Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating
Robots and Mechanisms
A thesis presented to
The faculty of
The Russ College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Master of Science
Xiao Xue
May 2013
© 2013 Xiao Xue. All Rights Reserved
2
This thesis titled
Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating
Robots and Mechanisms
by
XIAO XUE
has been approved for
the Department of Mechanical Engineering
and the Russ College of Engineering and Technology by
Robert L. Williams II
Professor of Mechanical Engineering
Dennis Irwin
Dean, Russ College of Engineering and Technology 3
ABSTRACT
XUE, XIAO, M.S., May 2013, Mechanical Engineering
Investigation into the Utility of the MSC ADAMS Dynamic Software for Simulating
Robots and Mechanisms (pp.143)
Director of Thesis: Robert L. Williams II
A Slider-Crank mechanism, a 4-bar mechanism, a 2R planar serial robot and a
Stewart-Gough parallel manipulator are modeled and simulated in MSC ADAMS/View.
Forward dynamics simulation is done on the Stewart-Gough parallel manipulator; Inverse dynamics simulation is done on Slider-Crank mechanism, 2R planar serial robot and 4- bar mechanism. In forward dynamics simulation, the forces are applied in prismatic joints and the Stewart-Gough parallel manipulator is actuated to perform 4 different expected motions. The motion of the platform is measured and compared with the results in
MATLAB. An example of inverse dynamics simulation is also performed on it. In the inverse dynamics simulation, the Slider-Crank mechanism and four-bar mechanism both run with constant input angular velocity and the actuating forces are measured and compared with the results in MATLAB. The 2R planar serial robot’s motions are defined with splines with driving torques measured and compared with Williams’s results in
MATLAB.
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ACKNOWLEDGEMENTS
Firstly I want to give my sincerest thanks to my advisor, R. L. Williams II. I will be thankful to him my whole life. He has been guiding and supporting me in my program.
He is always supportive and patient to my questions. He gave me constructive suggestions. Without him, I would not have finished my thesis.
I also want to thank my friend Yatin. He uses MSC ADAMS and I often discuss my problems with him. He is a very nice person and helps me a lot in the details of my robots.
I want to give thanks to Dr. Cotton, too. I work in his lab and he keeps the lab environment in a very positive and productive condition.
I would like to give sincere thanks to my girlfriend, Xiaowei, Zhu. She keeps me positive and patient during the whole thesis process.
At last, I want to thank my mum. Though she is in China, but I will always talk to her when I came to a problem. She always tries her best to help me both spiritually and financially. Without my mum, I will never finish this thesis.
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TABLE OF CONTENTS
Page
Abstract...... 3
Acknowledges ...... 4
List of Figures ...... 7
List of Tables ...... 12
Chapter 1: Introduction ...... 13
1.1 Statement of Purpose ...... 13
1.2 Background ...... 13
1.3 Literature Review...... 14
1.4 Thesis Organization ...... 25
Chapter 2: Project Information ...... 27
2.1 Planar Slider-Crank Mechanism ...... 27
2.2 Four-Bar Mechanism ...... 29
2.3 2R Planar Serial Robot...... 31
2.4 3D Stewart-Gough Parallel Manipulator ...... 32
2.5 Thesis Objectives ...... 35
Chapter 3 Kinematics ...... 37
3.1 Newton-Raphson Method ...... 37
3.2 Kinematic Analysis ...... 38
3.3 Slider-Crank Mechanism Kinematics ...... 39
3.4 Four-Bar Mechanism Kinematics ...... 42 6
3.5 2R Planar Serial Robot Kinematics ...... 46
3.6 Stewart-Gough Parallel Manipulator Kinematics ...... 49
Chapter 4 Dynamics ...... 56
4.1 Slider-Crank Mechanism Inverse Dynamics ...... 56
4.2 Four-Bar Mechanism Inverse Dynamics ...... 59
4.3 2R Planar Serial Robot Inverse Dynamics...... 62
4.4 Stewart-Gough Parallel Manipulator Forward Dynamics ...... 64
Chapter 5: Simulation Results ...... 68
5.1 Inverse Dynamics Simulation and Results for Slider-Crank Mechanism ...... 68
5.2 Inverse Dynamics Simulation and Results for Four-Bar Mechanism ...... 72
5.3 Inverse Dynamics Simulation and Results for 2R Planar Serial Robot ...... 78
5.4 Forward and Inverse Dynamics Simulation and Results for Stewart-Gough Parallel
Manipulator ...... 86
Chapter 6 Conclusions and Future Work ...... 106
6.1 Conclusions ...... 106
6.2 Future Work ...... 107
References ...... 108
Appendix A: 2R Planar Serial Robot Inverse Dynamics Input Rotational Tabular
Data…… ...... 112
Appendix B: Four Bar Mechanism Inverse Dynamics Splines ...... 117
Appendix C: Stewart-Gough Parallel Manipulator Forward Dynamics Splines...... 119
Appendix D: MSC ADAMS/View User Manual ...... 126 7
LIST OF FIGURES Page
Figure 1-1 Parallel Robot HEXA ...... 15
Figure 1-2 Surface Tracking Task ...... 15
Figure 1-3 Experimental Results of the Application of the Kinematics Control Algorithm to the CPR Prototype ...... 16
Figure 1-4 Mechanical System Aperture of the CPR ...... 17
Figure 1-5 AIM Frame with Attached Drill Press ...... 18
Figure 1-6 Schematic Detail of Experimental Apparatus Used to Measure the Accuracy of the AIM Frame in Free Space...... 19
Figure 1-7 SurgiScope in Action at the...... 20
Figure 1-8 ABB Flexible Automation's IRB 340 Flex-Picker ...... 20
Figure 1-9 NASA LaRC 8-axis 8R Spatial Serial Manipulator ...... 21
Figure 1-10 Schematic of the Suspension Mechanism ...... 24
Figure 1-11 ADAMS Model ...... 24
Figure 1-12 Frequency Response of the Seat Mass Center ...... 25
Figure 2-1 Slider-Crank Mechanism ...... 27
Figure 2-2 Four-Bar Mechanism ...... 29
Figure 2-3 2R Planar Serial Robot ...... 31
Figure 2-4 Stewart-Gough Parallel Manipulator ...... 33
Figure 2-5 Prismatic Joint ...... 34
Figure 2-6 Spherical Joint ...... 35
Figure 3-1 Slider-Crank Mechanism in MSC ADAMS ...... 39 8
Figure 3-2 Rotational Motion ...... 40
Figure 3-3 XA Position of Slider during Simulation ...... 41
Figure 3-4 XB Position of Slider during Simulation ...... 41
Figure 3-5 Four-Bar Mechanism in MSC ADAMS ...... 43
Figure 3-6 Fixed Joint ...... 43
Figure 3-7 XA in Four-Bar Mechanism Simulation ...... 45
Figure 3-8 XB in Four-Bar Mechanism Simulation ...... 45
Figure 3-9 2R Planar Serial Robot in MSC ADAMS ...... 47
Figure 3-10 YA of 2R Planar Serial Robot in Simulation ...... 48
Figure 3-11 Input Translational Displacement Numerical Data ...... 50
Figure 3-12 Stewart-Gough Parallel Manipulator in MSC ADAMS/View...... 51
Figure 3-13 Side View of Stewart-Gough Parallel Manipulator in MSC ADAMS/View 52
Figure 3-14 Ball Joint Position on Base ...... 53
Figure 3-15 Ball Joint Position on Platform ...... 54
Figure 3-16 Velocity in Z Direction of the Platform ...... 55
Figure 4-1 Simulation of Slider-Crank Mechanism, t=0.1s ...... 57
Figure 4-2 Link2 and Link 3 Align at t=0.0158s ...... 57
Figure 4-3 SForceX, SForceY vs. time ...... 58
Figure 4-4 Four-Bar Mechanism in Simulation in MSC ADAMS ...... 59
Figure 4-5 SForceX1 vs. time ...... 61
Figure 4-6 SForceX2 vs. time ...... 61
Figure 4-7 Shaking Force X in Four-Bar Mechanism Simulation ...... 62 9
Figure 4-8 2R Planar Serial Robot at the End of Simulation ...... 63
Figure 4-9 Driving Torques Measured in MSC ADAMS (with Gravity) ...... 64
Figure 4-10 the Stewart-Gough Platform in MSC ADAMS/View at t=0.4s of Forward
Dynamics Simulation ...... 65
Figure 4-11 Euler Angles of a Limb ...... 66
Figure 4-12 Free-Body Diagram of a Typical Limb ...... 67
Figure 5-1 Theta2 vs. time in MSC ADAMS ...... 68
Figure 5-2 Acceleration of Link 3(m/s2) vs. time(s) ...... 69
Figure 5-3 AG3Y (G), AG3X(R) (m/s2) vs. time(s) in MSC ADAMS ...... 69
Figure 5-4 Singularity Occur at t=0.2650s, Link2 and Link3 Fold ...... 70
Figure 5-5 Shaking Force Fs(N) vs. time(s) ...... 71
Figure 5-6 Fsx(r), Fsy(g)(N) vs. time(s) Plotted in MSC ADAMS ...... 71
Figure 5-7 θ3, θ4 and µ(deg) vs. θ2(deg) ...... 73
Figure 5-8 θ3, θ4 and µ(deg) vs. θ2(deg) in MSC ADAMS ...... 73
Figure 5-9 ω3, ω4(rad/s) vs.θ2(deg) ...... 74
Figure 5-10 ω3, ω4(rad/s) vs.θ2(deg) in ADAMS ...... 74
Figure 5-11 α3, α4 (rad/s2)vs.θ2(deg) ...... 75
Figure 5-12 α3, α4(rad/s2) vs.θ2 (deg)in ADAMS ...... 75
Figure 5-13 Measured FSX and FSY(N)vs.θ2 (deg) plotted in MATLAB up, Williams’s
Shaking Forces down ...... 77
Figure 5-14 X, Y phi vs. time(s) Measured in MATLAB ...... 79
Figure 5-15 x, y(m) phi(rad) vs. time(s) Measured in MSC ADAMS ...... 79 10
Figure 5-16 Joint Angles(deg) vs time(s) Measured in MATLAB...... 80
Figure 5-17 Joint Angles(deg) vs time(s) Measured in MSC ADAMS...... 80
Figure 5-18 Angular Velocity(rad/s) vs time(s) Measured in MATLAB ...... 81
Figure 5-19 Angular Velocity(rad/s) vs time(s) Measured in MSC ADAMS ...... 81
Figure 5-20 Angular Acceleration (rad/s2) vs time(s) Measured in MATLAB ...... 82
Figure 5-21 Angular Acceleration (rad/s2) vs time(s) Measured in MSC ADAMS ...... 82
Figure 5-22 Driving Torques(Nm) vs time(s) Measured in MATLAB (without Gravity) 84
Figure 5-23 Driving Torques (Nm) vs time(s) Measured in MSC ADAMS (Ignoring
Gravity) …………………………………………………………………………………84
Figure 5-24 Driving Torques (Nm) vs time(s)Measured in MATLAB (with Gravity) .... 85
Figure 5-25 Driving Torques (Nm) vs time(s) Measured in MSC ADAMS (with
Gravity)………………………………………………………………………………….85
Figure 5-26 Top view of Stewart-Gough Parallel Manipulator in MSC ADAMS/View . 87
Figure 5-27 Side view of Stewart-Gough Parallel Manipulator in MSC ADAMS/View 88
Figure 5-28 Actuating Forces(N) vs time(s) ...... 89
Figure 5-29 the Stewart-Gough Parallel Platform at t=0.4s ...... 90
Figure 5-30 CM Acceleration(m/s2) vs. time(s) in X Direction ...... 91
Figure 5-31 Actuating Forces(N) vs time(s) ...... 92
Figure 5-32 the Stewart-Gough Parallel Platform at t=0.4s ...... 93
Figure 5-33 CM Acceleration(m/s2) vs. time(s) in Z Direction ...... 94
Figure 5-34 Actuating Forces(N) vs. time(s) ...... 95
Figure 5-35 The Stewart-Gough Parallel Platform at t=0.4s ...... 96 11
Figure 5-36 CM Angular Acceleration(rad/s2) vs. time(s) around Y Axis ...... 97
Figure 5-37 Actuating Forces(N) vs. Time(s) ...... 98
Figure 5-38 the Stewart-Gough Parallel Platform at t=0.4s ...... 99
Figure 5-39 CM Angular Acceleration(rad/s2) in Z Direction vs. time(s) ...... 100
Figure 5-40 Inverse Dynamics Translational Motion Input ...... 101
Figure 5-41 Inverse Dynamics on Stewart-Gough Parallel manipulator, t=0s ...... 102
Figure 5-42 Inverse Dynamics on Stewart-Gough Parallel manipulator t=0.4s ...... 102
Figure 5-43 Platform Acceleration(m/s2) in Z Direction vs. time(s) ...... 103
Figure 5-44 Actuating Forces(N) vs time(s) ...... 104
Figure 5-45 Actuator Forces(N) vs. time(s) ...... 104
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LIST OF TABLES Page
Table 2-1 Slider-Crank Mechanism Parameters ...... 28
Table 2-2 Four-Bar Mechanism Parameters ...... 30
Table 2-3 2R Planar Serial Robot Parameters ...... 32
Table 2-4 3D Stewart-Gough Parallel Manipulator Parameters ...... 34
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CHAPTER 1: INTRODUCTION
1.1 Statement of Purpose
The purpose of this thesis is to investigate into the methods of simulation of robots and mechanisms. Dynamic parameters will be used from literature and comparison of the dynamic results will be made. Parallel Robot models will be built using MSC
ADAMS/View. Both forward and inverse pose dynamic simulations will be performed.
Validation will be performed after the simulation. Joints’ variables will be obtained from dynamic analysis.
The goal is to simulate the four-bar mechanism, 2R serial robot, Slider-Crank mechanism and Parallel Robot with designed motions and make dynamic analysis.
1.2 Background
Robot is defined as, “An electromechanical device with multiple degrees-of- freedom (dof) that is programmable to accomplish a variety of tasks” by Williams (2012).
The earliest robot on record dates back to 350 BC, and it was invented by the Greek
Archytas of Tarentum. Though it was only a simple flying mechanism propelled by steam, since then interest in robots have not stopped. The first industrial robot driven by motor was built in 1961 by Unimation company. The robot acted more precisely and quickly with the help of the motor. In 1961, the first computer-controlled mechanical hand—MH-1 was built by Heinrich Ernst in Massachusetts Institute of Technology
(MIT). Later in 1969, the robot arm was defined by “Stanford Arm” and created by a student named Victor Scheinman. In 1996, Honda released P3-the first humanoid robot
(http://robotics.megagiant.com/history.html). 14
A parallel robot has several “parallel linkages” connecting the base and the working end. Unlike the serial robot, a parallel robot has several connections to the end platform.
Kinematics is the subject to analyze the translational and rotational position, velocity, and acceleration of different mechanisms and solves the related problems in real life. Dynamics analysis involves the translational and rotational position, velocity and acceleration of the studied mechanism, as well as the center of mass acceleration of the rigid body. Kinematics is the study of motion without the regard of force, but dynamics considers the effect of force and moment, Williams (2012). Kinematics equations are necessary before the dynamics analysis.
1.3 Literature Review
1.3.1 Parallel Robot
The parallel robot HEXA developed by Intelligent Machines Laboratory in Japan has six closed –loop chains. The chains provide enough stiffness and accuracy due to the designed computer control system. The motor DM1015B is placed inside the top base. It has a maximum torque of 15[Nm] and a top rotational speed of 2.4[rps]
( http://www.space.mech.tohoku.ac.jp/research/hexa/hexa-e.html). Each motor is controlling a chain. The HEXA thus has faster movement and higher precision.
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Figure 1-1 Parallel Robot HEXA,
Source: http://www.space.mech.tohoku.ac.jp/research/hexa/hexa-e.html
In order to realize the user interface in 3D and make the parallel robot carry out some task instead of human force, the control system for HEXA is newly developed.
( http://www.space.mech.tohoku.ac.jp/research/hexa/hexa-e.html )
Figure 1-2 Surface Tracking Task
Source: http://www.space.mech.tohoku.ac.jp/research/hexa/hexa-e.html 16
The climbing parallel robot developed at Miguel Hernandez University of Elche,
Spain is derived from Stewart-Gough parallel platform. The Stewart-Gough parallel is widely used in industrial since its debut in 1962, such as tire test machines, aircraft simulators, micro hands and large spherical radio telescopes, M. Almonacid (2003).
Figure 1-3 shows a climbing parallel robot doing maintenance on a palm tree.
Figure 1-3 Experimental Results of the Application of the Kinematics Control
Algorithm to the CPR Prototype
Source: Almonacid (2003)
The climbing robot is controlled by human on ground in a tele-operated way.
Before maintenance, the robot will be assembled around the tree on the ground. Each leg is consisted of a prismatic joint, a spherical and a universal joint. It has a total of 6-DOF.
The end effector on top serves as the platform for loads and the transmission between the ground operator and the motor in robot. 17
The development of the Parallel Robot creates a revolution in the maintenance of the palm trees. Maintenance work becomes more efficient and safe.
Figure 1-4 Mechanical System Aperture of the CPR
Source: Almonacid (2003)
Parallel robot is also proposed to be used to carry out the Percutaneous Cochlear
Implantation surgery. The prototype robot was developed at Vanderbilt University by
Labadie. Before surgery the pre-positioning frame will be attached to patient and CT scan will be obtained. After the optimized surgery route is planned, the robot will be controlled to head to the targeted location on the patient. The robot will be fixed on the location after arriving and drill to the tissue, Kratchman (2011). Compared to the traditional cochlear implantation surgery, the parallel robot device is expected to achieve higher accuracy and efficiency during surgery. The parallel robot also achieved lower error than serial robot in this research. 18
Figure 1-5 AIM Frame with Attached Drill Press
Source: Kratchman (2011)
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Figure 1-6 Schematic Detail of Experimental Apparatus Used to Measure the
Accuracy of the AIM Frame in Free Space
Source: Kratchman (2011)
The Delta parallel robot was first proposed by Reymond Clavel in early 80’s.
Unlike other parallel robots, the Delta parallel robot has three translational and one rotational degree of freedom. The delta parallel robot has many advantages such as high speed, rigidity and accuracy. It is widely used in package and medical industry.
The three links around the end-effector are made in the shape of parallelogram and can provide translational movement. The rotational movement is realized by the link between the center of the base and the center of the end-effector. 20
Figure 1-7 SurgiScope in Action at the Surgical Robotics Lab
Humboldt-University at Berlin
Source: http://www.parallemic.org/Reviews/Review002.html
Figure 1-8 ABB Flexible Automation's IRB 340 Flex-Picker
Source: http://www.parallemic.org/Reviews/Review002.html 21
Newer generations of technologies make computer simulation very feasible and useful in both education and industrial designs. Faster CPUs and more powerful GPUs can create many complicated simulations that used to be impossible. Aided by MSC
ADAMS, engineers can simulate their designs and get the test results. This revolution makes designing and research much more efficient, thus saves a lot of money and energy
FU (2003).
1.3.2 Parallel robot and Serial Robot
Figure 1-9 NASA LaRC 8-axis 8R Spatial Serial Manipulator
Source: Williams (2012)
Serial robots connect links one by one with revolute or prismatic joints. These rigid links transmit the control from actuator to the end effector. In the image above a spatial serial robot arm is shown. The end effector is a mechanical hand and the actuator is installed inside the arm. 22
Serial manipulators have some characters due to their structures. The advantages are they have fewer singularities and take up less room compared with parallel manipulators. But their disadvantages are also obvious. First, they have lower rigidity.
They have only one link between the actuator and end effector. Parallel robots have higher rigidity because they have multiple links. Second, serial manipulators have low accuracy due to less actuator. Parallel robots usually have multiple actuators for each links and thus have better control accuracy over the working platform. The number of actuators also makes serial robots act slowly. Third, serial robots have less load capacity because of less links on end effector. In contrast, parallel manipulators have multiple links connected to working platform thus are capable of higher load.
1.3.3 Virtual Prototyping
As technologies develop, Virtual Prototyping is playing a more and more important role in our lives. Virtual Prototyping is widely used in robotics research, industrial design, such as rail way design, car design and vibration. The software used in this thesis is MSC ADAMS/view.
Virtual prototyping is a model of a structure or apparatus used for testing and evaluating form, design fit, performance, and manufacturability or used for studying and training. Normally, a complete Virtual Prototyping consists of a 3D solid model, a human-product interaction model and perspective test related models Wang (2012)
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1.3.4 MATLAB
Around the world there are millions of scientists, engineers and students using
MATLAB. MATLAB is a high-level language which is capable of matrix manipulation, plotting and simulations.
In the late 1970, Cleve Moler, who was a professor at University of New Mexico, created the earliest MATLAB for the purpose of more convenient use of LINPACK and
EISPACK for students. Jack Little was the first person to commercialize MATLAB. He recreated MATLAB from a Fortran based MATLAB to a matrix based MATLAB, Moler
(2006). Later in 1984, Cleve Moler, Jack Little and Steve Bangert founded the
MathWorks company in California. Since the release of MATLAB 1.0 in 1984,
MATLAB got well developed and is released as many new editions. The latest edition
R2012b is released on Sep.11, 2012.
1.3.5 MSC ADAMS/View
ADAMS stands for “Automatic Dynamic Analysis of Mechanical Systems”.
ADAMS/ View software package is quite powerful and can build almost any multi-body mechanism model. One important application is the car suspension design. Car suspension system is important because it sustains the weight of car body and transmits the forces between the car and the ground. The car suspension system designed by MSC
ADAMS/View is shown in Figure 1-10. In traditional suspension system design method, much more time and money were wasted on the modifications after the initial design. In
Visual Prototyping and computer simulations, general information can be obtained prior to the construction of the system, which makes the designing process more convenient. 24
Additionally, in visual designing environment the potential safety concerns in the system will be eliminated, Li (2007).
This suspension system has twenty movable parts and twenty DOF. The seat and the chassis are connected by bushing forces, so are the tires and the ground. Li (2007).
Figure 1-10 Schematic of the Suspension Mechanism
Source: Li (2007)
Figure 1-11 ADAMS Model
Source: Li (2007) 25
The test results of the suspension system are shown in Figure.1-10. Plotting of acceleration, displacement and velocity versus frequency were created. These curves were obtained in the MSC ADAMS/Vibration. Though this result is not exactly the same with the real data, this research shows good combination of flexibility and vibration analysis in ADAMS, Li (2007).
Figure 1-12 Frequency Response of the Seat Mass Center
Source: Li (2007)
1.4 Thesis Organization
Chapter 2 of this thesis presents the project information including introduction of the target mechanisms and robots and thesis objectives. Chapter 3 is the kinematic introduction, includes the modeling process of the mechanisms and robots in MSC
ADAMS. Specific modeling steps and initial position are shown. Chapter 4 introduces 26 the dynamic simulation method; forward and inverse dynamic simulation methods are analyzed. Chapter 5 presents the comparison of the simulation results in this thesis and the results from literature; errors of results are calculated. At last, Chapter 6 concludes the thesis and proposes work to be done in the future.
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CHAPTER 2: PROJECT INFORMATION
In this project a Slider- Crank mechanism, a four-bar mechanism, a 2R serial robot and a Stewart-Gough parallel manipulator are built and simulated in MSC
ADAMS. These models are built according to their prototypes in MATLAB. Slider-
Crank mechanism, four-bar mechanism and 2R serial robot MATLAB prototypes are from Williams’ literature. Stewart-Gough parallel manipulator MATLAB prototype is from Tsai’s book. In this chapter the object robots and mechanisms models in MATLAB and their dimensions, structural properties, the initial conditions and designated motions are introduced.
2.1 Planar Slider-Crank Mechanism
A Slider-Crank mechanism is consisted of two links and a slider. Slider moves through a prismatic joint horizontally and link1 rotates the original in counterclockwise with known angular velocity.
The target Slider-Crank mechanism at initial position is shown as follows:
Figure 2-1 Slider-Crank Mechanism 28
In the image above, at initial position, is , -the angle between the ground and link 2 is .
The dimensions of the Slider-Crank mechanism in Fig 2-1are described as follows:
Table 2-1 Slider-Crank Mechanism Parameters
Dimension(m) Mass of Mass(kg)
Inertia(kgm2)
Link1 0.102×0.019×0.013
Link2 0.203×0.019×0.013
Link3 0.076 ×0.019×0.013 N/A
Where , , The slider is connected with the ground by prismatic joint with the initial x-coordinate x=0.29m. In the simulation, the slider is moving within the range: .
This Slider-Crank mechanism is consisted of R1, R2 and the Slider. Connecting joints are prismatic joints and rotational joints. R1 and R2, R1 and the ground, R2 and
Slider are connected by revolute joints; Slider and the ground are connected by prismatic joint. A constant 1N force is applied horizontally at the middle of the Slider. Friction is considered, given µ=0.2.
Mobility is calculated according to Kutzbach’s Planar Criterion:
( ) 29
M is the mobility, N is the total number of links, including ground, is the number of one-DOF joints, is the number of two-DOF joint.
In the Slider-Crank mechanism, N is 4(ground, R1, R2 and the Slider), is 4, is 0. According to the equation above, the calculated M is 1; the Slider-Crank mechanism has totally 1 DOF. Inverse dynamics simulation of Slider-Crank is performed in MSC
ADAMS/View.
2.2 Four-Bar Mechanism
Four-bar mechanism is a planar mechanism consisted of four links. The four links are connected by revolute joints. A Four-Bar mechanism is modeled and simulated in
MSC ADAMS in this thesis.
The general angle and structure information of four-bar mechanism is shown in
Figure 2-2:
Figure 2-2 Four-Bar Mechanism 30
The four-bar mechanism is activated by rotational motion of link2. The position of the four-bar mechanism can be analyzed by the following equations:
is the angle between and the ground. is the angle between the ground and . is the angle between the ground and . is the angle between the ground and . Link2 and Link4 are connected to the ground by Joint5 and Joint6. The dimensions of the target Four-Bar mechanism are described as follows:
Table 2-2 Four-Bar Mechanism Parameters
2 Length(m) RG Mass of Inertia(kgm ) Mass(kg)
Link1 N/A N/A N/A N/A
Link2 0
Link3
Link4 0
Besides, and =20 rad/s.
According to Kutzbach’s Planar Criterion, it has totally 1 DOF.
In the simulation, inverse dynamics will be performed in MSC ADAMS/View, angles, angular velocities, angular accelerations, driving torques and shaking forces between the mechanism and the ground will be measured. The results will be compared with Williams’s results in MATLAB. 31
2.3 2R Planar Serial Robot
2R serial robot is the planar robot consisted of two links. A 2R planar serial robot is modeled and simulated in MSC ADAMS in this thesis. The model to be built is shown as follows:
Figure 2-3 2R Planar Serial Robot
is the absolute angle between the and the ground; is the relative angle between and . L1 and ground, L1 and L2 are connected by revolute joints. Both links have their center of gravity at their geometry center. According to equation 2.1, it has totally 2 DOF. The parameters of the 2R planar serial robot in Fig 2-3 are shown in the table as follows:
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Table 2-3 2R Planar Serial Robot Parameters
Length(m) (degree) Velocity (m/s) Ρ(kg/m3)
Link1 1 10 0 7806
Link2 0.5 90 0.5 7806
In the inverse dynamics simulation in MSC ADAMS, the tip of L2 moves vertically upward with a constant speed of 0.5m/s; using this known general motion at the end of L2, the rotational motions at joint 1 and joint 2 will be obtained. Cartesian Pose, joint angles, joint rates, joint accelerations and driving torques will be measured in MSC
ADAMS. The results will be compared with Williams’s results in MATLAB.
2.4 3D Stewart-Gough Parallel Manipulator
Parallel robots’ most noticeable advantages are that they have higher swiftness and heavy load capacity. Stewart-Gough parallel manipulator consists of a moving platform, a base platform and connecting joints between them.
The 3D Stewart-Gough Parallel manipulator built in MATLAB is shown as follows:
33
Figure 2-4 Stewart-Gough Parallel Manipulator
Source: Tsai (1999)
The parallel robot in Figure 2-4 is a Stewart-Gough manipulator. Upper limbs and lower limbs are connected by prismatic joints; limbs and base, limbs and platform are connected by ball joints.
The dimensions of the Stewart-Gough parallel manipulator in Fig 2-4 are shown as follows:
34
Table 2-4 3D Stewart-Gough Parallel Manipulator Parameters
Mass(kg) Mass Moment of Inertia(kg*m2) Radius
Platform 32 [ ] 0.5
Upper Limbs 2 [ ] N/A
Lower Limbs 2 [ ] N/A
Base N/A N/A 1
The dimensions of limbs are: , .
All the parts have their center of gravity in their geometry center. Connecting joints in Stewart-Gough parallel manipulator are spherical joints S and prismatic joints P, shown as follows:
Prismatic Joint (P)
Prismatic joints are applied between the upper limbs and lower limbs and can provide translational movement as the slider slides along the axis. It has one-DOF.
Figure 2-5 Prismatic Joint
Source: http://www.mathworks.com/help/toolbox/physmod/mech/ref/prismatic.html 35
Spherical Joint (S)
Spherical joints are widely used in robots. They can provide roll, pitch and yaw movement. It has 3-DOF.
Figure 2-6 Spherical Joint
Source: http://www.mathworks.com/help/toolbox/physmod/mech/ref/spherical.html
According to the Kutzbach’s Spatial Criterion ( )
, the Stewart-Gough parallel manipulator has six-DOF.
2.5 Thesis Objectives
A Slider-Crank mechanism, a 4-bar mechanism, a 2R planar serial robot and a 3D
Stewart-Gough Parallel robot are modeled in MSC ADAMS. Inverse dynamic simulation is performed on them; actuating forces are measured and validated by results from literature. Forward dynamic simulation is also performed on Stewart-Gough parallel manipulator; the motion of the platform is measured and compared with Tsai’s results in his book. The results used for validation are from MATLAB. 36
Another objective will be exploring the MSC ADAMS software. The MSC
ADAMS software is the most widely used dynamics analysis software; it will be very helpful to the research in robotics, dynamics and manufacturing. It plays a key role in making efficient machines.
The specific objectives are:
1. Modeling of Slider-Crank mechanism, 4-bar mechanism, 2R planar serial robot
and 3D Stewart-Gough Parallel robot with dynamic parameters from literature.
2. Forward and inverse dynamic simulation of robots and mechanisms in MSC
ADAMS.
3. Validation on the Robots and mechanisms model with dynamics results measured in MSC ADAMS.
4. Compare MSC ADAMS results and MATLAB results.
5. Calculation error ratios and make conclusions.
37
CHAPTER 3 KINEMATICS
Kinematics is the study of motion without consideration of forces. The variables studied include displacement, velocity and acceleration. In this chapter we will discuss the kinematics of Slider-Crank mechanism, four-bar mechanism, 2R planar serial robot and Stewart-Gough parallel Manipulator. Both translational and rotational motions are analyzed. The modeling process of the mechanisms above in MSC ADAMS will be presented.
3.1 Newton-Raphson Method
The Newton-Raphson method is used in MSC ADAMS/View as solution for non- linear multi-vector equations in simulations. The Newton-Raphson method is a numerical iterative method. According to Williams (2013), the iterative method is introduced as follows:
{ ( )} { }
Where the n function can be written as
{ ( )} { ( ) ( ) ( ) ( )}
In function{ ( )}, the variable X represents the vectors array
{ } { }
Taylor series expansion of { ( )} at { } we get
({ } { }) ({ }) ∑ ({ }) i=1, 2,∙∙∙, n
Where the Newton-Raphson Jacobian Matrix can be defined as 38
[ ] [ ( )] [ ]
We take the first two terms of the expansion, the expansion becomes:
({ } { }) i=1, 2, ∙∙∙, n
Let ({ }
({ } { }) ({ }) ∑ ({ } [ ]{ }) { } i=1, 2, ∙∙∙, n
In each iteration step, the correction factor is calculated as follows:
{ } [ ] { ({ })}
As the correction factor is getting smaller, the correction factor should be calculated each step until ‖{ }‖
A success iteration should yield convergence results.
3.2 Kinematic Analysis
Motion in MSC ADAMS is defined by function with variables of coordinates and time, thus motion is directly relevant to time. D.Negrut(2004) did the general kinematic analysis and the motion differential equation is expressed as follows:
[( ) ] ( ) ̇
Where L is Lagrangian factor, Q is extremely applied forces, q is position coordinates. is the Lagrange multiplier.
Using Newton-Raphson method and Taylor expansion equation, the position in
MSC ADAMS can be expressed by D.Negrut(2004) as follows:
( ) ( ) ( )( ) 39
At each iteration, the correction ( ) is found as follows D.Negrut(2004):
( ) ( ) ( ) ( )
Where is the correction, the next coordinate is calculated by adding correction to the last coordinate D.Negrut(2004):
( ) ( ) ( )
The final position is obtained with the small enough correction found.
3.3 Slider-Crank Mechanism Kinematics
The inverse kinematics problem of Slider-Crank mechanism is stated as follows:
Given , , , , , , , , , , , angular velocity , find
.
A Slider-Crank mechanism is built in MSC ADAMS/View. Mass moment of inertias are applied to all three links as given, a rotational motion is applied on link1.
During the simulation, link1 will rotate around origin. The accelerations in x and y direction at the center of the slider are measured in MSC ADAMS/View.
The Slider-Crank mechanism is shown as follows:
Figure 3-1 Slider-Crank Mechanism in MSC ADAMS 40
Rotational motion is applied on link1, with counterclockwise, shown as follows:
Figure 3-2 Rotational Motion
The Slider-Crank Mechanism model is built in following steps:
First open MSC ADAMS/View, click “new model”, in the “Create New Model” dialogue, select “gravity”, “Units” and “Working Directory”, then press “OK”.
In the whole modeling process, the rigid bars are first built. Link 2 and link 3 are built horizontally according to the dimensions. Joint1 connects the ground and link2.
Links are built using “Bodies” tab, solids library. Then link3 is rotated around joint2 to the position as follows; link4 is built according to the known dimensions in other places and then is translated to joint3. At joint3 both revolute joint and prismatic joint are applied. All the joints can be found under the “connectors” tab. One marker is applied on the ground for angle measure purpose; the marker function can be found from “Bodies” tab, “Construction” library. Measures will be created on angles and accelerations. 41
At initial position, , x= 0.29m , h= 0.076m. is proportional to time as follows:
Where
During the simulation, ,
, the position of JointA (XA) and the position of slider(XB) in simulation are shown as follows:
Figure 3-3 XA Position of Slider during Simulation
Figure 3-4 XB Position of Slider during Simulation 42
Link 4 and ground are connected with prismatic joint. The Slider-Crank mechanism is driven by the rotational motion at Joint 1; Separate rotational motions are measured at Joint 2 and Joint 3; the slider is moving from right to left and then back according to the rotation. The rotational motions at joint 2 and joint 3 will be found in simulation with , and the two motions will be exported as numerical data and be saved as test files in computer. These text data are then imported into MSC
ADAMS and stored as splines. In the final simulation, the two motions will be defined by the splines and the shaking forces will be measured.
3.4 Four-Bar Mechanism Kinematics
The inverse kinematics problem of the four-bar mechanism is stated as follows:
Given mass , , , center of gravity CG2,CG3 and CG4, mass moment of inertia , and , length of links , , , , initial angle of link1 . Find θ3,
θ4 ,µ, ω3, ω4, α3 and α4.
In the four-bar mechanism, r2 and r3, r3 and r4 as well as r2, r4 with the ground are connected with revolute joints. Link2 rotates in the range of with a constant angular velocity =20 rad/s; . A rotational motion is applied on link2. During the simulation, θ3, θ4, µ, ω3, ω4, α3 and α4 will be measured.
In this section the modeling process of Four-Bar Mechanism in ADAMS with given initial conditions will be introduced. The four-bar mechanism during simulation in
MSC ADAMS is shown as follows:
43
Figure 3-5 Four-Bar Mechanism in MSC ADAMS
Two ends of Link1 are locked to the ground by fixed joints shown as follows:
Figure 3-6 Fixed Joint
The MSC ADAMS model is built according to the dimensions in Chapter 2.
Link2 is built horizontally from origin to positive x axis. Link3 is first modeled as bar shape from end of link2 to positive x axis horizontally. Link3 is then rotated around 44
Joint2 for 66˚. Link1 is created from origin to positive x axis then rotated around origin for 10.3˚ counterclockwise; two ends of Link1 are also locked to the ground by fixed joints. Link4 is created between Link1 and Link3. Two markers are applied to ground in reference to the origin and Joint6 for angle measurement. All rigid links are connected by revolute joints. Rotational motion = 20 rad/s is applied to Link2 at Joint1. In the
“Motions” tab, “Rotational Joint Motion” is used for this motion. After the motion is applied to Joint1, the motion can be modified by double clicking. In the “Type” dialogue box, “displacement” is selected; in the “function” dialogue box, “20*time” is entered.
The simulation can be run by clicking the “simulation” tab, “simulate control” button.
The Link3 is then changed into rectangle link for simulation. The total simulation time is
0.314s.
During the simulation, x and y coordinates are relevant to and expressed as follows:
[ ] [ ][ ]
The position of pointA and pointB are plotted in MSC ADAMS as follows: 45
Figure 3-7 XA in Four-Bar Mechanism Simulation
Figure 3-8 XB in Four-Bar Mechanism Simulation
In this simulation, I perform inverse dynamics simulation for the open branch of the 4-bar mechanism above. Given the rotation of the link2 ω2=20 rad/s, kinematic parameters will be measured. After the comparison of kinematic parameters with 46
Williams’s results, shaking force of Joint 5 and Joint 6 will be measured, the results will be presented in Chapter5.
3.5 2R Planar Serial Robot Kinematics
The inverse kinematics problem of the 2R planar serial robot is stated as follows:
Given the robot dimensions , , , density
, initial angle positions are , . Movement is ̇
. Find the Cartesian Pose x, y, , joint angles , , joint rates and joint accelerations.
During the modeling process, a positive y-direction translational motion is applied on end of link2. The translational motion motivates the whole 2R serial robot and produce rotational motion of link1 and link2 during simulation; the rotational motions at joint1 and joint2 are stored as numerical point data and created into splines. After the translational motion being removed, two rotational motions are applied on joint1 and joint2, defined by the previous stored numerical rotational data. During this simulation,
Cartesian Pose x, y, , joint angles , , joint rates and joint accelerations are measured.
The 2R planar serial robot model in MSC ADAMS is shown as follows:
47
Figure 3-9 2R Planar Serial Robot in MSC ADAMS
The 2R planar serial robot is first built from Link1. Link1 is created from origin to positive x axis.Link2 is then created from the end of Link1 to positive x axis. Link2 is rotated around Joint2 for counterclockwise. Link1 is then rotated around origin for
counterclockwise. The end of Link2’s location is (0.9, 0.67) globally. Link1 and
Link2, Link1 and the ground are connected by revolute joints. The known general motion is the translational motion of the end of Link2. The translational motion can be applied by clicking “Motions” tab, “Translational Joint Motion” button. In the dynamic part, the rotational motions of Joint1 and Joint2 will be measured and recorded; the measured results of the rotational motions will be created into splines; the rotational motions will be defined by these splines. Markers are created on ground for angles measuring purpose. A marker is applied on ground for measurement of Theta2.
is proportional to time as follows: 48
The initial position of the end of Link2 is as follows:
( ) ( ) m
During the simulation, the coordinate of the end effector can be expressed as:
( ) ( )m
Where
is calculated by:
Where is the angle between Link1 and Link2
The Y of point A during simulation in MSC ADAMS is shown as follows:
Figure 3-10 YA of 2R Planar Serial Robot in Simulation
49
The model is built based on the dimensions mentioned above and the motion is applied. After the comparison of the kinematic parameters, the shaking forces and driving torque will be measured and compared with Williams’ results in MATLAB in Chapter5.
3.6 Stewart-Gough Parallel Manipulator Kinematics
The inverse kinematics problem of the Stewart-Gough parallel manipulator is stated as follows:
Given mass , , , , ,mass moment of inertia , , , initial position , angles of limbs, accelerations of the center of the platform , find the translational displacement of upper limbs ( ).
The Stewart-Gough parallel robot has six legs, base and the working platform. Six prismatic motors were installed on the limbs. The working platform and the base are connected with the limbs by ball joints. It has totally 6-DOFs. When at rest, the working platform is 1 m above the base.
In simulation, a translational motion with a given acceleration is applied at the center of platform. During simulation the translational displacement of each prismatic joint is measured and stored as numerical point data. In the data both time and displacement are recorded as point data. After the translational motion being removed, six new translational motions are applied on prismatic joints and are defined by the stored numerical data accordingly.
The input Translational Displacement Tabular Data of the Stewart-Gough parallel manipulator inverse simulation in MSC ADAMS is shown as follows:
50
Time Displacement 8.000000E-003 1.262108E-004 1.600000E-002 5.048889E-004 2.400000E-002 1.136171E-003 3.200000E-002 2.020287E-003 4.000000E-002 3.157552E-003 4.800000E-002 4.548377E-003 5.600000E-002 6.193256E-003 6.400000E-002 8.092771E-003 7.200000E-002 1.024759E-002 8.000000E-002 1.265847E-002 8.800000E-002 1.532623E-002 9.600000E-002 1.825178E-002 1.040000E-001 2.143612E-002 1.120000E-001 2.488029E-002 1.200000E-001 2.858543E-002 1.280000E-001 3.255274E-002 1.360000E-001 3.678346E-002 1.440000E-001 4.127892E-002 1.520000E-001 4.604050E-002 1.600000E-001 5.106961E-002 1.680000E-001 5.636774E-002 1.760000E-001 6.193640E-002 1.840000E-001 6.777716E-002 1.920000E-001 7.389162E-002 2.000000E-001 8.028140E-002 2.080000E-001 8.694816E-002 2.160000E-001 9.389360E-002 2.240000E-001 1.011194E-001 2.320000E-001 1.086273E-001 2.400000E-001 1.164191E-001 2.480000E-001 1.244964E-001 2.560000E-001 1.328610E-001 2.640000E-001 1.415148E-001 2.720000E-001 1.504593E-001 2.800000E-001 1.596964E-001 2.880000E-001 1.692278E-001 2.960000E-001 1.790552E-001 3.040000E-001 1.891804E-001 3.120000E-001 1.996050E-001 3.200000E-001 2.103306E-001 3.280000E-001 2.213590E-001 3.360000E-001 2.326917E-001 3.440000E-001 2.443304E-001 3.520000E-001 2.562765E-001 3.600000E-001 2.685317E-001 3.680000E-001 2.810973E-001 3.760000E-001 2.939749E-001 3.840000E-001 3.071659E-001 3.920000E-001 3.206716E-001 4.000000E-001 3.344935E-001
Figure 3-11 Input Translational Displacement Numerical Data 51
The numerical data above is used to create the spline, which is used to define the motions of prismatic joints.
The Stewart-Gough parallel manipulator model in MSC ADAMS is shown as follows:
Figure 3-12 Stewart-Gough Parallel Manipulator in MSC ADAMS/View
The side view of the proposed Stewart-Gough parallel manipulator is shown as follows:
52
Figure 3-13 Side View of Stewart-Gough Parallel Manipulator in MSC
ADAMS/View
In MSC ADAMS, the model base of the manipulator is first built on x-y plane.
The base is also locked to the ground by three fix joints. On the base, the six limbs are built. According to Tsai’s image of the manipulator, the six lower limbs are connected to the base at 0˚, 110˚, 120˚, 230˚, 240˚ and 350˚ (Fig 3-14). The above points on the base are marked. Meanwhile the working platform is also created on the base. The upper limbs are connected to the platform at 50˚, 60˚, 170˚, 180˚, 290˚ and 300˚(Fig 3-16). These points are also marked by the virtual links from the center of the base to the edge of the platform. After the platform is translated from the base to 1 meter above along y axis, six long limbs are created between the marked points correspondingly. The marking links are removed after the limbs are created. These six long limbs can provide the accurate angles 53 of the upper limbs and lower limbs. The upper links are created 0.85m from the points on the platform downwards along the long limbs; the lower limbs are also created 0.3m along the long limb from the other end on the base upwards. Prismatic joints are applied between the upper limbs and lower limbs. The limbs are connected to the base and platform by revolute joints.
Figure 3-14 Ball Joint Position on Base 54
Figure 3-15 Ball Joint Position on Platform
The solid parts in MSC ADAMS can be modified by double clicking. The
“Modify Body” dialogue box will appear. In the “Define Mass By” category, pull down and select “Geometry and Density”. In the “Density” box, density can be input. In this model, the solid body properties are defined by mass of inertia matrix. In the “Modify
Body” dialogue box, “Define Mass By” box pull down and select “User Input”, then
and [ ] are input as mass and mass of inertia for the platform. The properties of the limbs and base are modified in the same way.
According to Tsai (1999), the velocity of each limb can be calculated as follows:
th Where [ ] describes the velocity of the i limb in x, y z directions. 55
In example2 simulation, the velocity of the platform in Z direction is shown as follows:
Figure 3-16 Velocity in Z Direction of the Platform
For the four designated motions, both forward and inverse dynamics simulation will be performed. In the forward dynamics simulation with the known motions of the working platform, the translational motion of the six prismatic joints will be measured, imported and saved as splines; with these splines, the translational motions of the prismatic joints are defined. With the translational motions defined, the actuating forces will be measured. In the inverse dynamics simulation of example 2, the platform translates along z axis with the acceleration of 5 m/s2. Initial velocity is zero. Actuating forces are applied as known at the prismatic joint; the motion of the working platform will be compared with the results in MATLAB.
56
CHAPTER 4 DYNAMICS
Dynamics is the study of motion with actuating forces and torques. Dynamics relates how actuating forces and torques affect the motion. In this chapter, the inverse dynamics simulation of Slider-Crank mechanism, four-bar mechanism 2R planar serial robot and Stewart-Gough parallel manipulator will be introduced. The forward dynamics of Stewart-Gough parallel manipulator is also introduced. In inverse dynamics simulation, the translational or rotational motions are known, shaking forces are to be found; in forward dynamics simulation, the actuating forces are known, the translational or rotational motions are to be found.
4.1 Slider-Crank Mechanism Inverse Dynamics
The inverse dynamics problem is stated as follows:
Given , , , , , , , , , , ,external constant force , angular velocity , find the total shaking force .
In the Slider-Crank Inverse Dynamics simulation link 2 will rotate around joint 1
within the range , with the angular velocity , . In a full simulation, the total time will be 0.419s. The Slider-Crank mechanism in simulation is shown as follows:
57
Figure 4-1 Simulation of Slider-Crank Mechanism, t=0.1s
Link2 and link3 align at t=0.0158s during simulation, shown as follows:
Figure 4-2 Link2 and Link 3 Align at t=0.0158s
In reality the singularities will also happen with θ3=90 , 270 , when link 2 moves to the vertical position. In these positions, the determinant of coefficient matrix becomes zero: | |=0, Williams (2012).
The displacement, velocity and acceleration of the slider can be expressed as follows:
, 58
̇ ̇ ̇
( )
The velocity and acceleration are the first and second derivatives of position. is angular velocity, is coordinate of slider, is the rotational transition.
Total shaking force Fsx is calculated as follows:
Fsx=Fsx1+Fsx2
Fsy=Fsy1+Fsy2
In x and y directions, slider acceleration and shaking forces are described as follows:
M3A3x=Fsx-f+1
M3A3y= Fsy
Figure 4-3 SForceX, SForceY vs. time
59
During the simulation with the singularity at t= , the shaking force is expected to be maximum when link 2 reaches lowest point at .
4.2 Four-Bar Mechanism Inverse Dynamics
The inverse dynamics problem of the four-bar mechanism is stated as follows:
Given mass , , , center of gravity CG2,CG3 and CG4, mass moment of inertia , and , length of links , , , , initial angle of link1 . Find total shaking forces and .
The inverse dynamics simulation will be performed on 4-bar mechanism. The four-bar mechanism in simulation in MSC ADAMS is shown in Figure 4-4.
Figure 4-4 Four-Bar Mechanism in Simulation in MSC ADAMS
60
The 4-bar mechanism is built in ADAMS according to Williams’s dimension.
Link 1 is connected to the ground at Point1 and Point2. With the constant angular
velocity , the link 2 will rotate in the range . The simulation runs with gravity, with g=9.806m/s2. The rotational motion is applied on joint5(1).
The displacement, velocity and acceleration of the four-bar mechanism at point A and B can be expressed as follows:
Position:
Velocity: ̇ ̇ ̇
̇ ̇ ̇ ̇ ̇
Acceleration: ( )
( ) ( )
( )
The total shaking force will be the sum of shaking force at joint 5(1) and joint
6(2):
Fsx=Fsx1+Fsx2
Fsy=Fsy1+Fsy2
Shaking forces of the four-bar mechanism at Joint5(1) and Joint6(2) are shown as follows:
61
Figure 4-5 SForceX1 vs. time
Figure 4-6 SForceX2 vs. time
62
The total shaking force X is shown as follows:
Figure 4-7 Shaking Force X in Four-Bar Mechanism Simulation
The shaking forces shown in Fig 4-5 and Fig 4-6 are added together to get the total shaking force in X direction, shown in Fig 4-7.
4.3 2R Planar Serial Robot Inverse Dynamics
The inverse dynamics problem of the 2R planar serial robot is stated as follows:
Given the robot dimensions , , , density
, initial angle positions are , . Movement is ̇
; Find joint torques and with gravity and without gravity.
The 2R planar serial robot simulation will last for 1 second. During the simulation the end of L2 will translate vertically along positive Y axis for 0.5 m. In MSC ADAMS the simulation has 100 steps, with 0.01s per step. The model at the end of the simulation in MSC ADAMS is shown as follows: 63
Figure 4-8 2R Planar Serial Robot at the End of Simulation
The end effector has a constant velocity in Y direction, acceleration is zero. There is no displacement in X direction at the tip of L2.
The displacement, velocity and acceleration of the end of L2 can be expressed as follows:
,
̇ ̇ ̇ ̇
( )
The simulation will be performed both with gravity and without gravity. With the rotational motions at joint 1 and joint 2 defined, the driving torques is measured, shown as follows(with gravity): 64
Figure 4-9 Driving Torques Measured in MSC ADAMS (with Gravity)
The measured results and comparison with MATLAB will be presented in
Chapter5.
4.4 Stewart-Gough Parallel Manipulator Forward Dynamics
The forward dynamics problem of the Stewart-Gough parallel manipulator is stated as follows:
Given mass , , , , ,mass moment of inertia , , , initial position , actuating forces , , , , and . Find platform acceleration .
Forward dynamics is method to apply known forces and torques to mechanism to get the expected motions in simulation with consideration of gravity. In Stewart-Gough parallel manipulator forward dynamics simulation, actuating forces are known and accelerations at the center of the platform will be measured. 65
The Stewart-Gough parallel manipulator model in example 2 simulation in MSC
ADAMS is shown as follows:
Figure 4-10 the Stewart-Gough Platform in MSC ADAMS/View at t=0.4s of
Forward Dynamics Simulation
The motion of the moving platform is described by three Euler angles,
. They are the rotational angles about x, y and z axis. The Euler angles of a limb of the Stewart-Gough parallel manipulator are shown as follows:
66
Figure 4-11 Euler Angles of a Limb
The rotation matrix for platform by Tsai(1999) becomes:
[ ]
The six limbs of the Stewart-Gough parallel manipulator have identical structure; they consist of upper limb and lower limb. The upper limb and the lower limb are connected by prismatic joint. The prismatic joint is actuated by translational actuators.
The limbs are connected with the working platform and the base by ball joints without actuators. Free-Body diagram of a typical limb is shown as follows:
67
Figure 4-12 Free-Body Diagram of a Typical Limb
The distance between the centers of the upper limb and the lower limb is not specified. Since the upper limb and the lower limb are connected by prismatic joint and are always align, also the mass moment of inertia of the limb is very small compared to the mass moment of inertia of the platform, the length between the limbs will not affect the simulation and its results.
With gravity applied on the parallel manipulator, the total actuating forces applied on the platform is given by:
( )
The Stewart-Gough parallel manipulator simulations include forward dynamics on all the four examples and inverse dynamics on example 2.
The simulation results will be presented and discussed in Chapter 5. 68
CHAPTER 5: SIMULATION RESULTS
In this chapter, the simulation and results of Slider-Crank mechanism, four-bar mechanism, 2R planar serial robot and Stewart-Gough parallel manipulator will be shown and discussed. The Slider-Crank mechanism, four-bar mechanism and 2R planar serial robot are simulated in inverse dynamics method; the measured shaking forces and torque in ADAMS are compared with the results from MATLAB. The Stewart-Gough parallel manipulator is simulated in forward dynamics method; the actuating forces are defined and the platform motion will be measured. Inverse dynamics simulation of example 2 is also demonstrated.
5.1 Inverse Dynamics Simulation and Results for Slider-Crank Mechanism
The inverse dynamics of the Slider-Crank mechanism is studied in this experiment. The Theta2 vs. Time plotting is created according to the formula:
. Simulation time is 0.419s. The measured Theta2 vs. time in MSC ADAMS is shown as follows:
Figure 5-1 Theta2 vs. time in MSC ADAMS 69
Since is constant and is proportional to time, the simulation results are plotted with independent axis being time. The results remain the same. The accelerations of link 3 in MSC ADAMS are shown in Figure 5-3, Figure 5-4, compared with
Williams’s results in MATLAB shown in Figure 5-2. The AG3X is in red, AG3Y is in green.
25
20
15
10
5 ) 2 0 s / m (
G3 -5 A
A -10 G3X A G3Y
-15
-20
-25
-30 0 50 100 150 200 250 300 350 (deg) 2
Figure 5-2 Acceleration of Link 3(m/s2) vs. time(s)
Source: Williams (2012)
Figure 5-3 AG3Y (G), AG3X(R) (m/s2) vs. time(s) in MSC ADAMS 70
Singularity at t=0.2650s is shown as follows. Link 2 and link 3 are folded together. At this time, extra shaking forces are produced between the slider and the ground.
Figure 5-4 Singularity Occur at t=0.2650s, Link2 and Link3 Fold
The shaking forces Fsx and Fsy measured in MSC ADAMS are shown in Figure
5-8, compared with the Williams’s results from MATLAB, shown in Figure 5-7(Fsx in red, Fsy in green).
71
1.5
1
F 0.5 SX F SY
0
-0.5 N) ( S F -1
-1.5
-2
-2.5
-3 0 50 100 150 200 250 300 350 (deg) 2
Figure 5-5 Shaking Force Fs(N) vs. time(s)
Source: Williams (2012)
Figure 5-6 Fsx(r), Fsy(g)(N) vs. time(s) Plotted in MSC ADAMS
72
During the Slider-Crank mechanism simulation, shaking forces and torques are measured and compared. The measured shaking forces have singularities around , when link2 is vertical down. This is caused by the physical singularity position.
5.2 Inverse Dynamics Simulation and Results for Four-Bar Mechanism
After the 0.314s simulation in ADAMS, θ3, θ4, ω3, ω4, α3, α4, Fsx, Fsy vs. time are measured. They are shown in Figure 5-11, Figure 5-13, Figure 5-15, Figure 5-23 and
Figure 5-25.
The measured θ3 (red), θ4 (green) and µ (blue) vs. θ2 of the four-bar in MATLAB by Williams is shown as follows:
73
Figure 5-7 θ3, θ4 and µ(deg) vs. θ2(deg)
Source: Williams (2012)
By comparison, the measured θ3, θ4 and µvs. θ2 in MSC ADAMS are shown as follows:
Figure 5-8 θ3, θ4 and µ(deg) vs. θ2(deg) in MSC ADAMS
74
The measured ω3, ω4 vs.θ2 in MATLAB by Williams is shown as follows:
Figure 5-9 ω3, ω4(rad/s) vs.θ2(deg)
Source: Williams (2012)
By comparison, the measured ω3, ω4 vs.θ2 of the four-bar mechanism in MSC
ADAMS is shown as follows:
Figure 5-10 ω3, ω4(rad/s) vs.θ2(deg) in ADAMS
75
The measured angular accelerations α3, α4 vs.θ2 of the four-bar mechanism in
MATLAB by Williams is shown as follows:
2 Figure 5-11 α3, α4 (rad/s )vs.θ2(deg)
Source: Williams (2012)
By comparison, the measured α3, α4 vs.θ2 of the four-bar mechanism in MSC
ADAMS is shown as follows:
2 Figure 5-12 α3, α4(rad/s ) vs.θ2 (deg)in ADAMS
76
All the kinematic parameters are matched with the author’s data.
Shaking forces measured in MATLAB by Williams is presented in Figure 5-13.
Force X is in red and Force Y is in green. The maximum of force X1 appears at around
0.155s (Appendix A), when the link two and link three fold together, meanwhile the minimum of force X2 appears (Appendix A). The shaking forces in Appendix A are measured in Joint 5 and Joint 6. The total shaking force is the sum of the two shaking forces. I added the two shaking forces in x direction and got the total shaking force in x direction. The result is created as spline in MSC ADAMS and is shown in Appendix
A.The SForce X is matching with Williams’s results in MATLAB (Figure 5-13, red).The shaking forces in Joint 5 and Joint 6 in Y direction are measured separately, as shown in
Appendix A.
In the same method, the two splines are added together into the total shaking force-SforceY. The shaking force Y is expressed in spline in MSC ADAMS shown in
Appendix A.The total shaking force in Y direction SForceY is matching with Williams’s results in MATLAB(Figure 5-13, green). The shaking force x and shaking force y measured in MSC ADAMS are also plotted in MATLAB as follows:
77
Figure 5-13 Measured FSX and FSY(N)vs.θ2 (deg) plotted in MATLAB up,
Williams’s Shaking Forces down, Williams(2012)
The shaking force X (SForce X) is the sum of the shaking forces at joint
5(SForceX2) and joint 6(SForceX1). The shaking force Y (SForce Y) is the sum of the shaking forces Y at joint 5(SForceY2) and joint 6(SForceY1). I first plotted the shaking 78 forces in Joint 5 and 6 separately; they are SForceX1, SForceX2, SForceY1 and
SForceY2. Then MSC ADAMS will plot the sum of them automatically in plot editor, as shown in Figure 5-16. The two shaking forces match with Williams’s results.
5.3 Inverse Dynamics Simulation and Results for 2R Planar Serial Robot
In the 2R planar serial robot simulation the inverse dynamics is performed in
MSC ADAMS. According to Williams’s (2012) literature, the motion at the end of link2 is known; the driving torques with the consideration of gravity and without the consideration of gravity are to be found. The Cartesian Pose of the link2 tip, including x, y and phi vs. Time are plotted in MSC ADAMS, compared with the results from
MATLAB are shown as follows:
79
Figure 5-14 X, Y phi vs. time(s) Measured in MATLAB
Source: Williams (2012)
Figure 5-15 x, y(m) phi(rad) vs. time(s) Measured in MSC ADAMS
80
The joint angles of Theta1 and Theta2 vs. time are measured and plotted MSC
ADAMS, compared with the results from MATLAB shown as follows:
Figure 5-16 Joint Angles(deg) vs time(s) Measured in MATLAB
Source: Williams (2012)
Figure 5-17 Joint Angles(deg) vs time(s) Measured in MSC ADAMS
81
In the simulation joint rates (angular velocity) of Joint1 and Joint2 are measured
MSC ADAMS, compared with the results from MATLAB shown as follows:
Figure 5-18 Angular Velocity(rad/s) vs time(s) Measured in MATLAB
Source: Williams (2012)
Figure 5-19 Angular Velocity(rad/s) vs time(s) Measured in MSC ADAMS
82
Angular accelerations of Joint1 and Joint2 are also measured in MSC ADAMS, compared with results from MATLAB, shown as follows:
Figure 5-20 Angular Acceleration (rad/s 2) vs time(s) Measured in MATLAB
Source: Williams (2012)
Figure 5-21 Angular Acceleration (rad/s 2) vs time(s) Measured in MSC ADAMS
83
According to the comparisons above, most of the motions in MSC ADAMS agree with the designated motion. The motions remain the same with or without gravity. The separate motions in Joint1 and Joint2 are defined based on the measure from the inverse dynamics simulation of the general motion: ̇ =0.5m/s.
In the inverse dynamics simulation with the separate motions defined, the driving torques are measured without gravity in MSC ADAMS, compared with Williams’s results in MATLAB, shown as follows:
84
Figure 5-22 Driving Torques(Nm) vs time(s) Measured in MATLAB (without
Gravity)
Source: Williams (2012)
Figure 5-23 Driving Torques (Nm) vs time(s) Measured in MSC ADAMS (Ignoring
Gravity)
85
The measured torques with gravity in MSC ADAMS are measured and plotted, compared with Williams’s results in MAMTLAB shown as follows:
Figure 5-24 Driving Torques (Nm) vs time(s)Measured in MATLAB (with Gravity)
Source: Williams (2012)
Figure 5-25 Driving Torques (Nm) vs time(s) Measured in MSC ADAMS (with
Gravity)
86
The torques at Joint1 and Joint2 are measured with gravity and without gravity in
MSC ADAMS. The results agree with Williams’s results in MATLAB. During the process of defining Motion1 and define of Motion 1, I measured the rotational motions separately and exported the measure results as text files; In MSC ADAMS imported the text files and created the corresponding splines for the motions. This method improves the accuracy greatly. With gravity, the measured Torque1 and Torque2 are same to
Williams’s results in MATLAB; the same method is used.
Forward dynamics simulation on 2R planar serial robot is not performed here because the driving torque failed to make the designated motions. The defining of the driving torques can be done by creating splines and save the splines as text format; then import the text file. This method can ensure the accuracy of the data. The driving torques is defined by the splines.
5.4 Forward and Inverse Dynamics Simulation and Results for Stewart-Gough
Parallel Manipulator
In this experiment the forward and inverse dynamics of Stewart-Gough parallel platform will be studied. Forward dynamics is studied in example1, 2, 3 and 4; Inverse dynamics simulation is performed in example 2. The data of my experiments are identical with Lung-Wen Tsai’s example simulated in MATLAB. All four of my experiments on the Stewart-Gough platform will run the parallel manipulator from at rest for 0.4s.
The same model of the Stewart-Gough parallel manipulator is built in MSC
ADAMS. The top view of model in MSC ADAMS is shown as follows: 87
Figure 5-26 Top view of Stewart-Gough Parallel Manipulator in MSC
ADAMS/View
88
The side view of the Stewart-Gough parallel manipulator in MSC ADAMS is shown as follows:
Figure 5-27 Side view of Stewart-Gough Parallel Manipulator in MSC
ADAMS/View
The difficulty is to define the actuating forces in prismatic joints. The author’s measured forces of the actuators are plotted in Figure 5-30, Figure 5-33, Figure 5-36,
Figure 5-39. The forces splines are input into ADAMS and the actuating forces are defined by these splines. The input forces are also shown in the above figures for comparison.
In Lung-Wen Tsai’s first simulation the working platform moves in the positive x direction. According to Lung-Wen Tsai, the platform starts from rest and has a constant
acceleration of ̈ . The platform has no motion in other directions. ̈ ̈ 89
̈ ̈ ̈ . The actuating forces of the six prismatic joints are shown in Figure 5-28 as below:
Figure 5-28 Actuating Forces(N) vs time(s)
̈ , Tsai (1999) (up), Forces vs. time Input in Example 1 plotted in
MATLAB (down)
The input forces are transferred into splines in MSC ADAMS according to their plotting in MATLAB, the six actuating force splines in MSC ADAMS are shown above. 90
In the first example, force splines are defined by eight points. This is enough for smooth curves. With the input actuating forces, the Stewart-Gough parallel manipulator at t= 0.4s in simulation is shown as follows:
Figure 5-29 the Stewart-Gough Parallel Platform at t=0.4s
The gap between the upper limbs and the lower limbs are due to the extension of the prismatic joints. After the simulation, the acceleration at the center of the working platform is measured and plotted as follows in MSC ADAMS:
91
Figure 5-30 CM Acceleration(m/s2) vs. time(s) in X Direction
The result is as expected at 5 m/s2, with tiny fluctuation.
In Lung-Wen Tsai’s second simulation, the working platform will translate along z axis with a constant acceleration ̈ . No motions are in other directions.
̈ ̈ ̈ ̈ ̈ . The input forces of the actuators 1 to 6 are shown as follows:
92
Figure 5-31 Actuating Forces(N) vs time(s)
̈ Tsai (1999) (up), Forces vs. time Input in Example 2 (down)
93
I use the same method to transfer the forces spline above into splines in ADAMS and define the forces with the splines. The force splines in MSC ADAMS are defined by eight points, as shown above.
The Stewart-Gough parallel manipulator at t=0.4s in simulation in MSC ADAMS is shown as follows:
Figure 5-32 the Stewart-Gough Parallel Platform at t=0.4s
The acceleration measured at the center of the platform along z axis is shown as follows: 94
Figure 5-33 CM Acceleration(m/s2) vs. time(s) in Z Direction
This result is compared to Tsai’s results, which is ̈ .
In Tsai’s 3rd simulation, the working platform will rotate around y axis with a constant angular acceleration ̈ . There are no motions in other directions.
̈ ̈ ̈ ̈ ̈ . The input forces of the actuators 1 to 6 were shown as follows:
95
Figure 5-34 Actuating Forces(N) vs. time(s)
̈ , Tsai (1999) up, the input for example 3 in ADAMS, plotted in
MATLAB down 96
The forces above are transferred into splines in MSC ADAMS and the actuating forces are defined by spline 1 to 6 accordingly. The input force splines in MSC ADAMS are shown as above:
During the simulation, the platform rotates around the Y axis with ̈
smoothly. The simulation at t=0.4s is shown as follows:
Figure 5-35 The Stewart-Gough Parallel Platform at t=0.4s
After the simulation, the measured angular acceleration around y axis is taken from the center of the platform. It is close to 2 rad/s, shown as follows: 97
Figure 5-36 CM Angular Acceleration(rad/s2) vs. time(s) around Y Axis
The measured angular acceleration is very close to 2 rad/s2. The spline is almost constant also matches the smooth movement in simulation.
In Tsai’s 4th simulation, the working platform will rotate around z axis with a constant angular acceleration ̈ . There are no motions in other directions.
̈ ̈ ̈ ̈ ̈ . The input forces of the actuators 1 to 6 are shown as follows:
98
Figure 5-37 Actuating Forces(N) vs. Time(s)
̈ Tsai (1999) up, Force Input of Actuator vs. time plotted in MATLAB
99
The splines of the force in MSC ADAMS are defined by eight points. The force splines from MATLAB are transferred into MSC ADAMS splines as in Figure 5-37
(down)
In this simulation platform rotate around z axis with angular acceleration ̈
. The Stewart-Gough parallel manipulator at t=0.4s in simulation is shown as follows:
Figure 5-38 the Stewart-Gough Parallel Platform at t=0.4s
The measured angular acceleration of the center of the platform for example 4 in
MSC ADAMS is shown as follows: 100
Figure 5-39 CM Angular Acceleration(rad/s2) in Z Direction vs. time(s)
In this experiment the measured angular acceleration is very close to 2 rad/s2; the result is constant as well.
In all the four forward dynamics simulations, the actuating forces are defined from the splines in Tsai’s (1999) literature. The translational accelerations and angular accelerations of the center of the platform are measured. After comparison, the results are within expectation. The differences are caused by the transfer of data between MSC
ADAMS and MATLAB.
In this section, the inverse dynamics simulation of the Stewart-Gough parallel manipulator is shown and discussed. The results will be compared with Example 2. The example 2 has the same forces splines of all six actuators, and has the simplest movements due to the same displacement of the six prismatic joints. The difficulty in inverse dynamics simulation is to specify the translational motions of the prismatic joints. 101
In the simulations that the general motions of the platform are applied, the actuating forces are separated into the 12 ball joints and the 6 prismatic joints. The specific translational motions on prismatic joints are necessary. The translational motions are measured in inverse dynamics, with the translational acceleration ̈ on platform.
The displacement of the prismatic joints in the translational motion in MSC
ADAMS is shown as follows:
Figure 5-40 Inverse Dynamics Translational Motion Input
The Stewart-Gough parallel manipulator at initial position in inverse dynamics simulation is shown as follows:
102
Figure 5-41 Inverse Dynamics on Stewart-Gough Parallel manipulator, t=0s
Based on the input translational motion splines above, the translational motions at prismatic joints are defined. The Stewart-Gough parallel manipulator at t=0.4s in inverse dynamics simulation is shown as follows:
Figure 5-42 Inverse Dynamics on Stewart-Gough Parallel manipulator t=0.4s 103
In this simulation, the translational acceleration at the center of the platform along z axis is measured. The result is plotted as follows:
Figure 5-43 Platform Acceleration(m/s2) in Z Direction vs. time(s)
104
The actuating forces plotted by MATLAB are shown as follows:
Figure 5-44 Actuating Forces(N) vs time(s), ̈
Source: Tsai(1999)
The measured actuating force in the prismatic joints in MSC ADAMS is shown as follows:
Figure 5-45 Actuator Forces(N) vs. time(s)
105
In the inverse dynamics simulation of the Stewart-Gough parallel robot, the actuating forces are measured and compared with Tsai’s results. The results are within expectation; meanwhile the platform’s motion is also ̈ .
106
CHAPTER 6 CONCLUSIONS AND FUTURE WORK
6.1 Conclusions
In this thesis, a 3D Stewart-Gough parallel manipulator, a Slider-Crank mechanism, a 2R planar serial robot and a four-bar mechanism are built. Theses robots and mechanisms are found from literature with good dimensions and input, output information. These robots and mechanisms are built and simulated in MSC ADAMS.
Both forward and inverse dynamics simulation are performed on Stewart-Gough parallel manipulator. Inverse dynamics simulation is performed on Slider-Crank mechanism, 2R planar serial robot and four-bar mechanism.
The Stewart-Gough parallel platform was found in Tsai’s book “Robot Analysis:
The Mechanics of Serial and Parallel Manipulators”, in the book it was simulated in
MATLAB in inverse dynamics, I built the same manipulator in ADAMS and succeed matching my results in forward dynamics, example 2 match the results in inverse dynamics. In inverse dynamics simulation, the results are within expectation.
In the inverse dynamics simulation of Slider-Crank mechanism, the Slider-Crank mechanism is built and simulated in MSC ADAMS, with ω2=15 rad/s. The shaking forces are measured and compared with Williams’s simulation results. The results match with singularities found. Singularities occur when the link2 and link 3 align and fold, at t=0.0158s and t=0.2650s separately. The singularities are shown as spikes in the results of shaking forces.
The inverse dynamics simulation of the 2R planar serial robot has the similar procedure with Stewart-Gough parallel manipulator. With the general motion at the end 107 of link three is known, the rotational motions at joint 1 and joint 2 are measured and used as the known motions in the inverse dynamics simulation. Driving forces are measured and compared with Williams’s results in MATLAB.
In the inverse dynamics simulation of Four-Bar mechanism, the four-bar mechanism is built and simulated in MSC ADAMS, with the angular velocity ω2=20 rad/s. The shaking forces are measured and compared with Williams’s simulation results in MATLAB. The results match. Singularities occur when the link 2 and link 3 align and fold, but due to the large mass of link 3, the singularities are not obvious during the simulation or in the results.
6.2 Future Work
Forward Dynamics simulation of Four-Bar mechanism, 2Rserial robot and Slider-
Crank mechanism are to be done in the future. The inverse dynamics of Stewart-Gough parallel robot is accomplished only on the example 2. It has all six prismatic joints applied with the same forces. The difficulty in forward dynamics simulation is to find the specific motion in each prismatic joint in MSC ADAMS. After each motion is determined, the motion of the working platform should match with the literature mentioned. Then each joint force can be measured and compared. To put the general motion on the center of the platform will not give out the correct force because the actuating force is separated into the prismatic joints and the ball joints as well. The force in the prismatic joints can be measured using the marker at the end of the upper limb.
108
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Cleve Moler, “The Growth of MATLAB® and The MathWorks over Two Decades”, The
MathWorks News & Notes, January 2006.
D.Negrut, A.Dyer, “ADAMS/Solver Primer”, Ann Arbor, August, 2004.
FU,TSENG-TI, “Applications of Computer Simulation in Mechanism Teaching”,
National Taiwan University, Taipei, Taiwan 10660, Republic of China, May 28, 2003.
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Robert J. Webster III, “Design of a Bone-Attached Parallel Robot for Percutaneous
Cochlear Implantation”, IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING,
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Commercial Vehicle Suspension System”, Proceedings of the 6th WSEAS International
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February 16-19, 2007, pp.203-207.
109
M. Almonacid, R. J. Saltarén, R. Aracil, and O. Reinoso, “Motion Planning of a
Climbing Parallel Robot”, IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION,
VOL. 19, NO. 3, JUNE 2003.
Miller.K, http://www.parallemic.org/Reviews/Review002.html, February. 17, 2013.
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January. 6, 2013.
M. Uchiyama, http://www.space.mech.tohoku.ac.jp/research/hexa/hexa-e.html,
February.25, 2013.
Tang, Chin Pei, “Lagrangian Dynamic Formulation of a Four-Bar Mechanism with
Minimal Coordinates”, March 2006.
Tsai, Lung-Wen, “Robot Analysis: The Mechanics of Serial and Parallel Manipulators”,
Wiley-Interscience Publication, 1999.
Tsai, Meng-Shiun, Wei-Hsiang Yuan, “Inverse dynamics analysis for a 3-PRS parallel mechanism based on a special decomposition of the reaction forces”, Mechanism and
Machine Theory, 45 (2010), pp.1491–1508.
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Wang, G. Gary,“Definition and review of virtual prototyping”, University of Manitoba.
Wilczyński D, Auguściński A, Dudziak M, Talaśka K, “CONTROL DESIGN OF
PARALLEL MANIPULATOR IN LABVIEW SYSTEM”, Chair of Basis of Machine
Design, Poznan University of Technology (Poland). 2008, Winnipeg, MB, R3T 5V6
Canada.
R.L. Williams II, “Engineering Biomechanics of Human Motion”, NoteBook self- published for: ME 467/BME 567 Biomechanics, Ohio University , pp.5,66.
R. L. Williams II, “An Introduction to Robotics”, EE/ME 4290/5290 Mechanics and
Control of Robotic Manipulators, p.13, p76, 2012.
R. L. Williams II, “Homework 8”, ME 301 Kinematics and Dynamics of Machines, 2012.
R. L. Williams II, “Robot Mechanics”, EE/ME 4290/5290 Mechanics and Control of
Robotic Manipulators, p.5, 2010.
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Mechanics and Control of Robotic Manipulators, p.5, 2010.
111
R. L. Williams II, “Mechanism Kinematics & Dynamics and Vibrational Modeling”, ME
3011 Kinematics & Dynamics of Machines, PP..84, 86, 2012, Bob Productions.
112
APPENDIX A: 2R PLANAR SERIAL ROBOT INVERSE DYNAMICS INPUT
ROTATIONAL TABULAR DATA
Joint1 rotational input data 1.000000E-002 4.924834E-003 2.000000E-002 9.851423E-003 3.000000E-002 1.478002E-002 4.000000E-002 1.971088E-002 5.000000E-002 2.464426E-002 6.000000E-002 2.958043E-002 7.000000E-002 3.451966E-002 8.000000E-002 3.946222E-002 9.000000E-002 4.440839E-002 1.000000E-001 4.935845E-002 1.100000E-001 5.431270E-002 1.200000E-001 5.927141E-002 1.300000E-001 6.423491E-002 1.400000E-001 6.920348E-002 1.500000E-001 7.417745E-002 1.600000E-001 7.915713E-002 1.700000E-001 8.414284E-002 1.800000E-001 8.913492E-002 1.900000E-001 9.413370E-002 2.000000E-001 9.913952E-002 2.100000E-001 1.041527E-001 2.200000E-001 1.091737E-001 2.300000E-001 1.142028E-001 2.400000E-001 1.192405E-001 2.500000E-001 1.242870E-001 2.600000E-001 1.293428E-001 2.700000E-001 1.344082E-001 2.800000E-001 1.394838E-001 2.900000E-001 1.445700E-001 3.000000E-001 1.496670E-001 3.100000E-001 1.547755E-001 3.200000E-001 1.598959E-001 3.300000E-001 1.650286E-001 3.400000E-001 1.701741E-001 3.500000E-001 1.753330E-001 3.600000E-001 1.805057E-001 3.700000E-001 1.856928E-001 3.800000E-001 1.908948E-001 3.900000E-001 1.961123E-001 113
4.000000E-001 2.013458E-001 4.100000E-001 2.065960E-001 4.200000E-001 2.118635E-001 4.300000E-001 2.171489E-001 4.400000E-001 2.224529E-001 4.500000E-001 2.277761E-001 4.600000E-001 2.331193E-001 4.700000E-001 2.384832E-001 4.800000E-001 2.438686E-001 4.900000E-001 2.492762E-001 5.000000E-001 2.547069E-001 5.100000E-001 2.601616E-001 5.200000E-001 2.656411E-001 5.300000E-001 2.711463E-001 5.400000E-001 2.766782E-001 5.500000E-001 2.822379E-001 5.600000E-001 2.878264E-001 5.700000E-001 2.934447E-001 5.800000E-001 2.990942E-001 5.900000E-001 3.047758E-001 6.000000E-001 3.104910E-001 6.100000E-001 3.162410E-001 6.200000E-001 3.220273E-001 6.300000E-001 3.278513E-001 6.400000E-001 3.337146E-001 6.500000E-001 3.396188E-001 6.600000E-001 3.455657E-001 6.700000E-001 3.515571E-001 6.800000E-001 3.575950E-001 6.900000E-001 3.636813E-001 7.000000E-001 3.698184E-001 7.100000E-001 3.760086E-001 7.200000E-001 3.822543E-001 7.300000E-001 3.885583E-001 7.400000E-001 3.949234E-001 7.500000E-001 4.013527E-001 7.600000E-001 4.078496E-001 7.700000E-001 4.144175E-001 7.800000E-001 4.210605E-001 7.900000E-001 4.277826E-001 8.000000E-001 4.345885E-001 8.100000E-001 4.414832E-001 8.200000E-001 4.484721E-001 8.300000E-001 4.555613E-001 114
8.400000E-001 4.627574E-001 8.500000E-001 4.700677E-001 8.600000E-001 4.775004E-001 8.700000E-001 4.850648E-001 8.800000E-001 4.927710E-001 8.900000E-001 5.006308E-001 9.000000E-001 5.086574E-001 9.100000E-001 5.168661E-001 9.200000E-001 5.252744E-001 9.300000E-001 5.339030E-001 9.400000E-001 5.427761E-001 9.500000E-001 5.519228E-001 9.600000E-001 5.613779E-001 9.700000E-001 5.711849E-001 9.800000E-001 5.813976E-001 9.900000E-001 5.920857E-001 1.000000E+000 6.033406E-001
Joint2 rotational input data 1.000000E-002 6.685570E-003 2.000000E-002 1.342144E-002 3.000000E-002 2.020794E-002 4.000000E-002 2.704538E-002 5.000000E-002 3.393411E-002 6.000000E-002 4.087450E-002 7.000000E-002 4.786692E-002 8.000000E-002 5.491176E-002 9.000000E-002 6.200942E-002 1.000000E-001 6.916033E-002 1.100000E-001 7.636493E-002 1.200000E-001 8.362367E-002 1.300000E-001 9.093705E-002 1.400000E-001 9.830555E-002 1.500000E-001 1.057297E-001 1.600000E-001 1.132100E-001 1.700000E-001 1.207470E-001 1.800000E-001 1.283414E-001 1.900000E-001 1.359937E-001 2.000000E-001 1.437045E-001 2.100000E-001 1.514745E-001 2.200000E-001 1.593044E-001 2.300000E-001 1.671948E-001 2.400000E-001 1.751466E-001 2.500000E-001 1.831604E-001 115
2.600000E-001 1.912370E-001 2.700000E-001 1.993774E-001 2.800000E-001 2.075822E-001 2.900000E-001 2.158524E-001 3.000000E-001 2.241889E-001 3.100000E-001 2.325927E-001 3.200000E-001 2.410647E-001 3.300000E-001 2.496060E-001 3.400000E-001 2.582176E-001 3.500000E-001 2.669008E-001 3.600000E-001 2.756565E-001 3.700000E-001 2.844861E-001 3.800000E-001 2.933908E-001 3.900000E-001 3.023719E-001 4.000000E-001 3.114307E-001 4.100000E-001 3.205687E-001 4.200000E-001 3.297874E-001 4.300000E-001 3.390882E-001 4.400000E-001 3.484729E-001 4.500000E-001 3.579431E-001 4.600000E-001 3.675005E-001 4.700000E-001 3.771470E-001 4.800000E-001 3.868845E-001 4.900000E-001 3.967149E-001 5.000000E-001 4.066405E-001 5.100000E-001 4.166634E-001 5.200000E-001 4.267858E-001 5.300000E-001 4.370103E-001 5.400000E-001 4.473392E-001 5.500000E-001 4.577754E-001 5.600000E-001 4.683215E-001 5.700000E-001 4.789805E-001 5.800000E-001 4.897556E-001 5.900000E-001 5.006498E-001 6.000000E-001 5.116668E-001 6.100000E-001 5.228100E-001 6.200000E-001 5.340834E-001 6.300000E-001 5.454909E-001 6.400000E-001 5.570368E-001 6.500000E-001 5.687258E-001 6.600000E-001 5.805625E-001 6.700000E-001 5.925521E-001 6.800000E-001 6.047001E-001 6.900000E-001 6.170123E-001 116
7.000000E-001 6.294948E-001 7.100000E-001 6.421542E-001 7.200000E-001 6.549978E-001 7.300000E-001 6.680330E-001 7.400000E-001 6.812681E-001 7.500000E-001 6.947118E-001 7.600000E-001 7.083737E-001 7.700000E-001 7.222640E-001 7.800000E-001 7.363940E-001 7.900000E-001 7.507757E-001 8.000000E-001 7.654224E-001 8.100000E-001 7.803485E-001 8.200000E-001 7.955700E-001 8.300000E-001 8.111044E-001 8.400000E-001 8.269709E-001 8.500000E-001 8.431911E-001 8.600000E-001 8.597888E-001 8.700000E-001 8.767910E-001 8.800000E-001 8.942278E-001 8.900000E-001 9.121334E-001 9.000000E-001 9.305470E-001 9.100000E-001 9.495134E-001 9.200000E-001 9.690846E-001 9.300000E-001 9.893213E-001 9.400000E-001 1.010295E+000 9.500000E-001 1.032093E+000 9.600000E-001 1.054817E+000 9.700000E-001 1.078597E+000 9.800000E-001 1.103592E+000 9.900000E-001 1.130010E+000 1.000000E+000 1.158123E+000
117
APPENDIX B: FOUR BAR MECHANISM INVERSE DYNAMICS SPLINES
Shaking forces of the four-bar mechanism at Joint5 and Joint6 are shown as follows:
SForceX1 vs. time
SForceX2 vs. time
118
SForceY1 vs. Time
SforceY2 vs. Time
119
APPENDIX C: STEWART-GOUGH PARALLEL MANIPULATOR FORWARD
DYNAMICS SPLINES
Example 1
The input actuating forces for example are shown as follows:
Input Force Spline for Actuator 1
Input Force Spline for Actuator 2
120
Input Force Spline for Actuator 3
Input Force Spline for Actuator 4
121
Input Force Spline for Actuator 5
Input Force Spline for Actuator 6
122
Example 3
The input actuating forces splines for example 3 in MSC ADAMS are shown as follows:
Input Force Spline for Actuator 1
Input Force Spline for Actuator 2 123
Input Force Spline for Actuator 3
Input Force Spline for Actuator 4 124
Input Force Spline for Actuator 5
Input Force Spline for Actuator 6
125
Example 4
The input actuating forces splines for example 4 are shown as follows:
Input Force Spline for Actuator 1, 3, 5
Input Force Spline for Actuator 2, 4, 6
126
APPENDIX D: MSC ADAMS/VIEW USER MANUAL
In this tutorial, I will introduce the modeling of 3D Stewart-Gough parallel manipulator in MSC ADAMS/View.
Double click the MSC ADAMS/View to enter ADAMS/View, the following dialogue box appears:
Click “New Model” Icon and come to the following interface: 127
The name is “model_1”, pull down the” Units” category, select “MKS-m, kg, N, s, degree”. Then click “OK”.
In the modeling process, the base and the platform are first created.
In the above “Bodies” Tab, select “Cylinder” in “Solids” class. 128
Then check the “Length” and “Radius” box, enter”5cm”, “100cm” separately, shown as follows:
Then pick the origin as one end of the base, drag cursor to locate the base at horizontal position, then click to confirm. Shown as follows:
In the same way, draw the platform with r=50cm on the same center on the base shown as follows: 129
During the modeling, hold “z” and drag cursor will zoom the object; hold “t” and drag the cursor will translate the object; hold “r” and drag the cursor will rotate the object.
In the “Bodies” tab, select “Link”, and enter the “Length” 100cm, as follows:
130
On the base, draw a link with L-100cm, shown as follows:
In the same method, build five more links with same length at the same place.
In the tool box select the rotate command as follows:
Enter 110˚ for the rotation. Select one of the link as object, the vertical vector through center of the base as axis to rotate. 131
The top view is shown as follows:
132
Rotate the rest of the links for 120˚, 230˚, 240˚ and 350˚ separately. Shown as follows:
Create six links of length=50cm from origin to the edge of the platform along negative x direction, then in the same method rotate them along the same axis for 50˚,
60˚, 170˚, 180˚, 290˚ and 300˚, Shown as follows: 133
The links on the platform and the base is used for marking the positions of the revolute joints.
Next translate the platform and the marking links along positive y axis for 1m.
In the tool bar select the “translate object” and set the distance into 100cm as follows: 134
When using the translate command, first select the object; then select the origin as the point to move from; at last select the point 1m above the origin as the point that move to. After the translation, the manipulator is shown as follows:
135
Next, connect the corresponding ends of the links between the platform and the base with links and remove the marking links(make sure all the links connect with the ends correctly), shown as follows: 136
To install upper limbs into the links, set the link length to 85cm and put the link from the upper ends on the platform along the long links, shown as follows:
137
Next connect the corresponding ends on the base with the end of the upper limbs.
After all the limbs are installed, remove the six long links leaving only upper limbs and lower limbs. Connect the platform and upper limbs, lower limbs and base with ball joints
; also connect upper limbs and lower limbs with prismatic joints . Both joints can be found under the “connectors” tab. At last, the base is locked to the ground by fixed connectors under the “connectors” tab. The IOS view of the complete Stewart-
Gough parallel manipulator is shown as follows:
138
How to use the measured results as input for creating splines
For example, I have the following measure to be transferred into splines in MSC
ADAMS:
139
First click the “export” as follows:
In the dialogue box, select “Numeric Data” in “File Type”, in the “Result Set Comp.
Name” box right click and select browse as follows:
140
In the following “Database Navigator” select the “time” and “Q” names for the measure. Then click “OK” as follows:
The measure spline data should show as follows:
141
Copy and paste the two columns data into text document as follows:
Save the text document and close it. In MSC ADAMS click “Import” as follows: 142
In the dialogue box, select “Test Data” , check “Create Splines” , and select the text file just saved as follows:
143
The created spline is shown in “Data Element” as follows:
The MSC ADAMS/View also includes a specific online tutorial with basic knowledge and examples.
On official website: http://www.mscsoftware.comthere is also academia forum that provides professional tips for MSC ADAMS.
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