Dynamic Modeling and Control of Flexible-Link

The Faculty of Postgraduate Studies and Scientific Research The German University in Cairo

Dynamic Modeling and Control of Flexible-Link Manipulators Subjected to Parametric Excitation

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechatronics Engineering By

Mohamed Walid Mehrez Elrafei Mohamed Said

Supervised by

Assoc. Prof. Ayman A. El-Badawy

March, 2010

The Faculty of Postgraduate Studies and Scientific Research The German University in Cairo

Dynamic Modeling and Control of Flexible-Link Manipulators Subjected to Parametric Excitation

A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechatronics Engineering By

Mohamed Walid Mehrez Elrafei Mohamed Said

Supervised by

Assoc. Prof. Ayman A. El-Badawy

March, 2010

Declaration This is to certify that: 1. the thesis comprises only my original work towards the Master Degree 2. due acknowledgment has been made in the text to all other material used

Signature

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Acknowledgments First, I would love to express my deepest thanks to Assoc. Prof. Ayman A. El Badawy, my supervisor, for his help and support along the time of this work. He was always putting me on the proper way and directing me right when I go wrong. I would like also to thank all members of the Mechatronics department of the German University in Cairo for their support and kindness. Last but not least, I would love to thank my great parents, Mehrez and Magda, my sisters Wedad and Walaa, for their love, encouragement, and support.

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Abstract The dynamics of flexible manipulators is complex due to the flexibility distributed along the mechanical structure. This, in return, makes an improved control is one of the important needs. This work is focused on two major subjects: flexible link manipulators, and parametrically excited flexible arm robots. For flexible link manipulators, the dynamic equations were derived based on the combined Lagrangian-Assumed modes method. The method was then used to derive the equations of motion of a two-link rigid-flexible arm. The response of the model was investigated under computed torque control to track a predefined trajectory. A singularly-perturbed model of flexible link manipulators was pursued. A control law based on the singularly-perturbed model has been developed. The performance of this system was investigated and compared with that of computed-torque law. Based on the study performed, computed torque control was found to perform equally well with composite control for flexible manipulators, as long as, the joint inertias are negligible. On the other hand, the introduction of singular perturbation theory to control flexible manipulator motion enhanced the system performance, over just computed torque method, in the presence of considerable joint inertias. For the case of flexible arm robots subjected to parametric excitation, a comprehensive nonlinear time-varying dynamic model of such systems was derived. The parametric excitation leads to equations of motion with rapidly varying coefficients. The response of such systems was then investigated under two control techniques, namely, computed torque and a composite control based on a developed singular perturbation method, while tracking a certain trajectory. Better response of the simulated arm was observed under the composite control than the computed torque control even with less applied control effort.

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Table of Contents Declaration ....................................................................................................................... i Acknowledgments...........................................................................................................ii Abstract ......................................................................................................................... iii Table of Contents .......................................................................................................... iv List of Abbreviations ....................................................................................................vii List of Symbols ........................................................................................................... viii List of figures .................................................................................................................. x Introduction and Literature Review ..................................................................... 1 1.1

Motivation ......................................................................................................... 1

1.2

Modeling of Flexible manipulators ................................................................... 3

1.3

Control of Flexible manipulators ....................................................................... 4

1.4

Objective ............................................................................................................ 7

1.5

Scope of the Thesis ............................................................................................ 8

Dynamic Modeling and Equations of Motion ...................................................... 9 2.1

Perspective on This Chapter .............................................................................. 9

2.2

Kinematics of beams ....................................................................................... 10

2.3

Lagrange's Equations of motion ...................................................................... 16

2.4

Final Simulation Equations ............................................................................. 16

2.5

Equations of Motion of a Two-link Rigid-Flexible Manipulator .................... 18

Manipulator Control Using Computed Torque Control ................................... 24 3.1

Perspective on This Chapter ............................................................................ 24

3.2

Inverse Kinematics .......................................................................................... 25

3.3

Forward Kinematics ........................................................................................ 27

3.4

Feedback linearization ..................................................................................... 28

3.5

Computed Torque Control ............................................................................... 32

iv

3.6

Trajectory Tracking Control of Flexible Manipulator using (CTM) ............... 38

Manipulator Control Using Singular Perturbation Approach ......................... 56 4.1


4.2

One-link Flexible Arm EOM ........................................................................... 57

4.3

Flexible Manipulator Dynamics revisited ....................................................... 60

4.4

A Singularly Perturbed Model ......................................................................... 61

4.5

Composite Control ........................................................................................... 64

4.6

Numerical Example ......................................................................................... 66

Flexible-Link Manipulators Subjected to Parametric Excitation .................... 81 5.1


5.2

Nonlinear Modeling of Flexible Manipulators Subjected to Parametric

Excitation ................................................................................................................... 82 5.2.1

Kinematics ................................................................................................ 82

5.2.2

Kinetic and potential energy..................................................................... 83

5.3

Fully Nonlinear Equations of Motion .............................................................. 84

5.4

A Singularly Perturbed Model ......................................................................... 85

5.5

Composite Control ........................................................................................... 88

5.6

Numerical Example ......................................................................................... 89

Conclusions and Recommendations .................................................................... 96 6.1

Conclusions ..................................................................................................... 96

6.2

List of Publications .......................................................................................... 97

6.3

Recommendations and Future Work ............................................................... 97

Appendix A ............................................................................................................ 99 Kinematics and Dynamics of Flexible Arms ............................................................. 99 A.1 Flexible Arm Kinematics .................................................................................... 99 A.2 System Kinetic Energy ...................................................................................... 104 A.3 System Potential Energy ................................................................................... 106

v

A.3.1 Elastic Potential Energy .............................................................................. 106 A.3.2 Gravity Potential Energy ............................................................................. 107 A.4 Assumed Mode Shapes ..................................................................................... 108 Appendix B .......................................................................................................... 111 References ................................................................................................................... 117

vi

List of Abbreviations CTM

Computed torque method

FJR

Flexible joint robot

LQR

Linear quadratic regulator

ODE

Ordinary differential equation

PDE

Partial differentia equation

PPF

Positive position feedback

VSC

Variable structure control

vii

List of Symbols u(x,t) v(x,t), w(x,t) θx(x,t)

Beam longitudinal deformation Beam transverse deformation components Beam twisting angle

θy(x,t),θz(x,t) Beam slope angles components Px*

Axial force

Py*,Pz*

Shear forces

My*,Mz*

Bending moments

Mx*

Twisting moment

σxx

Axial stress

εxx

Axial strain

e(x,t)

Axial stretch

Ai

Joint transformation matrix for joint i due to rigid motion

Ei

Link i transformation matrix due to its length and deflection

K

System kinetic energy

Ve

System elastic potential energy

Vg

System gravity potential energy

ρi

Link i density (kg/m)

E

Young’s modulus of elasticity

G

Shear modulus

Ix

Polar area moment of inertia

Iy, Iz

The area moment of inertia of the link cross section about the y and z axes

EI

Flexural rigidity

q

Generalized coordinate vector

θ

Joint variable

δ

Time varying modal variable



Shape function

K

Stiffness matrix

T

Vector of joint torques

J

The Jacobian matrix

viii

m

Link’s mass

L

Link’s length

Kv

Velocity gain matrix

Kp

Position gain matrix

ζ

Damping ratio

ωn

Natural frequency

Th

Hub kinetic energy

TL

Link kinetic energy

Tp

Payload kinetic energy

Mh

Hub mass

Jh

Hub mass moment of inertia

Mp

Payload mass

Jp

Payload mass moment of inertia

ζ

Vector of elastic forces

ε

Perturbation parameter

τ

Fast time scale

T

Slow control

Tf

Fast control

kpf

Position fast gain matrix

kvf

Velocity fast gain matrix

A

Parametric excitation amplitude

ω

Parametric excitation frequency

ix

List of figures Figure 2.1 Undeformed beam......................................................................................... 10 Figure 2.2 Deformation of beam axis ............................................................................. 10 Figure 2.3 Plane rotation ................................................................................................ 11 Figure 2.4 Deformed infinitesimal element of the beam ............................................... 11 Figure 2.5 projection of the differential element onto x*z* ........................................... 12 Figure 2.6 projection of the differential element onto x*y* ........................................... 12 Figure 2.7 Deformed beam axis ..................................................................................... 14 Figure 2.8 Deformed differential element ...................................................................... 14 Figure 2.9 Two revolute joint planar manipulator ......................................................... 18

Figure 3.1 A two-link planar arm ................................................................................... 25 Figure 3.2 inverse kinematics......................................................................................... 26 Figure 3.3 Forward kinematics....................................................................................... 27 Figure 3.4 Inner Loop/Outer-loop control architecture for feedback linearization. ...... 30 Figure 3.5 Inner-Loop/Outer-loop control architecture ................................................ 35 Figure 3.6 Robot Mechanism ........................................................................................ 38 Figure 3.7 Manipulator end point desired trajectory ..................................................... 39 Figure 3.8 Desired trajectory scheme along x0 .............................................................. 40 Figure 3.9 Desired position of point C along x0 ............................................................ 45 Figure 3.10 Desired velocity of point C along x0 .......................................................... 45 Figure 3.11 Desired acceleration of point C along x0 ................................................... 46 Figure 3.12 Desired position of joint variable  1 ......................................................... 47 Figure 3.13 Desired velocity of joint variable  1 ......................................................... 47 Figure 3.14 Desired acceleration of joint variable  1 ................................................... 48 Figure 3.15 Desired position of joint variable  2 ......................................................... 48 Figure 3.16 Desired velocity of joint variable  2 ......................................................... 49 Figure 3.17 Desired acceleration of joint variable  2 ................................................... 49 Figure 3.18 Computed torque control schematic .......................................................... 50 Figure 3.19 End point error along x0 ............................................................................. 52

x

Figure 3.20 End point error along y0 ............................................................................. 52 Figure 3.21 Control torque applied at A, T1. ................................................................. 53 Figure 3.22 Control torque applied at B, T2. ................................................................. 53 Figure 3.23 First modal coordinate  1 .......................................................................... 54 Figure 3.24 Second modal coordinate  2 ...................................................................... 54

Figure 4.1 One-link flexible-arm ................................................................................... 57 Figure 4.2 Composite Control ........................................................................................ 65 Figure 4.3 First modal variable  1 (Computed Torque Control) with joint inertia ....... 68 Figure 4.4 First modal variable  1 (Composite control) with joint inertia .................... 68 Figure 4.5 First modal variable δ1 (Computed Torque Control) without joint inertia ... 69 Figure 4.6 First modal variable δ1 (Composite control) without joint inertia ................ 69 Figure 4.7 Second modal variable δ2 (Computed Torque Control) with joint inertia .... 70 Figure 4.8 Second modal variable δ2 (Composite control) with joint inertia ................. 71 Figure 4.9 Second modal variable δ2 (Computed Torque Control) without joint inertia ........................................................................................................................................ 71 Figure 4.10 Second modal variable δ2 (Composite control) without joint inertia.......... 72 Figure 4.11 Tip position along X0 (Computed Torque Control) with joint inertia ........ 73 Figure 4.12 Tip position along X0 (Composite control) with joint inertia ..................... 73 Figure 4.13 Tip position along X0 (Computed Torque Control) without joint inertia ... 74 Figure 4.14 Tip position along X0 (Composite control) without joint inertia ................ 74 Figure 4.15 Control torque T (Computed Torque Control) with joint inertia ................ 75 Figure 4.16 Control torque T (Composite control) with joint inertia............................. 76 Figure 4.17 Control torque T (Computed Torque Control) without joint inertia ........... 76 Figure 4.18 Control torque T (Composite control) without joint inertia ....................... 77 Figure 4.19 Joint angle (Computed Torque Control) with joint inertia ......................... 78 Figure 4.20 Joint angle (Composite control) with joint inertia ...................................... 78 Figure 4.21 Joint angle (Computed Torque Control) without joint inertia .................... 79 Figure 4.22 Joint angle (Composite control) without joint inertia ................................. 79

xi

Figure 5.1 multi-link flexible manipulator subjected to parametric excitation .............. 83 Figure 5.2 One-link flexible-arm ................................................................................... 90 Figure 5.3 First modal variable  1 (Computed Torque Control) ................................... 91 Figure 5.4 First modal variable δ1 (Composite control) ................................................. 92 Figure 5.5 Second modal variable δ2 (Computed Torque Control)................................ 93 Figure 5.6 Second modal variable δ2 (Composite control) ............................................ 93 Figure 5.7 Control torque T (Computed Torque Control) .............................................. 94 Figure 5.8 Control torque T (Composite control) .......................................................... 94 Figure 5.9 Joint angle ..................................................................................................... 95

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Chapter 1 - Introduction and Literature Review

Chapter 1 Introduction and Literature Review 1.1 Motivation Robots have been widely used in a variety of applications, automotive manufacturing and tool manufacturing to name a few. Robots come in many shapes and sizes and have many different abilities. Basically, a robot is simply a mechanical system integrated with a computer system. Usually, it is able to move and has one or more electronic senses. Robotics is a very broad and interesting application, because like humans, robots have many fascinating aspects. In fact, robotic manipulators are undergoing a very fast evolution from relatively simple systems performing simple tasks to very complicated robots performing very demanding tasks. The use of such mechanical devices is justified primarily by their ability to accurately repeat tedious tasks, and by the ease with which they can reconfigured or reprogrammed. As a scope of applications of robots becomes wider, the performance requirements become more demanding. These requirements include high operating speeds and having manipulators with long and light weight members capable of handling large payloads [1]. Most existing robotic manipulators are designed to maximize stiffness in an attempt to minimize system vibration and to achieve good positional accuracy. High stiffness is achieved by using heavy material. As a consequence, such robots are usually heavy with respect to the operating payload. This, in turn, limits the speed of operation of the robot manipulation. In contrast, flexible robot manipulators exhibit many advantages over their rigid counterparts: they require less material, are lighter in weight, have

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higher manipulation speed, are more maneuverable and transportable, are safer to operate, have less overall cost and higher payload to robot weight ratio [2]. Due to their characteristics, this class of manipulators is suitable for a number of nonconventional robotic applications, including space applications. On the other hand, the study of link flexibility is enforced also for some kind of heavy manipulators such as large scale systems. In either case, it is no longer possible to assume that link deformation can be neglected [3]. In other words, one must explicitly consider the effects of structural link flexibility and properly deal with (active and/or passive) control of vibrational behavior. In this context, it is highly desirable to have an explicit, complete, accurate dynamic model at disposal. The model should be explicit to provide a clear understanding of dynamic interaction and coupling effects, to be useful for control and design, and to guide reduction and/or simplification based on terms relevance [4]. The problems due to the distributed link flexibility and the resulting vibrations often lead to long settling times and position inaccuracy in lightweight manipulators. Hence, it is essential to develop comprehensive dynamic models for this type of manipulator that consider all these factors. This will enable improved control techniques to be devised so that performance can be optimized [5]. Having a manipulator capable to perform well in different environment is an important objective in many situations. For example, when a robotic manipulator is mounted to a crane, mobile platform, or an autonomous vehicle, its performance is affected by the base excitation. This excitation causes problems with the accuracy of the tip positioning of the manipulator due to the vibration of the manipulator links. In such cases, the excitation appears as coefficients in the governing differential equations. And the system is said to be parametrically excited system. Here, a suitable control is required to damp out the oscillations of the manipulator links as they are commanded to follow a certain path.

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1.2

Modeling of Flexible manipulators

In the study of flexible manipulator, it is no longer possible to assume that link deformation can be neglected. So, it is important to have an accurate model that can adequately describe the dynamics of the manipulator [3]. A common way of modeling a flexible robot manipulator consists of using a combined Lagrange-assumed modes approach [6], where the deflection of the link is represented by a truncated series in terms of the mode shape functions and the time varying mode amplitudes. The shape functions can be obtained by solving the Euler-Bernoulli beam equation for flexural vibrations under a specified link boundary conditions [7]. A frequent way of modeling beams considers the well-known linear Euler-Bernoulli beam theory while modeling the flexible links where the longitudinal displacement of any point on the beam axis is neglected [8]. Hence, only a single displacement variable v(x,t) is used to represent the deflection of the beam. On the other hand, Nayfeh and Pai [9], and Baruh [10] regarded the nonlinear Euler-Bernoulli beam theory, where the shortening of the projection effect is affecting the position of the end point of a deformed beam. De Luca and Siciliano [4] presented the closed form equations of motion of planar multilink light-weight robots. Their kinematic model is based on standard frame transformation matrices describing both rigid rotation and flexible displacement under small deflection assumption. Zhang et al [11] presented and compared two modeling approaches of a flexible two link manipulator, namely, the assumed mode shape method (lumped model approach resulting in ordinary differential equations, ODE), and the distributed model approach (resulting in partial differential equations, PDE). The assumed mode shape approach forms a set of ODE’s that is finite dimensional. On the other hand, the distributed model approach results in a model that is infinite dimensional. The two approaches are then compared from stability and control perspectives. It is found that the PDE modeling and control strategy reported is requiring design insights for simplification, but has the advantage that modeling accuracy does not affect control as much as in the ODE case.

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Morris and Taylor [12] used a variational calculus approach to obtain the equations of motion and the associated boundary conditions for a flexible beam, where the authors chose a non-inertial frame of reference and drove expressions for the kinetic and potential energies in this frame. Then, they used separation of variables to derive the characteristic equation of the system and hence the natural frequencies of the system, which are the roots of this characteristic equation. Yoshikawa and Hosoda [13] proposed a modeling scheme using virtual rigid links and passive joints. Here, each link is modeled individually, and by connecting models of links, a model of the whole flexible arm is obtained. Ider and Amirouche [14] presented the analysis of flexible link manipulators as multibody systems, where the equations of motion, based on Kane’s equations, the strain energy, and modal analysis, is developed for three-dimensional beams that exhibit longitudinal and transversal deformations. When a flexible robot is mounted on an oscillating base, the dynamics of this system is coupled with the dynamics of the arm adding a new degree-of-freedom. On the other hand, the base of the robot arm is modeled as attached to a point under the influence of a prescribed acceleration. In this case, the oscillation of the base is considered as parametric excitation to the arm, where the resulting dynamic equations of motion are with rapidly varying coefficients without adding a new degree-of-freedom.

The

difference between the two modeling strategies is that, in the first one, the oscillating base is modeled as a spring-mass-damper system and the control objective is to achieve suppression of base vibration as well as design of control inputs that induce minimum vibrations [15]. This is in contrast to the case considered in this thesis, where one cannot reduce the base excitation as in the case of wave excitation of shipmounted robot manipulators, the control objective is to reduce the vibrations of the flexible link and maintain the accuracy of the tip position in the presence of sustained parametric excitation.

1.3

Control of Flexible manipulators

Position control problem associated with flexible manipulators is an important issue, as the dynamics of such systems is much more complex than that for rigid manipulators.

4


For robots with flexible links, the end-effector trajectory tracking problem is solved in [16] based on the iterative computation of the link deformations associated with the desired output motion, combined with a state trajectory regulator. Morris and Madani [17] presented a modified computed torque control, where the authors designed the control signal so that it, firstly, compensates for static deflection of the flexible links under gravity forces and, secondly, it acts to reduce both magnitude and time duration of link oscillations which arise naturally out of its flexibility. The problem of active damping of flexural vibrations of the robot links by only the joint control inputs have been addressed by Theodore and Ghosal [18], where the authors developed a controller based on end-point sensing and the rigid Jacobian of the manipulator. A model of N-link flexible manipulators is derived and reformulated in the form of singular perturbation theory and an integral manifold is used to separate fast dynamics from slow dynamics. A composite control algorithm is proposed for the flexible robots, which consists of two main parts. Fast control, uf, guarantees that the fast dynamics remains asymptotically stable. Slow control, us, consists of a robust PD designed based on the rigid model [19]. Sang-Ho Lee et al [20] proposed the hybrid control scheme to stabilize the vibration of a two-link flexible manipulator while the robustness of Variable Structure Control (VSC) developed for rigid manipulators is maintained for controlling the joint angels. Aoustin and Formal [21] proposed a method for synthesizing the feedforward torques and reference trajectory for a flexible two-link planar manipulator. The control law for each drive torque consists of two parts: a commanded feedforward torque and a linear angular position and velocity feedback. A full-state feedback control after linearizing the dynamic equations of a two-link robot with flexible links is developed, by Ider et al [22]. A control strategy based on input-output linearization of a flexible link system is developed by Moallem et al [23]. Here, the output redefinition concept is employed to choose points near the tip outputs such that stable zero dynamics are achieved. An observation strategy is developed and incorporated in the control scheme in order to estimate the rates of change of flexible modes. The problem of modeling and control of flexible joint robots (FJR) is considered by Osgoli and Taghirad [24], where the authors present a literature survey on control of

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such systems. Lozano et al [25] proposed a simple PD controller for robot manipulators, simultaneously, considering joint flexibility, actuator dynamics and friction. The controller synthesis is carried out using passivity and only requires angular position and velocity measurements. The control scheme presented is proved to converge globally in the sense that all the variables remain bounded and the robot angular position converges to its desired value. Modeling, simulation and control of a manipulator of flexible links and joints is presented by Subudhi and Morris [26] in which the authors have to control the flexure of both the joint and the link attached to it. A Modified Transpose Jacobian (MTJ) control, based on feedback linearization approach, is presented by Moosavian and Papadopoulos [27]. Here, the control algorithm employs stored data of the control command in the previous time step, as a learning tool to yield improved performance. The gains of this algorithm can be selected systematically, and do not need to be large; hence the noise rejection characteristics of the algorithm are improved. Based on Lyapunov’s theorems, it is shown that both the standard and the MTJ algorithms are asymptotically stable. Positive position feedback (PPF) controller is analyzed and applied to the PZT actuators for suppressing multi-mode vibrations while slewing a single-link flexible manipulator [28]. Experimental investigations into the development of feedforward and feedback control schemes for vibration control of a very flexible and high-friction manipulator system was developed by Mohamed et al [2], where a feedforward control scheme based on input shaping and low-pass filtering techniques and a strain feedback control scheme are examined. The performances of the controllers are assessed in terms of the input tracking capability and vibration reduction as compared to the response with PD control. Sanz and Exbarria [29] presented an experimental study of a robust control scheme for flexible-link robotic manipulators, where the design is based on a simple strategy for trajectory tracking which exploits the two-time scale nature of the flexible part and the rigid part of the dynamic equations of this kind of robotics arms.

6


For rigid manipulators subjected to base excitation, a simple robust control strategy that reduces mechanical vibrations of the base and enables better tip positioning is used by Young and Moon [30]. The control algorithm uses the sensory feedback of the base oscillation to modulate the manipulator actuator input to induce the inertial damping forces. An active damping control problem of robot manipulators with oscillatory bases is considered by Lin et al [15], where a first investigation of two-time scale fuzzy logic controller with vibration stabilizer for such structures has been proposed. Lin and Huang [31] proposed a new hierarchical fuzzy logic controller of mechanical systems with oscillatory bases, where the dynamics of a robotic system is strongly affected by disturbances due to the base oscillation. In the above literature, the robotic manipulator mounted on the oscillating-base is considered to be rigid.

1.4 Objective The dynamics of flexible manipulators is complex due to the flexibility distributed along the mechanical structure. This, in return, makes an improved control is one of the important needs. Conventional industrial manipulators use links and drives, made stiff, to minimize vibrations at the expense of lower speed and higher actuator power. A flexible manipulator control must achieve the same motion objectives as rigid manipulators, and must also stabilize the vibrations which are naturally excited, damping them out as fast as possible at the end of the path. The main objective of this work is to show a straight forward procedure of modeling and control of flexible link manipulators subjected to parametric excitation, and the effect of parameter changes on the response of such systems. In the case of rigid manipulators, the tip position is recognized by knowing the joint angles as well as the lengths of the manipulator links. And the control of such systems is achieved by feeding back the joint angles as in the case of the computed torque control technique. In the case of flexible manipulators, the joint angle feedback will not be enough to give a good manipulation performance as this feedback gives no information about the vibrations that the manipulator links exhibit.

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1.5

Scope of the Thesis

The scope of this thesis is concentrated basically on two major subjects: flexible link manipulators, and parametrically excited flexible arm robots. For the case of flexible link manipulators, a dynamic modeling of such systems is presented in Chapter 2, where the kinematics and dynamics of flexible manipulators, based on the combined Lagrangian-Assumed modes method, are computed. As an example of the modeling technique shown, a dynamic model of a two link flexible arm is presented. In Chapter 3, a control technique based on inverse dynamics approach is considered. Then, this technique is used and applied to the dynamical model presented in chapter 2 so that the manipulator follows a desired trajectory. Chapter 4 is devoted to control flexible robots by using a singularly perturbed model of the dynamical system. This model divides the system dynamics into two subsystems, namely, a rigid (slow) subsystem and a flexible (fast) subsystem. A control algorithm is then derived and applied to a single-link flexible arm. For the case of flexible arm robots subjected to parametric excitation, a model of Nlinks, parametrically excited manipulator that moves in the horizontal plane, is derived in Chapter 5. Two control techniques, computed torque control and a composite control based on singular perturbation theory are applied to a single-link parametrically exited manipulator. Finally, the concluding remarks on the study made and recommendations for future work will be shown in Chapter 6.

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Chapter 2 – Dynamic Modeling and Equations of Motion

Chapter 2 Dynamic Modeling and Equations of Motion

2.1 Perspective on This Chapter In this chapter, we introduce an efficient, complete, and conceptually straight forward modeling approach using 4×4 matrices that are familiar to workers in the field of robotics. It uses 4×4 matrices to represent the both joint and deflection motion. The deflection transformation is represented in terms of summation of modal shapes. The equations of motions are resulting from Lagrangian formulation. The kinematics of flexible beams is shown in section 2.2. In section 2.3, the concept of the Lagrangian formulation is presented. The final form of the simulation equations is presented in section 2.4. In section 2.5, the equations of motion of a two-link planar rigid-flexible arm are derived.

9


2.2 Kinematics of beams Consider an axially long structure, such as a straight beam of length L. we use an inertial xyz coordinate system, where the x axis is along the axis of the beam, going through the centroid of the cross section, as shown in Figure 2.1. This axis is referred to as the beam axis. As a result of deformation, the beam axis moves and a point A on the beam axis moves to a point A* as shown in Figure 2.2. The position of A* can be expressed as [10] r(x,t) = (x + u(x,t))i + v(x,t) j+ w(x,t)k

(2.1)

Where the u(x,t)), v(x,t) , and w(x,t) denote the components of the deformation in the x, y, and z directions, respectively. The distance along the beam axis to point A* is denoted by s. z

y

z x (beam axis)

o

ِ o

y

A

A* w

s

x

v

A x

L

Figure 2.1 Undeformed beam

x

u

Figure 2.2 Deformation of beam axis

In addition to the deformation of the beam axis, the plane shown in Figure 2.1 rotates, and a differential element around the beam axis assumes an orientation x*y*z*, shown in Figure 2.3. We will quantify this orientation by angles  x ( x, t ), y ( x, t ), and  z ( x, t ) . We hence have six variables at each point x to describe the deformation of the beam, the three displacements and the three rotations. Figure 2.3 shows the free-body diagram of the cross section, where there are three internal forces and three internal moments, shown along their components in x*y*z* coordinate system. These are referred to as

10


Px*: Axial force My*,Mz*: Bending moments

Py*,Pz*: Shear forces Mx*: Twisting moment

We now introduce additional assumptions that simplify the problem: all deformations u, v, and w are small compared to the length of the beam, all angles  x , y , and z are small. And all shear deformation effects can be ignored. The first two assumptions are justified by limiting the analysis to beams that deform very little. The assumption of no shear deformation effects is justified analytically by taking a full-blown model of a beam and analyzing the effects of shear deformation. For slender beams, it can be shown that the shear deformation effects become negligible compared to the bending and torsion. To examine the effects of these assumptions, consider an infinitesimal element of the beam that has an undeformed length of dx whose deformed length becomes ds (Figure 2.4). The coordinate axes x* y* z* and x** y** z** at the two ends of the differential element are obtained by rotations d x , d y , and d z . Because these angles are very small, the rotation sequence to go from x* y* z* to the x** y** z** coordinate is immaterial.

z* Mz*

z

My* Py*

y* y**

y Mz*

Pz*

Px* A*

x**

z** x*

y*

A**

x*

x z*

A*

Figure 2.3 Plane rotation

ds

Figure 2.4 Deformed infinitesimal element of the beam

Consider now the projections of the differential element onto the x*z* and x*y* planes, shown in Figure 2.5 and Figure 2.6. One can relate the rotations angles d z and d y to

11


ds by  y and  z , the radii of curvature of projections of the curve on the xy and xz planes, by

1

y



d z ds

1

z

z*



d y

(2.2)

ds y*

Beam axis

y

z ds A*

Beam axis

ds

 d y A*

ِx*

-y*

d z

x*

z*

Figure 2.5 projection of the differential element onto x*z*

Figure 2.6 projection of the differential element onto x*y*

The radii of curvature along xy and xz planes are given by

1

y



v' ' (1  v'2 )3 / 2

1

z



w' ' (1  w'2 )3 / 2

(2.3)

In which the prime denotes partial differentiation with respect to the spatial variable x. that is, v' ( x, t )  v( x, t ) / x. Because of small angels and deflections, we approximate the expressions 1 /  y and 1 /  z in the above equation by v'' and w'', respectively, and we approximate the differentiation with respect to s with differentiation with respect to x in Equations 2.2. Equating the different expressions for the radii of curvature and integrating both sides over x, we obtain

y  

w( x, t )   w' ( x, t ) x

z 

v( x, t )  v ' ( x, t ) x

(2.4)

Hence, the six variables used for the description of the deformation are reduced to four: u, v, w, and  x . The relations from the force and moment balances corresponding to Equations 2.4 indicate that the shear force is the derivative of the bending moment. We

12


next assume that the torsional motion of the beam can be separated from the axial motion and bending motions. Axial and bending motions are coupled to torsional motions only through higher-order terms. Having separated torsion from the rest of the motion, we now consider the axial stretch and bending in the y and z directions. Thus, the only component of the stress and the strain that we consider are  xx and  xx . Effect that will be included in the results from the curvature in the formulation are the shortening of the projection of the deformed member, which results from the curvature of the beam, and the effects that will not be included are rotary inertia, and shear strain. A beam modeled this way is referred to as an Euler-Bernoulli beam. Table 2.1 summarizes the commonly used beam models and the assumptions they make. The validity of the assumptions we use depends not only on the physical dimensions of the cross section of the beam, but also on the boundary conditions and the loading. The significance of the shortening of projection actually depends not so much on the amplitude of the elastic motion, but on the presence of large axial loads or rapid rigid body motions. Rotational motion can be viewed as generating a centrifugal force, which can be treated as an axial load. The rotary inertia is usually significant when there is rapid rotation about the x axis. We also ignore temperature effects. Table 2.1 Comparison of beam models Effects Included Model

Rotary inertia

Euelr-Bernoulli No Rayleigh Yes Timoshenko Yes Shear No

Bending Strain Yes Yes Yes No

Shear Strain No No Yes Yes

Shortening of projection Yes or no Yes or no No No

To illustrate the shortening of the projection [9], we consider a deformed beam. Figure 2.7 shows the projection of the deformation of the beam axis on the xy plane. The slope of the deformation along xy and xz planes is given by Equations 2.4. It follows that the slope of the beam axis at any point x is

 ( x, t )  v' 2 ( x, t )  w' 2 ( x, t )

(2.5)

13


Because shear effects are ignored, we neglect any other rotational deformation of the beam. We denote by s( x, t )  x  e( x, t )  Distance traversed along the beam axis from

x=0 to the deformed position A* e( x, t )  How much the beam has stretched

 ( x, t )  x  u( x, t )  Projection of the beam axis position onto the x axis

y Beam axis

 z (x)

A*

ds

e s A x

z

v

x

x dx

u

du

d

ζ

Figure 2.7 Deformed beam axis

Figure 2.8 Deformed differential element

Consider now a differential element of the beam, whose undeformed length is dx and whose deformed length is ds. Figure 2.8 shows the projection of this differential element on the xy plane. The projection of ds onto the s axis is d and it can be expressed as

 v   w  d  ds cos   ds cos       x   x  2

2

(2.6)

The difference between d and dx, d -dx=du, is referred to as the shortening of the projection. The nomenclature can be explained by noting that if there is no stretch, du will be negative quantity. Because the deformations are small, we approximate cos  in the above equation by a Taylor series expansion, so that

14


2 2 1 2 1  v   w   d  ds(1   )  ds        ds 2 2  x   x  

(2.7)

Integrating Equation 2.7, we obtain

1  ( x, t )  x  u ( x, t )  s ( x , t )  2

s ( x ,t )

 0

 v  2  w  2      d        

(2.8)

We introduce the definition of s and  into this equation. Also, because the stretch e(x,t) is small we replace the upper limit of the integral, s  x  e , by x. Hence, we have an expression relating the deformation in x direction to the stretch and the curvature in the y and z directions as 2 2 x 1  v   w   u ( x, t )  e( x, t )       d 2 0       

(2.9)

Of course, when e(x,t) is neglected, the deformation in x direction will be written as

u ( x, t )  

2 2 x 1  v   w        d  2 0       

(2.10)

15


2.3 Lagrange's Equations of motion Using the total kinetic energy of the system as well as the total potential energy (see appendix A), the Lagrange's equations are given by [4] The joint-j equation:

d  K dt  q j

  K    q   j

  Ve    q   j

  Vg    q   j

   Fj  

(2.11)

  Vg       j

 0  

(2.12)

The mode-j equation:

d  K dt   j

  K     j  

  Ve       j

Where K, Ve, and Vg are as outlined in appendix A. Fj is the external generalized forces corresponding to the generalized coordinates. The generalized force corresponding to the joint variable (qj) is the joint torque Fj. For the deflection variables (δj), the corresponding generalized forces will be zero if the corresponding modal deflections or rotations have no displacement at those locations where external forces are applied.

These equations are in the "inverse dynamic" form. To convert them to the simulation form, one must extract the coefficients of the second derivatives of the generalized coordinates to compose the inertia matrix for the system. The second and the first derivatives together make up the derivative of the state vector, which can be used in one of the available integration schemes, e.g. Runga-Kutta, to solve for the state as a function of time for given initial conditions and inputs Fj.

2.4 Final Simulation Equations Referring to appendix A, we define the vector of the generalized coordinates to be

q  11112 1m  n n1 n 2  nm 

(2.13)

Then, the final equations of motion have the form:

  C(q, q )  K e (δ)  g(q)  u M(q) q

(2.14)

16


where M rr M(q)   er M

M re   M ee 

(2.15)

is the inertia matrix, in particular, M rr q  is associated to the rigid coordinates,

M re q  and M er q  take into account the relationship between the flexible and rigid coordinate, and M ee q  is associated to the flexible coordinates, C r  C(q, q )   e  C 

(2.16)

contains Coriolis, and centrifugal terms, superscript r and e refer to rigid body and elastic degrees of freedom, respectively,

0  K e (δ)    Kδ

(2.17)

And K is a diagonal matrix whose diagonal submatrices are the stiffness matrices of links in terms of the modal variables,

g(q) contains the gravitational terms,

T  u  0

(2.18)

And T is the vector of joints torques.

17


2.5 Equations of Motion of a Two-link Rigid-Flexible Manipulator In this section, we find the equations of motion of a two-link flexible manipulator. Consider the manipulator shown in Figure 2.9 which is a two revolute planar manipulator that we want to find its equations of motion. As shown previously, to find the equations of motion, we need to compute the total kinetic energy of the system as well as the total potential energy, then we find the equations of motion using Lagrange formulation, for this manipulator the potential energy due to gravity will not be considered since the motion here is in the horizontal plane.

The upper arm of the manipulator is considered to be rigid while the forearm is a uniform Euler-Bernoulli beam modeled in accordance with assumed modes method, fixed free boundary conditions are assumed for the forearm, since the axis x2 is always tangent to the beam centerline at B. 1 and  2 are the joint angles. The deformation of the forearm is described by the first two bending modes since the higher bending modes are observed to be negligible.

C

x2

yc y0

y1

2

y2 x1

A

B

1 x0 xc

Figure 2.9 Two revolute joint planar manipulator In this system, n  2 (number of joints), mi  m2  2 (number of considered modes), so this system is a four degree of freedom system. 18


The generalized coordinate vector is q  1  2

1

 2 . We start now by finding the

total kinetic energy of the system, and to do this we have to find an expression for the position of an infinitesimal element on each link with respect to the local non-rotating frame attached to this link.

For the first link, it is assumed to be rigid. The position of an arbitrary point on this link expressed in frame 1 is given by (refer to appendix A)

 x1  0 i 1 h i ( xi ) h1 ( x1 )    0   1

(2.19)

Also, the transformation matrix between the inertial frame and frame 1 which is required to compute the kinetic energy of the first link is given as Ti  T1  T0 E 0 A1  A1 cos 1  sin  1   0   0

 sin 1 cos 1 0 0

0 0 0 0 1 0  0 1

(2.20)

Now, differentiating Equations (2.19) and (2.20) with respect to time to obtain 1 h 1 ( x1 ) and T 1 , then using 1 h1 ( x1 ) , 1 h 1 ( x1 ) , T1 , and T 1 plugging them in Equation (A.28 appendix A) and performing the required integrations using a symbolic manipulation package like Mathematica, we find an expression for the kinetic energy of the first link.

For the second link, which is flexible, the position of an arbitrary point on it with respect to frame 2 is again given as

19


 x2   x 2 j  0 2   i 2   h i ( xi ) h 2 ( x 2 )    2 j  y2 j   0  j 1  0      1  0 

(2.21)

where

 x2 j is the shortening of the projection due to the deformation of the second link as shown in section 2.2, and this shortening when neglecting the axial strain e is given in Equation 2.10.

For our case, Equation 2.10 can be written as

u ( x, t )  

2 x 1 2  v     d  2 0    

(2.22)

where

d ( x ) v   2 j y 2 j 2   2 jx 2 j  dx2

(2.23)

 y2 j denotes the mode shape vector, i.e.  y 2 j ( x2 )   y 21 ( x2 )  y 22 ( x2 )T where x2 is the position of an arbitrary point along x2-axis and the fixed-free deflection modes are [7] 

a x

a x 

sin a

 sinh a



a x

a x 

j 2 j 2 j j    cos j 2  cosh j 2   y 2 j   sin  sinh L2 L2  cos a j  cosh a j   L2 L2  

for j  1,2

(2.24)

where a1  1.875 and a2  4.694

20


Now Equation 2.21 can be written as 2 x   1 2  v   d   x2    2 0      2 h 2 ( x 2 )    21 y 21   22 y 22    0     1

(2.25)

At this point, we will proceed to find the equations of motion by first neglecting the shortening effect. Second, we will consider the shortening effect.

For both cases, we have to find the transformation matrix between the frame attached to the second link and the inertial frame, which is given again using Equation ( A.3 appendix A) as T2  T1E1 A 2

(2.26)

where T1 = is as given in Equation 2.20

1 0 E1   0  0

0 0 L1  1 0 0  transformation matrix of link 1 0 1 0  0 0 1

cos  2  sin  2 A2    0   0

 sin  2 cos  2 0 0

0 0 0 0 transformation due to joint 2 1 0  0 1

(2.27)

(2.28)

Again, differentiating Equations (2.27) and (2.28) with respect to time to obtain 2

h 2 ( x2 ) and T 2 , then using 2 h 2 ( x2 ) , 2 h 2 ( x2 ) , T2 , and T 2 , plugging them in Equation

21


(A.28 appendix A) and performing required integrations, we find an expression for the kinetic energy of the second link.

Now, adding the two expressions for the kinetic energy of the two links, we obtain the total kinetic energy of the manipulator.

The potential energy of the whole manipulator is considered now. We have shown that the potential energy comes out of two sources, which are gravity and elasticity. Since the problem considered here is planar, the gravitational potential energy will not be considered and we are left only with the elastic potential energy. Since the first link is considered to be rigid, then it does not share the elastic potential energy and the only source of the potential energy here is the elasticity of the second link. Using Equation (A.32 appendix A), the potential energy along an incremental length dx2 is 2 2  1    v( x2 , t )      dve 2  dx2 EI z  2 2   x2    

(2.29)

where

v( x2 )   2 j (t ) y 2 j ( x2 )

(2.30)

And the integration presented in Equation 2.43 is given now as

2

  2v( x2 )  12  dx2 Ve 2   EI z  2 20  x2  l

(2.31)

This gives the elastic potential energy of link two, which represents the total potential energy of the system.

Using the total kinetic energy as well as the total potential energy, plugging these expressions in Lagrange's Equations (2.11) and (2.12) and performing the required differentiations, we obtain the equations of motion of the manipulator in the form given by Equation (2.14) which can be written in the form

22


M rr  er M

M re  θ C r   0  T            M ee  δ C e  Kδ  0 

(2.32)

At this step, simplifying assumptions are valid. Since the elastic deflections are small, quadratic terms of  i and their derivatives will be omitted.

The elements of the matrices and vectors presented in Equation (2.32) are given in appendix B.

23

Chapter 3 – Manipulator Control Using Computed Torque Control

Chapter 3 Manipulator Control Using Computed Torque Control

3.1 Perspective on This Chapter This chapter is devoted to present the idea of inverse dynamics control derived for rigid manipulator, and the application of this control technique on a flexible-link manipulator. In most robotic applications, the reference commands are given in the Cartesian space. In the case of planar two-link manipulator, the position of the end point of the second link is defined by x and y coordinates. In order to obtain the joint angles that lead the end point of the two-link manipulator to the desired position, the inverse kinematics of the corresponding two-link rigid manipulator has to be carried out. The inverse kinematics of a two-link rigid manipulator has been extensively covered in the literature. In sections 3.2 and 3.3, brief descriptions of derivations leading to the equations used to perform the inverse kinematics and the forward kinematics are included for completeness. The idea of feedback linearization of nonlinear systems is discussed in section 3.4. The computed torque control algorithm derived for rigid manipulators is presented in Section 3.5. In Section 3.6, the control of the end point of a two-link rigid-flexible manipulator along a pre-defined path using computed torque method is presented. Simulation results and discussions about the effectiveness of the controllers are presented in this chapter.

24


3.2 Inverse Kinematics Consider the manipulator shown in Figure 3.1.; the desired position of the end point in Cartesian space is given by rc (t )  xc

yc  . The corresponding joint angles are given T

by [32]:

yc

l2

2

l1

1 xc Figure 3.1 A two-link planar arm

 yc  xc

1  tan 1 

  l sin  2   tan 1  2  l1  l 2 cos  2 

  1  D2  D 

 2  tan 1 

  

   

(3.1)

(3.2)

Where

xc2  y c2  l12  l 22 D 2l1l 2

(3.3)

Note that the expressions used include both the elbow-up and elbow-down solutions The velocity of the end point in Cartesian space is given by:

rc  J θ

(3.4)

Where J is the Jacobian matrix of the manipulator and given by:  l s  l s J   1 1 2 12  l1 c1  l 2 c12

 l 2 s12  l 2 c12 

(3.5)

25


The symbols s1 , c1 , sij , and cij refer to s1  sin 1 , c1  cos 1 , s12  sin(1   2 ) , and c12  cos(1   2 )

Provided that J is non singular matrix, the joint velocities can be obtained as:

θ  J 1rc

(3.6)

Differentiating Equation 3.4 yields and expression for the acceleration

d rc  Jθ   J θ  dt 

(3.7)

Thus, given a vector rc of end-effector accelerations, the instantaneous joint acceleration vector θ is given as:

θ  J 1 r   d J θ   c   dt   

(3.8)

Inverse kinematics can be approximated as shown in Figure 3.2.

d

rd

Inverse

rd

d

Kinematics

d

rd end-effector desired

Joints variables desired

position, velocity, and


acceleration.

acceleration.

Figure 3.2 inverse kinematics

In next sections, we can use this scheme to show that we will do inverse kinematics.

26


3.3 Forward Kinematics Consider again the rigid manipulator shown in Figure 3.1; the problem of forward kinematics is much easier than the problem of inverse kinematics. Here the joint variables are given and we need to determine the end-effector position. In the case of a two-link rigid manipulator, the end-position is given as [32]:

xc  l1 cos 1  l 2 cos(1   2 )

(3.9)

y c  l1 sin 1  l 2 sin(1   2 )

(3.10)

Of course, the output of the two-link flexible manipulator is given in terms of the joint angles, the deflections and the slopes at the end of each link, but we will not consider the case of flexible manipulator here and we will derive an expression for forward kinematics of flexible manipulator in a subsequent section. Forward kinematics can be approximated as shown in Figure 3.3.

d

rd

Forward

d

rd

Kinematics

d

rd

Joints variables

end-effector position,


velocity, and

acceleration.

acceleration.

Figure 3.3 Forward kinematics In next sections, we can use this scheme to show that we will do forward kinematics.

27


3.4 Feedback linearization The equations of motion of robotic manipulators weather considering their flexibility or not are multivariable, highly nonlinear, coupled equations. Multivariable means that the equations include more than one variable which are the joint variables in case of rigid manipulator. Nonlinear means that the equations have nonlinear terms like sines and cosines which change the system output in a nonlinear way. Finally, coupled systems means that the set of equations used to describe the system dynamics are interconnected. i.e. more than one variable of the system variables exist in one equation of motion of the equations describing the whole system. Consider the equations of motion given by Equation 3.11, which are the equations of motion of a two-link rigid planar manipulator

1 / 3m1 L12  m2 L12  1 / 3m2 L22  m2 L1 L2 cos( 2 ) 1 / 3m2 L22  0.5m2 L1 L2 cos( 2 ) 1        1 / 3m2 L22  0.5m2 L1 L2 cos( 2 ) 1 / 3m2 L22    2   m2 L1 L212 sin  2   0.5m2 L1 L222 sin  2  T1     0.5m2 L1 L212 sin  2    T2 

(3.11) The previous equations of motion are multivariable since they include two variables 1and 2 . Also, the equations are nonlinear since we have nonlinear terms like sin( 2 ) ,

and also they are coupled because when we write the equation of each joint individually we find that, for example, the joint variable  2 exists in the first equation. If we can linearize and decouple the system, then we can control the system using linear controllers like PID controller for example. Decoupling means that we simplify the coupled system into multi uncoupled equations. So, in the end of linearization and decoupling, we have independent linear system instead of having nonlinear coupled system.

28


Of course, there are several ways to linearize the nonlinear system; we mention two of them now. As Lyapunov proved, if a small-signal linear model is valid near an equilibrium position and is stable, then there is a region (which may be small, of course) containing the equilibrium within which the nonlinear system is stable. So, we can safely make a linear model and design a linear control for it such that, at least in the neighborhood of the equilibrium, our design will be stable. An alternative approach to obtain a linear model for use as the basis of control system design is to use part of the control effort to cancel the nonlinear terms and to design the remainder of the control based on linear theory. This approach-linearization by feedback-is popular in the field of robotics, where it is called the method of computed torque [33]. In this section, we discuss the idea of feedback linearization, and in the next section, we explain the method of computed torque. To introduce the idea of feedback linearization, consider the following simple system, x1  a sin( x2 )

(3.12)

x 2   x12  u

(3.13)

Note that we cannot simply choose u in the above system to cancel the nonlinear term a sin( x2 ) . However, if we first change variables by setting y1  x1

(3.14)

y2  a sin( x2 )  x1

(3.15)

then, by the chain rule, y1 and y2 satisfy y1  y2

(3.16)

y 2  a cos( x2 )( x12  u)

(3.17)

We see that the nonlinearities can now be cancelled by the control input

u

1 v  x12 a cos( x2 )

(3.18)

29


which result in the linear system in the ( y1 , y2 ) coordinates y1  y2

(3.19)

y 2  v

(3.20)

The term v has the interpretation of an outer-loop control and can be designed to place the poles of the second order linear system given by Equations (3.19) and (3.20) in the coordinates ( y1 , y2 ) . For example, the outer-loop control v  k1 y1  k2 y2

(3.21)

applied to the Equations (3.19) and (3.20) results in the closed-loop system y1  y2

(3.22)

y 2  k1 y1  k2 y2

(3.23)

This has the characteristic polynomial

p(s)  s 2  k2 s  k1

(3.24)

and hence, the closed-loop poles of the system with respect to coordinates ( y1 , y 2 ) are completely specified by the choice of k1 and k 2 . Figure 3.4 illustrates the innerloop/outer-loop implementation of the above control strategy. Linearized System





 k1 y1  k2 y2

v

1 v  x12 a cos( x2 )

 y1     y2 

u

x1  a sin( x2 )

 x1     x2 

x2   x12  u Inner Loop

Outer Loop

y1  x1 y2  a sin( x2 )

Figure 3.4 Inner Loop/Outer-loop control architecture for feedback linearization.

30


The response in the y variables is easy to determine. The corresponding response of the system in the original coordinates ( x1 , x2 ) can be found by inverting the transformation given by Equations (3.14), and (3.15). The result is x1  y1

x2  sin 1 (

(3.25)

y2 ) a

 a  y2   a

(3.26)

This example illustrates important features of feedback linearization. The first thing to note is the local nature of the result. We see from Equations (3.14) and (3.15) that the transformation and the control make sense only in the region    x1   ,



 2

 x2 

 2

. Second, in order to control the linear system given by Equations (3.19)

and (3.20), the coordinates ( y1 , y2 ) must be available for feedback. This can be accomplished by measuring them directly if they are physically meaningful variables, or by computing them from the measured ( x1 , x2 ) coordinates using the transformation given by Equations (3.14) and (3.15). In the latter case, the parameter a must be known precisely.

31


3.5 Computed Torque Control We now consider the computed torque method (CTM) which is a nonlinear control technique developed for rigid manipulators. In the literature, this method is also known as the inverse dynamics and this method is a special case of feedback linearization discussed in the previous section. To show the idea of this technique, we consider the dynamics equations of an n-link rigid robot which is given in the matrix form in Equation (3.27)   C(q, q )  g(q)  u M(q) q

(3.27)

The idea of inverse dynamics is to seek a nonlinear feedback control law u  f (q, q , t )

(3.28)

which, when substituted in Equation (3.27), results in a linear closed-loop system. For general nonlinear systems, such a control law may be quite difficult or impossible to find. In the case of manipulator dynamics given by Equation (3.27), the problem is actually easy. By inspecting Equation (3.27), we see that if we choose the control u according to the equation [32]

u  M(q)a q  C(q, q )  g(q)

(3.29)

then, since the inertia matrix M is invertible, the combined system given by Equations (3.27)-(3.29) reduces to

  a q q

(3.30)

The term a q represents a new input that is yet to be chosen. Equation (3.30) is known as the double integrator system as it represents n uncoupled double integrators. Equation (3.29) is called the inverse dynamics control (computed torque control) and achieves a rather remarkable result, namely that the system given by Equation (3.30) is linear and decoupled. This means that each input a qk can be designed to control a SISO

32


linear system. Moreover, assuming that a qk is a function only of q k and q k , then the closed-loop system will be decoupled. Since a q can now be designed to control a linear second order system, an obvious choice is to set ~  K q ~  d (t )  Κ v q aq  q p

(3.31)

~  q d  q , K , K are diagonal matrices with diagonal elements ~  qd  q , q Where q p v consisting of position and velocity gains, respectively. The reference trajectory

 d (t )) t  (q d (t ), q d (t ), q

(3.32)

defines the desired time history of joint positions, velocities, and accelerations. Note that Equation (3.31) is nothing more than a PD control with feedforward acceleration. Substituting Equation (3.31) into Equation (3.30), results in ~ ~ ~(t )  K q q v  K pq  0

(3.33)

~  q  d  q  Where q A simple choice for the gain matrices Kp, and Kv is 12  0 Kp       0

0



2 2

 0

0 21   0  0 , Kv            n2   0 

0 2 2  0

0  0       2 n   

(3.34)

This results in a decoupled closed-loop system with each joint response equal to the response of a critically damped linear second order system with frequency  i . The natural frequency  i determines the speed of response of the joint, or equivalently the rate of decay of the tracking error.

33


The inverse dynamics approach is extremely important as a basis for control and worthwhile to examine it from alternative viewpoints. We can give a second interpretation of the control law Equation (3.29) as follows. Consider again the manipulator dynamics Equation (3.27). Since M(q) is invertible for q  n , we may solve for the acceleration q of the manipulator as

  M1(q){u  C(q, q )  g(q)} q

(3.35)

Suppose we were able to specify the acceleration as the input to the system. That is, suppose we had actuators capable of producing directly a commanded acceleration (rather than indirectly by producing the force or torque). Then the dynamics of the manipulator, which is after all a position control device, would be given as

(t )  a q (t ) q

(3.36)

where a q (t ) is the input acceleration vector. This is again the familiar double integrator system. Note that Equation (3.36) is not an approximation in any sense; rather it represents the actual open-loop dynamics of the system provided that the acceleration is chosen as the input. The control problem for (3.36) is now easy and the acceleration input is a q can be chosen as before according to Equation (3.31). In reality, however, "such acceleration actuators" are not available to us and we must be content with ability to produce a generalized force (torque) u i at each joint i. Comparing Equations (3.35) and (3.36) we see that the torque input u and the acceleration input a q of the manipulator are related by

M1(q){u  C(q, q )  g(q)}  aq

(3.37)

Solving Equation (3.37) for the input torque u(t ) yields

u  M(q)a q  C(q, q )  g(q)

(3.38)

34


this is the same as the previously derived expression (3.29). Thus, the inverse dynamics can be viewed as an input transformation that transforms the problem from one of choosing torque input commands to one of choosing acceleration input commands. Note that, the implementation of this control scheme requires the real time computation of the inertia matrix and the vectors of Coriolis, centrifugal, and gravitational generalized forces. An important issue, therefore, in the control system implementation is the design of the computer architecture for the above computations. Linearized System

Trajectory Planner

qd

Outer Loop a q Controller

u

Inner Loop Controller

q

Robot

Inner Loop

Outer Loop Figure 3.5 Inner-Loop/Outer-loop control architecture

Figure 3.5 illustrates the so-called inner-loop / outer-loop control architecture. The inner-loop control computes the vector u of input torques as a function of the measured joint positions and velocities and the given outer-loop control in order to compensate the nonlinearities in the plant model. The outer-loop control designed to track a given reference trajectory can then be based on a linear and decoupled plant model. By this, that the computation of the nonlinear control law, Equation (3.29) is performed in an inner-loop with vectors q, q and a q as its inputs and u as output. The outer-loop in the system is then the computation of the additional input a q . Note that, the outer-loop control a q is more in line with the notion of a feedback control in the usual sense of being error driven.

35


The design of the outer-loop feedback control is in theory greatly simplified since it is designed for the plant represented by the dotted lines in Figure 3.5, which is now a linear system. Now, if we combine Equations (3.29) and (3.31), we obtain the overall robot arm input as ~  K q ~ d (t )  K vq  u  M(q)( q p )  C(q, q)  g(q)

(3.39)

~ and q ~ by e and e , respectively, Equation (3.39) can be written as If we replace q d (t)  K ve  K pe)  C(q, q )  g(q) u  M(q)( q

(3.40)

Now, we can write the closed-loop error dynamics as

e  K v e  K p e  0

(3.41)

Or in state-space form, d e  0  dt e   K p

I  e  K v  e 

(3.42)

The closed-loop characteristic polynomial is

 c ( s)  s 2 I  K v s  K p

(3.43)

Choice of PD Gains. It is usual to take the n n gain matrices diagonal so that

K v  diag k vi ,

K p  diag k pi 

(3.44)

Then n

 c ( s)  ( s 2 I  kvi s  k pi ) i 1

(3.45)

36


and the error system is asymptotically stable as long as k vi and k pi are all positive. It is important to note that although selecting the PD gain matrices diagonal results in a decoupled control at the outer-loop level, it does not result in a decoupled joint-control strategy. This is because multiplication by M(q) and addition of the nonlinear terms C(q, q ) and g(q) in the inner-loop includes the signal a q among all the joints. Thus,

information on all joint positions q(t ) and velocities q (t ) is generally needed to compute the control torque u(t ) for any one given joint. The standard form of the second-order characteristic polynomial is

P(s)  s 2  2 n s  n2 with 

(3.46)

the damping ratio and  n the natural frequency. Therefore, desired

performance in each component of the error e(t ) may be achieved by selecting the PD gains as

k pi   n2 ,

k vi  2 n

(3.47)

With  ,  n the desired damping ratio and natural frequency for joint error i. it may be useful to select the desired responses at the end of the arm faster than near the base, where the masses that must be moved are heavier. It is undesirable for the robot to exhibit overshoot, since this could cause impact if, for instance, a desired trajectory terminates at the surface of a workpiece. Therefore, the PD gains are usually selected for critical damping   1. In this case,

kvi  2 k pi ,

kvi2 k pi  4

(3.48)

Selection of the natural frequency; the natural frequency  n governs the speed of response in each error component. It should be large for fast responses and is selected depending on the performance objectives. Thus the desired trajectories should be taken into account in selecting  n . We discuss now some additional factors in this choice.

37


There are some upper limits on the choice for  n . The links of robots may have some flexibility. In this case, the natural frequency of the error should be chosen so that the vibration modes are not excited. We should select  n 

r 2

, where  r is the first

vibration mode frequency.

3.6 Trajectory Tracking Control of Flexible Manipulator using (CTM) In this section, we use the computed torque technique to control a flexible link manipulator to track a predefined trajectory. Consider the manipulator shown in Figure 2.13, which is a two link rigid-flexible arm moving in the horizontal plane. The equations of motion of this arm are given by Equation (2.80). We can re-write this equation as   C(q, q )  K e (δ)  u M(q) q

(3.49)

Since in this section we will simulate the motion of the manipulator, a more generic diagram that represents the robot mechanism given by Equation (3.49) is shown in Figure 3.6.

u 



M 1(q)

 q





q



q

C(q, q )  K e (δ)

Figure 3.6 Robot Mechanism It is shown in Figure 3.6 that the robot mechanism receives the input signal u which is the control torque applied to the manipulator. This signal is added to the vector of Corliolis and centrifugal terms and the vector of the stiffness matrix multiplied by the modal variables vector. The output signal from the summer is then multiplied by the inverse of the mass matrix to give the second derivative of the generalized coordinate

38


vector q this signal is then integrated twice to give q and q , which will be again used as feedback signals. In order to simulate the response of the manipulator under the computed torque control, a straight line trajectory of the end point is being used. Figure 3.7 shows the two-link flexible manipulator end point desired trajectory, which is a straight line along x 0 starting from rest xc 0  1 m and ending to rest at xcf  1 m , with yc  0.75 m . It is composed of an acceleration part, a constant velocity part (with 0.56 m/s velocity), and a deceleration part. For the acceleration and deceleration parts, the function of the trajectory is continuous up to the jerk to avoid impulsive forces. The acceleration and deceleration periods are 1.5 s each, and the constant velocity period is 2 s. The data used are m1  m2  1 kg , L1  1 m, and L2  1.5 m .

Constant velocity part Deceleration part

Acceleration part Desired trajectory

C

B

C

y0

0.75 m

L2

A

x0

L1 B  0.56 m

1m

Figure 3.7 Manipulator end point desired trajectory

To achieve this trajectory, we will specify the trajectory in three parts. In Figure 3.8, the desired trajectory schematic is presented. We wish to evaluate some unknowns like the

39


slope of the trajectory in the constant velocity interval, and the positions at which the constant velocity period starts and ends.

1m Trajectory along x0

1 m t f  tb

tb Blend time

tf

Time

Figure 3.8 Desired trajectory scheme along x0 To evaluate the unknowns mentioned above, the interval of the constant velocity will have a special interest. This interval starts at time t b (blend time) which is equal to 1.5 s, and ends at time t f  tb which is equal to 3.5 s. This interval is a linear segment with slope V represents the velocity of the manipulator end point through the constant velocity period, and the trajectory equation is given by x(t )  x(tb )  V (t  tb )

(3.50)

where V  0.56 m/s

tb  1.5 s

40


So, it is remaining now to determine the equation of the trajectory in the interval of constant velocity, to determine the value x(tb ) we use the fact of the symmetry of the trajectory that gives

tf x 2

 x0  x f   2 

(3.51)

Also, from Equation (3.50) we have

t t  x f   x(tb )  V ( f  tb ) 2 2

(3.52)

Substituting by Equation (3.51) into Equation (3.52), we get x0  x f 2

 x(tb )  V (

tf 2

 tb )

(3.53)

Solving for x(tb ) x(tb ) 

x0  x f 2

V (

tf 2

 tb )

(3.54)

Using the values of x0 , x f ,V , t b , and t f , the position of the end point at the end of acceleration interval is 1  (1) 5  (0.56)(  1.5) 2 2 x(t b )  0.56 m x(t b ) 

(3.55)

Of course, due to the symmetry of the trajectory, the position at which the end point starts the deceleration interval is x(t f  t b )  0.56 m , and this position is shown in Figure 3.7. Now Equation (3.50) is given as

41


x(t )  0.56  0.56(t  1.5) x(t )  1.4  0.56t

(3.56)

That represents the spline function in the constant velocity interval. Considering now the acceleration interval, which starts at t  0 s and ends at t  tb  1.5 s , and the position at the end of this interval is as shown previously

x(tb )  0.56 m . What we need to find now is an equation that describes the spline

function in this interval. Since the spline function in this interval should be continuous up to the jerk, then we have 8 constraints on the motion, two for the position, two for velocity, two for acceleration, and two for the jerk. These constraints are given as At t  0 s

and

x ( 0)  1 m

at t  tb  1.5 s

v(0)  0 m/s

x(1.5)  0.56 m v(1.5)  0.56 m/s

a (0)  0 m/s 2

a(1.5)  0 m/s 2

a (0)  0 m/s 3

a (1.5)  0 m/s 3

(3.57)

Since we have 8 constrains for the acceleration part, the spline function used here is a polynomial of order 7, and the spline function is given as [34] x(t )  c0  c1t  c2t 2  c3t 3  c4t 4  c5t 5  c6t 6  c7 t 7

(3.58)

where c0 c7 are constant coefficients to be determined. By differentiating this expression three times, we obtain expressions for velocity, acceleration, and jerk as follows v(t )  c1  2c2t  3c3t 2  4c4t 3  5c5t 4  6c6t 5  7c7t 6

(3.59)

a(t )  2c2  6c3t  12c4t 2  20c5t 3  30c6t 4  42c7t 5

(3.60)

a (t )  6c3  24c4t  60c5t 2  120c6t 3  210c7t 4

(3.61)

It is clear from the four constraints at t  0 s that c0  1, c1  0, c2  0, and c3  0 . Using the constraints remaining and imposing them into the proper equations, we get four equations in four unknowns, which can be written in the matrix form as

42


11.4 17.1  c 4   0.44 5.06 7.6  0.56 13.5 25.31 45.6 79.73  c5     0   27 67.5 151.88 318.94  c6       135 4.5 1063.125 c7   0   36

(3.62)

Solving the previous set of equations using MATLAB, we obtain the values of the coefficients as c4  0.5531, c5  0.5531, c6  0.1967, and c7  0.0234 . Now plugging the values of the coefficients into Equation (3.58) we obtain the spline function for the acceleration part as x(t )  1  0.5531t 4  0.5531t 5  0.1967t 6  0.0234t 7

(3.63)

Now, we consider the deceleration interval, which starts at t  t f  t b  3.5 s and ends at t  t f  5 s . The position at the beginning of this interval is as shown previously x(t f  t b )  0.56 m . In this interval, we have 8 constraints on the motion, two for the

position, two for velocity, two for acceleration, and two for the jerk. These constraints are given as At t  t f  t b  3.5 s

and

at t  t f  5 s

x(3.5)  0.56 m v(3.5)  0.56 m/s

x(5)  1 m v(5)  0 m/s

a(3.5)  0 m/s 2

a (5)  0 m/s 2

a (3.5)  0 m/s 3

a (5)  0 m/s 3

(3.64)

Here, we have 8 constraints for the deceleration part. The spline function used here is a polynomial of order 7 and it is given as shown in Equation (3.58). Also, expressions for velocity, acceleration, and jerk are given as shown in Equations (3.59), (3.60), and (3.61). By using the constraints given by Equation (3.64) and imposing them into the trajectory equations, we obtain 8 equations in 8 unknowns, which are the coefficients of the spline polynomial. We can write these equations in the matrix form as

43


6433.92  c0   0.56 1 3.5 12.25 42.9 150.1 525.22 1838.27   1  1 5 25 125 625 3125 15625 78125   c1      0.56 0 1 7 36.75 171.5 750.3 3151.31 12867.9  c 2       10 75 500 3125 18750 109375  c3   0   0 1  0  0 0 2 21 147 857.5 4501.875 22059.2  c 4       2 30 300 2500 1875 131250  c5   0  0 0  0  0 0 0 6 84 735 5145 31513.125 c6       0 6 120 1500 15000 131250  c7   0  0 0

(3.65) Again solving the previous set of equations using MATLAB, we obtain the values of the coefficients as

c0  140, c1  325.19, c2  301.12, c3  147.69, c4  41.95, c5  6.944, c6  0.6227, and c7  0.0234 Now plugging the values of the coefficients into Equation (3.58), we obtain the spline function for the deceleration part as x(t )  140  325.19t  301.12t 2  147.69t 3 41.95t 4  6.944t 5  0.6227t 6  0.0234t 7

(3.66)

So far, we know the functions that describe the end point trajectory in x direction. And by differentiating these function with respect to time, we obtain equations for velocity and acceleration, but that was in x direction only. If we consider the y direction now, the spline function is very simple since the trajectory along y0 axis is required to be constant and y(t )  0.75 during all intervals. So, along the y0 axis, the trajectory equations are given as y(t )  0.75 m

(3.67)

v(t )  0 m/s

(3.68)

a(t )  0 m/s 2

(3.69)

Figures (3.9), (3.10), and (3.11) show the desired trajectory (position, velocity, and acceleration) of the end point along x0 axis. We draw these curves using the Equations (3.56), (3.63), and (3.66) and their derivatives.

44


1.5

Displacement Along x0 (m)

1 0.5 0 -0.5 -1 -1.5

0

1

2 3 Time (sec.)

4

5

Figure 3.9 Desired position of point C along x0

0.1

Velocity Along x0 (m/s)

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0

1

2 3 Time (sec.)

4

5

Figure 3.10 Desired velocity of point C along x0

45


0.8

Acceleration Along x0 (m/s2)

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 0

1

2 3 Time (sec.)

4

5

Figure 3.11 Desired acceleration of point C along x0 The end point trajectory is now known. The corresponding trajectory of the manipulator joints is also required. The joints trajectory will be obtained using inverse kinematics of a two-link rigid manipulator shown previously in section 3.2. Of course, the trajectory of the joints will be used in the simulation process, especially, in computing the torque required to derive the manipulator given by Equation (3.39). So, using Equations (3.1) and (3.2) in conjunction with Equations (3.56), (3.63), and (3.66) we obtain the desired trajectory of the manipulator joints. Figures (3.12), (3.13), and (3.14) show the desired position, velocity, and acceleration of the first joint angle  1 . Similarly, figures (3.15), (3.16), and (3.17) show the desired position, velocity, and acceleration of the first joint angle  2 .

46


1.5

1 (rad)

1

0.5

0

-0.5

-1 0

1

2 3 Time (sec.)

4

5

Figure 3.12 Desired position of joint variable  1

1.2 1

1 (rad/s)

0.8 0.6 0.4 0.2 0 -0.2 0

1

2 3 Time (sec.)

4

5

Figure 3.13 Desired velocity of joint variable  1

47


1

2

1 (rad/s )

0.5

0

-0.5

-1 0

1

2 3 Time (sec.)

4

5

Figure 3.14 Desired acceleration of joint variable 1

2.9 2.8 2.7

2 (rad)

2.6 2.5 2.4 2.3 2.2 2.1 2 0

1

2 3 Time (sec.)

4

5

Figure 3.15 Desired position of joint variable  2

48


0.4 0.3

2 (rad/s)

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 0

1

2 3 Time (sec.)

4

5

Figure 3.16 Desired velocity of joint variable  2

0.6 0.4

2

2 (rad/s )

0.2 0 -0.2 -0.4 -0.6 -0.8 0

1

2 3 Time (sec.)

4

5

Figure 3.17 Desired acceleration of joint variable  2

49


What we did so far is typically the function of the trajectory planner shown in Figure 3.5, which acts to find the desired trajectory of the joint variables knowing the desired trajectory of the manipulator end point (endeffector) Now, we will use the desired values for (position, velocity, and acceleration) obtained into the control law. The control law (computed torque control) which is given by Equation 3.40 can now be written as

 d (t)  K v e  K p e)  C(q, q ) u  M(q)( q

(3.70)

Figure 3.18 shows a schematic diagram for the computed torque control given by Equation (3.70). It presents the transfer of signals between the trajectory planner, the controller, the robot mechanism, and the endeffector position.

xd xd xd

yd y d

Inverse kinematics assuming two - links

forward

θ  d

M(q)







u



Robot Mechanism

δ θ

act

θ act

rigid robot

yd

θ d  θd

kp

C(θ, θ )

kv







kinematics

xact

assuming two - links y act flexible robot

Figure 3.18 Computed torque control schematic Now, we need to compute the gains of the computed torque control. These gains are obtained based on the smallest natural frequency of the flexible arm, which is computed using the following equation [7]

1  (a1 ) 2

EI mL3

(3.71)

50


Where a1 is the first eigenvalue, m is the mass of the beam, EI is the flexural rigidity of the beam, and L is the beam length. The length of the beam L  1.5 m , its mass m2  1 kg , and its flexural rigidity

EI  50 N.m 2 . This gives the smallest natural frequency of the beam to be approximately equal to 13.5 rad/s . And to avoid excitation of any of the natural frequency, the control gains are obtained by setting  n in the following equations to be 6.76 rad/s

k p  n2 ,

kv  2n

(3.72)

And the control gains are given as kp = 45.7 and kv = 13.52. It is left now to simulate the manipulator motion under the control of the (computed torque method). The forward kinematics equations of the manipulator, when flexible links are involved, are given as;

 xact   L1 cos 1  L1 cos(1   2 )  [11 ( L2 )   22 ( L2 )] sin(1   2 )  y    L sin   L sin(   )  [  ( L )    ( L )] cos(   )  1 1 1 2 1 1 2 2 2 2 1 2   act   1

(3.73)

Figures (3.19) and (3.20) show the manipulator endeffector error while it tracks the desired trajectory. In these figures, the error in x-direction starts by a larger value than the starting value of the error in y-direction. Also, the ending error value in y-direction is larger than that for the value in x-direction. This is due to the fact, that the starting orientation of the second link is almost vertical and the ending orientation is almost horizontal. Figures (3.21) and (3.22) show the torques used to derive the manipulator. In these figures, it is clear that the torque required at joint A is higher than that for joint B. This is because the actuator at A carries two links inertias and the actuator at B only carries one link inertia. Figures (3.23) and (3.24) show the modal coordinates 1 and  2 , where their order of magnitude is small, and in return, the error of the end point is also small.

51


-3

5

x 10

Error Along x0 (m)

4 3 2 1 0 -1 -2 0

1

2

3 Time (sec.)

4

5

6

5

6

Figure 3.19 End point error along x0 -3

2.5

x 10

Error Along y0 (m)

2 1.5 1 0.5 0 -0.5 -1 0

1

2

3 Time (sec.)

4

Figure 3.20 End point error along y0

52


0.6

Control Torque T 1 (N.m)

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

1

2

3 Time (sec.)

4

5

6

Figure 3.21 Control torque applied at A, T1.

0.5

Control Torque T 2 (N.m)

0.4 0.3 0.2 0.1 0 -0.1 -0.2 0

1

2

3 Time (sec.)

4

5

6

Figure 3.22 Control torque applied at B, T2.

53


-3

1

x 10

Modal Variable  1 (m)

0.5 0 -0.5 -1 -1.5 -2 0

1

2

3 Time (sec.)

4

5

6

5

6

Figure 3.23 First modal coordinate  1

-5

3

x 10


2 1 0 -1 -2 -3 -4 -5 0

1

2

3 Time (sec.)

4

Figure 3.24 Second modal coordinate  2 54


In this chapter, the response of a two-link flexible manipulator under the (computed torque control) has been investigated. The flexibility of the forearm was expressed in terms of two modal variables. Four variables were used to describe the motion of the manipulator. Two of these variables are the angles representing the joint motion of the equivalent rigid link. The reference commands for the two link manipulator were given in Cartesian space. In order to obtain the joint angles corresponding to the desired end point position in Cartesian space, the inverse kinematics of a rigid two link manipulator was carried out. The trajectory was a straight line between two points. The computed torque control was successful to control the manipulator, since the order of magnitude of the error is observed to be small. But, it should be kept in mind that, when links with lower flexural rigidity are used, the order of magnitude of the error will be higher and higher.

55

Chapter 4 – Manipulator Control Using Singular Perturbation Approach

Chapter 4 Manipulator Control Using Singular Perturbation Approach

4.1 Perspective on This Chapter Singular perturbation theory has been shown to be a convenient strategy for "reducedorder modeling". It is well known that the dynamics of a singularly perturbed system can be approximated by the dynamics of the corresponding reduced-order (slow) and boundary layer (fast) subsystems for sufficiently small values of the singular perturbation parameter. The goal is to simplify the implementation of control algorithms. A control strategy for each subsystem is utilized so that the two subsystems stabilize [35]. In this chapter, two control techniques, namely computed torque law and composite control based on the singular perturbation approach, are used to control the motion of a one flexible-link manipulator. The effect of manipulator joint inertia on the selection of the control technique used has been investigated. It is found that, unlike composite control, the joint inertia increases the oscillations of the flexible arm when the arm is commanded to follow a certain trajectory under the computed torque control law alone. On the other hand, when we exclude the joint inertias, while modeling the flexible arm, there is no significant difference between the system response based on singular perturbation approach and that based on the computed toque method. The equations of motion of a one-link flexible arm are derived in section 4.2. Section 4.3 is devoted to rearrange the equations of motion to be in a form suitable for time

56


scale separation. Section 4.4 includes the application of time scale separation to transform the whole system into a singularly perturbed system. A control strategy to stabilize the system is presented in section 4.5. The response of a one-link flexible arm is presented in section 4.6, where the effect of the joint inertia has been investigated.

4.2 One-link Flexible Arm EOM In this section, the equations of motion of a one-link flexible arm, shown in Figure 4.1, are derived. To find the equations of motion, we need to compute the total kinetic energy of the system as well as the total potential energy of the system, and then we find the equations of motion using Lagrange formulation. For this manipulator, the potential energy due to gravity will not be considered since the motion here is in the horizontal plane. Y0

Y1

M

X1

 O

Xe

X0

Figure 4.1 One-link flexible-arm The arm is a uniform Euler-Bernoulli beam modeled in accordance with assumed modes method. fixed free boundary conditions are assumed for the arm since the axis X1 is always tangent to the beam centerline at O.  is the joint angle. The deformation of the forearm is described by the first two bending modes since the higher bending modes are observed to be negligible. Here, n  1 (number of joints), mi  m1  2 (number of considered modes).

57


As shown in chapter 2, the equations of motion of a flexible arm can be written in the form;

  C(q, q )  K e (δ)  g(q)  u M(q) q where M(q) Is the inertia matrix, q  1   n

(4.1)

11  1m1  21   2mn T is

the vector of the generalized coordinates,  is the joint variable,  is the deflection variables, C(q, q ) Contains Coriolis and centrifugal terms, K e (δ) is a diagonal matrix whose diagonal sub-matrices are the stiffness matrices of links in terms of the modal variables,, g(q) contains the gravitational terms, and u  T

0 , Where T is the T

vector of joints torques. Here, the generalized coordinate vector is q  θ δ1

δ2  . T

We start now by finding the total kinetic energy of the system. The kinetic energy is given by [4]; T  Th  TL  Tp

(4.2)

Where, Th is the kinetic energy of the joint inertia J h located at O.

Th 

1 2 J h 2

(4.3)

TL is the link kinetic energy. To find it, we have to find an expression for the position of an infinitesimal element on the link with respect to the local non-rotating frame attached to this link. This is given by;

x  x    vx, t   t  x    t  x  2 2 1  1 1  h 1 ( x, t )    0    0     1  1   

(4.4)

1 and  2 denote the mode shapes of the link deformation. The fixed-free deflection modes are [7];

58




ajx



L

 j x    sin

 sinh

a j x  sin a j  sinh a j   a x a x    cos j  cosh j  , for j  1, 2 . L  cos a j  cosh a j   L L 

(4.5) For 0.5 payload to link mass ratio, we have a1  1.42 and a2  4.11 . Then, we find this position with respect to the inertial frame 0 h1 ( x, t )  h1 ( x, t ) by multiplying the position obtained by the transformation matrix between frame 1 and frame 0 which is given as;

cos   sin  T  0   0

 sin  cos  0 0

0 0 0 0 , and 1 0  0 1

(4.6)

h1 ( x, t )  T1 h1 ( x, t )

(4.7)

Now, the kinetic energy of the link is given by L

TL 

1  h 1T x, t h 1 x, t dx 2 0

(4.8)

T p is the kinetic energy associated with the payload of mass mp and moment of inertia

Jp, and is given as; 1 1    T p  M p h 1T L, t h 1 L, t   J p    vx, t   2 2  t  x 

2

(4.9)

Now we consider the potential energy of the system. Here, the manipulator is moving in the horizontal plane, and the gravitational potential energy is not included. So, we are left only with the elastic potential energy. The potential energy along an incremental length dx is 2 1    2v( x, t )   dve  dx EI 2   x 2    

(4.10)

59


where, EI is the flexural rigidity. Integrating over the length of the link, we obtain the elastic potential energy Using the total kinetic energy, as well as the total potential energy, plugging these expressions in Lagrange's Equations, we obtain the Equations of motion of the manipulator.

4.3 Flexible Manipulator Dynamics revisited Consider now the equations of motion given by Equation (4.1). These Equations can be written in the form [19]

θ  f (θ, θ )   g1 (θ, θ , δ, δ )   0  T M(θ, δ)     1               δ f 2 (θ, θ) g 2 (θ, θ, δ, δ) Kδ  0 

(4.11)

where M is the inertia matrix, f1 and f2 are the vectors containing gravitational (only in f1), Coriolis, and centrifugal terms, g1 and g2 are the vectors which account for the interaction of joint variables and their time derivatives with deflection variable and their time derivatives. Since the inertia matrix M is positive definite, then it can be inverted and denoted by H, which can be partitioned as follows:

 H11[ nn ] H12[ nm]  M 1  H    H 21[ mn ] H 22[ mm] 

(4.12)

Now, equations (4.11) become

θ  H f  H f  H g  H g  H Kδ  H T 11 1 12 2 11 1 12 2 12 11

(4.13a)

δ  H f  Η f  H g  H g  H Kδ  H T 21 1 22 2 21 1 22 2 22 21

(4.13b)

The flexible dynamic system (4.13a and 4.13b) is characterized by n+m generalized coordinates but only n control inputs. Therefore the synthesis of a nonlinear feedback control is not as easy as for rigid arm (a control input for each joint).

60


A model order reduction is made by the use of singular perturbation theory leading to a composite control for the full order system [36]. The reduced order system is presented in the next section.

4.4 A Singularly Perturbed Model A singularly perturbed model of the dynamic system (4.13) can be obtained as follows. Assume that the orders of magnitude of the K matrix elements (ki's) are comparable. It is the appropriate to extract a common scale factor k (the smallest spring constant, for instance) such that

~ ki  kki ,

i = 1, …, m.

(4.14)

The following new variables (elastic forces) can be defined: ~ ζ  k kδ

(4.15)

~ ~ ~ k  diag (k1  km )

(4.16)

The next step is to define   1/ k and obtain

θ  H (θ, ζ)f (θ, θ )  H (θ, ζ)f (θ, θ ) 11 1 12 2    H11 (θ, ζ)g1 (θ, θ, ζ, ζ)  H12 (θ, ζ)g 2 (θ, θ , ζ, ζ )

(4.17a)

 H12 (θ, ζ)ζ  H11 (θ, ζ)T

ζ  H 21 (θ, ζ)f1 (θ, θ )  H 22 (θ, ζ)f 2 (θ, θ )  H 21 (θ, ζ)g1 (θ, θ , ζ, ζ )  H 22 (θ, ζ)g 2 (θ, θ , ζ, ζ )

(4.17b)

 H 22 (θ, ζ)ζ  H 21 (θ, ζ)T

which is a singularly perturbed model of the flexible arm. Notice that all quantities on ~ the right hand side of (4.17b) have been conveniently scaled by k . It can be shown that as   0 , the model of the rigid manipulator is obtained from (4.17a and 4.17b). Formally, setting   0 and solving for ζ in (4.17b) one obtains

61


ζ  H 221 (θ,0)[H 21(θ,0)f1 (θ, θ )  H 21(θ,0)g1 (θ, θ ,0,0)  H 21(θ,0) T ]  f (θ, θ )  g (θ, θ ,0,0) 2

(4.18)

2

where the overbars are used to indicate that the system with   0 is considered. Plugging (4.18) into (4.17a) yields

θ  [H (θ,0)  H (θ,0)H 1 (θ,0)H (θ,0)]  [f (θ, θ )  T] 11 12 22 21 1

(4.19)

It can be easily checked that 1 H11(θ,0)  H12 (θ,0)H 221 (θ,0)H 21(θ,0)  M11 (θ )

(4.20)

where M11 (θ ) is the [n  n] positive definite matrix for the rigid-link arm. Hence, f1 is exactly the function of θ and θ appearing in the rigid-arm model. choosing x1  θ , x 2  θ , and z1  ζ , z 2   ζ with    gives the state-space form of the system (4.17a and 4.17b);i.e.

x 1  x 2 , x 2  H11 (x1 ,  2 z1 )f1 (x1 , x 2 )  H12 (x1 ,  2 z1 )f 2 (x1 , x 2 )  H11 (x1 ,  2 z1 )g1 (x1 , x 2 ,  2 z1 , z 2 )  H12 (x1 ,  2 z1 )g 2 (x1 , x 2 ,  2 z1 , z 2 )

(4.21a)

 H12 (x1 ,  2 z1 )z1  H11 (x1 ,  2 z1 )T

 z 1  z 2 ,  z 2  H 21(x1 ,  2 z1 )f1 (x1 , x 2 )  H 22 (x1 ,  2 z1 )f 2 (x1 , x 2 )  H 21(x1 ,  2 z1 )g1 (x1 , x 2 ,  2 z1 , z 2 )  H 22 (x1 ,  2 z1 )g 2 (x1 , x 2 ,  2 z1 , z 2 )

(4.21b)

 H 22 (x1 ,  2 z1 )z1  Η 21(x1 ,  2 z1 )T

62


At this point, singular perturbation theory requires that the slow subsystem and the fast (boundary layer) subsystem be identified. The slow subsystem is formally obtained by setting   0 , i.e., the rigid model of the arm of the arm obtained above

x 1  x2 1 x 2  M11 (x1 )[f1 (x1 , x 2 )  T]

(4.22)

To derive the fast subsystem, we introduce the fast time scale   t /  . Then it can be recognized that the system (4.21a and 4.21b) in the fast time scale becomes

dx1   x2 , d dx 2   [H11(x1 ,  2 ( η1  ζ ))f1 (x1 , x 2 ) d  H12 (x1 ,  2 ( η1  ζ ))f2 (x1 , x 2 )  H11(x1 ,  2 ( η1  ζ ))g1 (x1 , x 2 ,  2 ( η1  ζ ), η2 )

(4.23a)

 H12 (x1 ,  2 ( η1  ζ ))g 2 (x1 , x 2 ,  2 ( η1  ζ ), η2 )  H12 (x1 ,  2 ( η1  ζ ))(η1  ζ )  H11(x1 ,  2 ( η1  ζ ))T]

dη1  η2 , d dη2  H 21(x1 ,  2 ( η1  ζ ))f1 (x1 , x 2 ) d  H 22 (x1 ,  2 ( η1  ζ ))f 2 (x1 , x 2 )  H 21(x1 ,  2 ( η1  ζ ))g1 (x1 , x 2 ,  2 ( η1  ζ ), η2 )

(4.23b)

 H 22 (x1 ,  2 ( η1  ζ ))g 2 (x1 , x 2 ,  2 ( η1  ζ ), η2 )  H 22 (x1 ,  2 ( η1  ζ ))(η1  ζ )  H 21(x1 ,  2 ( η1  ζ ))T

Where the new fast variables η1 and η 2 are defined as

η1  z1  ζ  z1  z1 ,

η2  z 2

Now setting   0 in (4.23a and 4.23b) gives

(4.24)

dx1 dx 2   0 ; i.e., x1 and x2 are constant d d

in the boundary layer. Furthermore, it can be recognized that g1 (x1 , x 2 ,0,0)  0 and

g 2 (x1 , x 2 ,0,0)  0 , since, by definition, those terms are representative of products of the

63


components of x1 and/or x2 with the components of  2 z1 and/or  z 2 . Therefore, the fast subsystem can be found to be

dη1  η2 , d dη2   H 22 (x1 ,0) η1  H 21 ( x1 ,0)(T  T ) d

(4.25)

Which is a linear system parameterized in the slow variables x1 .

4.5 Composite Control As evidenced by the two reduced-order subsystems (4.22) and (4.25), a composite control strategy [36] can be pursued. The design of a feedback control for the full system can be split into two separated designs of feedback controls T and T f for the two reduced-order systems; formally

T  T(x1, x2 )  Tf (x1, η1, η2 )

(4.26)

where T(x1 , x 2 ) is the slow control, and Tf (x1 , η1 , η2 ) is the fast control. As far as the slow control is concerned, the computed torque control developed for rigid manipulators as shown previously in chapter 3 can be utilized and the slow control is given by

T  M11[θd  k v (θ d  θ )  k p (θd  θ)]  f1

(4.27)

where kp and kv are the position and velocity feedback gain matrices respectively. At this point, singular perturbation theory requires that the boundary layer system (4.25) be uniformly stable along the equilibrium trajectory ζ given by Equation (4.18). This can be accomplished if the pair

I  0 A ,  H 22 0

 0  B  H 21 

(4.28)

64


is uniformly stabilizable for any slow trajectory x1 (t ) . Assuming that this holds, a fast state feedback control of the type Tf  k pf (x1 )η1  k vf (x1 )η2

(4.29)

will stabilize the boundary layer system (4.25) to η1  0 (z1  ζ ) and η2  0 (z 2  0) . Since the main purpose of flexible manipulator control is to damp out the deflections at steady state as fast as possible, the fast control (4.29) can be designed as an optimal control for the boundary layer. Under the above conditions, a fundamental result in singular perturbation theory ensures that the state vectors of the full system can be approximated by

x1  x1  O( ) ,

x 2  x 2  O( ) ,

(4.30a)

z1  ζ  η1  O( ) ,

z 2  η2  O( ) .

(4.30b)

where the operator O( ) means the order of magnitude of  . Under the slow control (4.17), θ and θ will tend respectively to θ d and θ d (the desired values). The fast control (4.29) will derive η1 and η 2 to zero. The goal of following a reference model for the joint variables and stabilizing the deflections around the equilibrium trajectory, naturally set up by the rigid system under the slow control, is then achieved by an O( ) approximation. This is the typical result of a singular perturbation approach. Composite control block diagram is shown in Figure 4.2. Robot Mechanism

T Trajectory planner

θ  d



θ d  θd



M rr



kp



kv





T



 Tf

 

k pf

η1  z 1  ζ

k vf

η2  z 2

δ, δ θ θ

f1 (θ, θ )

Figure 4.2 Composite Control 65


4.6 Numerical Example A one-link flexible arm shown in Figure 4.1 is used to investigate the effect of the joint inertia on the selection of the controller type. The composite control strategy is compared with the computed torque method. Step input of the manipulator is commanded. A step change from   0  to   90  is assigned. Two sets of simulations are presented. In the first set, the joint inertia of the arm is considered. In the second set, the joint inertia of the manipulator is considered to be zero. The performance of the manipulator under the composite control and under the computed torque control, in both cases including and excluding the joint inertia, is discussed. The gains of the computed torque control are obtained based on the smallest natural frequency of the flexible arm which is computed using the following equation [7]

1  (a1 )2

EI mL4

(4.31)

Where a1 is the first eigenvalue, m is the mass per unit length of the beam, EI is the flexural rigidity of the beam, and L is the beam length. The beam considered here is made of aluminum alloy with density   2500 kg/m 3 and 5 2 E  82 GPa [37], and its cross sectional area A  6.67 10 m . This leads to

m  0.167 kg/m . The length of the beam L = 1.2 m, and its flexural rigidity

EI  60 N.m2 . This gives the smallest natural frequency of the beam to be approximately equal to 26 rad/s . And to avoid excitation of any of the natural frequencies, the control gains are obtained by setting  n in the following equations [32] to be 4 rad/s

k p  n2 ,

kv  2n

(4.32)

And the control gains are given as kp = 16 and kv = 8.

The fast control (4.29) can be chosen according to the Linear Quadratic Regulator LQR technique [38]. The weighing matrices of the regulator Q and R are chosen as Q  30I and R  5I , where the larger Q is the more emphasis the optimal control places on returning the system to zero. And increasing R has the effect of reducing the amount of

66


the control effort allowed. As far as the joint inertia is considered, the gain matrices of the fast sub-system k pf and k vf are respectively

[0.1772 2.8513] and

[2.9389 2.4436] . And as the system of zero joint inertia is considered, the gain matrices of the fast sub-system k pf and k vf are respectively [0.4380 17.6052] and

[2.9425 2.0143] . The performance of the two controllers on both systems (with and without joint inertia) is shown in the following figures. The first modal variable δ1 of the arm under the computed torque control and the composite control, where the joint inertia is included, are shown in Figure 4.3 and Figure 4.4, respectively. As the joint inertia is not included, the first modal variable δ1 of the arm under the computed torque control and the composite control, are shown in Figure 4.5 and Figure 4.6, respectively.

67


0.03


0.02 0.01 0 -0.01 -0.02 -0.03 -0.04 0

1

2 Time(sec.)

3

4

Figure 4.3 First modal variable  1 (Computed Torque Control) with joint inertia

0.01


0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 0

1

2 Time(sec.)

3

4

Figure 4.4 First modal variable  1 (Composite control) with joint inertia 68


0.005


0 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 0

1

2 Time(sec.)

3

4

Figure 4.5 First modal variable δ1 (Computed Torque Control) without joint inertia

0.01


0.005 0 -0.005 -0.01 -0.015 -0.02 0

1

2 Time(sec.)

3

4

Figure 4.6 First modal variable δ1 (Composite control) without joint inertia

69


The maximum value reached by δ1, under the computed torque control, as the joint inertia is considered is  0.038 m , where the oscillation of the arm extends beyond 4 sec. Under the composite control, and consideration of the joint inertia, the maximum value reached by δ1 is  0.027 m , and the oscillation of the arm disappears after 1.8 sec. Figures 4.5 and 4.6 show that the maximum value of δ1 is  0.027 m , under the computed torque control, while under the composite control δ1 reaches a maximum value of  0.018 m . The oscillation of the arm disappears after 1.8 sec. under both control techniques and neglecting the joint inertia. The second modal variable δ2 of the arm under the computed torque control and the composite control, where the joint inertia is included, are shown in Figure 4.7 and Figure 4.8, respectively. As the joint inertia is not included, the second modal variable δ2 of the arm under the computed torque control and the composite control are shown in Figure 4.9 and Figure 4.10, respectively. -4

1.5

x 10


1 0.5 0 -0.5 -1 -1.5 -2 0

1

2 Time(sec.)

3

4

Figure 4.7 Second modal variable δ2 (Computed Torque Control) with joint inertia

70


-5

2

x 10


0 -2 -4 -6 -8 -10 -12 0

1

2 Time(sec.)

3

4

Figure 4.8 Second modal variable δ2 (Composite control) with joint inertia -3

1.5

x 10


1 0.5 0 -0.5 -1 -1.5 0

1

2 Time(sec.)

3

4

Figure 4.9 Second modal variable δ2 (Computed Torque Control) without joint inertia

71


-5

2

x 10

1 Modal Variable  2 (m)

0 -1 -2 -3 -4 -5 -6 -7 0

1

2 Time(sec.)

3

4

Figure 4.10 Second modal variable δ2 (Composite control) without joint inertia The maximum value reached by δ2, under the computed torque control, as the joint inertia is considered is  1.4 10 4 m , where the oscillation of the arm extends after 4 sec. Under the composite control, and consideration of the joint inertia, the maximum value reached by δ2 is  11105 m , and the oscillation of the arm disappears after 1.8 sec. The joint inertia is not considered now. The maximum value of δ2 is  1.110 3 m , under the computed torque control. Under the composite control δ2 reaches a maximum value of  6.5 105 m . The oscillation of the arm disappears after 1.8 sec. under both control techniques and neglecting the joint inertia. As the joint inertia is considered, Figure 4.11 and Figure 4.12 show the tip position along X0 under the computed torque control and the composite control, respectively. Figure 4.13 and Figure 4.14 show the tip position along X0 under the computed torque control and the composite control, where the joint inertia is not considered, respectively.

72


Tip Position in X-Direction (m)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0

1

2 Time(sec.)

3

4

Figure 4.11 Tip position along X0 (Computed Torque Control) with joint inertia


1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 4.12 Tip position along X0 (Composite control) with joint inertia 73



1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 4.13 Tip position along X0 (Computed Torque Control) without joint inertia


1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 4.14 Tip position along X0 (Composite control) without joint inertia

74


As the joint inertia is considered, and using computed torque control, the arm tip oscillates about the desired position, while using the composite control, the tip position reaches the desired value without oscillations. On the other hand, when the joint inertia is not included, the arm tip reaches the desired position without oscillations under both control strategies. Figure 4.15 and Figure 4.16 show the control torques under the two control techniques used with consideration of the joint inertia. Under the computed torque control, the control torque value keeps oscillating about the zero value for more than 4 seconds and this is happened as a consequence of the arm oscillations. Under the composite control, the torque value reaches zero after 1.8 seconds where the link oscillations disappeared. The control torques used for both control techniques, as the joint inertia is neglected are shown in Figure 4.17 and Figure 4.18. Here, the torques values under both control techniques used are almost the same except for the starting value in the case of the composite control. This is due to the fact that when the joint inertia is not included the oscillations of the arm disappear quickly and the arm behaves approximately like a rigid one.

60

Control Torque T (N.m)

50 40 30 20 10 0 -10 0

1

2 Time(sec.)

3

4

Figure 4.15 Control torque T (Computed Torque Control) with joint inertia

75


60


50 40 30 20 10 0 -10 0

1

2 Time(sec.)

3

4

Figure 4.16 Control torque T (Composite control) with joint inertia

7


6 5 4 3 2 1 0 -1 0

1

2 Time(sec.)

3

4

Figure 4.17 Control torque T (Computed Torque Control) without joint inertia 76


20


15

10

5

0

-5 0

1

2 Time(sec.)

3

4

Figure 4.18 Control torque T (Composite control) without joint inertia Figure 4.19 and Figure 4.20 show the joint angle step response, when the joint inertia is considered. Figure 4.21 and Figure 4.22 show the joint angle step response, when the joint inertia is considered. As the joint inertia is considered, the (10% to 90%) rise time in the case of the computed torque control is equal to 0.835 sec. and in the case composite control equals 0.803 sec. When the joint inertia is neglected, the (10% to 90%) rise time in the case of the computed torque control is equal to 0.834 sec. and in the case composite control equals 0.775sec. Of course, the rise time does not change dramatically under both controllers as the deflection variables do.

77


1.6 1.4

Joint Angle  (rad)

1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 4.19 Joint angle (Computed Torque Control) with joint inertia

1.6 1.4


1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 4.20 Joint angle (Composite control) with joint inertia

78


1.6 1.4


1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 4.21 Joint angle (Computed Torque Control) without joint inertia

1.6 1.4


1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 4.22 Joint angle (Composite control) without joint inertia 79


In this chapter, the response of a one-link flexible arm while considering the effect of joint inertia has been investigated. Based on the results, computed torque law would perform equally well with composite control for flexible manipulators as long as the joint inertias are negligible. On the other hand, the introduction of singular perturbation theory, to control flexible manipulator motion, enhanced the system performance, over just computed torque method, in the presence of considerable joint inertias. Also, as long as the joint inertias are negligible, it is preferable to just use computed torque law to control the motion of the flexible arm as its implementation would be easier than that of the composite control.

80

Chapter 5 – Flexible-Link Manipulators Subjected to Parametric Excitation

Chapter 5 Flexible-Link Manipulators Subjected to Parametric Excitation 5.1 Perspective on This Chapter In this chapter, the nonlinear dynamic modeling and control of flexible-link manipulators subjected to parametric excitation is presented. The equations of motion are obtained using the Lagrangian-assumed modes method. Here, the resulting dynamic equations of motion are with rapidly varying coefficients. Singular perturbation methodology is developed for the nonlinear time varying equations of motion to obtain a reduced-order set of equations. Control strategies, computed torque control and a composite control based on the singular perturbation formulation developed, are utilized to reduce mechanical vibrations of the flexible-link and enable better tip positioning, as the manipulator loses its accuracy due to parametric excitation. It is observed that the singular-perturbation control methodology developed offers better system response over the computed torque control as the manipulator is commanded to follow a certain trajectory. The equations of motion of parametrically excited flexible-link manipulators are derived in section 5.2. In Section 5.3, the general form of the equations of motion is shown and rearranged to be suitable for time scale separation. Section 5.4 includes the application of time scale separation to transform the system into a singularly perturbed model. A control strategy to stabilize the system is presented in section 5.5. The response of a one-link flexible arm subjected to parametric excitation is presented in section 5.6.

81


5.2 Nonlinear Modeling of Flexible Manipulators Subjected to Parametric Excitation 5.2.1 Kinematics The kinematics of flexible manipulators subjected to parametric excitation is presented in this section. Consider the manipulator shown in Figure 5.1. It is a multi-link flexible arm with rotary joints that moves in the horizontal plane. The following coordinate frames are established: the inertial frame (X0,Y0), the rigid body moving frame associated to link i (xi,yi), and the flexible body moving frame associated to link i ( xˆ i , yˆ i ) . The rigid motion is described by the joint angle  i , while v( xi , t ) denotes the

transversal deformation of link i that has the length li. The position of a point on the i deformed link i expressed in frame i is given as h i ( xi , t )  [ xi

v( xi , t ) 0 1]T and

i expressed in the inertial frame as h i  Ti h i where Ti is the transformation matrix

between frame i and the inertial frame and given as [4] Ti  Ti 1E i 1 A i

(5.1)

where A i is the joint transformation matrix for joint i due to rigid motion.

E i 1 is the link transformation matrix for link i-1 between joints i-1 and i due to

its length and deflection.

cos( i )  sin( i )  sin( ) cos( ) i i Ai    0 0  0  0 where vie ( xi , t ) 

vi ( xi , t ) xi

0 0 0 0 , 1 0  0 1

vie ( xi , t )  1 v  ( x , t ) 1 E i   ie i  0 0  0  0

0 li  0 0  1 0  0 1

(5.2)

, and the linear approximation arctan( vie ( xi , t ))  vie ( xi , t ) xi li

valid for small deflections. As far as the manipulator is subjected to parametric excitation, the transformation matrix T1 that relates frame 1 to the inertial frame is given as

82


cos(1 )  sin(1 )  sin( ) cos( ) 1 1 T1    0 0  0  0

0 0 1 0

0  A cos(at )  0  1 

(5.3)

where the term A cos(at ) is added due to the excitation of the base as shown in Figure 5.1. yn

yi

Y0

yˆ i 1

Mp

xˆ i

yˆ i

xn

xˆ1

y1

i

yˆ 1

xî 1

1

xi

x1

A cos(at )

X0 Figure 5.1 multi-link flexible manipulator subjected to parametric excitation

5.2.2 Kinetic and potential energy The dynamic equations of motion of the manipulator shown in Figure 5.1 can be derived following the standard Lagrangian approach, i.e. by computing the system kinetic energy T as well as the system potential energy U. The total kinetic energy is given by the sum of the following contributions: n

n

i 1

i 1

T   Thi   Tli  T p

(5.4)

where Thi, Tli, Tp stand for hub i, link i, and payload kinetic energies, respectively. The kinetic energy of the rigid body located at hub i of mass Mhi and moment of inertia Jhi is

Thi 

i i 1 1 1 M hih Ti1 (li 1 , t )h i 1 (li 1 , t )  J hi (i   vk ( xk , t )) 2 2 2 j 1 k 1

(5.5)

83


where the upper dot denotes time derivative. The kinetic energy of the first hub due to the parametric excitation is given as

Thi 

1 d 1 M hi ( ( A cos(at )))2  J h1 (1 )2 2 dt 2

(5.6)

The kinetic energy Tli pertained by link i of mass per unit length  i is given as

Tli 

1 li  ( xi )h Ti ( xi , t )h i ( xi , t )dxi  0 2

(5.7)

The kinetic energy of the payload of mass Mp and mass moment of inertia Jp is given as

Tp 

i 1 1 M p h Tn (l n , t )h n (l n , t )  J p (i  vi ( xi , t )) 2 2 2 j 1

(5.8)

The potential energy of the system which is due to the elasticity only is given as 2

 v ( x , t )  1 li U  U i    ( EI ) i ( xi )  i i  dxi 0 i 1 2  xi  n

(5.9)

Where Ui is the elastic energy stored in link i, being (EI)i its flexural rigidity

5.3 Fully Nonlinear Equations of Motion Using total kinetic energy and total potential energy of the flexible manipulator, the equations of motion of the manipulator can be obtained using a combined Lagrangeassumed modes approach [19]. The equations of motion of an n-link flexible arm with mi being the number of modes used to describe the deflection of link i can be written in the form. θ  f (θ, θ , t )   g 1 (θ, θ , δ, δ , t )   0  T M(θ, δ)     1               δ f 2 (θ, θ, t ) g 2 (θ, θ, δ, δ, t ) Kδ  0 

M rr

where M  

er M

(5.10)

M re   is the inertia matrix, superscripts r and e refer to rigid body M ee 

T (joint) and elastic degrees of freedom respectively. θ  [1   n ] is the vector of

joint variables, δ  [ 11   1m1

 21   2mn ]T is the vector of deflection

variables, f1 and f2 are the vectors containing gravitational (only in f1), Coriolis, and centrifugal terms, g1 and g2 are the vectors which account for the interaction of joint

84


variables and their time derivatives with deflection variables and their time derivatives, K is a diagonal matrix whose diagonal sub-matrices are the stiffness matrices of links in terms of the deflection variables, and u  [T 0] T , where T is the vector of joints torques.. As an effect of parametric excitation, the vectors f1, f2, g1, and g2 became functions of time (t) besides being functions of θ and δ and their time derivates, which is not the case when the excitation is not considered. Since the inertia Matrix M is positive definite, then it can be inverted and denoted by H, which can be partitioned as follows:  H11[ nn ] M 1  H   H 21[ mn ]

H12[ nm]  H 22[ mm] 

(5.11)

Now, equations (5.10) become

θ  H f  H f  H g  H g  H Kδ  H T 11 1 12 2 11 1 12 2 12 11

(5.12a)

δ  H f  Η f  H g  H g  H Kδ  H T 21 1 22 2 21 1 22 2 22 21

(5.12b)

As shown previously, the flexible dynamic system (5.12a and 5.12b) is characterized by n+m generalized coordinates but only n control inputs. Therefore, the synthesis of a nonlinear feedback control is not as easy as for rigid arm (a control input for each joint). A model order reduction is made by the use of singular perturbation theory. Leading to a composite control for the full order system [36].The reduced order system is presented in the next section.

5.4 A Singularly Perturbed Model A singularly perturbed model of the dynamic system (5.12) can be obtained as follows. Assume that the orders of magnitude of the K matrix elements (ki's) are comparable. It is appropriate to extract a common scale factor k (the smallest spring constant, for instance) such that

~ ki  kki ,

i = 1, …, m.

(5.13)

85


The following new variables (elastic forces) can be defined:

~ ζ  k kδ

(5.14)

~ ~ ~ k  diag (k1  km )

(5.15)

The next step is to define   1/ k and obtain

θ  H (θ, ζ)f (θ, θ , t )  H (θ, ζ)f (θ, θ , t )  H (θ, ζ)g (θ, θ , ζ, ζ, t ) 11 1 12 2 11 1  H (θ, ζ)g (θ, θ , ζ, ζ , t )  H (θ, ζ)ζ  H (θ, ζ)T 12

2

12

(5.16a)

11

ζ  H 21 (θ, ζ)f1 (θ, θ , t )  H 22 (θ, ζ)f 2 (θ, θ , t )  H 21 (θ, ζ)g 1 (θ, θ , ζ, ζ , t )  H 22 (θ, ζ)g 2 (θ, θ , ζ, ζ , t )  H 22 (θ, ζ)ζ  H 21 (θ, ζ)T (5.16b) which is a singularly perturbed model of the flexible arm. Notice that all quantities on

~

the right hand side of (5.16b) have been conveniently scaled by k . It can be shown that as   0 , the model of the rigid manipulator is obtained from (17a and 17b). Formally, setting   0 and solving for ζ in (17b) one obtains

ζ  H 221 (θ,0)[H 21(θ,0)f1 (θ, θ , t )  H 21(θ,0)g1 (θ, θ ,0,0, t )  H 21(θ,0) T]  f (θ, θ , t )  g (θ, θ ,0,0, t ) 2

(5.17)

2

where the overbars are used to indicate that the system with   0 is considered. Plugging (18) into (17a) yields θ  [H (θ,0)  H (θ,0)H 1 (θ,0)H (θ,0)]  [f (θ, θ , t )  T] 11 12 22 21 1

(5.18)

1 H11(θ,0)  H12 (θ,0)H 221 (θ,0)H 21(θ,0)  M11 (θ)

(5.19)

where

where M11 (θ) is the [n  n] positive definite matrix for the rigid-link arm. Choosing x1  θ , x 2  θ , and z1  ζ , z 2   ζ with    gives the state-space form of the system (5.16a and 5.16b);i.e. x 1  x 2 ,

86


x 2  H 11 (x 1 ,  2 z 1 )f1 (x 1 , x 2 , t )  H 12 (x 1 ,  2 z 1 )f 2 (x 1 , x 2 , t )  H 11 (x1 ,  2 z 1 )g 1 (x 1 , x 2 ,  2 z 1 , z 2 , t )  H 12 (x 1 ,  2 z 1 )g 2 (x 1 , x 2 ,  2 z 1 , z 2 , t )

(5.20a)

 H 12 (x 1 ,  2 z 1 )z 1  H 11 (x 1 ,  2 z 1 )T  z 1  z 2 ,

 z 2  H 21 (x 1 ,  2 z 1 )f 1 (x 1 , x 2 , t )  H 22 (x 1 ,  2 z 1 )f 2 (x 1 , x 2 , t )  H 21 (x 1 ,  2 z 1 )g 1 (x 1 , x 2 ,  2 z 1 , z 2 , t )  H 22 (x 1 ,  2 z 1 )g 2 (x 1 , x 2 ,  2 z 1 , z 2 , t )

(5.20b)

 H 22 (x 1 ,  2 z 1 )z 1  Η 21 (x 1 ,  2 z 1 )T

At this point, singular perturbation theory requires that the slow subsystem and the fast (boundary layer) subsystem be identified. The slow subsystem is formally obtained by setting   0 , i.e., the rigid model of the arm of the arm obtained above

x 1  x2 1 (5.21) x 2  M11 (x1 )[f1 (x1 , x 2 , t )  T] To derive the fast subsystem, we introduce the fast time scale   t /  . Then it can be recognized that the system (5.21a and 5.21b) in the fast time scale becomes

dx1   x2 , d

dx 2   [H 11 (x 1 ,  2 ( η1  ζ ))f1 (x1 , x 2 , t )  H 12 (x 1 ,  2 ( η1  ζ ))f 2 (x 1 , x 2 , t ) d  H 11 (x1 ,  2 ( η1  ζ ))g 1 (x1 , x 2 ,  2 ( η1  ζ ), η 2 , t )

(5.22a)

 H 12 (x1 ,  ( η1  ζ ))g 2 (x 1 , x 2 ,  ( η1  ζ ), η 2 , t ) 2

2

 H 12 (x1 ,  2 ( η1  ζ ))(η1  ζ )  H 11 (x 1 ,  2 ( η1  ζ ))T]

dη1  η2 , d dη 2  H 21 (x 1 ,  2 ( η1  ζ ))f1 (x 1 , x 2 , t )  H 22 (x1 ,  2 ( η1  ζ ))f 2 (x1 , x 2 , t ) d  H 21 (x 1 ,  2 ( η1  ζ ))g 1 (x 1 , x 2 ,  2 ( η1  ζ ), η 2 , t )

(5.22b)

 H 22 (x 1 ,  ( η1  ζ ))g 2 (x1 , x 2 ,  ( η1  ζ ), η 2 , t ) 2

2

 H 22 (x 1 ,  2 ( η1  ζ ))(η1  ζ )  H 21 (x 1 ,  2 ( η1  ζ ))T

Where the new fast variables η1 and η2 are defined as

87


η1  z1  ζ  z1  z1 ,

η2  z 2

Now setting   0 in (23a and 23b) gives

(5.23)

dx1 dx 2   0 ; i.e., x1 and x2 are constant in d d

the boundary layer. Furthermore, it can be recognized that g1 (x1 , x 2 ,0,0)  0 and g 2 (x1 , x 2 ,0,0)  0 , since, by definition, those terms are representative of products of

the components of x1 and/or x2 with the components of  z1 and/or  z 2 . Therefore, 2

the fast subsystem can be found to be

dη1  η2 , d dη2   H 22 (x1 ,0) η1  H 21 ( x1 ,0)(T  T ) d

(5.24)

Which is a linear system parameterized in the slow variables x1 .

5.5 Composite Control As evidenced by the two reduced-order subsystems (5.23) and (5.24), a composite control strategy [36] can be pursued. The design of a feedback control for the full system can be split into two separated designs of feedback controls T and T f for the two reduced-order systems; formally

T  T(x1 , x 2 , t )  T f (x1 , η1 , η 2 )

(5.25)

where T(x1 , x 2 , t ) is the slow control, and T f (x1 , η1 , η 2 ) is the fast control. As far as the slow control is concerned, the computed torque control developed for rigid manipulators as shown previously can be utilized and the slow control is given by

T  M11[θd  k v (θ d  θ )  k p (θd  θ)]  f1

(5.26)

where kp and kv are the position and velocity feedback gain matrices respectively. At this point, singular perturbation theory requires that the boundary layer system (25) be uniformly stable along the equilibrium trajectory ζ given by Equation (5.17). This can be accomplished if the pair  0 A  H 22

I , 0

 0  B  H 21 

(5.27)

88


is uniformly stabilizable for any slow trajectory x1 (t ) . Assuming that this holds, a fast state feedback control of the type

Tf  k pf (x1 )η1  k vf (x1 )η2

(5.28)

will stabilize the boundary layer system (5.24) to η1  0 (z1  ζ ) and η2  0 (z 2  0) . Since the main purpose of flexible manipulator control is to damp out the deflections at steady state as fast as possible. The fast control (5.28) can be designed as an optimal control for the boundary layer. Under the above conditions, a fundamental result in singular perturbation theory ensures that the state vectors of the full system can be approximated by x1  x1  O( ) ,

x 2  x 2  O( ) ,

(5.29a)

z1  ζ  η1  O( ) ,

z 2  η2  O( ) .

(5.29b)

Where the operator O( ) means the order of magnitude of  . Under the slow control (5.26), θ and θ will tend respectively to θ d and θ d . The fast control (5.28) will derive

η1 and η 2 to zero. The goal of following a reference model for the joint variables and stabilizing the deflections around the equilibrium trajectory, naturally set up by the rigid system under the slow control, is then achieved by an O( ) approximation.

5.6 Numerical Example A one-link flexible arm shown in Figure 5.2, subjected to parametric excitation, is commanded to follow a certain trajectory under the control of both computed torque and composite control. The arm is modeled in accordance with the assumed modes method, where the deformation of the arm is described by the first two bending modes since the higher bending modes are observed to be negligible. The base of the manipulator is excited with the function A cos(at ) , where a is the excitation frequency coefficient, ω is equal to the smallest natural frequency of the beam, A is the amplitude of excitation A = 0.05 (m). The equations of motion of the arm are given in Appendix A. From the f and g terms, the excitation terms ( A cos(at ) ) is multiplied by the generalized coordinates (  and 1,2 ) and thus it is considered to be parametric

89


excitation. The other excitation term is the external input Torque required for the motion of the robot arm to follow the desired trajectory. A step change from   0  to   90  is assigned to the joint angle. The performance of the manipulator under the composite control and under the computed torque control is examined as the coefficient a = 2 (case of principal parametric resonance where a small parametric excitation can produce a large response [39]). The gains of the computed torque control are obtained based on the smallest natural frequency of the flexible arm which is computed using the following equation [7]

1  (a1 ) 2

EI mL4

(5.30)

Where a1 is the first eigenvalue, m is the mass per unit length of the beam, EI is the flexural rigidity of the beam, and L is the beam length. Y0 y1

Mp

x1

 A cos(at ) O

Xe

X0

Figure 5.2 One-link flexible-arm The beam considered here is made of aluminum alloy with density   2500 kg/m 3 and E  82 GPa [37], and its cross sectional area A  6.67 10 5 m 2 . This leads to m  0.167 kg/m . The length of the beam L  1.2 m and its flexural rigidity

EI  60 N.m2 . This gives the smallest natural frequency of the beam to be equal to 26.53 rad/s . And to avoid excitation of any of the natural frequencies, the control gains

are obtained by setting  n in the following equations [32] to be 4 rad/s

90


k p  n2 ,

kv  2n

(5.31)

And the control gains are given as kp =16 and kv=8. The gains used to calculate the fast control torque (5.28) can be determined according to the well-known Linear Quadratic Regulator LQR technique [38]. The weighing matrices of the regulator Q and R are chosen as Q  30I and R  5I , where the Larger Q is the more emphasis the optimal control places on returning the system to zero. And increasing R has the effect of reducing the amount of the control effort allowed. Accordingly, the gain matrices of the fast sub-system k pf and k vf are respectively

[ 0.3445  8.8036] and [ 2.9362  2.3510] . The performance of the two controllers is shown in the following figures. In Figures (5.3 and 5.4), the first modal variable δ1 amplitude of oscillations in the case of composite control is in the order of magnitude of one-half that of computed torque. Similar conclusions was also shown when the excitation frequency was equal to onetimes the natural frequency (a = 1).

0.15


0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0

1

2 Time(sec.)

3

4

Figure 5.3 First modal variable  1 (Computed Torque Control)

91


0.06


0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 0

1

2 Time(sec.)

3

4

Figure 5.4 First modal variable δ1 (Composite control) In Figures (5.5 and 5.6), the second modal variable δ2 amplitude of oscillations in the case of composite control is again in the order of magnitude of one-half that of computed torque. The response also shows that δ2 takes longer time to decay in the case of the computed torque control over that in the case of the composite control. This result is important especially if the excitation frequency is near that of the second mode.

Figures (5.7 and 5.8) show that the required torque is less in the case of composite control than that for computed torque control except in the value of the starting torque which is still within the reasonable levels.

92


-3

4

x 10


3 2 1 0 -1 -2 -3 -4 0

1

2 Time(sec.)

3

4

Figure 5.5 Second modal variable δ2 (Computed Torque Control) -3

2

x 10


1.5 1 0.5 0 -0.5 -1 -1.5 -2 0

1

2 Time(sec.)

3

4

Figure 5.6 Second modal variable δ2 (Composite control) 93


40


30 20 10 0 -10 -20 -30 -40 0

1

2 Time(sec.)

3

4

Figure 5.7 Control torque T (Computed Torque Control)

150


100 50 0 -50 -100 -150 0

1

2 Time(sec.)

3

4

Figure 5.8 Control torque T (Composite control)

94


Figures (5.9) show that not much difference in the response of the joint angle under the computed torque control, or the composite control although the deflection of the arm was dramatically reduced using composite control.

1.6 1.4 composite control computed torque control


1.2 1 0.8 0.6 0.4 0.2 0 0

1

2 Time(sec.)

3

4

Figure 5.9 Joint angle

In this chapter, the nonlinear time-varying dynamic equations of motion of flexible-link manipulators subjected to parametric excitation were derived. In contrast with the external excitation, the parametric excitation leads to equations of motion with rapidly varying coefficients. The response of such systems while tracking a certain trajectory is investigated under the control of both computed torque and a composite control based on a developed singular perturbation method. Better response of the simulated arm is observed under the composite control than the computed torque control even with less applied control effort.

95

Chapter 6 – Conclusions and Recommendations

Chapter 6 Conclusions and Recommendations 6.1 Conclusions This thesis is focused on two major subjects: flexible link manipulators, and parametrically excited flexible arm robots.

In this work, the dynamic equations, for flexible link manipulators, were derived based on the combined Lagrangian-Assumed modes method. Computed torque control was shown to be successful as long as the manipulators are with high flexural rigidity links and are commanded to follow slow trajectories. These results were obtained as a result of controlling a two link flexible arm, while it is commanded to follow a straight line trajectory.

The singular perturbation theory was then introduced, where the dynamics of a flexiblelink manipulator is approximated by the dynamics of slow and fast subsystems. A control signal for each subsystem was then pursued. The effect of joint inertia on the selection of the control technique has been investigated. Based on the study performed, computed torque control was found to perform equally well with composite control for flexible manipulators, as long as, the joint inertias are negligible. On the other hand, the introduction of singular perturbation theory to control flexible manipulator motion enhanced the system performance, over just computed torque method, in the presence of considerable joint inertias. For the case of flexible arm robots subjected to parametric excitation, where the manipulator base is subjected to sustained oscillations, a comprehensive nonlinear

96


time-varying dynamic model of such systems was derived. The parametric excitation led to equations of motion with rapidly varying coefficients. The response of such systems was investigated under two control techniques, namely, computed torque and a composite control based on a developed singular perturbation method, while tracking a certain trajectory. Better response of the simulated arm was observed under the composite control than the computed torque control even with less applied control effort.

6.2 List of Publications The following list of scientific articles is a result of the research conducted in the thesis. 1. Mohamed W. Mehrez and Ayman A. El-Badawy, Effect of the joint inertia on selection of under-actuated control algorithm for flexible-link manipulators, Mechanism and Machine Theory, vol. 45, no. 7, pp. 967-980, 2010 [40]. 2. Ayman A. El-Badawy, Mohamed W. Mehrez, Amir R. Ali, Nonlinear Modeling and Control of Flexible-Link Manipulators Subjected to Parametric Excitation, Nonlinear Dynamics, vol. 62, no. 4, pp. 769-779, 2010 [41]. 3. Ayman A. El-Badawy, Mohamed W. Mehrez, Multibody Dynamic Analysis and Nonlinear Control of Flexible-Manipulators, Proceedings of the 17th International Congress on Sound and Vibration (ICSV17), Paper No. 42, Egypt, 2010 [42]. 4. Mohamed. W. Mehrez, A. A. El-Badawy, Aeroelastic Analysis of Wind Turbines Using Free General-Purpose Multibody Dynamics Software, Proceedings of AES-ATEMA‘2011 Seventh International Conference, pp. 79 – 86, Italy, 2011 [43].

6.3 Recommendations and Future Work The Manipulator configuration considered in this study includes the cases when the robotic arm moves in the horizontal plane. Other configurations responses are needed to be investigated. Also, to investigate the effectiveness of the control techniques shown, experimental work needs to be conducted and compared. For future work, some research work related to this study can be organized as follows;

97


•

Experimental verification of the results should be conducted. The experiments will include installing a flexible-link robot on a Stewart platform or an electrodynamic shaker to simulate the parametric excitation of the base of the robot.

•

Approximate analytical solution for the developed systems should be investigated using perturbation and nonlinear dynamics techniques.

•

Comparison of the performance of control techniques with its performance using the approximate analytical equations of motion.

•

Advanced control algorithms such as adaptive control should be tested and compared with results obtained.

98

Appendix A

Appendix A Kinematics and Dynamics of Flexible Arms A.1 Flexible Arm Kinematics Consider the kinematics of the flexible link manipulator shown in Figure A.1. Define the position of a point in Cartesian coordinates by a vector: [X-component Y-component

Z-component

1]T

Define the coordinate system [ x y z]i on the link i with origin Oi at the proximal end (nearest the base) oriented so that the x-axis is coincident with the neutral axis of the beam in its un-deformed condition. A point on the neutral axis at x  xi when the beam is un-deformed is located at i h i ( xi ) under a general condition of deformation, in terms of system i. By a homogeneous transformation of coordinates, the position of a point can be described in any other coordinate system j if the transformation matrix

j

Ti is

known. The form of this matrix is

  jR j Ti   i   0 

x j  component of Oi  y j  component of Oi  z j  component of Oi   1 

(A.1)

Where j

R i  a 3  3 rotation matrix.

0  a 1 3 vector of zeros.

99

Appendix A

Ti 1

y0

x0 z0

î 1 x

frame i

xî link i

Ti

dm

Ai 1

xi 1

xi

Ai

Figure A.1 flexible manipulator serial chain As shown in Figure A.1, the position of a point on link i in terms of the inertial coordinates of the base. h i  0 Ti i h i  Ti i h i

(A.2)

Where the special case of 0 Ti  Ti . It is useful to separate the transformations due to the joint from the transformation due to the flexible link as follows

ˆ A Ti  Ti 1E i 1 A i  T i 1 i

(A.3)

Where

A i  the joint transformation matrix for joint i due to rigid motion. E i 1  the link transformation matrix for link i-1 between joints i-1 and i due to its length and deflection.

ˆ  the cumulative transformation from base coordinates to Oˆ T i 1 i 1

The transformation matrix A i can be computed just like in the case of rigid robot manipulators [32]. On the other hand, the transformation E i deserves special attention. Firstly, consider the general form of a rotation matrix j R i between two coordinate frames of common origin (see Figure A.2)

100

Appendix A

 xiT x j  j R i   xiT y j  xiT z j 

y iT x j y iT y j y iT z j

z iT x j   cos(θxi x j ) cos(θyi x j ) cos(θz i x j )     z iT y j   cos(θxi y j ) cos(θyi y j ) cos(θz i y j ) z iT z j   cos(θxi z j ) cos(θyi z j ) cos(θz i z j ) 

(A.4)

where x, y, and z denote the unit vectors of the respective axes. Then, the relationship between j pi and i pi is given by

j

pi  j R i i pi .

(A.5)

zi zj

 xi

zî

pi

 yi

yˆ i

yi

zi

 zi

xi yj

 xi

 yi

 zi

yi

xî

xj

xi

Figure A.2 Rotation of coordinate frame

Figure A.3 Rotation of a coordinate frame due to deformation of flexible link

From Equation A.4, it can be easily understood that the knowledge of the angles θxi x j ...θzi z j is enough to compute j R i . With this background, the matrices Ei can be

determined as follows. Consider Figure A.1 again and assume that the x-axis of frame i is along the link. Assuming small link deformation, Ei can be expressed as

101

Appendix A

1 cos( / 2   zi ) cos( / 2   yi ) li  u i    cos( / 2   ) 1 cos( / 2   xi ) vi  zi  Ei  cos( / 2   yi ) cos( / 2   xi ) 1 wi    0 0 0 1  

(A.6)

Where  xi , yi , and  zi are the angles of rotation, and ui, vi, and wi represent the link i deformation along x, y, z, respectively, being li the length of the link without deformation. The angles  xi , yi , and  zi are depicted in Figure A.3. By taking into account the fact cos( / 2   )   sin( ) and assuming small angles, so that sin( )   is valid, the matrix E i can be approximated as

 1  E i   zi   yi   0

  zi 1

 yi   xi

 xi

1 0

0

li  u i  vi  wi   1 

(A.7)

ui, vi, wi,  xi , yi , and  zi can be expressed according to the so-called assumed modes method as will be presented later in this chapter.

mi

ui    xij ij j 1

mi

mi

 xi   xij  ij mi

vi   yij  ij

 yi   yij  ij

mi

 zi   zij ij

j 1

wi   zij ij j 1

(A.8)

j 1

(A.9)

j 1

mi

(A.10)

j 1

where

 xij ,  yij ,  zij , xij , yij , and  zij = the spatial mode shapes used to model the deflection (torsion) of link i.

 ij = the time varying amplitude of mode j of link i.

mi = the number of modes used to describe the deflection of link i.

102

Appendix A

By using the homogenous transformation Ai and Ei, the position of any point along the robot manipulator can uniquely be determined from Equations (A.2), (A.3), and (A.7). Consider now the position of an arbitrary point on link i, which is given as follows:

 xi   xij   0  mi   i h i ( xi )       ij  yij   0  j 1  yij      1  0 

(A.11)

To find the velocity of a point on link i, take the time derivative of the position

d  i h  T i h h i  h i  T i i i i dt

(A.12)

Due to the serial nature of the kinematics chain, it is computationally efficient to relate the position of a point and its derivatives to preceding members in the chain. By differentiating Equation A.3, one obtains:

ˆ  T ˆ  T i i 1 A i  Ti 1 A i

(A.13)

ˆ ˆ    T ˆ  T i i 1 A i  2Ti 1 A i  Ti 1 A i

(A.14)

A i  Ui qi

(A.15)

  U q 2  U q  A i 2i i i i

(A.16)

where

Ui 

U 2i 

A i qi

 2Ai qi2

(A.17)

(A.18)

qi = the joint variable of joint i.  ˆ , its derivatives, and the partials  can be computed recursively from T Thus, Ti and T i 1 i

ˆ with respect to the variables of link i-1 and joint i. Here, one additionally needs T i 1 and its derivatives. These can be computed recursively from Ti 1 and its derivatives:

103

Appendix A

ˆ  TE T i i i

(A.19)

ˆ T  E TE  T i i i i i

(A.20)

ˆ     T i  Ti E i  2Ti E i  Ti E i

(A.21)

E E i  i t

(A.22)

2    E i E i t 2

(A.23)

A.2 System Kinetic Energy In this part, an expression for the system kinetic energy is developed to be used in Lagrange's equations. First, the kinetic energy for a differential element is written. Then, integration of the differential kinetic energy over the link gives the link total contribution. Summation over the links provides the total kinetic energy [4].

The kinetic energy of a point on the i-th link is

dk i 

1 dmTr{h i h Ti } 2

(A.24)

Where dm is the differential mass of the point, and

Tr{.} is the trace operator. Expanding Equation A.24 and using the fact that Tr{AB T }  Tr{BA T } , the expression for

dk i

becomes

dk i 

1  h hT T  T  2T  h h T T  T  T h h T TT } dmTr{T i i i i i i i i i i i i 2

(A.25)

104

Appendix A

where

 xij    mi i   h i    ij  yij   yij  j 1    0 

(A.26)

By integrating over the link, one can obtain the total link kinetic energy. Here, it is assumed that the links are slender beams because it makes central development clearer. For slender beams dmi = ρi dxi, and one can integrate over xi from 0 to li. Only the terms in ihi and its derivatives are functions of xi for link i. Thus, the integration can be performed without knowledge of Ti and its derivatives. Summing over all n links, one finds the system kinetic energy to be

n li

K    dki

(A.27)

i 1 0 n

T B T T  T K   Tr{T i 3i i  2Ti B 2i Ti  Τ i B1i Ti }

(A.28)

i 1

Where

l

B1i 

1 i  T  i h i h i dxi 2 0

(A.29)

l

1 i B 2i    i h i h Ti dxi 20

(A.30)

l

B 3i 

1 i  i h i h Ti dxi  20

(A.31)

105

Appendix A

A.3 System Potential Energy The potential energy of the system arises from two sources: In both cases, they are included by first writing the potential energy contribution of a differential element, integrating over the length of the link, and then summing over all links.

A.3.1 Elastic Potential Energy Consider a point on the i-th link undergoing small deflections. First restrict the link of the slender beam type. The elastic potential energy is accounted for to good approximation by bending about the transverse y and z and twisting about the longitudinal x axis. Compression is not initially included since it is generally much smaller. Along an incremental length dxi the elastic potential energy is 2 2 2   2 w( xi )     x ( xi )   1      2v( xi )     I y     GI x    dvei  dxi E  I z  2 2 2    xi   xi    xi     

(A.32)

Where

 x ( xi ),

w( xi ) v( xi ) are the rotations of the neutral axis at the point xi about the x, , and xi xi

y, and z directions respectively. E = Young’s modulus of elasticity of the material. G = the shear modulus of the material Ix = the polar area moment of inertia of the link cross section about the neutral axis. Iy, Iz = the area moment of inertia of the link cross section about the y and z axes, respectively Now, to obtain the total elastic potential energy of link i , we integrate over the length of the link 2 2 2 l   2 w( xi )     x ( xi )   1 i     2v( xi )           Vei   E I z   Iy  GI x  dxi 2 2    2 0    xi  xi   xi        

(A.33)

106

Appendix A

Where li = the length of the link i. And, the total elastic potential energy of the manipulator is given as n

Ve  Vei

(A.34)

i 1

Note that the Ve is independent on qi, the joint variables.

i.e.

Ve 0 qi

(A.35)

A.3.2 Gravity Potential Energy For a differential element on the i-th link of length dxi the gravity potential is

dv gi    i g T Ti i h i dxi

(A.36)

where

 i = is the density of the link i (kg/m) g = is the gravity vector which has the form

g x  g  g   y gz    0

(A.37)

gx, gy, and gz are the gravity components in x, y, and z directions, respectively.

When integrated over the length of the beam and summed over all beams, the gravity potential energy becomes n

Vg  g T  Ti ri

(A.38)

i 1

Where mi

ri  M i rri    ik  ik

(A.39)

k 1

107

Appendix A

Mi = the total mass of link i.

rri  rxi

0 0 1

T

(A.40)

A vector to the center of gravity of the undeformed link i from joint i.

 ik   i  xij  yij  zij 0 dxi li

(A.41)

0

It is the distance from the undeformed center of gravity to the center of gravity when all δ’s are zeros except δik, which is one Upon taking the partial derivatives required by Lagrange’s equations, we find for the joint variables Vg q j

n

 g T  i j

Ti ri q j

(A.42)

For deflection variables, for 1  j  n  1 Vg  jf

 g T

n

Ti

 

i  j 1

ri  g T Tj  jf

(A.43)

jf

For j = n Vg  nf

 g T Tn  nf

(A.44)

A.4 Assumed Mode Shapes Links are modeled as Euler-Bernoulli beams of uniform density ρi and constant flexural rigidity (EI)i, with deformation vi(xi,t) satisfying the partial differential equation [7]

EI i  vi ( x4i , t )  i  vi ( x2i , t )  0 4

2

xi

t

(A.45)

where v and xi are as shown in Figure A.4.

108

Appendix A

y

v x xi

Figure A.4 Deformed link In order to solve this equation, proper boundary conditions have to be imposed at the base and the end of each link. In particular, we assume each link to be clamped (fixed) at the base.

vi (0, t )  0

vi (0, t )  0 ,

i  1,, n.

(A.46)

Concerning the remaining boundary conditions, it is usually considered that the link end is free of dynamic constrains, and it is more correct to consider mass boundary conditions representing balance of moment and shearing force.

 2 vi ( xi , t ) EI i xi2

xi li

 v ( x , t )   i i   xi  xi li   d2  MD i 2 vi ( xi , t ) x l i i dt

d2   J Li 2 dt



EI i 

3

vi ( xi , t ) xi3

 M Li xi li



d2 vi ( x i , t ) x l i i dt 2

 MD i





d 2  vi ( xi , t ) dt 2  xi 

xi li

i  1,, n.

   

(A.47)

109

Appendix A

Where M Li and J Li are the actual mass and moment of inertia at the end of link i. Moreover,

MDi accounts

for the contributions of masses of distal links,i.e.

noncollocated at the end of link i.

A finite-dimensional model (of order mi) of link flexibility can be obtained by the assumed mode technique, and the link deflection can be expressed as

mi

vi ( xi , t )    ij xi ij xi 

(A.48)

j 1

Where  ij xi  the time-varying variables (modal variables) are associated with the spatial mode shapes ij xi  of link i.

110

Appendix B

Appendix B

B.1 Equations of Motion of a Two Link Planar Manipulator with Flexible Forearm The equations of motion of the manipulator which is given as shown in Equation (2.32) can be written as

 M 11rr  rr  M 21  M 11er  er  M 21

M 12rr

M 11re

rr M 22

re M 21

M 12er

M 11ee

er M 22

ee M 21

 T  M 12re  1  C1r    0   1  r re     M 22   2  C2    T  e   0    2 ee     M 12  1 C1    1    0   e  K    ee     M 22   2  C2    2    0 

(B.1)

Where the elements of the previous set of equations are given as 1 1 rr M 11  m1 L12  m2 L12  m2 L22  m2 L1 L2 cos 2   2.13 m2 L1 1 sin  2  3 3  0.852 m2 L1 2 sin  2 

(B.2)

1 rr rr M 12  M 21  m2 L22  0.5m2 L1 L2 cos 2   1.07m2 L1 1 sin  2  3  0.43 m2 L1 2 sin  2 

(B.3)

rr M 22 

1 m2 L22 3

re er M 11  M 11  0.775 m2 L2  1.07 m2 L1 cos 2 

(B.4) (B.5)

re er M 12  M 21  0.1 m2 L2  0.43 m2 L1 cos 2 

(B.6)

re er M 21  M 12  0.77 m2 L2

(B.7)

111

Appendix B

re er M 22  M 22  0.1 m2 L2

(B.8)

ee M 11  1.86 m2

(B.9)

ee ee M 12  M 21 0

(B.10)

ee M 22  m2

(B.11)

C1r   2.13 m2 L111 sin  2   0.85 m2 L112 sin  2   2.13 m L   sin    0.85 m L   sin   2

1 2 1

2

2

1 2

2

2

 m2 L1 L212 sin  2   2.13 m2 L1 112 cos  2   0.85 m L    sin    0.5m L L  2 sin   2

1 2 1 2

2

2

1

2 2

(B.12)

2

 1.06 m2 L1 122 cos  2   0.43 m2 L1 222 cos  2 

C 2r  0.5m2 L1 L212 sin  2   1.07 m2 L1 112 cos 2   0.43 m L   2 cos 

(B.13)

C1e 1.07 m2 L112 sin  2   1.86 m2 112  3.71 m2 112  1.86 m   2

(B.14)

C 2e  0.43 m2 L112 sin  2   m2 212  1.93 m2 212  m   2

(B.15)

EI  22.94 L3 2 K  0 

(B.16)

2

1

2 1

2

2 1 2

2

2

2

   EI 468.04 3  L2  0

112

Appendix B

B.2 Equations of Motions of a One Link Flexible Arm The fixed-free deflection modes are; 

ajx



L

 j ( x)   sin

a j x  (sin a j  sinh a j )  a x a x    cos j  cosh j  , for j  1, 2 . L  (cos a j  cosh a j )  L L 

 sinh

(B.17)

For 0.5 payload to link mass ratio, we have a1  1.42 and a2  4.11 .

 M 11 M 12 M 13      f1   g1   0  T1                  M 21 M 22 M 23  1    f 21    g 21   K  1     0   M 31 M 32 M 33  2   f 22   g 22    2    0    

(B.18)

Where the elements of the previous equation are given as:

1 M 11  J h  J p  mL2  M p L2 3 M 12  2.33741

Jp L2

M 13  6.89773

 0.444843m  1.6002M p

Jp

 0.381995m  0.525115M p

L2

M 21  M 12 M 22  5.46349

(B.19)

(B.20)

(B.21)

(B.22) Jp

 0.615m  2.56M p

L2

M 23  16.1228

Jp L2

 0.4236m  0.8403M p

(B.23)

(B.24)

M 31  M 13

(B.25)

M 32  M 23

(B.26)

M 33  47.5786

Jp L2

 0.87754m  0.275745M p

(B.27)

113

Appendix B

 f 1  0   f   0   21     f 22  0

(B.28)

g1  0

(B.29)

g 21  0.615m1 2  2.5072M p 1 2  0.42357m 2 2  0.8403M p 2 2

(B.30)

g 22  0.4236m 1 2  0.8403M p 1 2  0.8775m 2 2  0.2757M p 2 2

(B.31)

EI  7.7055 L3 K  0  

(B.32)

   EI  290.059 3  L  0

where beam mass m  0.2 kg , payload mass M p  0.1kg , payload inertia J p  0.001kg.m 2 joint inertia J h  2 kg.m 2 , beam length L  1.2 m , flexural rigidity EI  60 N.m2

114

Appendix B

B.3 Equations of Motions of a One Link Flexible Arm Subjected to Parametric Excitation The fixed-free deflection modes are; 

ajx



L

 j ( x)   sin

 sinh

a j x  (sin a j  sinh a j )  a x a x    cos j  cosh j  , for j  1, 2 . L  (cos a j  cosh a j )  L L 

(B.33)

For 0.5 payload to link mass ratio, we have a1  1.42 and a2  4.11 . The equations of motions of a one link flexible arm can be written in the following form,

 M 11 M 12   M 21 M 22  M 31 M 32 

M 13      f1   g1   0  T1    M 23  1    f 21    g 21   K  1     0     M 33  2   f 22   g 22    2    0 

(B.34)

Where the elements of the previous equation are given as: M 11  J h  J p 

1 mL2  M p L2 3

Jp

M 12  2.33741

L2

M 13  6.89773

 0.444843m  1.6002M p

Jp

 0.381995m  0.525115M p

L2

M 21  M 12

(B.35) (B.36) (B.37) (B.38)

M 22  5.46349

Jp

 0.615m  2.56M p

L2

M 23  16.1228

Jp L2

 0.4236m  0.8403M p

(B.39) (B.40)

M 31  M 13

(B.41)

M 32  M 23

(B.42)

M 33  47.5786

Jp L2

 0.87754m  0.275745M p

(B.43)

115

Appendix B

f1  f1  0.5a 2 ALm 2 cos(at ) cos( )  0.5a 2 ALM p 2 cos(at ) cos( )

(B.44)

f  f 2   21   f 22 

(B.45)

where f 21  0.607 a 2 A m 2 cos(at ) cos( )  1.6 a 2 A M p 2 cos(at ) cos( ) f 22  0.751a 2 A m 2 cos(at ) cos( )  0.525 a 2 A M p 2 cos(at ) cos( )

g1  g1  0.608 a 2 A m 2 cos(at ) sin( )  1  1.6 a 2 A M p  2 cos(at ) sin( )  1  0.7513 a 2 A m 2 cos(at ) sin( )  2  0.525 a 2 A M p  2 cos(at ) sin( )  2

g  g 2   21   g 22 

(B.46) (B.47)

(B.48)

(B.49)

where g 21  0.615 m1 2  2.5072 M p 1  2  0.42357 m  2  2  0.8403 M p  2  2

(B.50)

g 22  0.4236 m  1  2  0.8403 M p  1  2  0.8775 m  2  2  0.2757 M p  2  2

(B.51)

EI  7.7055 L3 K  0  

(B.52)

   EI  290.059 3  L  0

where beam mass m  0.2 kg , payload mass M p  0.1kg , payload inertia J p  0.001kg.m 2 joint inertia J h  0.5 kg.m 2 , beam length L  1.2 m , flexural rigidity EI  60 N.m2

116

References

References [1] [2]

[3] [4]

[5]

[6]

[7] [8] [9] [10] [11]

[12]

[13]

[14]

[15]

[16]

Amor Jenfine, Dynamics and Control of light weight flexible manipulator, Ph.D. Thesis, University of Ottawa,1996. Z. Mohamed, J.M. Martins, M.O. Tokhi, J. da Costa, M. Botto. Vibration control of a very flexible manipulator system Control Engineering Practice, Vol.13, pp.267277, 2005. F. Wang, Yanqing Gao, Advanced Studies of Flexible Robotic Manipulators: Modeling, Design, Control and Applications, volume 4, World Scientific, 2003. De Luca, Siciliano, Closed-form dynamic model of planar multilink lightweight robots, Systems, Man and Cybernetics, IEEE Transactions, Volume 21, Issue 4, pp.826 – 839, 1991. B. Subudhi and A. S. Morris, On the singular perturbation approach to trajectory control of a multilink manipulator with flexible links and joints, Proceedings of the Institution of Mechanical Engineers, Journal of Systems & Control Engineering, Vol. 215, Issue 6, pp.587-598, 2001. Wayne J. Book, Recursive Lagrangian Dynamics of Flexible Manipulator Arms via Transformation Matrices, The International Journal of Robotics Research, Vol. 3, No. 3, 87-101, 1984. L. Meirovitch, Fundamentals of Vibrations, McGraw Hill, 2001. S. Dwivedy, P. Eberhard, Dynamic analysis of flexible manipulators, a literature review, Mechanism and Machine Theory, vol. 41, pp. 749–777, 2006. A. Nayfeh, P. Pai, Linear and Nonlinear Structural Mechanics, Wily Interscience, 2004. H. Baruh, Analytical Dynamics, McGraw Hill, 1999. X. Zhang, W. Xu, S. S. Nair, Comparison of some modeling and control issues for a flexible two link manipulator, The Instrumentation, Systems, and Automation Society – transactions, Vol. 43, pp. 509–525, 2004. K. A. Morris, K. J. Taylor, A Variational Calculus Approach to the Modeling of Flexible Manipulators, Society for Industrial and Applied Mathematics, Vol. 38, No. 2, pp. 294-305, 1996. Tsuneo Yoshikawa, Koh Hosoda, Modeling of Flexible Manipulators Using Virtual Rigid Links and Passive Joints, The International Journal of Robotics Research, Vol. 15, No. 3, pp. 290-299,1996. S. K. Ider, F. M. L. Amirouche, Influence of Geometric Nonlinearities in the Dynamics of Flexible Treelike Structres, Journal of Guidance, Control, and Dynamics, Vol. 12, pp. 830-837, 1989. J. Lin, Z.Z. Huang, P.H. Huang, An active damping control of robot manipulators with oscillatory bases by singular perturbation approach, Journal of sound and vibration, vol. 304, pp. 345-360, 2007. A. De Luca, Trajectory Control of Flexible Manipulators, Lecture notes in Control and Information Sciences, Springer Berlin, Vol. 230, pp. 83-104, 1998.

117

References

[17]

[18] [19]

[20]

[21]

[22] [23]

[24] [25]

[26]

[27] [28]

[29]

[30]

[31]

[32] [33]

A.S. Morris, A. Madani, Computed torque control applied to a simulated twoflexible-link robot, Transactions of the Institute of Measurement and Control, Vol. 19, No. 1, pp. 50-60, 1997. R. J. Theodore, A. Ghosal, Robust control of multilink flexible manipulators, Mechanism and Machine Theory, Vol. 38, pp. 367-377, 2003. B. Siciliano, W. J. Book, A Singular Perturbation Approach to Control of Lightweight Flexible Manipulators, The International Journal of Robotics Research, Vol. 7, No. 4, pp.79-90, 1988. Sang-Ho Lee, Chong-Won Lee, Hybrid Control Scheme for Robust Tracking of Two-Link flexible Manipulator, Journal of Intelligent and Robotic Systems, Vol. 34, pp. 431-452, 2002. Y. Aoustin, A Formal’sky, On the Feedforward Torques and Reference Trajectory for Flexible Two-link Arm, Multibody System Dynamics, Vol. 3, pp. 241265,1999. S. Ider, M. Ozgoren, V. Ay, Trajectory Tracking Control of Robots with Flexible Links, Mechanism and Machine Theory, Vol. 37, pp. 1377-1394, 2002. M. Moallem, R. V. Patel, K. Khorasani, Nonlinear tip-position tracking control of a flexible-link manipulator theory and experiments, Automatica, Vol. 37, pp. 1825 1834, 2001. S. Ozgoli and H. D. Taghirad, A Survey on the Control of Flexible Joint Robots, Asian Journal of Control, Vol. 8, No. 4, pp. 1-15, 2006. R. Lozano, A. Valera, P. Albertos, S. Arimoto, T. Nakayama, PD control of robot manipulators with joint flexibility, actuatorsdynamics and friction, Automatica, Vol. 35, pp. 1697-1700, 1999. B. Subudhi and A. S. Morris, Dynamic modelling, simulation and control of a manipulator with flexible links and joints, Robotics and Autonomous Systems, Vol. 41, Issue 4, pp. 257-270, 2002. S. A. A. Moosavian, E. Papadopoulos, Modified transpose Jacobian control of robotic systems, Automatica, Vol. 43, pp. 1226 – 1233, 2007. J. Shan, Hong-Tao Liu, D. Sun, Slewing and vibration control of a single-link flexible manipulator by positive position feedback (PPF), Mechatronics, Vol 15, pp. 487–503, 2005. A.Sanz, V. Exebarria, Experimental Control of a Two-Dof Flexible Robot Manipulator by Optimal and Sliding Methods, Journal of Intelligent and Robotic Systems, Vol. 46, pp. 95-110, 2006 Jae Young Lew, Suk-Min Moon, A Simple Active Damping Control for Compliant Base Manipulators, IEEE/ASME transaction on mechatronics, vol. 6, no. 3, pp. 305-310, 2001. J. Lin, Z.Z. Huang, A hierarchical fuzzy approach to supervisory control of robot manipulators with oscillatory bases, Mechatronics, vol. 17, no. 10, pp. 589-600, 2007. Mark W.Spong, Seth Hutchinson, M.Vidyasagar, Robot Modeling and Control, John Wiley& Sons, Inc., 2006. G. Franklin, J. David Powell, A. Emami-Naeini, Feedback Control of Dynamic

118

References

[34] [35]

Systems, 5th edition, Prentice-Hall, 2006. Robert L. Norton, Design of Machinery, third ed., McGraw-Hill, 2004.

[37]

Jonqlan Lin, Dynamic Modeling, Estimation, and Control of Flexible Structural systems, PhD thesis, 1994. Petar V. Kokotovic, Applications of Singular Perturbation Techniques to Control Problems, Society for Industrial and Applied Mathematics, Vol. 26, No. 4, (Oct., 1984), pp. 501-550. Michael F.Ashby, Materials Selection in Mechanical Design, Elsevier, 2007.

[38]

K. Ogata, Modern Control Engineering, Prentice-Hall, 2002.

[39]

A. H. Nayfeh, D. T. Mook, Nonlinear Oscillations, Wiley, 1995.

[40]

Mohamed W. Mehrez and Ayman A. El-Badawy, Effect of the joint inertia on selection of under-actuated control algorithm for flexible-link manipulators, Mechanism and Machine Theory, vol. 45, no. 7, pp. 967-980, 2010. Ayman A. El-Badawy, Mohamed W. Mehrez, Amir R. Ali, Nonlinear Modeling and Control of Flexible-Link Manipulators Subjected to Parametric Excitation, Nonlinear Dynamics, vol. 62, no. 4, pp. 769-779, 2010. Ayman A. El-Badawy, Mohamed W. Mehrez, Multibody Dynamic Analysis and Nonlinear Control of Flexible-Manipulators, Proceedings of the 17th International Congress on Sound and Vibration (ICSV17), Paper No. 42, Egypt, 2010. Mohamed. W. Mehrez, A. A. El-Badawy, Aeroelastic Analysis of Wind Turbines Using Free General-Purpose Multibody Dynamics Software, Proceedings of AESATEMA‘2011 Seventh International Conference, pp. 79 – 86, Italy, 2011.

[36]

[41]

[42]

[43]

119

120