Dynamic model based optimum design of robotic systems

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DYNAMIC MODEL BASED OPTIMUM DESIGN OF ROBOTIC SYSTEMS

VINAY GUPTA

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI

OCTOBER 2016

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© Indian Institute of Technology Delhi (IITD), New Delhi, 2016

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DYNAMIC MODEL BASED OPTIMUM DESIGN OF ROBOTIC SYSTEMS

by

VINAY GUPTA

Department of Mechanical Engineering

Submitted

in fulfillment of the requirements for the degree of Doctor of Philosophy to the

INDIAN INSTITUTE OF TECHNOLOGY DELHI

OCTOBER 2016

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Dedicated to

Late Professor S. Prasad

who had always been my source of inspiration, guidance and encouragement,

and my gratitude for him can never be put in words

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CERTIFICATE

This is to certify that the thesis entitled Dynamic Model Based Optimum Design of Robotic Systems being submitted by Mr. Vinay Gupta to the Indian Institute of Technology Delhi for the award of the degree of Doctor of Philosophy is a bonafide record of original research work carried out by him under our supervisions in conformity with rules and regulations of the institute.

The results contained in this thesis have not been submitted, in part or in full, to any other University or Institute for the award of any Degree or Diploma.

Dr. S. K. Saha Dr. H. Chaudhary

Professor Associate Professor

Department of Mechanical Engineering Department of Mechanical Engineering Indian Institute of Technology Delhi Malviya National Institute of Technology

New Delhi-110016, India Jaipur, Rajasthan-302017, India

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ACKNOWLEDGEMENTS

I am feeling an immense pleasure for getting the opportunity to show my gratitude to the people without whom this thesis would not have been possible.

I thank my supervisor Prof. S. K. Saha for his continuous guidance and motivation without which this thesis would not be completed. I am obliged to his patience with which he heard my queries, and provided simple solutions to them. His intuitive knowledge on the subject of multibody dynamics provides great deal of knowledge to me.

His art of giving life skills is also incredible which helped to relieve pressure throughout my Ph. D journey. I also convey my gratitude to my co-supervisor Dr. Himanshu Chaudhary from MNIT Jaipur for his timely critical comments which led to a lot of improvement in my work. His insightful guidance helped me throughout my Ph. D work.

I am thankful to the members of Student Research Committee (SRC), Prof. S. P.

Singh, Dr. S. V. Modak and Dr. Mashuq-Un-Nabi (Electrical Engineering Department) for evaluating my research plan and synopsis.

I am deeply grateful to members of Mechatronics and Programme for Autonomous Robotics (PAR) labs who made my study at IIT Delhi enjoyable and knowledgeable. To name a few Mr. D. Jaitly, lab technician, Mr. Paramanand Nandhihal for his technical discussions, and courteous gesture that cannot be expressed in words, Dr. Majid Hameed Koul for his friendly behavior and technical discussions which resulted in a collaborative work on parallel manipulators, Mr. Sasanka Sekahar Sinha for his valuable help in conceptualizing the shape optimization work, and technical discussions on several issues.

To name few more, Dr. Suril V. Shah, Mr. Rajeev lochana C. G., Mr. Arun Dayal Udai,

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Mr. Aamir Abdullah Hayat, Mr. Riby Abraham Boby, Mr. Anil Kumar Sharma, Ms.

Aparna Pandharkar, Mohammed Zubair, Mr. Sachin Kansal, Mr. Bhivraj Suthar, Mr.

Vishal Abhishek, Mr. Zubin Priyansh, Mr. Arvind S. A., Mr. Sidhartha Jatily, Ms. Sakshi, Mr. Vinoth Venketesan, Mr. Ravi Prakash Joshi, Mr. Ratan Sadanand P., Mr Ashutosh Kumar, and many more. My thanks are also due to all the office staff of the department for their kind support and co-operation. I am thankful to Simulator Development Division (SDD), Secunderabad for its financial support through a project under Prof. S. K. Saha.

I am also greatly indebted to my teachers in the past. I greatly appreciate the affection and guidance given by Late Prof. S. Prasad and his wife Mrs. Malti Prasad, my guide during master's thesis Prof. B. Sahay. They instigated research ability in me and motivated me to pursue Ph. D.

I would like to thank the management of IEC Group of Institutions for permission and relieving me from my duties. Especially, Executive Director Prof. B. N. Roy, Vice President Prof. A. K. Khare, Head of Department Prof. P. V. Kumar, faculty and friends Dr. S. S. Chauhan, Dr. Sudhir Singh, Mr. Nurul Hassan, Mr. Ashutosh Pandey, Mr. Anuj Gupta, Mr. Jatin Kumar Hasija and many more.

I am grateful to all my family members and friends for encouraging and supporting me. Especially, I thank my brothers Ajay, Vijay, loving Piyush, and my uncle Subodh for their constant encouragement throughout this period. Thanks to parents are as small as drops of water in an ocean as their support was invaluable. My friends Raghvendra and Amit are highly acknowledged for their motivation. My heartfelt thanks go to my wife Neha. Her constant support, patience and companionship have turned my journey of my Ph. D into a pleasure.

Vinay Gupta

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ABSTRACT

Robotic systems require driving torques/forces at joints to perform their tasks.

Minimization of driving torques/forces results in smaller motors. This saves a lot of energy and cost.

The thesis presents a methodology for the optimum design of the robotic systems based on minimum driving torques/forces by internal mass redistribution of links. The thesis has three major components, namely, dynamics formulation to compute torques/forces, optimization scheme to minimize torques/forces, and shape optimization of links for realizing the results of torques/forces minimization.

The recursive inverse dynamics was performed using the Decoupled Natural Orthogonal Complement (DeNOC) matrices for a point-mass system termed as DeNOC-P. The recursive algorithm computed the driving torques/forces for open-loop serial-chain robotic systems. This formulation was extended to perform dynamics of closed-loop and parallel robotic systems by converting them into equivalent serial-chain subsystems by cutting at the joints.

The minimization problem was formulated as a weighted sum of Root Mean Square (RMS) values of the driving torques/forces at the joints to follow a given trajectory or for a set of trajectories that may be required to perform typical tasks. An equimomental system of point-masses was used to replace a continuum link to exploit its inherent characteristics of linearization and discretization for optimum mass redistribution. The significant reduction in torques/forces was seen with practically realizable solutions. This was achieved by imposing practical constraints in the minimization problem. Finally, optimized mass and inertia parameters of the links obtained from minimization were realized through shape optimization based on material distribution method. The methodology presents the potential to realize any shape for given mass and inertia properties.

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LIST OF FIGURES

Fig. 1.1 Examples of robot applications ... 2

Fig. 1.2 Inverse and forward dynamics description ... 3

Fig. 1.3 Types of dynamic formulation ... 4

Fig. 1.4 Comparison of original and optimized torque of the PUMA robot ... 6

Fig. 2.1 Balancing devices ... 19

Fig. 2.2 Counter-rotary countermass balancing (CRCM) ... 20

Fig. 2.3 Counterweight balancing ... 20

Fig. 2.4 Small Element Superimposing method ... 24

Fig. 2.5 Two-views of optimized shape of Counterweights on crank ... 24

Fig. 2.6 Finite element mesh ... 25

Fig. 2.7 B-splines curve ... 25

Fig. 2.8 Boundary element ... 25

Fig. 2.9 Topology optimization of cantilever ... 26

Fig. 2.10 Optimized shape of a link ... 26

Fig. 3.1 A serial-chain robot ... 32

Fig. 3.2 The i^thrigid-link with its equimomental system of point-masses ... 32

Fig. 3.3 Three-DOF planar RRR robot with its three point-mass representations ... 41

Fig. 3.4 Inverse dynamics results for the three-DOF planar RRR robot ... 44

Fig. 3.5 The PUMA robot ... 46

Fig. 3.6 Inverse dynamics results of the PUMA robot ... 47

Fig. 3.7 Original and optimized torques of three-DOF planar RRR robot ... 53

Fig. 3.8 Original and the optimized torques of the PUMA robot for the single trajectory 54 Fig. 3.9 Original and the optimized torques of the PUMA robot for trajectory 1 ... 57

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Fig. 3.10 Original and the optimized torques of the PUMA robot for trajectory 2 ... 58

Fig. 3.11 Original and the optimized torques of the PUMA robot for trajectory 3 ... 58

Fig. 3.12 Location of the optimized mass centers of the three-DOF planar RRR robot for case a1 ... 60

Fig. 4.1 A closed-loop robotic system ... 63

Fig. 4.2 A four-link robotic system ... 66

Fig. 4.3 The two-DOF five-link planar parallel robot ... 68

Fig. 4.4 Inverse dynamics results of the five-link planar parallel robot ... 70

Fig. 4.5 The spatial RSUR four-link mechanism ... 72

Fig. 4.6 Inverse dynamics results of the spatial RSUR four-link mechanism ... 74

Fig. 4.7 Original and optimized torques of the two-DOF five-link planar parallel robot.. 77

Fig. 4.8 Mass center location after optimization of the two-DOF five-link planar parallel robot ... 77

Fig. 4.9 Original and optimized torques for the spatial RSUR four-link mechanism ... 80

Fig. 5.1 Six-DOF Stewart platform... 84

Fig. 5.2 Stewart platform cut joints and subsystems ... 93

Fig. 5.3 Active joint variables of the legs of the Stewart platform for the yaw motion .... 97

Fig. 5.4 Inverse dynamics results of the Stewart platform ... 98

Fig. 5.5 The three-RRR planar parallel robot ... 99

Fig. 5.6 Cut joints and subsystems of the three-RRR planar parallel robot ... 100

Fig. 5.7 Active joint variables of the three-RRR planar parallel robot ... 103

Fig. 5.8 Inverse dynamics results of the three-RRR planar parallel robot ... 104

Fig. 5.9 Original and optimized forces of the Stewart platform for the single trajectory 106 Fig. 5.10 Mass center locations of platform after optimization of the Stewart platform for the single trajectory ... 107

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Fig. 5.11 Original and optimized forces of the Stewart platform for trajectory 1 ... 108

Fig. 5.14 Mass center locations of platform after optimization of the Stewart platform for multiple trajectories ... 110

Fig. 5.15 Original and optimized of the three-RRR planar parallel robot ... 112

Fig. 5.16 Mass center locations after optimization of the three-RRR planar parallel robot ... 112

Fig. 6.1 Discretization of a planar link ... 119

Fig. 6.2 Discretization of a spatial link ... 119

Fig. 6.3 Two-link planar robot arm ... 123

Fig. 6.4 Original link #1 and #2 for two-link planar robot arm ... 123

Fig. 6.5 Optimized link shapes for two-link planar robot arm... 125

Fig. 6.6 Comparison of torques of the two-link planar robot arm ... 126

Fig. 6.7 RecurDyn model of the two-link robot arm with optimized shape ... 126

Fig. 6.8 Von Mises stress analysis of the two-link robot arm ... 126

Fig. 6.9 Original coupler dimensions ... 127

Fig. 6.10 Optimized shape of the coupler of spatial RSUR four-link mechanism ... 128

Fig. 6.11 Comparison of torques of spatial RSUR four-link mechanism ... 129

Fig. B.1 DH parameters and coordinate frames ... 151

Fig. E.1 Different motions of Stewart platform ... 157

Fig. E.2 Stewart Platform parts definition ... 159

Fig. E.3 Execution of AKISPL command ... 161

Fig. E.4 Mass and inertia properties of the platform ... 162

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LIST OF TABLES

Table 3.1 DH parameters of the three-DOF planar RRR robot ... 41

Table 3.2 Mass and inertia properties of the three-DOF planar RRR robot ... 42

Table 3.3 DH parameters, and the mass and inertia properties of the PUMA robot ... 45

Table 3.4 Equivalent point-masses of each link for the PUMA robot ... 45

Table 3.5 Point-masses coordinates of a link in spatial motion ... 46

Table 3.6 Optimized masses and inertia properties of the three-DOF planar RRR robot . 52 Table 3.7 Comparison of the RMS values of the torques for three-DOF planar RRR robot ... 52

Table 3.8 Optimized design variables of the PUMA robot for the single trajectory ... 53

Table 3.9 Optimized mass and inertia properties of the PUMA robot for the single trajectory ... 54

Table 3.10 Comparison of RMS values of torques of the PUMA robot for the single trajectory ... 55

Table 3.11 Initial and final positions of end-effector of the PUMA robot for multiple trajectories ... 56

Table 3.12 Optimized design variables of the PUMA robot for multiple trajectories ... 57

Table 3.13 Optimized mass and inertia properties of the PUMA robot for multiple trajectories ... 57

Table 3.14 Comparison of RMS values of torques of the PUMA robot for multiple trajectories ... 57

Table 4.1 Mass and inertia properties of two-DOF five-link planar parallel robot ... 68

Table 4.2 DH parameters of subsystems of spatial RSUR four-link mechanism ... 72

Table 4.3 Mass and inertia properties of spatial RSUR four-link mechanism ... 73

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Table 4.4 Equivalent point-masses of each link of spatial RSUR four-link mechanism ... 73 Table 4.5 Values of the coordinates of the point-masses of spatial RSUR four-link mechanism ... 73 Table 4.6 Optimized mass and inertia properties and design variables of two-DOF five- link planar parallel robot ... 76 Table 4.7 Comparison of the RMS values of the torques of two-DOF five-link planar parallel robot ... 78 Table 4.8 Optimized design variables for the spatial RSUR four-link mechanism ... 79 Table 4.9 Optimized mass and inertia properties of the spatial RSUR four-link mechanism ... 79 Table 4.10 Comparison of the RMS values of the spatial RSUR four-link mechanism ... 80 Table 5.1 The DH parameters of the k^th leg of Stewart platform ... 85 Table 5.2 Base and platform coordinates of the Stewart platform in body fixed frame .... 95 Table 5.3 Mass and inertia properties of the Stewart platform ... 96 Table 5.4 Equivalent point-masses of each link of the Stewart platform ... 96 Table 5.5 Values of the coordinates of the point-masses of the Stewart platform ... 96 Table 5.6 Base and platform coordinates of the three-RRR planar parallel robot in body fixed frame ... 102 Table 5.7 Mass and inertia properties of the three-RRR planar parallel robot ... 103 Table 5.8 Optimized design variables for the Stewart platform for the single trajectory 106 Table 5.9 Optimized mass and inertia properties of the Stewart platform for the single trajectory ... 106 Table 5.10 Comparison of the RMS values of forces of the Stewart platform for the single trajectory ... 106

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Table 5.11 Optimized design variables for the Stewart platform for multiple trajectories ... 108 Table 5.12 Optimized mass and inertia properties of the Stewart platform the multiple trajectories ... 108 Table 5.13 Comparison of RMS values of forces of the Stewart platform for multiple trajectories ... 108 Table 5.14 Optimized mass and inertia properties and design variables of the three-RRR planar parallel robot ... 111 Table 5.15 Comparison of the RMS values of the torques for the three-RRR planar parallel robot ... 111 Table 6.1 Desired mass and inertia properties of two-link planar robot arm ... 124 Table 6.2 Mass and inertia properties of optimized shape of two-link planar robot arm 125 Table 6.3 Desired mass and inertia properties of the coupler of spatial RSUR four-link mechanism ... 128 Table 6.4 Mass and inertia properties of optimized shape for the spatial RSUR four-link mechanism ... 128

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LIST OF SYMBOLS AND ACRONYMS

The important symbols and abbreviations are given below in alphabetical order, where each is explained. Notational rules followed in this thesis are as follows:

1) Italic Roman/Greek letters (lower case) refer to scalars

2) Boldface Roman/Greek lower case and upper case letters denote vectors and matrices, respectively.

1. Roman Letters

The letters are given in alphabetical order. Lower case letters appear first, followed by capital letters.

ai One of the DH parameters, representing distance between Z_iand

i1

Z axes

ak, bk The 3-dimensional vector from center to the k^thvertex of the base and moving platform of a parallel manipulator, respectively a_i The 3-dimensiaonal vector denoting distance from origin Oi to Oi+1

Aij The 6×6 twist propagation matrix from the i^th link to the j^th one bi One of the DH parameters in the direction of joint axis, i.e., Z_i-axis Ci Mass center of the i^th link

di The 3-dimensional vector denoting the mass center, Ci from origin Oi of the i^th link

d_ij The 3-dimensional vector denoting j^th point-mass of the i^th link from its originO_i,

Di The 9n6 and 21n6 point-mass matrix of i^thlink for planar and Spatial motion respectively

ei The 3-dimensional unit vector along the axis of the i^th joint fij The 3-dimensional vector of force acting on j^th point-mass

of i^th link

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xxii f_p

The 9- and 21-dimensional vector of external force on the point-masses platform of parallel robot

f_p^ The 9- and 21-dimensional vector of force on point-masses due to Lagrange multiplier on the platform of parallel robot

*

fi The 9- and 21-dimensional vector of inertia force associated to point- masses of the i^th link in planar and spatial motion, respectively ɦ Height of the initial link for shape optimization

iij ^j^th element of General Inertia Matrix of i^th link

Ii The moment of the i^th link about its mass center, Ci for planar link

, ,

Ii xx I_{i yy}_, ,I_{i zz}_, Moments of inertia, i.e., the diagonal components of 33 inertia tensor of the i^th link about its center of mass

, ,

Ii xy I_{i yz}_, , I_{i zx}_, Products of inertia, i.e.,off-diagonal components of 33 inertia tensor of the i^th link about its center of mass

J The constraint Jacobian matrix

Jk The constraint Jacobian matrix associated with k^th subsystem J The matrix contains elements of Jacobian matrix of subsystems

associated to unactuated joints used for the computation of  Ju The matrix contains elements of Jacobian matrix of legs

associated to unactuated joints in parallel manipulator

J__k The vector contains elements of Jacobian matrix of k^th subsystem associated to actuated joint in parallel robots to compute

torques/forces

łk The 3-dimensional length vector of the k^th subsystem of Stewart platform L The 33 transformation matrix between rate of Euler angle rotations and

angular velocity vector of the moving platform L Length of initial design link for shape optimization

mi Mass of the i^th link

mij Mass of the j^th point mass associated with the i^th rigid link M _i 9n9n and 21n21n mass matrix of the point-masses associated

with i^th rigid link for planar and spatial motion, respectively

n Number of rigid continuum bodies or links in the multibody system under study

N The 6nn Natural orthogonal complement (NOC) matrix ,

N_l N_d The 6nn and 6n6n Decoupled Natural Orthogonal Complement (DeNOC) matrices, respectively

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xxiii N

The 9nn and 21nn DeNOC matrices for point-masses (DeNOC-P) for planar and spatial motion, respectively Oi Origin of the i^th link where it is coupled with its previous link O Define the corresponding original value

P Prismatic

, ,

p p p  Linear position, velocity and acceleration of the center of the moving platform from the origin O of the parallel robot

Q_p The 33 rotation matrix from moving platform to base frame, in parallel robot

, ,

Q_x Q_y Q_z Elementary 33 rotation matrices between frames rotated about X, Y and Z axes, respectively

R Revolute

r_i The 3-dimensional vector from the mass center Ci to joint Oi+1 of i^thlink rij Distance of the j^th point mass associated with the i^th rigid link from its

origin, Oi+1

S Spherical

ti The 6-dimensional vector of twist of the i^th link T The 6n-dimensional generalized vector of twist

Tk The 44 homogeneous transformation matrix associated to the k^th subsystem

vi The 3-dimensional linear velocity of the origin of the i^th link, Oi

v_ij The 3-dimensional vector of acceleration of j^th point-mass of i^th link v_i The 9- and 21-dimensional vector of acceleration of point-masses of i^th

link

w^ Wrench due to Lagrange multipliers at cut joint

x Design vector for optimization

i, ,i i

x y z Mass center coordinated of thei^th link

0 Column vector of zeros with compatible length

O Matrix of Zeros or null matrix with the size compatible to the dimensions of the matrices where it appears

1 Identity matrix of compatible size

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i One of the DH parameters, the angles between, Zi and Zi+1, measured about the positive direction of Xi+1

ij Orientation of the j^th point-mass from Xi+1-axis of i^th link in three point-mass system

i ⁽¹⁾ Relative angle of the i^thlink with respect to its previous link, (i-1)^st one, that is measured about the positive direction of Zi between, Xi and Xi+1

in space motion

(2) One of the DH parameter

1, 2, 3

k k pk

  Joint variable for k^th legs of the Stewart platform

i_,

i Relative joint rate and acceleration of the i^th link, respectively

, ,

   Euler angle rotations of moving platform about X, Y, and Z axes respectively, in the parallel robot

, ,  Generalized joint angle, rate, and acceleration, respectively

* The n-dimensional vector of generalized inertia forces due to motion of links in the closed-loop system

*

p The vector of generalized inertia forces due to motion of platform, in the parallel robot

*

 The vector of inertia forces associated to unactuated joints for the computation of 

λ λ, _k Lagrange multipliers representing constraint forces/moment for k^th subsystem, i.e., constraint force/moment at cut-joint

 Width of the initial link for shape optimization

i Driving torques/forces at the i^th joint

i Generalized driving torques/forces



^sum

, i rms

 Root Mean Square torques/forces

p External forces/moments on the platform, in parallel robot

p

 Forces/Moments on the platform by the Lagrange multipliers, in the parallel robot

 Artificial density of the elements of initial link for the shape optimization

ωi, ω_i Angular velocity and acceleration of the i^th link, respectively

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Ω_p 33 skew symmetric matrix associated to vector ω_p , i.e., angular velocity of platform, in the parallel robot

 Fraction value ‗0.5‘

3. Acronyms

DeNOC Decoupled Natural Orthogonal Complement

DeNOC-P Decoupled Natural Orthogonal Complement for point-masses

DOF Degree of Freedom

DAE Differential Algebraic Equations

DH Denavit-Hartenberg

EL Euler-Lagrange

GIM Generalized Inertia Matrix

NE Newton-Euler

NOC Natural Orthogonal Complement

ODE Ordinary Differential Equations

RMS Root Mean Square

RSUR Revolute Spherical Universal Revolute

Dynamic model based optimum design of robotic systems

DYNAMIC MODEL BASED OPTIMUM DESIGN OF ROBOTIC SYSTEMS

VINAY GUPTA

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI

OCTOBER 2016

© Indian Institute of Technology Delhi (IITD), New Delhi, 2016

DYNAMIC MODEL BASED OPTIMUM DESIGN OF ROBOTIC SYSTEMS

VINAY GUPTA

INDIAN INSTITUTE OF TECHNOLOGY DELHI

OCTOBER 2016

Late Professor S. Prasad

CERTIFICATE

ACKNOWLEDGEMENTS

ABSTRACT

CONTENTS

LIST OF FIGURES

LIST OF TABLES

LIST OF SYMBOLS AND ACRONYMS

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