum upside. The impulse response simulation of the open loop system shows us that the
pendulum is unstable. The comparison of the closed loop impulse response simulation of the
pendulum with LQR and pole placement controllers results that both controllers have stabilized
the system but the pendulum with LQR controllers have a high overshoot with long settling time
than the pendulum with pole placement controller. Finally the comparison results prove that the
pendulum with pole placement controller improve the stability of the system.
The cubic root unscented kalman filter to estimate the position and orientat...IJECEIAES
In this paper we introduce a cubic root unscented kalman filter (CRUKF) compared to the unscented kalman filter (UKF) for calculating the covariance cubic matrix and covariance matrix within a sensor fusion algorithm to estimate the measurements of an omnidirectional mobile robot trajectory. We study the fusion of the data obtained by the position and orientation with a good precision to localize the robot in an external medium; we apply the techniques of kalman filter (KF) to the estimation of the trajectory. We suppose a movement of mobile robot on a plan in two dimensions. The sensor approach is based on the CRUKF and too on the standard UKF which are modified to handle measurements from the position and orientation. A real-time implementation is done on a three-wheeled omnidirectional mobile robot, using a dynamic model with trajectories. The algorithm is analyzed and validated with simulations.
Attitude Control of Quadrotor Using PD Plus Feedforward Controller on SO(3)IJECEIAES
This paper proposes a simple scheme of Proportional-Derivative (PD) plus Feedforward controller on SO(3) to control the attitude of a quadrotor. This controller only needs the measurement of angular velocity to calculate the exponential coordinates of the rotation matrix. With rotation matrix as an error variable of the controller, the simulation shows that the controller is able to drive the attitude of the quadrotor from hovering condition to desired attitude and from an attitude condition goes to the hovering condition, despite the system is disturbed. When the system is convergent, the rotation error matrix will be a 3 3 identity matrix.
ANALYSIS AND SLIDING CONTROLLER DESIGN FOR HYBRID SYNCHRONIZATION OF HYPERCHA...IJCSEA Journal
Hybrid synchronization of chaotic systems is a research problem with a goal to synchronize the states of master and slave chaotic systems in a hybrid manner, namely, their even states are completely synchronized (CS) and odd states are anti-synchronized. This paper deals with the research problem of hybrid synchronization of chaotic systems. First, a detailed analysis is made on the qualitative properties of hyperchaotic Yujun system (2010). Then sliding controller has been derived for the hybrid synchronization of identical hyperchaotic Yujun systems, which is based on a general hybrid result derived in this paper.MATLAB simulations have been shown in detail to illustrate the new results derived for the hybrid synchronization of hyperchaotic Yujun systems. The results are proved using Lyapunov stability theory.
Design and Simulation of Different Controllers for Stabilizing Inverted Pendu...IJERA Editor
The Inverted Pendulum system has been identified for implementing controllers as it is an inherently unstable system having nonlinear dynamics. The system has fewer control inputs than degrees of freedom which makes it fall under the class of under-actuated systems. It makes the control task more challenging making the inverted pendulum system a classical benchmark for the design, testing, evaluating and comparing. The inverted pendulum to be discussed in this paper is an inverted pendulum mounted on a motor driven cart. The aim is to stabilize the system such that the position of the cart on the track is controlled quickly and accurately so that the pendulum is always erected in its vertical position. In this paper the linearized model was obtained by Jacobian matrix method. The Matlab-Simulink models have been developed for simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using different control methods. The methods discussed in this paper are a double Proportional-Integral-Derivative (PID) control method, a modern Linear Quadratic Regulator (LQR) control method and a combination of PID and Linear Quadratic Regulator (LQR) control methods. The dynamic and steady state performance are investigated and compared for the above controllers.
Action Trajectory Reconstruction for Controlling of Vehicle Using SensorsIOSR Journals
Abstract: Inertial sensors, such as accelerometers and gyro-scopes, are rarely used by themselves to compute
velocity and position as each requires the integration of very noisy data. The variance and bias in the resulting
position and velocity estimates grow un-bounded in time. This paper proposes a solution to provide a de-biased
and de-noised estimation of position and velocity of moving vehicle actions from accelerometer measurements.
The method uses a continuous wavelet transform applied to the measurements recursively to provide reliable
action trajectory reconstruction. The results are presented from experiments performed with a MEMS accelerometer
and gyroscope.
Keywords: Action trajectory, continuous wavelet transform, inertial measurement unit.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Comparison of a triple inverted pendulum stabilization using optimal control ...Mustefa Jibril
In this paper, modelling design and analysis of a triple inverted pendulum have been done using
Matlab/Script toolbox. Since a triple inverted pendulum is highly nonlinear, strongly unstable
without using feedback control system. In this paper an optimal control method means a linear
quadratic regulator and pole placement controllers are used to stabilize the triple inverted
pendulum upside. The impulse response simulation of the open loop system shows us that the
pendulum is unstable. The comparison of the closed loop impulse response simulation of the
pendulum with LQR and pole placement controllers results that both controllers have stabilized
the system but the pendulum with LQR controllers have a high overshoot with long settling time
than the pendulum with pole placement controller. Finally the comparison results prove that the
pendulum with pole placement controller improve the stability of the system.
The cubic root unscented kalman filter to estimate the position and orientat...IJECEIAES
In this paper we introduce a cubic root unscented kalman filter (CRUKF) compared to the unscented kalman filter (UKF) for calculating the covariance cubic matrix and covariance matrix within a sensor fusion algorithm to estimate the measurements of an omnidirectional mobile robot trajectory. We study the fusion of the data obtained by the position and orientation with a good precision to localize the robot in an external medium; we apply the techniques of kalman filter (KF) to the estimation of the trajectory. We suppose a movement of mobile robot on a plan in two dimensions. The sensor approach is based on the CRUKF and too on the standard UKF which are modified to handle measurements from the position and orientation. A real-time implementation is done on a three-wheeled omnidirectional mobile robot, using a dynamic model with trajectories. The algorithm is analyzed and validated with simulations.
Attitude Control of Quadrotor Using PD Plus Feedforward Controller on SO(3)IJECEIAES
This paper proposes a simple scheme of Proportional-Derivative (PD) plus Feedforward controller on SO(3) to control the attitude of a quadrotor. This controller only needs the measurement of angular velocity to calculate the exponential coordinates of the rotation matrix. With rotation matrix as an error variable of the controller, the simulation shows that the controller is able to drive the attitude of the quadrotor from hovering condition to desired attitude and from an attitude condition goes to the hovering condition, despite the system is disturbed. When the system is convergent, the rotation error matrix will be a 3 3 identity matrix.
ANALYSIS AND SLIDING CONTROLLER DESIGN FOR HYBRID SYNCHRONIZATION OF HYPERCHA...IJCSEA Journal
Hybrid synchronization of chaotic systems is a research problem with a goal to synchronize the states of master and slave chaotic systems in a hybrid manner, namely, their even states are completely synchronized (CS) and odd states are anti-synchronized. This paper deals with the research problem of hybrid synchronization of chaotic systems. First, a detailed analysis is made on the qualitative properties of hyperchaotic Yujun system (2010). Then sliding controller has been derived for the hybrid synchronization of identical hyperchaotic Yujun systems, which is based on a general hybrid result derived in this paper.MATLAB simulations have been shown in detail to illustrate the new results derived for the hybrid synchronization of hyperchaotic Yujun systems. The results are proved using Lyapunov stability theory.
Design and Simulation of Different Controllers for Stabilizing Inverted Pendu...IJERA Editor
The Inverted Pendulum system has been identified for implementing controllers as it is an inherently unstable system having nonlinear dynamics. The system has fewer control inputs than degrees of freedom which makes it fall under the class of under-actuated systems. It makes the control task more challenging making the inverted pendulum system a classical benchmark for the design, testing, evaluating and comparing. The inverted pendulum to be discussed in this paper is an inverted pendulum mounted on a motor driven cart. The aim is to stabilize the system such that the position of the cart on the track is controlled quickly and accurately so that the pendulum is always erected in its vertical position. In this paper the linearized model was obtained by Jacobian matrix method. The Matlab-Simulink models have been developed for simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using different control methods. The methods discussed in this paper are a double Proportional-Integral-Derivative (PID) control method, a modern Linear Quadratic Regulator (LQR) control method and a combination of PID and Linear Quadratic Regulator (LQR) control methods. The dynamic and steady state performance are investigated and compared for the above controllers.
Action Trajectory Reconstruction for Controlling of Vehicle Using SensorsIOSR Journals
Abstract: Inertial sensors, such as accelerometers and gyro-scopes, are rarely used by themselves to compute
velocity and position as each requires the integration of very noisy data. The variance and bias in the resulting
position and velocity estimates grow un-bounded in time. This paper proposes a solution to provide a de-biased
and de-noised estimation of position and velocity of moving vehicle actions from accelerometer measurements.
The method uses a continuous wavelet transform applied to the measurements recursively to provide reliable
action trajectory reconstruction. The results are presented from experiments performed with a MEMS accelerometer
and gyroscope.
Keywords: Action trajectory, continuous wavelet transform, inertial measurement unit.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Comparative study of results obtained by analysis of structures using ANSYS, ...IOSR Journals
The analysis of complex structures like frames, trusses and beams is carried out using the Finite
Element Method (FEM) in software products like ANSYS and STAAD. The aim of this paper is to compare the
deformation results of simple and complex structures obtained using these products. The same structures are
also analyzed by a MATLAB program to provide a common reference for comparison. STAAD is used by civil
engineers to analyze structures like beams and columns while ANSYS is generally used by mechanical engineers
for structural analysis of machines, automobile roll cage, etc. Since both products employ the same fundamental
principle of FEM, there should be no difference in their results. Results however, prove contradictory to this for
complex structures. Since FEM is an approximate method, accuracy of the solutions cannot be a basis for their
comparison and hence, none of the varying results can be termed as better or worse. Their comparison may,
however, point to conservative results, significant digits and magnitude of difference so as to enable the analyst
to select the software best suited for the particular application of his or her structure.
This study deals with the active control of the dynamic response of a string with fixed ends and mass
loaded by a point mass. It has been controlled actively by means of a feed forward control method. A point mass of a
string is considered as a vibrating receiver which be forced to vibrate by a vibrating source being positioned on the
string. By analyzing the motion of a string, the equation of motion for a string was derived by using a method of
variation of parameters. To define the optimal conditions of a controller, the cost function, which denotes the dynamic
response at the point mass of a string was evaluated numerically. The possibility of reduction of a dynamic response
was found to depend on the location of a control force, the magnitude of a point mass and a forcing frequency
ABSTRACT : In this paper, the simulation of a double pendulum with numerical solutions are discussed. The double pendulums are arranged in such a way that in the static equilibrium, one of the pendulum takes the vertical position, while the second pendulum is in a horizontal position and rests on the pad. Characteristic positions and angular velocities of both pendulums, as well as their energies at each instant of time are presented. Obtained results proved to be in accordance with the motion of the real physical system. The differentiation of the double pendulum result in four first order equations mapping the movement of the system.
VIBRATION ANALYSIS OF AIRFOIL MODEL WITH NONLINEAR HEREDITARY DEFORMABLE SUSP...ijmech
In the present work, the vibratory behavior of an airfoil is discussed. The airfoil is considered as twodegree-of-freedom structure with hereditary deformable suspensions. The weak singular integrodifferential equation is numerically solved using numerical integration method. Finally, numerical results for the creep response and resonance behavior of the viscoelastic materials were analyzed. These results are obtained for the perfect elastic and viscoelastic suspensions with the nonlinearity feature. As demonstrated in the airfoil model, the equation of motion with hereditary and nonlinear terms successfully illustrate realistic vibratory characteristics of two-dimensional viscoelastic problems.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
ALL EYES ON RAFAH BUT WHY Explain more.pdf46adnanshahzad
All eyes on Rafah: But why?. The Rafah border crossing, a crucial point between Egypt and the Gaza Strip, often finds itself at the center of global attention. As we explore the significance of Rafah, we’ll uncover why all eyes are on Rafah and the complexities surrounding this pivotal region.
INTRODUCTION
What makes Rafah so significant that it captures global attention? The phrase ‘All eyes are on Rafah’ resonates not just with those in the region but with people worldwide who recognize its strategic, humanitarian, and political importance. In this guide, we will delve into the factors that make Rafah a focal point for international interest, examining its historical context, humanitarian challenges, and political dimensions.
In 2020, the Ministry of Home Affairs established a committee led by Prof. (Dr.) Ranbir Singh, former Vice Chancellor of National Law University (NLU), Delhi. This committee was tasked with reviewing the three codes of criminal law. The primary objective of the committee was to propose comprehensive reforms to the country’s criminal laws in a manner that is both principled and effective.
The committee’s focus was on ensuring the safety and security of individuals, communities, and the nation as a whole. Throughout its deliberations, the committee aimed to uphold constitutional values such as justice, dignity, and the intrinsic value of each individual. Their goal was to recommend amendments to the criminal laws that align with these values and priorities.
Subsequently, in February, the committee successfully submitted its recommendations regarding amendments to the criminal law. These recommendations are intended to serve as a foundation for enhancing the current legal framework, promoting safety and security, and upholding the constitutional principles of justice, dignity, and the inherent worth of every individual.
WINDING UP of COMPANY, Modes of DissolutionKHURRAMWALI
Winding up, also known as liquidation, refers to the legal and financial process of dissolving a company. It involves ceasing operations, selling assets, settling debts, and ultimately removing the company from the official business registry.
Here's a breakdown of the key aspects of winding up:
Reasons for Winding Up:
Insolvency: This is the most common reason, where the company cannot pay its debts. Creditors may initiate a compulsory winding up to recover their dues.
Voluntary Closure: The owners may decide to close the company due to reasons like reaching business goals, facing losses, or merging with another company.
Deadlock: If shareholders or directors cannot agree on how to run the company, a court may order a winding up.
Types of Winding Up:
Voluntary Winding Up: This is initiated by the company's shareholders through a resolution passed by a majority vote. There are two main types:
Members' Voluntary Winding Up: The company is solvent (has enough assets to pay off its debts) and shareholders will receive any remaining assets after debts are settled.
Creditors' Voluntary Winding Up: The company is insolvent and creditors will be prioritized in receiving payment from the sale of assets.
Compulsory Winding Up: This is initiated by a court order, typically at the request of creditors, government agencies, or even by the company itself if it's insolvent.
Process of Winding Up:
Appointment of Liquidator: A qualified professional is appointed to oversee the winding-up process. They are responsible for selling assets, paying off debts, and distributing any remaining funds.
Cease Trading: The company stops its regular business operations.
Notification of Creditors: Creditors are informed about the winding up and invited to submit their claims.
Sale of Assets: The company's assets are sold to generate cash to pay off creditors.
Payment of Debts: Creditors are paid according to a set order of priority, with secured creditors receiving payment before unsecured creditors.
Distribution to Shareholders: If there are any remaining funds after all debts are settled, they are distributed to shareholders according to their ownership stake.
Dissolution: Once all claims are settled and distributions made, the company is officially dissolved and removed from the business register.
Impact of Winding Up:
Employees: Employees will likely lose their jobs during the winding-up process.
Creditors: Creditors may not recover their debts in full, especially if the company is insolvent.
Shareholders: Shareholders may not receive any payout if the company's debts exceed its assets.
Winding up is a complex legal and financial process that can have significant consequences for all parties involved. It's important to seek professional legal and financial advice when considering winding up a company.
Responsibilities of the office bearers while registering multi-state cooperat...Finlaw Consultancy Pvt Ltd
Introduction-
The process of register multi-state cooperative society in India is governed by the Multi-State Co-operative Societies Act, 2002. This process requires the office bearers to undertake several crucial responsibilities to ensure compliance with legal and regulatory frameworks. The key office bearers typically include the President, Secretary, and Treasurer, along with other elected members of the managing committee. Their responsibilities encompass administrative, legal, and financial duties essential for the successful registration and operation of the society.
Car Accident Injury Do I Have a Case....Knowyourright
Every year, thousands of Minnesotans are injured in car accidents. These injuries can be severe – even life-changing. Under Minnesota law, you can pursue compensation through a personal injury lawsuit.
2. 318 J. WANG AND C.M. GOSSELIN
be in “static” equilibrium and the principle of virtual work is applied to derive the
input force or torque [16]. Since constraint forces and moments do not need to
be computed, this approach leads to faster computational algorithms, which is an
important advantage for the purposes of control of a manipulator.
2. Illustration of the Approach
The well known planar four-bar linkage is represented in Figure 1. It consists of
three movable links. The links of length l1, l2 and l3 are respectively the input link,
the coupler link and the output link and their orientation is described respectively
by angles θ, α and φ. If φ, ˙φ and ¨φ are known, the solution of the inverse dynamic
problem consists in finding the torque τ that is required to actuate the input link to
produce the specified trajectory. In this section, the dynamic analysis of this one-
degree-of-freedom mechanism using the approach of virtual work is performed in
order to illustrate the application of the approach.
2.1. COMPUTATION OF THE INERTIAL FORCES AND MOMENTS OF EACH
LINK
Following d’Alembert’s principle [17], the inertial force and moment on a body
are defined as the force and moment exerted at the center of mass of the body and
whose magnitude is given respectively by the mass of the link times the acceler-
ation of the center of mass and the inertial tensor of the link times the angular
acceleration of the body. These forces and moments are applied in a direction
opposite to the direction of the linear and angular accelerations. As it is well known,
introducing these virtual forces and moments in the system allows one to consider
it as if it were in “static” equilibrium. To this end, the acceleration of the center of
mass and the angular accelerations must first be computed.
2.1.1. Inverse Kinematics
From the geometry of the mechanism, one can write
l1 cos θ + l2 cos α = x0 + l3 cos φ, (1)
l1 sin θ + l2 sin α = y0 + l3 sin φ. (2)
Eliminating angle α from Equations (1) and (2), one obtains
A cos θ + B sin θ = C, (3)
where
A = x0 + l3 cos φ, B = y0 + l3 sin φ,
C = [(x0 + l3 cos φ)2
+ (y0 + l3 sin φ)2
+ l2
1 − l2
2]/(2l1).
3. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 319
The solution of Equation (3) then leads to
sin θ =
BC + KA
√
A2 + B2
,
cos θ =
AC − KB
√
A2 + B2
,
where K = ±1 is the branch index, and
= A2
+ B2
− C2
. (4)
Then, from Equations (1) and (2) one can obtain
sin α = (x0 + l3 cos φ − l1 cos θ)/l2,
cos α = (y0 + l3 sin φ − l1 sin θ)/l2.
2.1.2. Velocity Analysis
Differentiating Equations (1) and (2) with respect to time, one has
D˙t = e, (5)
where
D =
−l1 sin θ −l2 sin α
l1 cos θ l2 cos α
, e =
−l3
˙φ sin φ
l3
˙φ cos φ
, ˙t =
˙θ
˙α
. (6)
From Equation (5) one obtains
˙t = D−1
e. (7)
2.1.3. Acceleration Analysis
Differentiating Equation (5) with respect to time, one then obtains
D¨t = h, (8)
where
¨t =
¨θ
¨α
, h = ˙e − ˙Dt (9)
and the solution of Equation (8) leads to
¨t = D−1
h. (10)
4. 320 J. WANG AND C.M. GOSSELIN
r
C
C
l l
lC
m
m
1
1
2
2
1
3
3
3
2
0 0
(x , y )
x
y
O
3
A
B
O’
r
r
1
θ
2
m
α
φ
Figure 1. Geometric representation of the four-bar linkage.
Having obtained the angular velocity and acceleration of each link, one can
easily compute the acceleration of the centers of mass as
a1 = ¨θEr1 − ˙θ2
r1, (11)
a2 = ¨θEl1 − ˙θ2
l1 + ¨αEr2 − ˙α2
r2, (12)
a3 = ¨φEr3 − ˙φ2
r3, (13)
where ri and ai (i = 1, 2, 3) are respectively the position vector and the accel-
eration of the center of mass of the ith link, E is a rotation matrix written as
E =
0 −1
1 0
, l1 is the position vector from O to A, as represented in Figure 1,
and
r1 =
r1 cos θ
r1 sin θ
, r2 =
r2 cos α
r2 sin α
,
r3 =
r3 cos φ
r3 sin φ
, l1 =
l1 cos θ
l1 sin θ
. (14)
The orientation matrix of the ith (i = 1, 2, 3) moving link can be written as
Q1 =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
, Q2 =
cos α − sin α 0
sin α cos α 0
0 0 1
,
Q3 =
cos φ − sin φ 0
sin φ cos φ 0
0 0 1
. (15)
5. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 321
2 C
0 0(x , y )
x
y
O
3
τ
m1
C2
C1
m2
m3
w
w
w
1
3f1
2f
f3
Figure 2. The inertial force and moment and the gravity forces exerted on each link.
Therefore, the forces and moments acting at the center of mass of each mov-
ing link include the inertial force and moment and the gravity, as represented in
Figure 2. They can be written as
fi = −miai + wi, i = 1, 2, 3
mi = −I0i ˙ωi − ωi × (I0iωi), i = 1, 2, 3,
where
I0i = QiIiQT
i , ω1 =
0
0
˙θ
, ω2 =
0
0
˙α
, ω3 =
0
0
˙φ
, (16)
and where wi and Ii (i = 1, 2, 3) are respectively the weight and the inertia tensor
of ith link about its center of mass. Vectors mi (i = 1, 2, 3) are the inertial torques
acting at the center of mass of the links.
2.2. COMPUTATION OF THE VIRTUAL DISPLACEMENTS OF EACH LINK
Differentiating Equations (1) and (2), one obtains
Aδx = b, (17)
where
A =
l2 sin α −l3 sin φ
−l2 cos α l3 cos φ
, b =
−l1δθ sin θ
l1δθ cos θ
, δx =
δα
δφ
, (18)
6. 322 J. WANG AND C.M. GOSSELIN
where δα and δφ are the virtual angular displacements of the links of length l2 and
l3 caused by the virtual angular displacement δθ of the input link of length l1.
The virtual linear displacement of the center of mass of each link is then com-
puted as follows
δ1 = δθEr1,
δ2 = δθEl1 + δαEr2,
δ3 = δφEr3.
2.3. COMPUTATION OF THE GENERALIZED INPUT FORCES OR TORQUES
By application of the principle of the virtual work, one can finally obtain the gener-
alized input force, namely, the torque τ to actuate the four-bar linkage, i.e., letting
δθ = 1 one has
τ = m1 + m2δα + m3δφ +
3
i=1
(fiδi). (19)
The simple example presented above has illustrated the general application of the
principle of virtual work to the solution of inverse dynamic problems. Now, a
general formulation will be proposed for the application of this principle to the
dynamic analysis of parallel manipulators, which is the main purpose of this paper.
3. Spatial Six-Degree-of-Freedom Parallel Manipulator
The formulation proposed here will now be derived for a six-degree-of-freedom
Gough–Stewart platform. However, it should be kept in mind that this formulation
can be applied to any type of parallel mechanism.
The six-degree-of-freedom manipulator is represented in Figure 3. It consists
of a fixed base and a moving platform connected by six extensible legs. Each
extensible leg consists of two links and the two links are connected by a prismatic
joint. The moving platform is connected to the legs by spherical joints while the
lower end of the extensible legs is connected to the base through Hooke joints. By
varying the length of the extensible legs, the moving platform can be positioned
and oriented arbitrarily with respect to the base of the manipulator.
The base coordinate frame, designated as the Oxyz frame is attached to the base
of the platform with its Z-axis pointing vertically upward. Similarly, the moving
coordinate frame O x y z is attached to the moving platform. The orientation of
the moving frame with respect to the fixed frame is described by the rotation matrix
Q. The center of the ith Hooke joint is noted Oi while the center of the ith spherical
joint is noted Pi.
7. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 323
Figure 3. Spatial six-degree-of-freedom parallel mechanism with prismatic actuators
(Gough–Stewart platform).
If the coordinates of point Pi in the moving reference frame are noted (ai, bi, ci)
and if the coordinates of point Oi in the fixed frame are noted (xio, yio, zio), then
one has
pi =
xi
yi
zi
, pi =
ai
bi
ci
, for i = 1, . . . , 6, p =
x
y
z
, (20)
where pi is the position vector of point Pi expressed in the fixed coordinate frame –
and whose coordinates are defined as (xi, yi, zi) – pi is the position vector of point
Pi expressed in the moving coordinate frame, and p is the position vector of point
O expressed in the fixed frame. One can then write
pi = p + Qp i, i = 1, . . . , 6, (21)
where Q is the rotation matrix corresponding to the orientation of the platform of
the manipulator with respect to the base coordinate frame. This rotation matrix can
be written, for instance, as a function of three Euler angles. With the Euler angle
convention used in the present work, this matrix is written as
Q =
cφcθ cψ − sφsψ −cφcθ sψ − sφcψ cφsθ
sφcθ cψ + cφsψ −sφcθ sψ + cφcψ sφsθ
−sθ cψ sθ sψ cθ
, (22)
where sx denotes the sine of angle x while cx denotes the cosine of angle x.
8. 324 J. WANG AND C.M. GOSSELIN
α i
βi
i
O
Oi
Pi
rio
ρi
x
y
z
x
y
z
i
i
Figure 4. Vector ρi in spherical coordinates.
3.1. INVERSE KINEMATICS
The inverse kinematic problem is defined here as the determination of the position
and oriention of each link with respect to the base coordinate frame from the given
six independent Cartesian coordinates of the platform x, y, z, φ, θ and ψ. This
problem is rather straightforward and has been addressed by many authors.
One can write pi in terms of the ith leg’s coordinates which are represented in
Figure 4.
pi = ri0 + ρi, i = 1, . . . , 6 (23)
where ri0 and ρi are the vectors from O to Oi and from Oi to Pi respectively, i.e.,
ρi =
ρi cos αi sin βi
ρi sin αi sin βi
ρi cos βi
, ri0 =
xi0
yi0
zi0
, i = 1, . . . , 6. (24)
Since xi, yi and zi have been obtained from Equation (21), Equation (23) consti-
tutes a system of three equations in three unknowns ρi, αi and βi, which can be
easily solved as
ρi = (xi − xi0)2 + (yi − yi0)2 + (zi − zi0)2,
cos βi = (zi − zi0)/ρi,
sin βi = 1 − cos2 βi, (0 ≤ βi ≤ π),
cos αi = (xi − xi0)/(ρi sin βi),
sin αi = (yi − yi0)/(ρi sin βi).
9. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 325
Once ρi, αi and βi are known, the position and orientation of the two links of ith
leg are completely determined.
3.2. VELOCITY ANALYSIS
In this section, the linear and angular velocities of all moving links will be com-
puted from the given independent Cartesian velocities of the platform: ˙x, ˙y, ˙z, ωx,
ωy and ωz, where the latter three scalar quantities are the components of the angular
velocity vector of the platform, ω.
One can write ˙pi in terms of the angular velocity vector of the ith leg noted ωi,
i.e.,
˙pi = ˙ρir + ωi × ρi, i = 1, . . . , 6, (25)
where
˙ρir =
˙ρi cos αi sin βi
˙ρi sin αi sin βi
˙ρi cos βi
, ρi =
ρi cos αi sin βi
ρi sin αi sin βi
ρi cos βi
,
ωi =
− ˙βi sin αi
˙βi cos αi
˙αi
. (26)
Equation (25) can be expressed in matrix form as
Cipλip = ˙pi, i = 1, . . . , 6, (27)
where
Cip =
cos αi sin βi −ρi sin αi sin βi ρi cos αi cos βi
sin αi sin βi −ρi cos αi sin βi ρi sin αi cos βi
cos βi 0 −ρi sin βi
,
λip =
˙ρi
˙αi
˙βi
, (28)
and Equation (27) is easily solved for λip which leads to the determination of ˙ρi, ˙αi
and ˙βi. Once these quantities are known, the computation of the velocities of the
bodies of ith leg is straightforward.
3.3. ACCELERATION ANALYSIS
The linear and angular accelerations of each of the moving bodies will now be
determined from the given Cartesian accelerations of the platform, i.e., ¨x, ¨y, ¨z,
˙ωx, ˙ωy and ˙ωz, where the latter three scalar quantities are the components of the
vector of angular acceleration of the platform, ˙ω.
10. 326 J. WANG AND C.M. GOSSELIN
Differentiating Equation (25) with respect to time, one obtains
¨pi = ¨ρir + ˙ωi × ρi + ωi × (˙ρir + ωi × ρi), i = 1, . . . , 6, (29)
where
¨ρir =
¨ρi cos αi sin βi
¨ρi sin αi sin βi
¨ρi cos αi
, ˙ωi =
− ¨βi sin αi − ˙βi ˙αi cos αi
¨βi cos αi − ˙βi ˙αi sin αi
¨αi
. (30)
Equation (29) can be rewritten in matrix form as
Cip
˙λip = hi, i = 1, . . . , 6, (31)
where Cip is given in Equation (28) and where
˙λip = [ ¨ρi ¨αi
¨βi ]T
,
hi =
¨xi − 2 ˙ρi ˙βicαi cβi + 2 ˙ρi ˙αisαi sβi + 2ρi ˙αi ˙βisαi cβi + ρi ˙α2
i cαi sβi + ρi ˙β2
i cαi sβi
¨yi − 2 ˙ρi ˙βisαi cβi − 2 ˙ρi ˙αicαi sβi − 2ρi ˙αi ˙βicαi cβi + ρi ˙α2
i sαi sβi + ρi ˙β2
i sαi sβi
¨zi + cβi + 2 ˙ρi ˙βisβi + ρi ˙β2
i cβi
,
where sx denotes the sine of angle x while cx denotes the cosine of angle x.
Equation (31) is readily solved for ˙λip. Once the acceleration components are
known, the accelerations of the leg bodies are easily determined.
3.4. GENERALIZED INPUT FORCES
The generalized input forces will now be determined by first including the iner-
tial forces and moments in the system and considering it as if it were in “static”
equilibrium. The principle of virtual work will be applied.
3.4.1. Computation of the Force and Torque Acting on the Center of Mass of
Each Link
According to d’Alembert’s principle, the force acting on the center of mass of each
link consists of two parts: the inertial force and the gravity force. Similarly, the
moment acting on each link is the inertial moment.
In order to compute inertial forces, one must first determine the acceleration of
the center of mass of each link.
One can write
aiu = ¨pi + ˙ωi × Qiriu + ωi × (ωi × Qiriu), i = 1, . . . , 6, (32)
ail = ˙ωi × Qiril + ωi × (ωi × Qiril), i = 1, . . . , 6, (33)
where aiu and ail are respectively the acceleration of the center of mass of the upper
and lower links of the ith leg. They are expressed in the fixed coordinate system.
11. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 327
Vector riu and ril are the position vectors of the center of mass of the upper and
lower links of the ith leg in the local frame, and one would generally have
riu =
0
0
riu
, ril =
0
0
ril
. (34)
Moreover, matrix Qi is the rotation matrix from the fixed frame to the local frame
attached to the ith leg.
The acceleration of the center of mass of the platform can be computed as
follows
ap = ¨p + ˙ω × Qrp + ω × (ω × Qrp), (35)
where ap is the acceleration of the center of mass of the platform and rp is the po-
sition vector of the center of mass of the platform, expressed in the frame attached
to the platform.
Then, the force and moment acting on the center of mass of each link can be
directly computed as follows
fiu = −miuaiu + wiu, i = 1, . . . , 6, (36)
fil = −milail + wil, i = 1, . . . , 6, (37)
miu = −I0iu ˙ωi − ωi × (I0iuωi), i = 1, . . . , 6, (38)
mil = −I0il ˙ωi − ωi × (I0ilωi), i = 1, . . . , 6, (39)
where
I0iu = QiIiuQT
i , I0il = QiIilQT
i , (40)
and where fiu, miu, fil and mil denote the force acting on the upper link, the mo-
ment acting on the upper link, the force acting on the lower link and the moment
acting on the lower link of the ith leg. Iiu and Iil are the inertia tensor computed in
body reference frame of the upper and lower links of the ith leg respectively and
miu and mil are their masses. Vectors wiu and wil are the weight vectors, i.e.,
wiu =
0
0
−miug
, wil =
0
0
−milg
. (41)
Finally, one has
fp = −mpap + wp, (42)
mp = −I0p ˙ωp − ωp × (I0pωp), (43)
where
I0p = QIpQT
, (44)
12. 328 J. WANG AND C.M. GOSSELIN
6u
o
o
o
1
2
4
3
x’
z’
y’
3
2
o4
2
1
3
4
1
O’
platformP
P P
P
P
6
1u
2u
3u
4u1l
2l
3l
4l
p
p
1u
2u
2l
1l
4l
4u
3u
P
5u
5u
5l
5l
6
6
5
7
6u
6l 6l
x
y
z
m
m
m
m
m
m
m
m
m
m
m
m
3l
O
f
f
f
f
f
f
f
f
f
f
f
f
m
f
Figure 5. Inertial forces and moments on each of the links of the system.
and where fp and mp denote the force and moment acting on the platform and
where Ip, mp and wp are the inertia tensor computed in body reference frame, the
mass and the weight vector of the platform, respectively.
The inertial forces and moments acting on the center of mass of each link of the
manipulator are represented schematically in Figure 5.
3.4.2. Computation of the Virtual Displacements of the Links
From Equation (23) one has
ρi = pi − ri0, i = 1, . . . , 6. (45)
Taking the square of the norm of Equation (45) leads to
ρ2
i = (xi − xio)2
+ (yi − yio)2
+ (zi − zio)2
, i = 1, . . . , 6. (46)
Differentiating both sides of Equation (46) with respect to time, one then obtains
ρi ˙ρi = (xi − xio)˙xi + (yi − yio) ˙yi + (zi − zio)˙zi, i = 1, . . . , 6, (47)
where ˙xi, ˙yi and ˙zi are the three components of the velocity vector ˙pi of point Pi.
They can be computed using the following equation:
˙pi = ˙p + ω × Qpi, i = 1, . . . , 6, (48)
where ˙p and ω are respectively the velocity of point O and the angular velocity of
the platform, i.e.,
˙p =
˙x
˙y
˙z
, ω =
ωx
ωy
ωz
. (49)
13. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 329
Substituting Equation (48) into Equation (47) and writing it in matrix form, one
then obtains
B˙ρ = A˙x, (50)
where
˙ρ = [ ˙ρ1 ˙ρ2 ˙ρ3 ˙ρ4 ˙ρ5 ˙ρ6 ]T
,
˙x = [ ˙x ˙y ˙z ωx ωy ωz ]T
,
and where A and B are Jacobian matrices written as
B =
ρ1 0 0 0 0 0
0 ρ2 0 0 0 0
0 0 ρ3 0 0 0
0 0 0 ρ4 0 0
0 0 0 0 ρ5 0
0 0 0 0 0 ρ6
. (51)
and
A =
ρ1x ρ1y ρ1z (b1ρ1z − c1ρ1y) (c1ρ1x − a1ρ1z) (a1ρ1y − b1ρ1x)
ρ2x ρ2y ρ2z (b2ρ2z − c2ρ2y) (c2ρ2x − a2ρ2z) (a2ρ2y − b2ρ2x)
ρ3x ρ3y ρ3z (b3ρ3z − c3ρ3y) (c3ρ3x − a3ρ3z) (a3ρ3y − b3ρ3x)
ρ4x ρ4y ρ4z (b4ρ4z − c4ρ4y) (c4ρ4x − a4ρ4z) (a4ρ4y − b4ρ4x)
ρ5x ρ5y ρ5z (b5ρ5z − c5ρ5y) (c5ρ5x − a5ρ5z) (a5ρ5y − b5ρ5x)
ρ6x ρ6y ρ6z (b6ρ6z − c6ρ6y) (c6ρ6x − a6ρ6z) (a6ρ6y − b6ρ6x)
, (52)
where
ρix = xi − xio,
ρiy = yi − yio,
ρiz = zi − zio,
ai = q11ai + q12bi + q13ci,
bi = q21ai + q22bi + q23ci,
ci = q31ai + q32bi + q33ci,
i = 1, . . . , 6,
in which qij (i, j = 1, 2, 3) is the ith row and jth column element of matrix Q.
Let δα
j
i and δβ
j
i be the virtual angular displacements of the ith leg associated
with jth actuated joint corresponding to angles αi and βi (i, j = 1, . . . , 6), let δxj
,
δyj
, δzj
and δϕj
x, δϕj
y, δϕj
z be the virtual displacements of point O and the virtual
angular displacements of the platform associated with jth actuated joint and let
δρj
be the virtual displacement of the jth actuated joint.
14. 330 J. WANG AND C.M. GOSSELIN
Using Equation (50), one can compute the linear and angular virtual displace-
ments of the platform associated to the virtual displacement of the jth actuated
joint, i.e.,
δxj
= A−1
Bδρj
, j = 1, . . . , 6, (53)
where
δxj
= [ δxj
δyj
δzj
δϕ
j
x δϕ
j
y δϕ
j
z ] , j = 1, . . . , 6,
δρ1
= [ 1 0 0 0 0 0 ]T
,
δρ2
= [ 0 1 0 0 0 0 ]T
,
δρ3
= [ 0 0 1 0 0 0 ]T
,
δρ4
= [ 0 0 0 1 0 0 ]T
,
δρ5
= [ 0 0 0 0 1 0 ]T
,
δρ6
= [ 0 0 0 0 0 1 ]T
.
Having obtained the virtual displacements of the platform of the manipulator,
the virtual displacements of ith leg associated with the jth actuator can be easily
obtained from Equation (27), i.e.,
δλ
j
ip = C−1
ip δp
j
i , i, j = 1, . . . , 6, (54)
where
δλ
j
ip = [ δρj
δα
j
i δβ
j
i ] , i, j = 1, . . . , 6, (55)
and δp
j
i is the virtual displacement of point Pi associated with a unit virtual dis-
placement of the jth actuator. This virtual displacement can be computed from the
virtual displacements of the platform, i.e.,
δp
j
i = δpj
+ δϕj
× Qpi , i, j = 1, . . . , 6, (56)
where
δpj
=
δxj
δyj
δzj
, δϕj
=
δϕ
j
x
δϕ
j
y
δϕ
j
z
. (57)
Once the virtual linear and angular displacements of each link of the manipula-
tor are known, the virtual displacements of the center of mass of each link can be
computed as follows
δ
j
iu = δp
j
i + δϕ
j
i × Qiriu, i, j = 1, . . . , 6,
δ
j
il = δϕ
j
i × Qiril, i, j = 1, . . . , 6,
δj
p = δpj
+ δϕj
× Qrp, i = 1, . . . , 6,
15. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 331
where δ
j
iu, δ
j
il and δj
p are the virtual displacements of the center of mass of the links
of the ith leg associated with a virtual displacement of the jth actuator, and where
δϕ
j
i =
−δβ
j
i sin α
j
i
δβ
j
i cos α
j
i
δα
j
i
, i, j = 1, . . . , 6. (58)
3.4.3. Computation of Actuated Force/Torque
Using the principle of virtual work and letting τi (i = 1, . . . , 6) be the actuating
force of ith actuated joint, one then has
τj = fpδj
p + mpδϕj
+
6
i=1
[fiuδ
j
iu + filδ
j
il + (miu + mil)δϕ
j
i ], j = 1, . . . , 6
which thereby completes the procedure.
4. Example
In this section, an example is given in order to illustrate the results. The prescribed
trajectory is one in which the platform of the manipulator translates along a simple
sine trajectory while maintaining a fixed orientation.
The parameters used in this example are given as
xo1 = −2.120, yo1 = 1.374, xo2 = −2.380, yo2 = 1.224,
xo3 = −2.380, yo3 = −1.224, xo4 = −2.120, yo4 = −1.374,
xo5 = 0.0, yo5 = 0.15, xo6 = 0.0, yo6 = −0.15,
zoi = 0.0, (i = 1, . . . , 6),
a1 = 0.170, b1 = 0.595, c1 = −0.8, a2 = −0.6, b2 = 0.15,
c2 = −0.8, a3 = −0.6, b3 = −0.15, c3 = −0.8, a4 = 0.170,
b4 = −0.595, c4 = −0.8, a5 = 0.430, b5 = −0.845, c5 = −0.8,
a6 = 0.430, b6 = 0.445, c6 = −0.8,
l5 = 1.5, ρi max = 4.5, ρi min = 0.5, (i = 1, . . . , 6),
mp = 1.5, miu = mil = 0.1, riu = ril = 0.5, (i = 1, . . . , 6), rp = 0,
Iiu = Iil = I6 =
1/160 0 0
0 1/160 0
0 0 1/1600
, (i = 1, . . . , 6),
Ip =
0.08 0 0
0 0.08 0
0 0 0.08
,
16. 332 J. WANG AND C.M. GOSSELIN
çघNङ
!tघradङ
ç2
ç1
1 2 3 4 5 6
6
8
10
12
Figure 6. Input force at actuated joints 1 and 2.
çघNङ
!tघradङ
ç3
ç4
1 2 3 4 5 6
6.5
7.5
8
8.5
Figure 7. Input force at actuated joints 3 and 4.
where the lengths are given in meters, the masses in kilograms and the inertias in
kilograms meter square.
The specified trajectory can be expressed as
x = −1.5 + 0.2 sin ωt, y = 0.2 sin ωt, z = 1.0 + 0.2 sin ωt,
φ = 0, θ = 0, ψ = 0,
where
ω = 3.0, (0 ≤ ωt ≤ 2π).
The generalized input forces obtained at the six actuated joints are represented
in Figures 6–8.
This example has been verified by the classical Newton–Euler approach. The
two approaches lead to identical results and the approach based on the principle
of virtual work leads to a faster algorithm which is about 30% faster than the one
obtained using the Newton–Euler approach.
17. DYNAMIC ANALYSIS OF PARALLEL MANIPULATORS 333
çघNङ
!tघradङ
ç5
ç6
1 2 3 4 5 6
6
7
8
9
10
11
12
Figure 8. Input force at actuated joints 5 and 6.
5. Conclusion
A new approach for the dynamic analysis of parallel manipulators has been pro-
posed in this paper. This approach is based on the well known principle of virtual
work. The principle of virtual work has first been recalled and illustrated through
the dynamic analysis of the four-bar linkage. Then, the dynamic analysis of spa-
tial six-degree-of-freedom parallel manipulators with prismatic actuators has been
performed. The procedure described here leads to efficient algorithms and can be
applied to any type of parallel mechanism.
Acknowledgment
The authors would like to acknowledge the financial support of the Natural Sci-
ences and Engineering Research Council of Canada (NSERC) as well as the Fonds
pour la Formation de Chercheurs et l’Aide à la Recherche du Québec (FCAR).
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