Dynamic modelling and control of two link flexible arm robotic manipulator

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DYNAMIC MODELLING AND CONTROL OF TWO LINK FLEXIBLE ARM ROBOTIC

MANIPULATOR

NATRAJ MISHRA

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2020

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© Indian Institute of Technology Delhi (IITD), New Delhi, 2020

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DYNAMIC MODELLING AND CONTROL OF TWO LINK FLEXIBLE ARM ROBOTIC

MANIPULATOR

by

NATRAJ MISHRA

Department of Mechanical Engineering

Submitted

in fulfilment of the requirements of the degree of Doctor of Philosophy to the

INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2020

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Dedicated to Goswami Shree Tulsidas ji

नगर । र र र ॥ गर र र । ग अन ॥

( र र न : र )

अ : ग सब । स , ग , स ग

ब ब , आ ।

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Certificate

This is to certify that the thesis entitled “Dynamic Modelling and Control of Two Link Flexible Arm Robotic Manipulator” being submitted by Mr. Natraj Mishra to the Indian Institute of Technology Delhi for the award of the degree of Doctor of Philosophy is a record of bonafide work carried out by him under our supervision. This thesis is in conformity with the rules and regulations of the Indian Institute of Technology Delhi, New Delhi. We further certify that the thesis has attained a standard required for the degree of Doctor of Philosophy. The research reported and the results presented in the thesis have not been submitted, in part or full to any other institute or university for the award of any other degree or diploma.

Dr. S.P. Singh Professor

Department of Mechanical Engineering Indian Institute of Technology Delhi New Delhi-110016, India

Dr. (Late) B.C. Nakra Professor Department of Mechanical Engineering Indian Institute of Technology Delhi New Delhi-110016, India

Date:

Place: New Delhi

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Acknowledgements

वर्णानणमर्ासंघणनणं रसणनणं छन्दसणमपि ।

मंगलणनणं च कर्णारौ वन्दे वणर्ीपवनणयकौ ॥(श्री गोस्वणमी र्ुलसीदणसजी, श्रीरणमचररर्मणनस : बणलकणण्ड)

I invoke Lord Ganesh: well-wisher of all and Goddess Sarasvati who provides knowledge about alphabets and all literary skills. My ability to persevere and ability to grasp knowledge is solely due to them. I pay tribute to my teachers including my parents and circumstances faced by me, who have shaped my destiny.

वन्दे बोधमयं पनत्यं गुरं शंकररूपिर्म् ।

यमणपश्रर्ो पि वक्रोऽपि चन्द्रः सवात्र वन्द्यर्े ॥(श्री गोस्वणमी र्ुलसीदणसजी, श्रीरणमचररर्मणनस : बणलकणण्ड)

I salute to my teachers who are the embodiments of omniscient, eternal Lord Shiva whose asylum glorifies even the curved moon. It is because of Prof. (Late) B.C. Nakra that I embarked upon this mission of Ph.D. I was one of the youngest research scholar co- supervised by him. He put complete faith upon me. I have tried my best to live up to his expectations. It is because of him only; I could get the noblest person- Prof. S.P. Singh as my Ph.D supervisor. My research on Dynamic modelling and control of Two-Link Flexible arm robotic manipulator has enhanced my knowledge about robotics, vibrations, controls, finite elements and modelling of mechanical systems. It has increased my craving for doing further research in these areas. I hope that in coming future I shall strive for increasing my knowledge and utilize this knowledge for the benefit of my nation and society.

I thank my colleagues at my work place: UPES, Dehradun for their kind support. I thank the research scholars: Dr. Faisal Rahmani, Dr. Parmanand Nandihal, Mr. Anvesh Reddy, Mr.

Dinesh Kochar and Mr. Dharmender for providing me valuable suggestions and inputs for

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improving my work. For validation of mathematical model, an experimental set-up was built at Vibrations laboratory in IIT Delhi. This could not have been accomplished without the hardwork of Kumarmangalam Mishra (Intern from Thapar University, Patiala) and support of Mr. K. N. Madanasundaran (Technical Supdt., Vibration Research Laboratory, IIT Delhi) and my SRC panelists especially Prof. J.K. Dutt (Professor, IIT Delhi). This acknowledgement is incomplete if I do not express my gratitude towards my wife Mrs. Neelam Mishra. It is because of her patience and unconditional support that I could lead my research work towards completeness. There were moments when I needed great psychological help. During these tough times, my friend- Mr. Laxmi Narayan Joshi and my Chacha ji- Shri Jitender Kumar Pandey have always mentored me. At the last, I would ask for forgiveness from my supervisor for any kind of negligence or indiscipline caused by me.

र्स्मणत्प्रर्म्य प्रपर्धणय कणयं

प्रसणदये त्वणमिमीशमीड्यं । पिर्ेव िुत्रस्य सखेव सख्ुः

पप्रयः पप्रयणयणिापस देव सोढुं ॥ (श्रीमद्भगवद्गीर्ण, अध्यणय ११, श्लोक ४४) You are the God worshipable by all beings. Therefore, I bow before you with utmost respect and beg for your benevolence. Just like a father bears the impudence of his son, or a friend withstands his pragmatic friend, or an affectionate husband endures his beloved faithful wife;

in the same way, You please shower your kindness upon me and tolerate all my flaws.

Natraj Mishra

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Abstract

Robots are finding new applications in various fields. In industries, the increased rate of production within stipulated period of time along with high quality of products has become a prime requirement. This is achieved by increasing the speed of operation. This increase in speed of operation of robots combined with use of light weight structures causes problems of vibrations of links which are the major cause of positional inaccuracies at the end-effector.

Furthermore, less power consumption is another area of concern, which may be achieved by decreasing the inertia of robots. This results in lightweight links which are prone to vibrations. The present work is based on minimizing the vibrations of these lightweight robots, also known as flexible robots. The thesis focuses upon the dynamic modelling and control of a Two-Link Flexible robot having two revolute joints. For this, firstly a mathematical model of the flexible robot is prepared using Lagrangian dynamics. The mathematical model thus obtained involves coupling between the rigid and flexible motions exhibited by the flexible robot. The rigid motion is due to the motion of joints and is responsible for change in configuration of the robot while the flexible motion is due to the vibration of links. The links undergo two types of vibrations: flexural/ bending vibrations and torsional vibrations. The vibration analysis of the flexible links is done using both assumed modes method and finite elements method. A robotic system is an inertia-variant system because its configuration changes with time. As a result, the natural frequencies of the system change with time. The effect of this time-dependency of natural frequencies of links on Joint and Tip responses is taken care of during mathematical modelling. On the other hand, while using finite elements method this effect of time-dependency of natural frequencies is taken care of by continuously updating the mass and stiffness matrices of the system. Furthermore, it is also easy to take care of boundary conditions during ‘finite element analysis’. The

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control of vibrations of flexible links is achieved by using passive damping technique using viscoelastic material combined with active damping technique using piezo-ceramics. While using passive damping technique, the phenomenon of viscoelasticity is modelled using Kelvin-Voigt elements. The active vibration control of flexible links is achieved with the help of piezoelectric sensors and actuators applied in segmented fashion on the links. Direct velocity feedback is used. A hypothesis is presented for active vibration control of torsional vibrations along with the simulation results. To significantly reduce the vibration of flexible links, both the vibration control techniques are used together to achieve hybrid damping.

Since, a robot is to be used for the performance of specified tasks, trajectory planning is requirement. In the present work, trajectory planning is done using both ‘point-to-point’ and

‘continuous path’ trajectories. It is shown that through proper planning of trajectory, tip vibrations, initial jerk and joint torque requirements can be reduced significantly. While making a robot follow a certain trajectory with minimum deviation from the desired path, control techniques are required. A new control technique based on Coupled-error dynamics control technique is also presented and found quite effective. The performance of this control technique is compared with computed-torque and robust control schemes in terms of error in path followed, effect of uncertainties within the system like mass uncertainty and link flexibility and control torque output. It is found that the performance of newly developed control scheme is better than computed-torque control and close to robust control.

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साराांश

आधुपनक समय में रोबोट कण प्रयोग पवपिन्न क्षेत्रों में िोर्ण िै । रोबोट कण प्रमुख उियोग उयोगों में िोर्ण

िै। उयोग क्षेत्र की मुख् आवश्यकर्ण िोर्ी िै - त्वररर् दर से पनधणाररर् समयणंर्रणल में उच्च कोपट के

गुर्वत्तण वणले उत्पणदों कण उत्पणदन। इस आवश्यकर्ण की िूपर्ा उत्पणदन की गपर् में वृद्धि द्वणरण की जण सकर्ी िै। रोबोट की बढ़ी हुई गपर् के फलस्वरूि रोबोट की िुजणओं में कम्पन की समस्यण उत्पन्न िो

जणर्ी िै जोपक रोबोट के अंर्-प्रेरक में प्रस्र्णिनण सम्बन्धी अशुद्धियों के उत्पन्न िोने कण प्रमुख कणरर् िै।

िरन्तु, इसके िररर्णमस्वरूि रोबोट की िुजणयें िलकी िो जणर्ी िैं जोपक कम्पन उन्मुख िोर्ी िैं। इससे

रोबोट में प्रस्र्णिनण सम्बन्धी अशुद्धियणाँ और बढ़ जणर्ी िैं। यि प्रस्तुर् शोधकणया इन िलकी िुजणओं वणले

रोबोट अर्णार्् लचीले रोबोट के कम्पन को न्यूनर्म करके पदखणने िर आधणररर् िै। यि शोधप्रबंध एक पद्व-िुज लचीले रोबोट के गत्यणत्मक प्रपर्रूिर् एवं पनयंत्रर् िर केंपिर् िै। इस रोबोट में दो संपधयणाँ िैं

पजनिर केवल िररक्रमर् रूिी गपर् िी संिव िै। इसके पलए सवाप्रर्म लचीले रोबोट कण लणग्णंज की

गत्यणत्मक ििपर् द्वणरण एक गपर्र्ीय प्रपर्रूि र्ैयणर पकयण गयण िै। इस प्रकणर प्रणप्त हुए गपर्र्ीय प्रपर्रूि में लचीले रोबोट द्वणरण प्रदपशार् अनम्य एवं नम्य गपर्यों कण युग्मन शणपमल िै। अनम्य गपर् की

कणरक िैं- संपधयों की गपर्यणाँ और यि रोबोट के पवन्यणस में िररवर्ान के पलए पजम्मेदणर िै। जबपक नम्य गपर् िुजणओं में कम्पन की वजि से िोर्ी िै । िुजणओं में दो प्रकणर के कम्पन िोर्े िैं : बंकन से उत्पन्न कम्पन और मरोड़ से उत्पन्न कम्पन। इन लचीली िुजणओं में िोने वणले कम्पन कण पवश्लेषर् 'कद्धिर्

आवृपत्त ििपर्' एवं 'िररपमर् अवयव ििपर्', दोनों के द्वणरण पकयण गयण िै। 'कद्धिर् आवृपत्त ििपर्' कण उियोग करर्े समय र्ंत्र के शणसी समीकरर्ों को इस प्रिणव को सद्धम्मपलर् करने के उिरणन्त प्रणप्त पकयण गयण िै। इस ििपर् कण लचीली िुजणओं के पलए उियोग करर्े समय यर्णर्ा सीमण शर्ों कण प्रयोग पकयण गयण िै। दूसरी र्रफ 'िररपमर् अवयव ििपर्' कण प्रयोग करर्े समय प्रणकृपर्क आवृपत्तयों की इस समय-पनिारर्ण कण ध्यणन र्ंत्र के जड़त्व एवं कठोरर्ण सम्बन्धी आव्यूिों को सर्र् रूि से अयर्न करर्े

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हुए रखण गयण िै। इसके अपर्ररक्त 'िररपमर् अवयव पवश्लेषर्' के समय सीमण शर्ों कण ध्यणन रखनण िी

सरल िोर्ण िै। पद्व-िुज लचीले रोबोट के गपर्र्ीय प्रपर्रूि की िुपि सणपित्य में उिलब्ध प्रपर्फलों द्वणरण की गयी िै और सणर् िी में प्रयोगों द्वणरण िी। लचीली िुजणओं की कम्पन कण पनयंत्रर् पवस्को-इलणद्धिक

िदणर्ा के उियोग से उत्पन्न 'पनद्धिय अवमंदन प्रपवपध' द्वणरण एवं िीजोसेरणपमक के प्रयोग से उत्पन्न

'सपक्रय अवमंदन प्रपवपध' द्वणरण प्रणप्त पकयण गयण िै। 'पनद्धिय अवमंदन प्रपवपध' कण प्रयोग करर्े समय 'पवस्को-इलणद्धिपसटी' नणमक र्थ्य कण प्रपर्रूिर् केद्धिन-वॉइट अवयवों द्वणरण पकयण गयण िै। लचीली

िुजणओं कण सपक्रय कम्पन पनयंत्रर् िीजोइलेद्धरिक सेंससा एवं एक्चुएटसा को िुजणओं िर खद्धण्डर् पवधणन द्वणरण अनुप्रयुक्त कर पकयण गयण िै। प्रत्यक्ष वेग प्रपर्िुपि कण प्रयोग पकयण गयण िै। मरोड़ से उत्पन्न कम्पन के सपक्रय कम्पन पनयंत्रर् िेर्ु एक िररकिनण प्रस्तुर् की गयी िै। लचीली िुजणओं में कम्पन को

अपिव्यंजकर्णिूवाक कम करने के पलए दोनों िी प्रकणर की कम्पन पनयंत्रर् प्रपवपधयों कण एकसणर् प्रयोग कर संकर अवमंदन कण प्रयोग पकयण गयण िै। चूाँपक, एक रोबोट कण उियोग पनधणाररर् कणयों के पनष्पणदन

िेर्ु िोर्ण िै, प्रक्षेििर् आयोजन अत्यणवश्यक िै। प्रस्तुर् शोधकणया में प्रक्षेििर्ों कण आयोजन 'पबंदु-से- पबंदु' एवं 'अपवद्धिन्न मणगा' प्रक्षेििर् प्रपवपधयों द्वणरण पकयण गयण िै। यि प्रदपशार् पकयण गयण िै पक उपचर्

प्रकणर से आयोपजर् प्रक्षेििर्ों द्वणरण िुजणन्त कम्पनों, प्रणरंपिक झटकों एवं संपध आघूर्ा आवश्यकर्णओं

को अर्ािूर्ार्ण से कणम पकयण जण सकर्ण िै। पकसी रोबोट को एक पनपिर् प्रक्षेििर् िर न्यूनर्म पवचलन से चलणने के पलए पनयंत्रर् प्रपवपधयों की आवश्यकर्ण िोर्ी िै। 'अपिकपलर्-आघूर्ा पनयंत्रर्' एवं 'िुि

पनयंत्रर्' प्रपवपधयों कण प्रयोग इसी िेर्ु पकयण जणर्ण िै। एक अन्य पनयंत्रर् प्रपवपध : 'युद्धग्मर्-त्रुपट गपर्की' कण िी इस प्रस्तुर् कणया में प्रयोग पकयण गयण िै। प्रस्तुर् पनयंत्रर् प्रपवपध के पनष्पणदन की र्ुलनण

'अपिकपलर्-आघूर्ा पनयंत्रर्' एवं 'िुि पनयंत्रर्' व्यवस्र्णओं से िर् अनुसरर् में त्रुपट, िव्यमणन अपनपिर्र्ण व िुजण नम्यर्ण जैसी अपनपिर्र्णओं के प्रिणव और पनयंत्रक आघूर्ा की आवश्यकर्णओं जैसे

िौपर्क चरों के मणध्यम द्वणरण की गयी िै। यि िणयण जणर्ण िै पक नवीनर्ण से पवकपसर् इस पनयंत्रर्

व्यवस्र्ण की पनष्पणदन क्षमर्ण 'अपिकपलर्-आघूर्ा पनयंत्रर्' से बेिर्र िै और 'िुि पनयंत्रर्' के समीि िै ।

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Fig. 2.12: Tip deflection of Link-2 as obtained by Habib and Korayem 63 Fig. 2.13: Joint-1 angle (input) for Flexible and Rigid manipulators 65 Fig. 2.14: Joint-2 angle (input) for Flexible and Rigid manipulators 65 Fig. 2.15: Comparison between joint torques of rigid and flexible manipulators (The flexible

links are supposed to vibrate in first mode of vibration) 68

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Fig. 2.16: Comparison between joint torques of rigid and flexible manipulators (The flexible links are supposed to vibrate in first two modes of vibration) 69 Fig. 2.17: Comparison between joint torques of rigid and flexible manipulators (The flexible links are supposed to vibrate in first three modes of vibration) 71 Fig. 2.18: Comparison of responses of Joint-1 for constant and time-varying natural

frequency cases 73

Fig. 2.19: Comparison of responses of Joint-2 for constant and time-varying natural

frequency cases 74

Fig. 2.20: Tip response of flexible link-1 for time-varying and constant natural frequency

cases 74

Fig. 2.21: Tip response of flexible link-2 for time-varying and constant natural frequency

cases 75

Fig. 3.1: Dynamic analysis of Two-Link Flexible manipulator undergoing both bending and

torsional deformations. 79

Fig. 3.2: Dynamics modelling of Two-Link Flexible manipulator using two Space-frame

finite elements. 82

Fig. 3.3: Simulation results for manipulator with physical parameters described in Table 3.3 99 Fig. 3.4: Simulation results for manipulator with physical parameters described in Table 3.4

104 Fig. 3.5: Simulation results for manipulator with physical parameters described in Table 3.5.

109 Fig. 3.6: Simulation results for manipulator with physical parameters described in Table 3.6

114

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Fig. 3.7: Simulation results for manipulator with physical parameters described in Table 3.7 119 Fig. 3.8: Simulation results for manipulator with physical parameters described in Table 3.8

125 Fig. 3.9: Simulation results for manipulator with physical parameters described in Table 3.9.

130 Fig. 3.10: Simulation results in presence of gravity for manipulator with physical parameters

described in Table 3.10 136

Fig. 3.11: Calculation of excitation forces at the ends of links (the alternate model) 138 Fig. 3.12: Simulation results using the alternate model (Physical parameters are as per Table

3.3.) 140

Fig. 3.13: Simulation results using the alternate model (Physical parameters are as per Table

3.4.) 141

3.5.) 143

3.6) 144

Fig. 3.16: Effect of vibration on positional accuracy of tip of Two-Link Flexible manipulator 148 Fig. 3.17: Variation of first four natural frequencies of the Two-Link Flexible manipulator

with change in configuration of the links 151

Fig. 3.18: Comparison between Joint-1 angles obtained by AMM and FEM based approaches 153

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Fig. 3.19: Comparison between Joint-2 angles obtained by AMM and FEM based approaches 153 Fig. 3.20: Comparison between the tip deflections of Link-1 obtained by AMM and FEM

based approaches 154

Fig. 3.21: Comparison between the tip deflections of Link-2 obtained by AMM and FEM

based approaches 154

Fig. 3.22: Convergence study showing the effect of number of finite elements on slope rate of

tip of a Two-Link Flexible manipulator 155

Fig. 3.23: Convergence study showing the effect of number of finite elements on tip velocity

of a Two-Link Flexible manipulator along Y-axis 156

Fig. 4.1: Dynamic analysis of Two-Link Flexible manipulator undergoing both bending and

torsional deformations. 162

Fig. 4.2: Dynamics modelling of a Two-Link Flexible manipulator using Space-frame finite

elements. 163

Fig. 4.3: Schematic diagram for active vibration control of a smart beam/ link. 165

Fig. 4.4: Feedback control system using PD controller 169

Fig. 4.5: Diagram showing the relative placements of sensors and actuators on the flexible links of the Two-Link Flexible manipulator. (In the figure, S1 = Sensor on Link-1; S2 = Sensor on Link-2; A₁ = Actuator on Link-1 and A₂ = Actuator on Link-2.) 170

Fig. 4.6: Viscoelastic model 1 179

Fig. 4.9: Development of mathematical model of torsional piezo-actuator 186

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Fig. 4.10: Variation of desired hub angle and actual hub angle with time for Single Link

Flexible manipulator 188

Fig. 4.11: Time response of the deflection of tip of Single Link Flexible manipulator along X-

direction 189

Fig. 4.12: Time response of the deflection of tip of Single Link Flexible manipulator along Y-

direction 190

Fig. 4.13: Time-domain and frequency-domain responses of slope rates (rate of change of slope) of tip of second flexible link of Two-Link Flexible manipulator for viscoelastically

damped and undamped cases 194

Fig. 4.14: Comparison of slopes of tip of second flexible link of Two-Link Flexible manipulator between viscoelastically damped and undamped cases 196 Fig. 4.15: Comparison of velocity of tip of second flexible link of Two-Link Flexible manipulator between viscoelastically damped and undamped cases 196 Fig. 4.16: Comparison of slope rates (rate of change of slopes) of tip of second flexible link of Two-Link Flexible manipulator between undamped and actively damped cases for collocated

arrangement. 197

Fig. 4.17: Time-domain and frequency-domain responses of velocities of tip of second flexible link of Two-Link Flexible manipulator between undamped and actively damped cases

for collocated arrangement. 198

Fig. 4.18: Comparison of slope rates (rate of change of slopes) of tip of second flexible link of Two-Link Flexible manipulator for collocated and non-collocated sensor-actuator pairs. 199 Fig. 4.19: Time-domain and frequency-domain responses of tip velocities of second flexible link of Two-Link Flexible manipulator for collocated and non-collocated arrangement of

sensor-actuator pairs 201

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Fig. 4.20: Comparison of slope rates (rate of change of slopes) of tip of second flexible link of Two-Link Flexible manipulator at different values of PD gains. 202 Fig. 4.21: Comparison of linear velocities of tip of second flexible link of Two-Link Flexible

manipulator at different values of PD gains. 203

Fig. 4.22: Comparison of slope rates (rate of change of slopes) of tip of second flexible link of Two-Link Flexible manipulator between active damping control and hybrid damping control.

204 Fig. 4.23: Comparison of slopes of tip of second flexible link of Two-Link Flexible manipulator between active damping control and hybrid damping control. 204 Fig. 4.24: Comparison of bending rates of tip of second flexible link of Two-Link Flexible manipulator between active damping control and hybrid damping control. 205 Fig. 4.25: Comparison of torsional deformations of tip of second flexible link of Two-Link Flexible manipulator between active damping control case and undamped case at low values

of PD gains. 206

Fig. 4.26: Comparison of torsional deformations of tip of second flexible link of Two-Link Flexible manipulator between active damping control case and undamped case at high values

of PD gains. 206

Fig. 5.1: A Two-Link Rigid serial robot having two Revolute Joints in X-Y plane. 213 Fig. 5.2: Block diagram for Computed-Torque Control scheme 224 Fig. 5.3: Block diagram for Coupled-Error Dynamics Control scheme 226 Fig. 5.4: Trajectory formulation for the robotic manipulator shown in Fig. 5.1 231 Fig. 5.5: Comparison of paths traced by end-effector of Two-Link Rigid robot in X-Y plane as obtained by CED and CTC. (PD gains for both the controllers are as per the Kp and Kv

matrices provided in first column of Table 5.4.) 233

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Fig. 5.6: Comparison of paths traced by end-effector of Two-Link Rigid robot in X-Y plane as obtained by CED and CTC. (PD gains for Joint-1: K_p1 = 100; K_v1 = 20 and PD gains for

Joint-2: Kp2 = 49; Kv2 = 14.) 234

Fig. 5.7: Comparison of paths traced by end-effector of Two-Link Rigid robot in X-Y plane as obtained by CED and CTC. (PD gains for Joint-1: K_p1 = 289; K_v1 = 34 and PD gains for

Joint-2: Kp2 = 121; Kv2 = 22.) 234

Fig. 5.8: Comparison of paths traced by end-effector of Two-Link Rigid robot in X-Y plane as obtained by CED and CTC. (PD gains for Joint-1: Kp1 = 324; Kv1 = 36 and PD gains for

Joint-2: K_p2 = 144; K_v2 = 24.) 235

Fig. 5.9: Comparison between control torques provided by CED and CTC at Joint 1. (The control torque provided by CED is scaled up by multiplying with a factor of 100; PD gains for Joint-1: K_p1 = 100; K_v1 = 20 and PD gains for Joint-2: K_p2 = 49; K_v2 = 14.) 236 Fig. 5.10: Comparison between control torques provided by CED and CTC at Joint 2. (The control torque provided by CED is scaled up by multiplying with a factor of 20; PD gains for Joint-1: Kp1 = 100; Kv1 = 20 and PD gains for Joint-2: Kp2 = 49; Kv2 = 14.) 236 Fig. 5.11: Comparison of paths traced by end-effector of Two-Link Flexible robot in X-Y plane as obtained by CED and CTC. (PD gains for Joint-1: K_p1 = 289; K_v1 = 34 and PD gains

for Joint-2: Kp2 = 121; Kv2 = 22.) 238

Fig. 5.12: Error in X-coordinate of end-effector of Two-Link Rigid manipulator due to

uncertainty in mass of payload 239

Fig. 5.13: Error in Y-coordinate of end-effector of Two-Link Rigid manipulator due to

uncertainty in mass of payload 239

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Fig. 5.14: Comparison of paths traced by end-effector of Two-Link Flexible robot in X-Y plane as obtained by CED-based and Robust controllers. (PD gains for Joint-1: K_p1 = 64; K_v1

= 16 and PD gains for Joint-2: Kp2 = 25; Kv2 = 10.) 241

Fig. 5.15: Comparison of paths traced by end-effector of Two-Link Flexible robot in X-Y plane as obtained by CED-based and Robust controllers. (PD gains for Joint-1: K_p1 = 289;

Kv1 = 34 and PD gains for Joint-2: Kp2 = 121; Kv2 = 22.) 242 Fig. 5.16: Comparison of paths traced by end-effector of Two-Link Flexible robot in X-Y plane as obtained by CED-based and Robust controllers. (PD gains for Joint-1: Kp1 = 289;

K_v1 = 34; K_i1 = 256 and PD gains for Joint-2: K_p2 = 121; K_v2 = 22; K_i2 = 75.) 243 Fig. 5.17: Comparison between control torques provided by CED-based and Robust controllers at Joint 1. (The control torque provided by CED is scaled up by multiplying with a factor of 20; PD gains for Joint-1: K_p1 = 100; K_v1 = 20 and PD gains for Joint-2: K_p2 = 49;

Kv2 = 14.) 244

Fig. 5.18: Comparison between control torques provided by CED-based and Robust controllers at Joint 2. (PD gains for Joint-1: Kp1 = 100; Kv1 = 20 and PD gains for Joint-2: Kp2

= 49; Kv2 = 14.) 244

Fig. 5.19: Comparison of positions of end-effector in X-Y Plane of Two-Link Flexible manipulator obeying Continuous path trajectory: obtained by using different controllers (PD gains used are: K_p1 = 64; K_v1 = 16; K_p2 = 25; K_v2 = 10) 246 Fig. 5.20: Comparison between control torques provided by different controllers at Joint 1 of Two-Link Flexible manipulator obeying Continuous path trajectory (PD gains used are: Kp1 =

64, K_v1 = 16; K_p2 = 25; K_v2 = 10) 246

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Fig. 5.21: Comparison between control torques provided by different controllers at Joint 2 of Two-Link Flexible manipulator obeying Continuous path trajectory (PD gains used are: K_p1 =

64, Kv1 = 16; Kp2 = 25; Kv2 = 10) 247

Fig. 5.22:Calculation of Joint torque requirement for Two-Link Flexible manipulator 250 Fig. 5.23: Joint torque requirements for Two-Link Rigid and Flexible manipulator for cubic

polynomial trajectory with final time 2 second. 251

Fig. 5.24: Joint torque requirements for Two-Link Rigid and Flexible manipulator for 5-

degree polynomial trajectory with final time 2 second. 252

Fig. 5.25: Joint torque requirements for Two-Link Rigid and Flexible manipulator for 5-

degree polynomial trajectory with final time 20 second. 253

Fig. 5.26: FFT of Joint 1 torque for Two-Link Rigid manipulator for cubic polynomial

trajectory with final time 2 second. 253

Fig. 5.27: FFT of Joint 2 torque for Two-Link Rigid manipulator for cubic polynomial

Fig. 5.28: FFT of Joint 1 torque for Two-Link Flexible manipulator for cubic polynomial

Fig. 5.29: FFT of Joint 2 torque for Two-Link Flexible manipulator for cubic polynomial

Fig. 5.30: FFT of Joint 1 torque for Two-Link Rigid manipulator for 5-degree polynomial

Fig. 5.32: FFT of Joint 1 torque for Two-Link Flexible manipulator for 5-degree polynomial

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Fig. 5.38: Tip deflections of second link of Two-Link Flexible manipulator at different

trajectories 261

Fig. 6.1: Experimental set-up for Single Link Flexible manipulator (without payload) at

Vibrations laboratory in IIT Delhi. 265

Fig. 6.2: Amplitude spectrum for tip vibration of Single Link Flexible manipulator without

payload. 268

Fig. 6.3: Experimental set-up for Single Link Flexible manipulator (with payload) at

Vibrations laboratory in IIT Delhi 269

Fig. 6.4: Amplitude spectrum for tip vibration of Single Link Flexible manipulator with

payload. 269

Fig. 6.5: Square wave input voltage provided to the motor at the joint. 271 Fig. 6.6: Comparison between Joint angles of Single Link Flexible manipulator (without

payload) obtained through experiment and simulation 272

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Fig. 6.7: Comparison between tip accelerations of Single Link Flexible manipulator (without

Fig. 6.8: Comparison between Joint angles of Single Link Flexible manipulator (with

Fig. 6.9: Comparison between tip accelerations of Single Link Flexible manipulator (with

payload) obtained through experiment and simulation. 273

Fig. 6.10: Sine wave input voltage provided to the motor at the joint. 274 Fig. 6.11: Comparison between Joint angles of Single Link Flexible manipulator (with

Fig. 6.12: Comparison between tip accelerations of Single Link Flexible manipulator (with

payload) obtained through experiment and simulation. 275

Fig. 6.13: Experimental set-up of Two Link Flexible manipulator developed at Vibrations

laboratory in IIT Delhi. 276

Fig. 6.14: Input voltages provided to both the motors at Joint-1 and Joint-2 of the Two-Link

Flexible manipulator 279

Fig. 6.15: Comparison between Joint-1 angles of Two-Link Flexible manipulator obtained

experimentally and through simulation 280

Fig. 6.16: Comparison between Joint-2 angles of Two-Link Flexible manipulator obtained

Fig. 6.17: Comparison between tip accelerations of Two-Link Flexible manipulator obtained

Fig. 6.18: Tip response using rectangular pulse input of duty cycle 5% 284 Fig. 6.19: Comparison between Joint angles of Single-Link Flexible manipulator obtained

through experiment and simulation 285

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Fig. 6.20: Comparison between tip accelerations of Single-Link Flexible manipulator

obtained through experiment and simulation 286

Fig. 6.21: Comparison between Joint 1 angles of Two-Link Flexible manipulator obtained

Fig. 6.22: Comparison between Joint 2 angles of Two-Link Flexible manipulator obtained

Fig. 6.23: Comparison between tip accelerations of Two-Link Flexible manipulator obtained

Fig. 6.24: Comparison between Joint-1 angles obtained through experiment under the

presence and absence of viscoelastic damping 291

Fig. 6.25: Comparison between Joint-2 angles obtained through experiment under the

Fig. 6.26: Comparison between tip accelerations obtained through experiment under the

Fig. 6.27: Comparison between Joint-1 angles obtained through simulation under the

Fig. 6.28: Comparison between Joint-1 angles obtained through simulation under the

Fig. 6.29: Comparison between tip accelerations obtained through simulation under the

Fig. A.1: A Two-Link Flexible manipulator with three revolute joints 323

Fig. B.1: The viscoelastic bar element 335

Fig. C.1: Schematic diagram of the experimental set-up of Two-Link Flexible manipulator 338

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Fig. C.2: Fabrication and assembly of experimental test-rig for flexible manipulator 339

Fig. C.3: The complete experimental test-rig 340

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List of Tables

Table 1.1: Major breakthroughs in the field of flexible robotics since 1975 24 Table 1.2: Design methods used by various researchers for modelling the flexible

manipulators 26

Table 1.3: Control approaches used by various researchers for control of flexible

manipulators 28

Table 2.1: Physical parameters of single flexible link 57

Table 2.2: Link parameters for Two-Link Flexible manipulator [Habib and Korayem, 2015]

59 Table 2.3: Link parameters for Two-Link Flexible manipulator 64 Table 2.4: Variation of natural frequencies (Eigen values) of links of the Two-Link Flexible manipulator at different payloads attached at the tip of the second link 66 Table 2.5: Table showing the effect of inclusion of higher modes of vibration on joint torque

requirement for the flexible manipulator 72

Table 2.6: Physical parameters of Two-Link Flexible manipulator to study the effect of time-

varying frequency 73

Table 3.1: Physical parameters of Two-Link Flexible manipulator used for validation

[Karagulle et al., 2017] 88

Table 3.2: First mode natural frequencies of Two-Link Flexible manipulator at different

configurations 89

Table 3.3: Physical parameter for Rigid-Flexible manipulator 94 Table 3.4: Physical parameters for Rigid-Flexible manipulator 99 Table 3.5: Physical parameters for Rigid-Flexible manipulator 105

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Table 3.6: Physical parameters for Flexible-Flexible manipulator 110 Table 3.7: Physical parameters for Flexible-Flexible manipulator 115 Table 3.8: Physical parameters for Flexible-Rigid manipulator 120 Table 3.9: Physical parameters for Flexible-Rigid manipulator 125 Table 3.10: Physical parameters for Two-Link Flexible Manipulator under the effect of

gravity 131

Table 3.11: Physical parameters for Two-Link Flexible manipulator undergoing combined

bending-torsional vibrations 146

Table 3.12: Physical parameters of both the links of Two-Link Flexible manipulator 149 Table 3.13: Physical parameters of Two-Link Flexible manipulator used for comparison

between AMM and FEM based approaches. 152

Table 3.14: Time required for executing the Finite Element program with different number of

finite elements 157

Table 4.1: Table describing the relative positions of sensors and actuators placed on the

flexible links 171

Table 4.2: Model representation of viscoelasticity 175

Table 4.3: Stored and dissipated energy functions 177

Table 4.4: System parameters for validation for single flexible link 188 Table 4.5: Parameters table for simulation of Two-Link Flexible manipulator 192 Table 4.6: Change in natural frequencies of the Two-Link Flexible manipulator due to the

presence of viscoelastic and piezoelectric layers 207

Table 5.1: Comparison between CTC and CED 227

Table 5.2: Physical parameters for Two-Link manipulator 228

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Table 5.3: Boundary conditions for formulation of joint-space trajectories 229 Table 5.4: Finding the control gains for CED-based controller and CTC-based controller for

critically damped response 232

Table 5.5: Trajectories used in present study 249

Table 6.1: Physical parameters for single-link flexible manipulator 266 Table 6.2: Physical parameters for Two Link Flexible manipulator 277 Table 6.3: Specifications of electronic instruments used during experiment 278

Table 6.4: Specifications of viscoelastic material 283

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Nomenclature

AMM Assumed modes method

FEM Finite elements method

EB Euler-Bernoulli

DH Denavit-Hartenberg

CMS Component mode synthesis

LQR Linear quadratic regulator

ERLS Equivalent rigid link system

PD Proportional-derivative

PZT Lead zirconate titanate

IRC Integral resonant control

ANN Artificial neural network

MLCC Maximum load carrying capacity

SMC Sliding mode control

MPC Model predictive control

PSO Particle swarm optimization

FLC Fuzzy logic control

DDC Decomposed dynamic control

VEPSO Vector evaluated particle swarm optimization

MGA Modified genetic algorithm

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ASTM American Society for Testing and Materials MIMSC Modified independent modal space control

ACL Active constrained layer

ADF Anelastic displacement field(s)

S/As Sensors and actuators

ACLD Active constrained layer damping

CP Continuous path

PTP Point-to-point

CTC Computed torque control

CED Coupled-error dynamics

C.G./ CG Center of gravity

N.A. Neutral axis

PID Proportional-integral-derivative

SISO Single input/single output

T.F. Transfer function

i Link number

j Joint number; √−1

pi Position of any point on Link-i with respect to inertial frame

Li Length of Link-i

θi Joint angle pertaining to Link-i

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xi Distance measured along un-deformed neutral axis of Link-i wi or wi(xi,t) Elastic deflection of Link-i undergoing flexural deformation with

respect to x_i

wi* Elastic deflection of end-point of Link-i undergoing flexural deformation with respect to x_i

𝑤_𝑖^∗′ Bending angle at end-point of Link-i = ^𝑑𝑤^𝑖

∗ 𝑑𝑥𝑖

Superscript (′) Differentiation w.r.t. space x, i.e. ^𝑑

𝑑𝑥 or transpose of a matrix ri Position coordinate of any point on Link-i measured w.r.t. local

frame attached to Link-i

ri* Position coordinate of end point of Link-i measured w.r.t. local frame attached to Link-i

𝑝̇_𝑖 Velocity of any point on Link-i measured w.r.t. inertial frame T, L, Lpf Transformation matrix

K.E. Kinetic energy

P.E. Potential energy

S.E. Strain energy

G.P.E. Gravitational potential energy

g Acceleration due to gravity (9.81 m/s²)

ρi Density of Link-i

Ai Area of cross-section of Link-i

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Ei Young’s modulus of elasticity of Link-i

Ii Area moment of inertia of Link-i

p Position coordinates w.r.t. inertial frame or distributed load

v Velocity coordinates w.r.t. inertial frame; voltage; bending deflection Subscript- (p) Payload or piezo

ℒ Lagrangian

qj Generalized coordinate for Joint-j

Qj Generalized external force applied at Joint-j [M(q)] Inertia matrix for robotic system

[H(q, q̇)] Centrifugal and Coriolis force/torque vector for robotic system [G(q)] Gravity vector for robotic system

[C(q)] Matrix associated with velocity dependent terms for robotic system [K(q)] Stiffness matrix for robotic system

n Number of modes in AMM; number of flexible degrees of freedom

Wn Mode shape function for nth mode

Tn Time-dependent function for nth mode

m Number of assumed modes; mass; mass per unit length Mpi Effective mass of payload attached at end-point of Link-i

Jpi Effective mass moment of inertia attached at end-point of Link-i z Root of the frequency equation obtained during application of AMM

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xxxiv Subscript- (h) Hub

ωn Natural angular frequency of vibration of nth mode ωdn Damped angular frequency of vibration of nth mode

ξ Damping ratio of nth mode; intrinsic or isoparametric coordinate in FEM

𝜓(𝑡) A time-dependent function

Ω Angular frequency

D Differential operator, ^𝑑

𝑑𝑡

ϕi or ϕi(xi, t) Torsional deformation of any point on Link-i ϕi* Torsional deformation of end point of Link-i

bi Width of Link-i

τ Torque; shear stress

Subscript- (r) Rigid

Subscript- (f) Flexible; final

N Rigid degrees of freedom; Shape function along axial direction in FEM;

[Mff], 𝒎_𝑮^𝒆 Global mass matrix [Kff], 𝒌_𝑮^𝒆 Global stiffness matrix

m^e Element mass matrix (global)

k^e Element stiffness matrix (global)

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Ff Global force vector

𝑓_𝐺^𝑒 Element load vector (global)

Cff Global damping matrix

E Young’s modulus of elasticity; spring constant

G Modulus of rigidity

Iy Area moment of inertia about y-axis

Iz Area moment of inertia about z-axis

J Polar moment of inertia; complex compliance

Jm; Im Mass moment of inertia

le Length of element in FEM

𝒎_𝒆^𝒍 Local mass matrix

𝒌_𝒆^𝒍 Local stiffness matrix

𝑓^𝒍 Local force vector

py Distributed load along y-direction pz Distributed load along z-direction

𝝀 Matrix of direction cosines

l, m, n Direction cosines along x-, y- and z-axes respectively

Q Global degrees of freedom

Ql Local degrees of freedom

vʹ Slope

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α; 𝜙 Twist

θ Angular rotation of outermost fibre of beam in torsion

qb Bending degree of freedom in FEM

qT Torsion degree of freedom in FEM

H Flexural shape function in FEM

U Strain energy

Ɖ Electric displacement

ℰ Electric field

𝜀 Dielectric constant

𝜖 Strain

𝜎 Stress

𝑠 Compliance

𝑑 Piezoelectric constant

𝑣_𝑠 Voltage generated by piezo-sensor

𝐶_𝑓, 𝐶 Capacitance of piezo-sensor 𝑅_𝑓, 𝑅 Resistance of piezo-sensor

bps Width of piezo-sensor

bpa Width of piezo-actuator

𝑣_𝑎 Voltage applied at actuator

h Thickness of Link

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hp Thickness of piezo-ceramic

𝑤̇ Rate of change of bending deflection

𝑤̇^′ Slope rate or rate of change of slope

M Bending moment

Superscript T Transpose

Kp Proportional gain

Kv Derivative gain

η Dynamic viscosity; modal coordinate; parameter in robust control

u Extension; control torque

Y Complex modulus

V Volume

𝜙 Dissipation function

ω Angular velocity

𝒄_𝒆^𝒍 Local damping matrix

𝒄_𝑮^𝒆 Global damping matrix

T Twisting torque

𝛾 Shear strain

r Radius

[W] Modal matrix

Subscript- (d) Desired

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t Time

tf Final time of trajectory

𝜃_𝑒 Joint error

w Disturbance

𝛿 Parameter in robust control

Δ Parameter in robust control

In Identity matrix of order n

eq. Equation

Exp Experimental

Sim Simulation

Dynamic modelling and control of two link flexible arm robotic manipulator

DYNAMIC MODELLING AND CONTROL OF TWO LINK FLEXIBLE ARM ROBOTIC

MANIPULATOR

NATRAJ MISHRA

DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2020

© Indian Institute of Technology Delhi (IITD), New Delhi, 2020

DYNAMIC MODELLING AND CONTROL OF TWO LINK FLEXIBLE ARM ROBOTIC

MANIPULATOR

by

NATRAJ MISHRA

Department of Mechanical Engineering

INDIAN INSTITUTE OF TECHNOLOGY DELHI

JANUARY 2020

Dedicated to Goswami Shree Tulsidas ji

Certificate

Acknowledgements

Abstract

साराांश

Contents

List of Figures

List of Tables

Nomenclature