Acceptance Certificate

Departement of Instrumentation and ControlCollege of Engineering, Pune

(An Autonomous Institute of Government of Maharashtra)

The project entitled ”Comparison of Different Controllers for Flexible Joint” sub-

mitted by

Omkar A. Harshe 110809033

Amrut M. Chavan 110809032

Sayed Awais Daimi 110809015

is accepted for being evaluated

Dr. P. D. Shendge Dr. S. L. Patil

Guide Head

Department of Instrumentation Department of Instrumentation

and Control and Control

Date: Date:

i

Project Approval Certificate

Departement of Instrumentation and ControlCollege of Engineering, Pune

(An Autonomous Institute of Government of Maharashtra)

The project entitled ”Comparison of Different Controllers for Flexible Joint” sub-

mitted by

Omkar A. Harshe 110809033

Amrut M. Chavan 110809032

Sayed Awais Daimi 110809015

is approved for the degree of Bachelor of Technology in Instrumentation and Control

Dr. P. D. Shendge Dr. S. L. Patil Examiner

Guide Head

Department of Instrumentation Department of Instrumentation

and Control and Control

Date: Date:

ii

Acknowledgement

We would like to take this opportunity to express our gratitude to Dr. P. D. Shendge for

his guidance, encouragement and motivation during the development of this project, without

which this project would not have developed to its present form and complexity.

We are very thankful to Prof. S. B. Phadke for his guidance and help in understanding the

theoretical aspects of this project. We would not have been able to apply our logic and thinking

without his help.

We are also thankful to Mr. Divyesh Ginoya and Mr. Ashish for their help in solving the

hardware related issues.

Finally we are thankful to Dr. S. L. Patil for his tremendous support, help and for providing

us an opportunity to avail all the lab facilities.

iii

Abstract

The problem of joint flexibility was of critical importance since the use of robot started in

the fields such as space science and surveillance. All these robots required light weight and

high load to weight ratio. The implementation of these new concepts proposes the problem of

inherent flexibility. Previously state linearization controller was used to address this problem

however this technique was inefficient as it required the detail knowledge of robot parameters

and extensive computation. Also this technique was susceptible to noise and model uncertainties.

In order to overcome these limitations we have proposed the use of sliding mode controller

with Proportional-Integral Observer for flexible joint which offers robustness and immunity to

uncertainty and disturbances. The controller does not require the information of plant uncer-

tainty and disturbance. The proposed reference model is to track the plant states according

to this model. The close loop stability for this model with uncertainty and disturbance is also

proposed.

In the second part of the project various control strategies such as Linear Quadratic regula-

tor, Proportional- Integral- Derivative Controller, State-Linearization Vector and Proportional-

Integral Observer based Sliding mode controller are implemented on 2-DOF Serial Flexible Joint

(2DSFJ)Robot. The response of the controller with plant are plotted and then compared.

Advantages of using PI Observer based SMC over other techniques concludes the project.

iv

Contents

Acceptance Certificate i

Project Approval Certificate ii

Acknowledgements iii

Abstract iv

1 Introduction 1

1.1 Review of related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Organisation of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Plant Description 4

2.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Component Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Harmonic Drive #1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Harmonic Drive #2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.3 Digital motor and joint position measurement: Optical encoders . . . . . 5

2.2.4 Joint Position Limit Switches . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.5 External DC Power Supply . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.6 Flexible joint springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.7 Quanser Current Linear Amplifier Package . . . . . . . . . . . . . . . . . 6

2.2.8 Analog current measurement: Current sense resistor . . . . . . . . . . . . 6

2.3 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Analysis of Plant 9

3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.2 Modeling of stage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.3 Modeling of stage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Alternative representation of Plant . . . . . . . . . . . . . . . . . . . . . . . . . . 14

v

4 Control Strategies 16

4.1 Proportional Integral Derivative Control . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1 Proportional Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.2 Integral Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.3 Derivative Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Controller Design in State Space using State Linearization Vector . . . . . . . . . 19

4.2.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2.2 Design Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.3 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3.2 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3.3 Sliding mode Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4 Proportional-Integral Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Experimentation and Results 29

5.1 Controller design for 2-DOF Serial Flexible Joint (2DSFJ) . . . . . . . . . . . . . 29

5.2 Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2.1 Proportional-Integral-Derivative controller . . . . . . . . . . . . . . . . . . 31

5.3 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4 Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 Conclusion 43

References 44

vi

List of Figures

2.1 Two degree of freedom Serial Flexible Joint . . . . . . . . . . . . . . . . . . . . . 4

2.2 Two-channel linear current amplifier package from the Quanser AMPAQ series . 6

3.1 Schematic of Two Degree of Freedom Flexible Joint (2DSFJ) System . . . . . . . 9

3.2 Schematic of 2DSFJ Robot Stage 1 System . . . . . . . . . . . . . . . . . . . . . 10

3.3 Schematic of 2DSFJ Robot stage 2 System . . . . . . . . . . . . . . . . . . . . . 12

4.1 Block Diagram of Proportional Controller . . . . . . . . . . . . . . . . . . . . . . 17

4.2 Block Diagram of Integral Controller . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Block Diagram of Derivative Controller . . . . . . . . . . . . . . . . . . . . . . . 18

4.4 Block Diagram of PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 Representation of plant in State Space . . . . . . . . . . . . . . . . . . . . . . . 19

4.6 Sliding and reaching Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.7 PI Observer -Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 2-DOF Serial Flexible Joint (2DSFJ) . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Interface to the Actual First And Second Stages Of The 2-DOF Serial Flexible

Joint (2DSFJ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3 Simulink model for PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.4 Plant response for parameter set #1 and #2 . . . . . . . . . . . . . . . . . . . . 32

5.5 Control signal for parameter set #1 and #2 . . . . . . . . . . . . . . . . . . . . 33

5.6 Simulink Model for Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . 34

5.7 State plot of stage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.8 State plot of stage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.9 Control Signal for Stage 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.10 Control Signal for Stage 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.11 Implementation of PI observer based SMC . . . . . . . . . . . . . . . . . . . . . . 38

5.12 Simulink Model for Sliding Mode Controller . . . . . . . . . . . . . . . . . . . . . 39

5.13 Analysis of Joint 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.14 Analysis of Joint 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.15 State Estimation by Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

vii

5.16 Implementation of PI Observer based SMC . . . . . . . . . . . . . . . . . . . . . 42

viii

List of Tables

2.1 Useful Information regarding two motors . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Spring constants for extension springs . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Flexible Joint #1 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Linear current amplifier (each channel) . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Flexible Joint #2 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Drive and Joint Optical Encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 External Power Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 Nomenclature of the symbols used in 2DSFJ stage 1 systems mathematical modeling 11

3.2 Nomenclature of the symbols used in 2DSFJ stage 2 systems mathematical modeling 13

5.1 Parameters for PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ix

Chapter 1

Introduction

1.1 Review of related Literature

The development of robotics in the past few years goes from the earlier standard applications

of industrial robots to the new fields such as space and service robotics and force-feedback sys-

tems. A common feature that all robots suitable for this application must share is a light weight

construction with high load to weight ratio.

A main problem which is specific for the implementation of these new robot concepts is the

issue of joint flexibility. It has received considerable attention as the major source of compli-

ance in most present day manipulator design. This joint flexibility typically arises due to gear

elasticity, shaft wind up etc., and is important in the derivation of the control law.

Joint flexibility must be taken into account in both modeling and control in order to achieve

better tracking performance, for practical applications. Unwanted oscillations due to joint flexi-

bility, imposes bandwidth limitations on all algorithm designs and may create stability problems

for feedback control loops that neglect joint flexibility.

The importance of joint flexibility in the modeling, control and performance evaluation of

robot manipulators has been established by several researchers. Spong used a singular per-

turbation model of the elastic joint manipulator dynamics and showed force control techniques

developed for rigid manipulators can be extended to the flexible joint case [1]. A complete linear

algorithm is proposed for composite robust control of flexible joint robots. Moreover, the robust

stability of the closed loop system in presence of structured and unstructured uncertainties is

modeled and converted into single perturbation form [2].

In literature, a number of feedback control schemes have been proposed to address the issue

1

of joint flexibility. A dynamic feedback controller for trajectory tracking control problem of

robotic manipulators with flexible joints is proposed in [3]. The design requires position mea-

surements on the link as well as the motor side and the velocities required in the controller are

estimated through a reduced order observer. Further robustness of the closed loop is established

by Assuming that the uncertainties satisfy certain conditions. A singular perturbation approach

is employed for the same task [4], wherein the controller needs measurement of position and elas-

tic force.

A nonlinear sliding mode state observer is used for estimating the link velocities and elastic

force time derivatives. A feedback linearization (FL) based control law made implementable

using extended state observer (ESO) is proposed for the trajectory tracking control of flexible

joint robotic system in [5]. Controller design based on the integral manifold formulation [6],

adaptive control [7], adaptive sliding mode [8], and back stepping approach [9] are some other

approaches reported in the literature.

Most of the schemes that appear in literature have certain issues that require attention.

Firstly many of them require measurements of all state variables or at least position variables

on link and motor side. Next robustness whenever guaranteed, is often highly model dependent.

Also some need knowledge of certain characteristics of the uncertainties, such as its bounds.

A variable structure observer that requires only measurement of link positions to estimate the

full state of a flexible joint manipulator is proposed in [10]. Additionally a reduced adaptive

observer that required the measurement of link and motor positions is reported in [11] and a

MIMO design for the strongly coupled joints in [12].

The design of robust, model following, sliding mode, and load frequency controller for single

area power system based on uncertainty and disturbance estimator (UDE) is discussed in [13].

The literature on UDE also mentions control of uncertain LTI systems [14], model following

sliding mode controller [15], Ackermanns formula for reaching phase elimination [16], and ro-

bust model following based on UDE [17]. The control proposed does not require the knowledge

of bounds of uncertainty and disturbance.

In this paper we introduced a Proportional-Integral observer based Sliding Mode Controller

[18] which provides robustness properties with respect to perturbations. This observer is charac-

terized by the use of two corrective actions proportional and integral instead of the convenient

proportional correction frequently employed in the Luenberger observer and/or its classical

extensions. SMC is proposed to control flexible joint manipulator with uncertainty and dis-

turbance. A non-linear disturbance is considered here and reaching phase is eliminated for

robustification.

2

1.2 Organisation of Thesis

Chapter 2 describes the system hardware, its major components along with the sensors and

transducers which are installed for measurement. It describes in detail about the parameters of

the 2-DOF Serial Flexible Joint (2DSFJ) Robot.

Chapter 3 deals with mathematical modeling of the plant based on electrical and mechanical

principles.

Chapter 4 discuss various control strategies which can be implemented on 2-DOF Serial

Flexible Joint (2DSFJ) Robot in order to efficiently control the position.

Chapter 5 lists all the experimentation and simulations performed along with the results.

The results of simulation and the plots are also included in it.

3

Chapter 2

Plant Description

2.1 An Overview

The two degree of freedom Serial Flexible Joint is depicted in figure.

Figure 2.1: Two degree of freedom Serial Flexible Joint

This robot system consists of two dc motors driving via gear harmonic gearboxes and a two

bar serial linkage. Both links are rigid. The primary link is coupled to first drive by means of

a flexible joint. It carries at its ends the second harmonic drive which is coupled to the second

rigid link via another flexible joint. Both motors and both flexible joints are instrumented with

quadrature optical encoders. Each flexible joint uses two springs. Also a thumbscrew mechanism

is available to move each spring to different anchor points along its support bars.

4

2.2 Component Description

2.2.1 Harmonic Drive #1

The harmonic drive 1 uses the harmonic gearhead CS-14-100-1U-CC-SP from harmonic drive

LLC. It offers zero backlash for a gear ratio of 100:1. Also it is coupled to a Maxon 273759

precision brush motor (90 watts).

2.2.2 Harmonic Drive #2

The harmonic drive 2 uses the harmonic gearhead CS-8-50-1U-CC-SP from harmonic drive LLC.

It offers zero backlash for a gear ratio of 50:1. Also it is coupled to a Maxon 118752 precision

brush motor (20 watts).

Some useful information regarding 2 motors

Property Value

Input voltage 27 V

Maximum peak current 3 A

Maximum continuous current 1.2 A

Table 2.1: Useful Information regarding two motors

2.2.3 Digital motor and joint position measurement: Optical encoders

Digital angular position measurement of both motors and both flexible joints are obtained by

using high-resolution quadrature optical encoders from US digital. All encoders have 1024 lines

per revolution.

2.2.4 Joint Position Limit Switches

As a safety precaution, two limit switches are installed at the minimum and maximum rotational

positions of each of the two flexible joints. They are magnetically operated position sensors

powered by an external 15 VDC power supply. Specifically, they are the Hamlin 55100 Mini

Flange Mount Hall Effect Sensors.

2.2.5 External DC Power Supply

The external DC power supply provides the system with a maximum output power of 42 W at

15 V DC. It supplies power to the four joint position limit switches.

5

2.2.6 Flexible joint springs

The 2DSFJ is provided with three pairs of extension springs, each of which has a different

stiffness. All linear springs are from the Associated Spring Raymond.

Spring model Spring constant

E0240-031-1500S 3.5 lb/in

E0240-029-1500S 2.42 lb/in

E0240-026-1500S 0.249lb/in

Table 2.2: Spring constants for extension springs

2.2.7 Quanser Current Linear Amplifier Package

The 2DSFJ robot is powered by a two-channel linear current amplifier package from the Quanser

AMPAQ series.

Figure 2.2: Two-channel linear current amplifier package from the Quanser AMPAQ series

2.2.8 Analog current measurement: Current sense resistor

A series load resistor is connected to the output of each of the linear current amplifiers. Such

a current measurement is used to monitor the current the current and in a feedback loop to

control the current in the motor. Current control is an effective way of eliminating effects of

back-EMF as well as a means of achieving force and torque control.

6

2.3 System Parameters

Description Value Unit

Motor #1 Torque constant 0.119 N.m/A

Motor #2 Back EMF constant 0.119 V.s/rad

Motor #1 maximum continuous current 0.944 A

Motor #1 armature resistance 11.5 Ohm

Harmonic drive #1 gear ratio 100

Drive #1 armature inductance 3.16 mH

Motor #1 rotor moment of inertia at motor shaft 6.28E-6 Kg.m2

Moment of inertia of drive #1 transition system 930.91E-6 Kg.m2

Moment of inertia of compounded load transition system 0.23041858 Kg.m2

Flexible joint #1 torsional stiffness constant 9.0 N.m/rad

Motor #1 mechanical time constant 5 Ms

Table 2.3: Flexible Joint #1 System

Description Value Unit

Linear amplifier maximum continuous current 3 A

Linear amplifier peak current 5 A

Linear amplifier maximum continuous voltage 28 V

Linear amplifier peak power 300 W

Linear amplifier bandwidth (current mode) 10 KHz

Linear amplifier gain 0.5 A/V

Table 2.4: Linear current amplifier (each channel)

7

Description Value Unit

Motor #2 Torque constant 0.0234 N.m/A

Motor #2 Back EMF constant 0.0234 V.s/rad

Motor #2 maximum continuous current 1.21 A

Motor #2 Armature resistance 2.32 Ohm

Drive #2 armature inductance 0.24 mH

Harmonic drive #2 gear ratio 50

Motor #2 rotor moment of inertia at motor shaft 1.03*E-6 Kg.m2

Moment of inertia of drive #2 transition system 930.91*E-6 Kg.m2

Moment of inertia of load #2 transition system 0.010724 Kg.m2

Flexible joint #2 torsional stiffness constant 4.0 N.m/rad

Motor #2 mechanical time constant 4 Ms

Table 2.5: Flexible Joint #2 System

Description Value Unit

Encoder line count 1024 Lines/rev

Encoder resolution (in quadrature) 4096 Counts/rev

Encoder angular resolution (in quadrature) 0.0015 Rad/count

Drive #1 encoder sensitivity (in quadrature) 1.534E-5 Rad/count

Drive #2 encoder sensitivity (in quadrature) 1.918E-5 Rad/count

Flexible joint encoder sensitivity (in quadrature) 23.968E-5 Rad/count

Encoder type TTL

Encoder signals A, B, Index

Table 2.6: Drive and Joint Optical Encoders

Description Value Unit

Power supply power 42 W

Power supply voltage 15 VDC

Table 2.7: External Power Supply

8

Chapter 3

Analysis of Plant

3.1 Modeling

3.1.1 Introduction

A schematic of Two-Degree-Of-Freedom Serial Flexible Joint (2DSFJ) system is represented

in figure. It depicts two flexible joints connected in series and each actuated by its own drive

system.

Figure 3.1: Schematic of Two Degree of Freedom Flexible Joint (2DSFJ) System

Where Ks1 and Ks2 are the first and second flexible joint torsional stiffness constants, Im1

and Im2 the drive currents, Ji(for i = 1, 2, 3, 4) the intermediary load moments of inertia, and

Bi(for i = 1, 2, 3, 4) the intermediary load viscous damping coefficients.

9

Sign Convention The positive direction of rotation, illustrated in fig 3.1.1 for all four load

angles ei(i = 1, 2, 3, 4) is chosen to be Counter clockwise when looking at the robot from top.

In the controller design procedure, the 2DSFJ system is considered decoupled and split into two

separate and independent stages: Stage 1 and Stage 2.

3.1.2 Modeling of stage 1

The schematic for stage 1 is represented in figure below.

Figure 3.2: Schematic of 2DSFJ Robot Stage 1 System

Table provides a nomenclature of the symbols used in 2DSFJ stage 1 system mathematical

modeling.

The Lagranges method is used to obtain the dynamic model of the system. X1 is chosen to

include the generalized co-ordinates as well as their first-order time derivatives. It is defined by

its transpose, shown below:

X1T =

[θ11(t), θ12(t),

d(θ11(t))

dt,d(θ12(t))

dt

](3.1)

The system input, U1, is the current to the first motor

Ui = Im1 (3.2)

The state-space matrices A1 and B1 are defined to give a dynamic representation of the 2DSFJ

stage 1 system, such that:

δX1

δt= A1X1 +B1U1 (3.3)

10

Symbol Description Units

Im1 First shoulder armature current A

Kt1 First (shoulder) drive torque constant N.m/A

T1 Torque produced by Drive #1, at the load shaft N.m

θ11 First (shoulder) driving shaft absolute angular position rad

d(θ11(t))

dtFirst (shoulder) driving shaft absolute angular velocity rad/s

θ12 First rigid link absolute angular position rad

d(θ12(t))

dtFirst rigid link absolute angular velocity rad/s

J11 First flexible joint actuated transition kg.m2

equivalent moment of inertia

B11 First flexible joint actuated transition N.m.s/rad

equivalent viscous damping co-efficient

J12 First flexible joint load transition equivalent kg.m2

moment of inertia (compounded with the stage 2 system)

B12 First flexible joint load transition equivalent N.m.s/rad

viscous damping co-efficient (compounded with stage 2 system)

Ks1 First flexible torsional stiffness constant N.m/rad

Table 3.1: Nomenclature of the symbols used in 2DSFJ stage 1 systems mathematical modeling

From the systems two equations of motion, the A1 matrix can be determined as follows:

A =

0 0 1 0

0 0 0 1

−Ks1

J11

Ks1

J11

−B11

J110

Ks1

J12

−Ks1

J120

−B12

J12

.

The transpose of the B2 matrix characterizing the system is as below:

B1T =

[0 0

Kt1

J110

](3.4)

11

3.1.3 Modeling of stage 2

The schematic fo stage 2 system of the 2DSFJ plant is represented in fig 3.3.1

Figure 3.3: Schematic of 2DSFJ Robot stage 2 System

The Lagranges method is used to obtain the dynamic model of the system. In the systems

state vector, X2 , is chosen to include the generalized co-ordinates as well as their first order

derivatives. It is defined by its transpose as shown below:

X2T =

[θ21(t), θ22(t),

d(θ21(t))

dt,d(θ22(t))

dt

](3.5)

The system input, U2, is the current to the second motor:

U2 = Im2 (3.6)

The state space matrices of A2 and B2 are defined to give a dynamic representation of the

2DSFJ stage 2 system, such that:

δX2

δt= A2X2 +B2U2 (3.7)

From the systems two equations of motion, the A2 matrix can be determined as follows:

A =

0 0 1 0

0 0 0 1−Ks2

J21

Ks2

J21

−B21

J210

Ks2

J22

−Ks2

J220

−B22

J22

.

The transpose of the B2 matrix characterizing the system is as below:

12

B2T =

[0 0

Kt2

J210

](3.8)

Table provides a nomenclature of the symbols used in 2DSFJ stage 2 systems mathematical

modeling.

Symbol Description Units

Im2 Second (elbow) motor armature current A

Kt2 Second (elbow) drive torque constant N.m/A

T2 Torque produced by Drive #2, at the load shaft N.m

θ21 Second (elbow) driving shaft angular position relative to link #1 rad

d(θ21(t))

dtSecond (elbow) driving shaft angular velocity relative to link #1 rad/s

θ22 Second rigid link angular position relative to link #1 rad

d(θ22(t))

dtSecond rigid link angular velocity relative to link #1 rad/s

J21 Second flexible joint actuated transition kg.m2

equivalent moment of inertia

B21 Second flexible joint actuated transition equivalent N.m.s/rad

viscous damping co-efficient

J22 Second flexible joint load transition equivalent moment of inertia kg.m2

B22 Second flexible joint load transition equivalent N.m.s/rad

viscous damping co-efficient

Ks2 Second flexible torsional stiffness constant N.m/rad

Table 3.2: Nomenclature of the symbols used in 2DSFJ stage 2 systems mathematical modeling

13

3.2 Alternative representation of Plant

For controlling the plant using by sliding mode control or pole placement method, we need

to ensure that the plant is in Phase Variable Form. When the plant is not in phase variable

form, Similarity Transformation technique can be used to convert it to the phase variable form.

Consider the given system as

A =

0 0 1 0

0 0 0 1

−141.2188 141.2188 −70.6094 0

39.0594 −39.0594 0 −0.3054

.

B =

0

0

140.0419

0

.

The method consists of transforming the system to phase variables, designing the feedback

gains, and transforming the designed system back to its origional state-variable representation.

This method requires that we first develop the transformation between system and it’s

transformation in phase-variable form. Consider our plant which is not represented in phase

variable form.

z = Az +Bu (3.9)

y = Cz +Du (3.10)

Controllability matrix is

CMz =[Bz AzBz Az

2Bz Az2Bz

](3.11)

The system can be transformed into phase variable form with the transformation

z = Px (3.12)

Substituting this transformation into plant equation we get,

x = P−1APx+ P−1Bu (3.13)

y = CPx (3.14)

14

The transformation matrix for plant is obtained as

P = CMzCMx−1 (3.15)

For flexible arm joint

A =

0 1 0 0

0 0 1 0

0 0 0 1

0 −2801.1 −201.8404 −70.9148

.

B =

0

0

0

4750

.

C = [ 1 0 0 0 ] (3.16)

CMx =[Bx AxBx Ax

2Bx Ax2Bx

](3.17)

P = CMzCMx−1 (3.18)

P =

1.1515 0.0090 0.0295 0

1.1516 0 1 0

0.0043 1.1515 0.009 0.0295

0 1.1516 0 0

.

15

Chapter 4

Control Strategies

4.1 Proportional Integral Derivative Control

Applying a PID control law consists of applying properly the sum of three types of control

actions: a proportional action, an integral action, and a derivative one.

4.1.1 Proportional Action

Proportional controller is used when the controller action is to be proportional to the size of the

error signal

e(t) = r(t)− y(t) (4.1)

u(t) = Kpe(t) (4.2)

where Kp is the proportional gain. It implements the typical operation of increasing the

control variable when the control error is large. The transfer function of a proportional controller

can be derived as:

C(s) = Kp (4.3)

The main drawback of using a pure proportional controller is that it produces a steady-state

error.

16

The block diagram for proportional controller is as below:

Figure 4.1: Block Diagram of Proportional Controller

4.1.2 Integral Action

The integral action is proportional to the integral of error, i.e. it is

u(t) = Ki

∫[e(t)]dt (4.4)

Where Ki is the integral gain. Integral action is related to the past values of the control error.

Integral controller overcomes the shortcomings of proportional controller by eliminating offset

without the use of excessive large gain.

The corresponding transfer function is:

C(s) =Ki

s(4.5)

The presence of a pole at the origin of the complex plane allows the reduction of the steady-

state error to zero when a step reference signal is applied. However it also induces oscillations

in the plant response.

The block diagram for integral controller is as below:

Figure 4.2: Block Diagram of Integral Controller

4.1.3 Derivative Action

While the proportional action is based on the current value of the control error and the integral

action is based on the past values of the control error, the derivative action is based on the

17

predicted future values of the control error. Derivative controller uses the rate of change of an

error signal. An ideal derivative law can be expressed as:

u(t) = Kdde(t)

dt(4.6)

WhereKd is the derivative gain. The corresponding controller transfer function is

C(s) = Kds (4.7)

The derivative action is also called as anticipatory control, or rate action, or pre act.

The block diagram for derivative controller is as below:

Figure 4.3: Block Diagram of Derivative Controller

The block diagram of PID controller is represented as follows:

Figure 4.4: Block Diagram of PID Controller

Disadvantages:

The PID controller is suitable for processes with almost monotonic step response. It cannot be

used when process is highly oscillatory and the requirements are too extreme.

18

4.2 Controller Design in State Space using State Lineariza-

tion Vector

In conventional pole placement method, a controller is designed such that the dominant closed

loops have a desired damping ratio and an un-damped natural frequency. In this approach, the

order of the system may be raised by 1 or 2 unless pole-zero cancellation takes place. In this

approach, it is assumed that the effects on the responses of non-dominant closed-loop poles to

be negligible.

4.2.1 An Overview

Consider a plant represented in state space as:

x = Ax+Bu (4.8)

y = Cx+Du (4.9)

Figure 4.5: Representation of plant in State Space

where

x = state vector

y = output signal

u = control signal

A = n x n matrix

B = n x 1 matrix

C = 1 x n matrix

D = constant

The control signal is

U = −Kx (4.10)

19

where

K = [k1k2...kn] (4.11)

This means that the control signal u is determined by an instantaneous state. This is called

state feedback. The 1 x n matrix K is called the state feedback gain matrix.

The design of state-variable feedback for closed loop pole placement consists of equating the

characteristic equation of a closed-loop system to a desired characteristic equation and then

finding the values of the feedback gain K

4.2.2 Design Steps

To apply feedback pole-placement methodology to plants represented in phase variable form,

following steps are followed:

1) Check whether the plant is in phase variable form.

2) If it is then skip to step no.7 If it not continue step 3.

3) Check whether the plant is controllable by finding out controllability matrix.

4) If the system is controllable we can assume that we can convert it in to phase variable form.

5) Use similarity transformation and convert the system to phase variable form.

6) Represent the plant state in phase variable form.

7) Feedback each phase variable to the input of the plant through a gain,K.

8) Find the characteristics equation for the closed-loop system represented in step 2.

9) Decide upon all closed-loop locations and determine an equivalent characteristic equation.

10) Equate like coefficients of the characteristic equations from step 3 and step 4 and solve for

K.

4.2.3 Controllability

The necessary and sufficient condition for arbitrary pole placement is that the system be com-

pletely state controllable. If the system is not completely state controllable, then there are eigen

vaules of matrix that cannot be controlled by state feedback.

If an input to a system can be found that takes every state variable from a desired initial state

to a desired final state, the system is said to be controllable; otherwise, the system is uncontrol-

lable.

Pole placement is viable design technique only for systems that are controllable.

An nth order plant whose state equation is

x = Ax+Bu (4.12)

is completely controllable if the matrix

CM = [B AB A2B ... An−1B] (4.13)

is of rank n, where CM is called the controllability matrix

20

4.3 Sliding Mode Control

4.3.1 Background

The use of sliding mode control is mainly due to the modeling inaccuracies. Non-linear system

model imprecision may come from actual uncertainty in the plant (e.g. unknown plant parame-

ters), or from the purposeful choice of a simplified representation systems dynamics. Modeling

inaccuracies can be classified into two major kinds

1) Structured Uncertainties: They are also called as parametric uncertainties. They correspond

to inaccuracies on the terms actually included in the model

2) Unstructured Uncertainties: They are also called as un-modeled uncertainties. They corre-

spond to inaccuracies on the system order.

Modeling inaccuracies can have strong adverse effect on non-linear control systems. One of the

most important approaches to dealing with model uncertainty is Robust Control.

4.3.2 An Overview

Sliding mode control is a robust control approach. Sliding mode controller provides a system-

atic approach to the problem of maintaining stability and consistent performance in the face of

modeling imprecision.

Consider the variable structure control as a high-speed switched feedback control resulting

in sliding mode. The gains in each feedback path switch between two values according to a rule

that depends on the value of the state at each instant. The purpose of the switching control law

is to drive the nonlinear plants state trajectory into a pre-specified surface in the state space

and to maintain the plants trajectory on this surface for subsequent time. The surface is called

switching surface. When the plant trajectory is above the surface, a feedback path has one gain

and a different gain if the trajectory drops below the surface. This surface defines the rule for

proper switching. This surface is also called as sliding surface. The switched control maintains

the plants state trajectory on the surface for all subsequent time and the plants state trajectory

slides along this surface.

Sliding mode control utilizes discontinuous feedback control laws to force the system state

to reach, and remain on a specified surface within the state space, i.e. the sliding surface. The

system dynamic, when confined to the sliding surface, is described as an ideal sliding motion

and represent the controlled system behavior.

The advantages of obtaining such a motion are two-fold: firstly the system behaves as a

system of reduced order with respect to the original plant; and secondly the movement on

the sliding surface of the system is insensitive to a particular kind of perturbation and model

21

uncertainties.

4.3.3 Sliding mode Controller Design

Consider the following nonlinear system in state space.

x = f(t, x) + g(t, x)u(t) (4.14)

The component of the discontinuous feedback are given by

ui = u+i (t, x), if σi(x) > 0 (4.15)

ui = u−i (t, x), if σi(x) < 0 (4.16)

where i(x) = 0 is the ith sliding surface, and

σ(x) = [σ1(x), σ2(x), ... σm(x)]T

(4.17)

is the (n - m) dimensional sliding manifold.

The control problem consists in developing continuous function ui+,ui- and the sliding surface

σ(x) = 0 so that the closed loop system mentioned above exhibit a sliding mode on the (n - m)

dimensional sliding manifold σ(x) = 0.

The design of the sliding mode control law can be divided in two phases:

1) Phase 1 consists in the construction of a suitable sliding surface so that the dynamic of the

system confined to the sliding manifold produces a desired behavior.

2) Phase 2 entails the design of a discontinuous control law which forces the system trajectory

to the sliding surface and maintains it there.

The switching surface is designed such that the system response restricted to (x) = 0 has a

desired behavior.

General non-linear switching surfaces are possible but the linear switching surfaces of the form:

σ(x) = Sx(t) = 0 (4.18)

After switching surface design, the next important aspect of sliding mode control is guarantee-

ing the existence of a sliding mode. A sliding mode exists, if in the vicinity of the switching

surface,σ(x) = 0, the velocity vectors of the state trajectory is always directed towards the

switching surface. Consequently, if the state trajectory intersects the sliding surface, the value

of the state trajectory remains within a neighborhood of (x|σ(x) = 0). If a sliding mode exists

on σ(x) = 0, then σ(x) is termed as a sliding surface.

An ideal sliding mode exists only when the state trajectory x(t) of the controlled plant sat-

isfies σ[x(t)] = 0 at every t >= t0 for some t0. Starting from some time instant t0, the system

is constrained on the discontinuity surface, which is an invariant set after the sliding mode has

22

been established.

This requires infinitely fast switching. In real systems there are practical imperfections such

as delay, hysteresis, etc., which forces switching to occur at a finite frequency. The system state

then oscillates within neighborhood of the switching surface. This oscillation is called chattering.

If the frequency of switching is very high compared with the dynamic response of the system,

the imperfections and the finite switching frequencies are often but not always negligible.

Advantages and Limitations

Advantages

1) Low Sensitivity to plant parameter uncertainty.

2) Finite- time convergence due to discontinuous control law.

Limitations

1) Chattering due to implementation imperfections.

2) Controller tuning becomes very difficult when the bounds of uncertainty are unknown.

Reduction of Chattering

Figure 4.6: Sliding and reaching Phase

In a system where modeling imperfections, parameter variations and disturbances are more,

value of K must be large enough to obtain a satisfactory tracking performance with sliding mode

controller. But larger values of K lead to more chattering of the control variable and system

states. A boundary layer of definite width on both sides of switching line is introduced to reduce

chattering. If is the width of the boundary layer on either side of the switching line, we modify

the control law such that the control gain is less inside the boundary layer which results in a

smooth control signal.

The plant in phase variable form is

23

A =

0 1 0 0

0 0 1 0

0 0 0 1

0 −2801.1 −201.8404 −70.9148

.

B =

0

0

0

4750

.

C = [ 1 0 0 0 ] (4.19)

x1 = x2; (4.20)

x2 = x3; (4.21)

x3 = x4; (4.22)

x4 = −a10x1 − a20x2 − a30x3 − a40x4 + dela1x1 + dela2x2 + dela3x3 + dela4x4 + bu; (4.23)

Sigma is defined as the tracking error i.e difference between error in plant states and sliding

surface.

σ = c1x1 + c2x2 + c3x3 + x4 (4.24)

The uncertainties assumed in the plant are ±20%. These are parametric uncertainties.

The control signal given to the plant is sum of continuous and discontinuous signal

u = un + ueq (4.25)

where,

ueq = −c1x2 + c2x3 + c3x4 − a10x1 − a20x2 − a30x3 − a40x4b

(4.26)

Values of c1 , c2 , c3 are chosen so that the speed of response is optimized.

un = − [ dela1 max dela2 max dela3 max dela4 max ]abs(x)signum(σ)

b(4.27)

24

Where, output of signum function is given as

y = 1 for x ≥ 0

= 0 for x ≤ 0 (4.28)

(4.29)

25

4.4 Proportional-Integral Observer

Controller design relies upon access to the state variables for feedback through adjustable gains.

This access can be provided through hardware, in our case the optical encoder. In certain cases

some of the state variables are not available because of the system configuration then these

states can be estimated.

Estimated states rather than the actual states are then fed back to the controller. PI observer is

Figure 4.7: PI Observer -Controller

an observer which feeds back the measurement error as a proportional and integral combination.

The observer yields:

v = −Lv

∫[y − y]dt (4.30)

˙x = Ax+Bu+ Lx(y − y) +NLv

∫[(y − y)]dt (4.31)

Where y − y is the measurement error.

26

4.5 Linear Quadratic Regulator

Consider a linear time invariant system having a state space model, as defined below:

dx(t)

dt= Ax(t) +Bu(t) (4.32)

y(t) = Cx(t) +Du(t) (4.33)

Where,

x is the state vector

u is the input

y is the output

x0 is the state vector at time t = t0 and A, B, C and D are the matrices of appropriate

dimensions. We aim at driving the initial state x0 to the smallest possible value as soon as

possible in the interval [t0, tf ] but without spending too much control effort to achieve that

goal. Then the optimal control program is defined as the problem of finding an optimal control

u (t) over the interval [t0, tf ] such that the following cost function is minimized.

Ju(x(t0), t0) =

∫[x(t)

TΨx(t) + u(t)

Tφu(t)]

Tdt+ x(tf )

TΨfx(tf ) (4.34)

where,

Ψ and Ψf are symmetric non-negative definite matrices and is a symmetric positive definite

matrix.

To solve this problem, following connections between general optimal problem and the LQR

problem are made.

f(x(t), u(t), t) = Ax(t) +Bu(t) (4.35)

V (x, u, t) = x(t)T

Ψx(t) + u(t)Tφu(t) (4.36)

g(x(tf )) = x(tf )T

Ψfx(tf ) (4.37)

The optimal control can be expressed as,

u0(t) = −Ku(t)x(t) (4.38)

where, Ku(t) is a time varying gain, given by

Ku(t) = φ−1BTP (t) (4.39)

Assumptions

There are two key assumptions

1) The system (A, B) is stabilizable from u(t). if the system isnt stabilizable, then optimal

27

control will not help recover the situation.

2) The system states are adequately seen by the cost function. This is stated as requiring that

(Ψ1/2, A) be detectable.

Advantages

1) It gives a robust system by guaranteeing stability margins.

2) The optimal signal u(t) is obtainable from full state feedback.

Disadvantages

1) Obtaining an analytical solution for Ricatti equation is quite difficult. Hence it has to be

done offline

2) The standard LQR does not put any restrictions on the input signal u(t) amplitude. Opti-

mizing input might have amplitudes that are well above the signal generation capacity of the

system.

3) Robustness offered by LQR is limited and it is highly state dependent.

28

Chapter 5

Experimentation and Results

5.1 Controller design for 2-DOF Serial Flexible Joint (2DSFJ)

Figure 5.1 shows the Simulink model for 2-DOF Serial Flexible Joint (2DSFJ).

Figure 5.2 shows the interfacing between the two stages of 2-DSFJ .

In next section, we will discuss the various control strategies for position control of the

flexible arm joint.

29

Figure 5.1: 2-DOF Serial Flexible Joint (2DSFJ)

30

Figure 5.2: Interface to the Actual First And Second Stages Of The 2-DOF Serial Flexible Joint

(2DSFJ).

5.2 Control strategies

5.2.1 Proportional-Integral-Derivative controller

Simulink Model PID Controller Response for Joint #1

31

Figure 5.3: Simulink model for PID controller

Figure 5.4: Plant response for parameter set #1 and #2

32

Control Signal for Joint #1

Figure 5.5: Control signal for parameter set #1 and #2

Parameter sets are as follows

Parameter set no. Proportional gain (KP ) Integral gain (KI) Derivative gain (KD)

1 1 1 0.005

2 5 1 1.5

Table 5.1: Parameters for PID Controller

From amplitude versus time plot, we observe that improvement in transient response results

in degradation of steady state response.

With parameter set 1, we can achieve settling time of 3.8 second. However, it happens at the

cost of 23 % overshoot. This is not acceptable for 2DOF Serial Flexible Joint , as it may result

in saturation of control signal.

With parameter set 2, we have dampened those oscillations at the cost of settling time. Thus,

the plant responds within its limits with settling time of 9 seconds.

33

5.3 Linear Quadratic Regulator

Simulink Model

Figure 5.6: Simulink Model for Linear Quadratic Regulator

Stage 1 State Plot

Figure 5.7: State plot of stage 1

34

Stage 2 State Plot

Figure 5.8: State plot of stage 1

Control Signal For Stage 1

35

Figure 5.9: Control Signal for Stage 1

Control Signal For Stage 2

As can be seen from the figures LQR makes the plant states to oscillate aggressively.

LQR tries to reduce the time integral of quadratic form in state vector X and input vec-

tor U which is the quadratic cost function. For the given LQR, the cost is sum of deviation

of all states and as can be seen from the figures, it always tries to minimize the sum of deviation.

The advantages of using LQR are that it increases the accuracy of state variables by esti-

mating the states. In pole placement method, we need to specify where Eigen values should be

placed. However, in LQR a set of performance weights can be specified which has more intuitive

appeal.

LQR does not offer robust response. The controller will not work efficiently if the bounds

of uncertainty in the states are high. Major disadvantage of LQR is that it is very sensitive to

high frequency noise and high sample rates.

36

Figure 5.10: Control Signal for Stage 2

5.4 Sliding Mode Control

PI Observer based SMC

37

Figure 5.11: Implementation of PI observer based SMC

38

Simlulink Model

Figure 5.12: Simulink Model for Sliding Mode Controller

Simulation of Joint 1

39

Figure 5.13: Analysis of Joint 1

Simulation of Joint 2

State Estimation by Observer

40

Figure 5.14: Analysis of Joint 2

Figure 5.15: State Estimation by Observer

41

Implementation of PI Observer based SMC

Figure 5.16: Implementation of PI Observer based SMC

As can be seen from the plots the effect of discontinuous control is more compared to contin-

uous in the early stages of the response. As the control signal enters the bounds the continuous

control signal becomes more dominant.

For joint one control signal becomes zero in 5.32 seconds while for joint 2 it becomes zero in

7.32 seconds.

As can be seen for PIO based sliding controller the value of sigma is very high initially.

This happens as a result of positioning of observer poles far away from imaginary axis. However

it is necessary to place these poles away in order to improve tracking performance of the observer.

The sliding action of the states can be observed from the sigma vs. time plot which shows

that plant exhibit sliding behavior in less than 2 seconds for joint 1 and in 2.5 seconds for joint

2.

42

Chapter 6

Conclusion

A proportional-Integral Observer (PIO), on the basis of a decoupled multiple model, is proposed

in this contribution in order to cope with the state estimation problem of a non-linear system in

presence of delay and perturbed measurements. The decoupled multiple model is an interesting

structure for modeling non-linear systems with variable structure because the dimensions of

the employed sub-models can be different. On the other hand, the PIO offers more degree of

freedom for robust state estimation with respect to classic Proportional Observer. The robust

stability problem of the estimation error is investigated using the Lyapunov functional method

and delay-independent sufficient conditions.

In this thesis the theory of sliding mode controller is studied in detail. The equations of the

2-DOF Serial Flexible Joint (2DSFJ) Robot are reorganized so as to apply the control tech-

nique. The controller gain and band width are designed, considering various factors such as

model inaccuracies, load torque disturbance and also to have ideal position tracking.

A sliding mode controller based on Lyapunov function offers robust design and reduces the

impact of uncertainty and plant disturbances. The controller is less aggressive compared to

other techniques such as Linear Quadratic Regulator or PID controller and offers better steady

state response and stability.

As a future work this controller can be applied to flexible link system. Fuzzy logic principle

can be incorporated to this controller to make it more efficient and robust.

43

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