Sliding-mode control of multi-link flexible manipulators · Sliding-mode control of multi-link flexible ... systematic and is used for the one link flexible manipulator. For the two

Sliding-mode control of multi-link flexible manipulators

Verdaasdonk, R.J.F.M.

Published: 01/01/1994

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T T

SLIDING-MODE CONTROL OF MULTI-LINK FLEXIBLE MANIPULATORS

A N IMPLEMENTATION WITH SIMULINIS’

Ruud Verdaasdonk

WFW stagereport nr. 94.150

SLIDING-MODE CONTROL OF MULTI-LINK FLEXIBLE

MANIPULATORS A N IMPLEMENTATION WITH SIMULINK

Ruud Verdaasdonk WFW stagereport nr. 94.150

Coach : dr.ir. M.J.G van de Molengraft Chair : prof. d r h . J.J. Kok

Eindhoven University of Technology (TUE) Department of Mechanical Engineering

Division of Mechanical Engineering Fundamentals (WFW)

Eindhoven, January 25, 1995

SUMMARY

This paper is a continuation of the research efforts reported in Yeung and Chen (1989) and Chen and Yeung (1991). This present work presents an extension of the sliding-mode control design method to multi-link flexible manipulators. The paper gives an overview of the methodology of designing the flexible model for multilink manipulators and the control designing.

Sliding mode technique is applied to achieve a robust feedback linearization of the high non-linear dynamic equation of the arm. Pole placement is used to attain good dynamic response.

The software package SIMULINK 1.3 is used to design the simulator. It is easy to design a controller in this package. Simulation studies have been conducted to demonstrate the use of SIMULINK.

The model used in Chen and Yeung (1991) is tested by using only the rigid part of the dynamic equation as cited in [i]. The results from the dynamic equations used by Slotine and Li (1991) differs from the equations used by Chen and Yeung (1991). The derivation following the procedure of Low (1987) is systematic and is used for the one link flexible manipulator. For the two link flexible manipulator it is difficult to derive the dynamic equation.

i

.. 11 SUMMARY

CONTENTS

SUMMARY

1 INTRODUCTION

2 THE FLEXIBLE MANIPULATOR 2.1 Introduction 2 . 2 The cantilever beam 2.3 The dynamic equations

3 CONTROL DESIGN 3.1 Theory of the slide-mode control

4 SIMULATION 4.1 4.2 The two-link flexible manipulator

An one link l-mode flexible link with sliding mode control

5 CONCLUSIONS AND RECOMMENDATIONS 5.1 Conclusions 5.2 Recommendations

REFERENCES

A THE EIGENVALUEPROBLEM

B PARAMETERS OF THE TWO LINK FLEXIBLE MANIPULATOR B. 1 The dynamical equation B.2 The dynamical equation in state space notation B.3 Parameters of the links

C CONTROL PARAMETERS C. l Expansion of T*

D CONTROL DESIGN FOR SEPARATE CASES D.l Case 1: First-link ramp-tracking of the manipulator on a horizontal plane D.2 Case 2: First-link parabola-tracking of the manipulator on a horizontal plane

1

9 9

15 15 17

21 21 21

23

25

27 27 28 29

31 31

33 33 34

... 111

i V

D.3 Case 3: First-link regulation of the manipulator in a vertical plane

E PROGRAMS E. l Program of an one link 1-mode flexible manipulator E.2 Program of the two link l-mode flexible arm

CONTENTS

34

37 37 38

1 INTRODUCTION

Current industrial manipulators often possess heavy structures. Not only are the material requirements and the power consumption expensive but also the operating speed becomes relatively slow. Attempting to eliminate these drawbacks investigators have paid their attention to lightweight flexible manipulators. If it is constructed as a lightweight manipulator practically only the two-DOF rotary long links will be deformed under heavy loading and fast motion. Recent researchers involved the flexibility in multi-link manipulators by using a two-link flexible model [2, Boo 751 and [3, Kle 861 as shown in figure 1.1. This paper also uses this approach.

Figure 1.1 Two-link flexible manipulator

As for the dynamic model analysis a clamped-loaded Euler-Bernouilli beam is selected as an approximate model for each link in the presence of lateral deformation. Neglecting coupling effects caused by flexibilities between the two links an analytic form of the deformation of each link under loading can be easily obtained by using the approach introduced by [4, Sak 851. Finally using Lagrangian formulation [5, Low 871 the dynamic equations of the overall system are obtained displaying the rigid and flexible parts.

In order to succesfully drive the flexible manipulator to track desired trajectories a variety of control techniques have been developed such as the conventional feedback control [6, Sin 851, optimal control [4, Sak 851 and model reference adaptive control [7, Wen 861. However there still exist some challenging problems to the designer:

1

2 CHAPTER 1

1. the requirement of an exact model

2. payload forecast

3. inadequancy in the suppression of disturbances

4. a large amount of the calculations due to the tedious inverse of the inertia matrix

In view of the success of eliminating the above drawbacks by applying the sliding-mode technique to a single-link flexible manipulator ([8, Yeu 881 and [9, Yeu 891) an extension of this technique is used in this paper to the control of multi-link flexible manipulators.

The goal of this paper is to give a view of the used method in the paper of [i, Che 911 and make this technique available in the software package SIMULINK (ver 1.3).

In the next chapter, the dynamic equations of a two-link flexible manipulator with r dominant modes will be derived. Chapter 3 introduces the sliding-mode controller designed together with the conventional PD/PID compensations. Numerical simulation results are shown in Chapter 4 to determine this robust sliding-mode control. Chapter 5 gives conclusions and recommendations.

THE FLEXIBLE MANIPULATOR

2.1 INTRODUCTION

In studying the dynamics of beams, the well-known Euler-Bernoulli theory is frequently used by researchers. The classical theory does not include the effect of both shear deformation and rotary inertia. For short beams or for higher modes, this contribution cannot be neglected. The Timenshenko theory of beams is then introduced as a means of accounting for the affect of both transvere shear and rotatory inertia [5, Low 871. In this work, the Euler-Bernouilli theory is employed since we assume the slenderness ratio of elastic link is so large that the effect just mentioned can be ignored. In this chapter the cantilever beam with the Euler-Bernoulli theory will be defined in section 2.2. In section 2.3 the dynamic equations of a two-link r-mode flexible manipulator will be obtained.

2.2 THE CANTILEVER BEAM

The two-link r-mode flexible manipulator (figure 2.1) operates on a planar work space. Only lateral deformation is considerd. In addition, we assume that the deformation ui(zi,t) is relatively small, say less then one-tenth of the length Li of the link.

to link-2 at O

Figure 2.1 Two-link flexible manipulator

3

4 CHAPTER 2

The arm shown in figure 2.1 is modelled as a continuous cantilever beam of length Li a mass mi and an additional pointmass ML for the payload at the end of arm 2. The deflections ui(xi,t) are described by an infinite series of separable modes.

o

ui(lc;, t ) = ~i(xi)qi(t) for i = i, 2 i=l

where 6, = [d>ilBi2.. . are the assumed mode shapes qi = [qilqi2 . . . qirIT are the modal functions (generalized coordinates)

The 6 , i j 's are admissible functions which statisfy the geometry boundary conditions of the system un- der consideration. Next, from kinematics, the partial differential equations together with the boundary conditions for the two links can be expressed as follows. For the first link:

For the second link:

E 2 1 2 ~ y ( ~ 2 , t ) + pA2U2(22,t) = O u2(O,t) = o U i ( 0 , t ) = o

UY(L2,t) = o E212?&(L2,t) = Mt,U,(L2,t)

where Li = Length of the i-th link IiEi = Flexural rigidity of the ith link Mtz mi = Total mass of the i t h link Pz = Density of the i t h link

= Mass of the concentrated tip-body of the i t h link

(2.3a) (2.3b) (2 .3~) (2.3d) (2.3e)

Note that dots denote differentiations with respect to the time and primes denote differentiations with respect to the spatial coordinate xi. The interactions between the links are assumed negligible. The mode shapes can be found by solving the eigenvalueproblem of relation 2.2- 2.3 (see appendix A), The mode shapes can be expressed as:

Bi(zi) = Ci(cosh(/3,zi) - cos(,ûixi) - yi(sinh(/?izi) - sin(/?izi) for i = i, 2 (2.4)

where /?i = r; = Dominant mode of the i t h link

e o s h ( p i l i ) t e o s ( B , l , ) "li = s i n h ( p i l i ) + s i n ( B , l , )

In next section the dynamic equations of a cantileaver beam will be discussed.

The flexible manipulator 5

Figure 2.2 Two-link one mode flexible manipulator

2.3 THE DYNAMIC EQUATIONS

The equations of motion of the two arm r-mode flexible-link are derived by using the Lagrangian approach, as treated in [5, Low 871. We first obtain the expressions for kinetic and potential energy.

With reference to figure 2.2, the system has four generalized coordinates in which &(t) and &(t) represent the joint angular positions; wheras ul (z1 , t ) and u2(z2, t ) denote elastic displacements. Assume the mass center of the i th link is located at zi away from the left end of the i-th link. The position vectors of the mass center of the two links are,

The position vectors of the tip mass of the links is defined as follows,

1 Llcoso1 - u1,(t)sin81 L1sinB1 + ule(t)cosB1 Ti t =

where rim = the position vector of the mass center of the i-th link Ti t

ui(xi, t ) uie(t)

= = =

the position vector of the joint or tip mass of the i-th link the elastic displacement of link i as defined in section 2.2 the elastic displacement of the tip of the link i as defined in section 2.2

By using the appropiate orthogonality on the mode functions as defined in section 2.2, the kinetic and potential energies can be determind as follows. The kinetic energy of the two elastic links can be expressed

6

in a quadratic form,

CHAPTER 2

Li Li

body b i d y O O

where pi = the mass density of the i t h link [Ic~.m-~] the cross section area of the i t h link [m2] Ai =

The kinetic energy of each tip mass can be expressed,

1 . T . 1 Tt = 5Mltrltr1t + ?M~t+?t+~t

The potential energy due to the gravitational effect is of the form,

(2.10)

(2.11) J L1

vg = - p i ~ i J gTrim - p2A2 g T r m - MitgTrit - ~ 2 t g ~ r z t

where gT = [O - g ] the gravitational force [ll"gm.~-~] . The strain energy of the elastic links can be written in the form,

O O

(2.12) O O

where Ei = the Young's modulus of i th link [ N . T ~ - ~ ] li @(xi) = the eigenmode function of the i th link

= the area moment of inertia of i t h link [m4]

Here, the Euler beam theory is adopted. In addition we assume that the coupling between torsonial and bending deformation is small and can therefore be ignored. By using the Lagrange's equation,

d dT dT dV dt(z)-di+z=- (2.13)

v = v, + v, The dynamic equations of the two arm l-mode flexible link can be written by using 2.6 , 2.8 in 2.13 as follows,

~ 7 i - t ~ 5 i - t f r ( I 1 , ) + f g r ( 8 , o ) = T S ~ (2.14a)

M r f i + ~ f y + ff (8, i) + f g j ( 8 , ~ ) + ~4 = O (2.14b)

The flexible manipulator 7

Note that equation functions of Mr , M f tions 3 . l a , 3. lb can

3.1a is called the rigid part and that equation 3.lb is called the flexible part. The

be written in a state space notation, as described in appendix B, , Mrf , H , f r ( & i), f g r (e, 9) 3 ff (e, i) and f g f (8, i) can be found in appendix B. Equa-

x = f(z> + + P(Z)b (2.15) -

where = = = =

is an n x 1 vector of genera’ized coordinates is an n x 1 vector of generalized forces applied at each joint is an n x 1 vector of generalized disturbances is n x 1 vector representing the Corolis, centripetal and gravitational forces

- U

- d f(g)

There are two significant properties regarding to the positive definite inertia matrix M , as described in appendix B.

Property 1 All elements of M are independent of the $first lank angular position 81

Property 2 Mr - M ; M ~ - ~ M ~ ! is also a symmetric positive definite matris:

Property 1 can be interpreted from the fact that the base axis of 81 can arbitrarily selected (If M were to depend on 81, the kinetic energy f g T M X would vary with the choice of the base axis, which is obviously untrue). Utilizing properties 1 and 2, a sliding mode control is developed for a two link flexible manipulator in the next chapter.

8 CHAPTER 2

CONTROL DESIGN

3.1 THEORY OF THE SLIDE-MODE CONTROL

Broadly speaking, sliding mode control involves a non-linear feedback control law that switches disconti- nously on a specified surface, which is embedded in the state space of a dynamical system, in such a way that when the natural trajectory of the open-loop system deviates from this surface, a force is applied in order to return the state of the system back to it. In this manner a trajectory originating on this surface is confined to moving or sliding along the surface. Clearly such a trajectory has to satisfy the algebraic relation which describes the surface. Conversely, a desired algebraic relationship can be induced on the state space of the open-loop system by restricting the dynamics on an appropriate surface. For example, the surface may be desired as a linear algebraic equation in the position error and its velocity. This rela- tionship, in turn, defines a different equation governing the dynamics on the surface. If the homogeneous solution to this differential equation is asymptotical stable then the position error and its velocity must tend to zero. Note that this relationship between states is independent of any disturbances or modelling errors, provided the trajectory remains on the surface. In summary, the application of sliding mode control involves choosing a surface and a control that directs the vector field toward the surface.

As the preceding discussion indicates, sliding mode control is particularly well suited for the manipulator control problem for the following reasons. First, the dynamics, provided there is no unmodeled structural uncertainty. This property is desirable since the complexity of the manipulator dynamics makes the exact calculation of the dynamics infeasible if not impossible. Second, when a sliding mode control is applied, the performance of the system can be made insensitive to bounded disturbances. This property is important in rejecting effects due to Coulomb and viscous friction. It is also important when the manipulator is carrying payloads because the force a payload exerts on the gripper of the manipulator can be translated into forces or disturbances at each of the joints. Thus the application of sliding mode control results in a performance that is robust with respect to disturbances and modelling errors, while i t provides accurate tracking. Consider the dynamic equation,

where e = [el &Is : the angular coordinates - 4 = [ql q2IT : the flexible coordinates - T = [TI T2IT : the input torques d = [dl d2]S : the input disturbances

9

(3.la)

(3.lb)

10 CHAPTER 3

First we linearize equation 3.1 by using the sliding mode technique. This technique replace the rigid part of the equation by a set of linear equations which are determined by the switching surfaces. For convenience we define the following variables,

Bt; Bdi = desired trajectory of the tip angular position e; et;

= tip angular position of the ith link = Bi + 9 deviatiori between Bi and Bdi of ith !ink = 8; - Bdi

deviation between $ti and Ba; of ith link = 6’ti - Bdi=ei + = =

LZ

We generate the following variables from measurement ,

(3.2a)

(3.2b)

(3 .2~)

(3.2d) (3.2e)

Differentiating the variables 3.2a, 3.2b and 3 . 2 ~ and combining with 3.1, we obtain the overall system,

Mre + M s q + f r + f g r = T + d - M r O d (3.3a)

Mrfë + M r j i + f f + f g j + ~q = -Mrfid (3.3b)

iil = cl& + Elel +PlQl +PlQl +P242 +Pa@ + kv3 (3.3c)

L1 L 2

which has [e i q 4 v1 v2 v3IT as its state.

A choise of switching variables is,

s1 = e , + V I

s2 = et2 + v2

(3.3d)

(3.3e)

(3.4a) (3.4b)

The designed parameters determine the rate of response of the system and the aim of the control is to force the motion of the system to be along the intersection of the switching planes S ( t ) = 0. To guarantee stable error dynamics, the scalar coefficients are chosen such that the polynomial in p (Laplace operator) is Hurwitz. The coefficients will be obtained as follows. Differentiating S( t ) = 0 once yields from 3.4,

(3.5a) (3.5b)

The parameters c2 and E2 are chosen according to the desired time response of et2 in the sliding motion. The fixed coeffecients in 7 1 will be determined to stabilize the manipulator. The equations 3.5a - 3.5b are the desired linear equations which, being valid during sliding motion, are to replace 3.3a in all subsequent

Control design 11

analysis. In order to force the switching variables g ( t ) to reach and stay at zero in finite time, we invoke a sufficient condition. Considering system 3.3a together with the Lyapunov function candidate,

1 , V = -S AS 2

where A = Mr - M : f M f l M r f is a symmetric positive matrix. The condition,

V5--rlSl , Y > O (3.7)

will be sufñcient to guarantee the reaching and sustaining of sliding motion. Condition 3.7 can be realized as follows, solving i in 3.3b and substituting the result in 3.3a yields,

A i = T + d + M $ M ~ ~ (ff + fgf + H Q ) - f r - f g r (3.8)

We now derive a reaching condition for the switching planes using the stability theorem of Lyapunov. Since A is positive definite, V ( s ) is a positive semi-definite function with respect to s. Moreover, V ( s ) vanish only for s = O. Differentiating V ( s ) with respect to time and using the symmetry of A and 3.8 yields,

1 2

V ( s ) = STAs+-s tÀs

= sT(r+r*) (3.9)

where

(3.10)

The above equation can be expanded into components. The expanded expression is tedious and long and can be found in appendix C. The expession has the following structure,

O T* = d + ~5 ~ j - 1 (ff + fgf + ~ 4 ) - f T - f gT + ~ ( 1 - k) + A [ ] + $.As

T,* =q;+q;+q;+q>, i = 1 , 2 (3.11)

where

q; = Computed torque = Computed flexible torque

q; = Computed stabilization term q> = Computed distortion term

We consider the following control law,

(3.12)

where,

1 S i > O

-1 S i < O

12 CHAPTER 3

j =3

2 2r+2

j=i j=r+3

2 2rC2

j=1 k=l

Substitution of the control law 3.12 into 3.9 yields,

As result of 3.11 through 3.12, the terms in parentheses in 3.13 are positive. Let,

(3.15)

Thus, the choise of control law 3.12 guarantees condition 3.7 which is a sufficient condition for the reaching and sustaining of sliding motion. Once in sliding motion, 3.3a will be valid. By selecting suitable values for ca and E2 in 3.5b, the tip angle ût2 can be regulated at the desired value in an arbitrarily short time. With et2 = O and 3.5a replacing 3.3a, the overall system takes on the simpler form,

(3.16d)

We shall employ PD/PID compensations to determine the coefficients of TI in order to stabilize 3.16a. The coefficients in question are cl, El, pi, j71 , p2, $2 and k . Where the P-gain coefficients are El, j71 and 5 2 . The D-gain coefficients are cl,pl and p ~ . The I-gain coefficient is represented by k .

Once the system is in sliding motion and after et2 approaches zero, the overall system behaviour is given by 3.16. However this equation is highly non-linear. In order to employ linear techniques to stabilize 3.16, this equation must be linearized. First, due to small deflections (Qi,qi < O . i ) , all non-linear flexible terms can be neglected and because e2 N et2 N O the variable 82 can be approximated by 8 d . Thus 3.16b becomes,

mrflël + q q + H q + fy, = -Mrf_d 0 ; (3.17)

where superscript' denotes that all the non-linear flexible terms have been eliminated and that 82 has been replaced by Bd2. When 8d2 is assumed to be constant. We can simlify equation 3.17 further to,

m,flëi 0 + M j q + Hq + fjf = -M:fBdi (3.18)

Control design 13

The overall linearized system can be expressed as a characteristic equation and can be obtained as,

X2 + C l X + El PlX +F1;p2X +Pa fk

3.k; 0 X det [ mrfiA2 + h M j X 2 + H

Li -1 (3.19)

If we select stable poles for the characteristic equation, the overall system will stabilize. Hence that the degree of X equals the number of unknown parameters. One could specify the pole locations and determine these unknown parameters. If the roots are selected to be stable, w3 = ûti - ûdl -+ O as t + 00, i.e. the angular tip position of the first link will follow the desired trajectory. The controller design will be discussed for three seperate cases based on the trajectory types and is discussed in appendix D.

Note firstly that the control design is based on a regulation of the second link and a desired trajectory for the first link. It is possible to design the controller so that the first link is a regulation and the second link must follow a desired trajectory. The designed controller is discussed in [9] and used in chapter 4.

Secondly that the control law is related to parameter bounds in a simple fashion so that parameter variations in the plant can be taken into account easily. Thirdly, the control law is given for any degree of freedom n of the plant.

14 CHAPTER 3

4 SIMULATION

4 , l AN ONE LINK 1-MODE FLEXIBLE LINK WITH SLIDING MODE CONTROL

In this section we develop a one link flexible 1-mode manipulator as decribed in chapter 2. The dynamical equations will be obtained in SIMULINK. A system-file (S-function) is developed as described in the SIMULINK manual, see appendix E. The final simulink program is figured in figure 4.1 Figures 4.2 - 4.3

F+ equation 2.14

U I Scope

I

equation 3.22 eauation 3.2 traiprtnni

Figure 4.1 The used SIMULINK program

shows results with a desired final arm position of ûd = 1 [rad]. Figure 4.2 shows the closed loop system responses with control which takes no account of flexibility and no payload (Curve a) and control using the sliding mode technique with payload and no payload ( Curve c and b). As can be seen the tipangular

15

16

-1 o.

CHAPTER 4

1 1.5 2 2.5 3 0.5

position of the link which takes no account of flexibility is still vibrating on its desired position. When we use the controller the tip is on its position and stays on its position without any vibration (Curve b and c). The performance is superb and robustness of control against payload change is evident. Notice that no payload forecast is required, nor is any estimation necessary.

Angular position of middle of beam

- - c u t w c

Angular posiüon of tht lip of beam

05 1 15 2 25 lime [Seconds]

-0 2

Figure 4.2 The angular position of the one link 1-mode flexible manipulator

Figure 4.3 shows the torque responses with the different closed loop systems. As can be seen the control torque of the controller which takes no account of flexibility is still working and is trying to get the tip position of the link to its desired position.

15

10

- 5 E z 0 ro al 3

O

-5

Control torques , I I 4

I

. .

!i I ti I I c

Figure 4.3 The torque of the one link i-mode flexible manipulator

Simulation 17

4.2 THE TWO-LINK FLEXIBLE MANIPULATOR The sliding-mode controller developed by Yeung and Chen (1991) is implemented in SIMULINK as figured in figure 4.4 and 4.5. The dynamic equations are implemented in a S-File and can be found in appendix E.

n 3 . 2 ~

uation I I

3.26- 3.2e

Equatioi 3.2a-3.2b

Equation 3.12 Equation 3.4

t ia i 3.2a-3.2b

Equation 3.12 Equation 3.4

Figure 4.4 SIMULINK program for the two-link control

Figure 4.5 Implemented equations in SIMULINK

18 CHAPTER 4

The simulation of the implemented program gives inherent results. We now first consider a planar two-link manipulator whose position can be described by a 2 vector q of joint angles and whose actuator inputs consists of a vector í- of torques applied at the manipulator joints. The dynamics of this manipulator is strongly non-linear and can be written in the general form,

where H ( q ) , = a 2 x 2 manipulator inertia matrix

C(q,) g(q ) = a 2 - vector of gravitational torques

= a 2 x 2 matrix which contains centripetal and corolis torques

From Slotine and Li (1991) we find the dynamic equation described as follows,

Note that for a detailed description of the different values of the equation we refer to Slotine and Li [lo] (page 211-212). In order to test the dynamical equation of Yeng and chen [i] we use equation 3.1 without the flexibility terms. The used data for both systems can be found in appendix B. Both models are programmed in SIMULINK as figured in figure 4.6.

Demux

out Yeung 1 L I . ... 4

w I 4 I theta1

U I I

Functional I I Phi(x1) Dlant

I -1 I - tTh Demux8

Clock Mux3

tau2

Figure 4.6 Program used for testing the rigid parts

Simulation 19

As can be seen in figure 4.6 we use a simple PD controller to get the two links to the position qd = O. The initial conditions used are qo = [-; , 9 , O , O I T . The results can be seen in figure 4.7. The curves

theta2 3- I I I I I

2-

I

E 1 - I

n I' I

' '\ I I

I I

I

- _ - O O 1 O 2 O 3 O 4 O 5 O 6 07 O 8 O 9

-3

time in seconds

Figure 4.7 The angular position of the two Link rigid manipulator

shows different responses. The used dynamic equation of Slotine and Li (1991) gives a good response. The response of Yeung and Chen (1991) is not statisfactory.

20 CHAPTER 4

5 CONCLUSIONS AND RECOMMENDATIONS

5.1 CQNCLUSIONS

The rigid part of the dynamic equations of the two-link l-mode flexible link doesn’t fit with the dynamic equations of rigid links as discussed in Slotine and Li (1991). The dynamic equations, in this case the rigid part, of Yeung and Chen (1991) are not correct in their article.

The used sliding-mode controller for the l-mode flexible one link manipulator is robust in his payload variations.

The systematic derivation by Low (1987) is easy to practice. The derivation should be used in MAPLE. The derivation of the position vector is difficult because it contains muItiple variables which are dependent on time.

The controller is only used for the control of one link with the second link in a given position. It is possible to extend the controller for multi-link manipulators.

It is easy to design a simulator in SIMULINK. The block building of SIMULINK is easy to practice.

5.2 RECOMMENDATIONS

Test the controller when the desired trajectories are dependent on time. The used controller is only tested for a regulation on the second link with a desired trajectory on the first link. Extend the controller for multiple trajectories.

Derive the mathematical equations of the two link one-mode flexible manipulator with MAPLE. The derivation with MAPLE will be more accurate in comparance with the derivation by hand.

Further research should be done with the simulator by coupling it to a real system. The real system in question is a robot with two flexible links and well-known parameters.

Usage of the translate code MATLAB to Cs+ should be tested. The used method in MATLAB is relative slow, translation to a C++ program makes the program more faster.

21

22 CHAPTER 5

REFERENCES

[l] Y.P. Chen and K.S. Yeung, “Sliding mode control of multi link flexible manipulators”, znternatzonal Journal of cuntrol, vol. 54, no. 2, pp. 257-278, Dec. 1991.

[a] O. Maizza-Net0 W.J. Book and D.E. Whitney, “Feedback control of two beams, two joint systems with distributed flexibility” , Journal of Dynamac Systems, vol. 97, pp. 424-431, 1975.

[3] U. Kleemann, “Dynamics and control of a robotic system with elastic arms”, Robotzcs: theory and appkcatzons, vol. 3, pp. 487-492, dec 7 1986.

[4] F. Matsuno Y. Sakawa and S. Fukushima, “Modeling and feedback control of a flexible arm”, Journal of Robotzc systems, vol. 2, no. 4, pp. 453-472, Aug. 1985.

[5] K.H. Low, “A systematic formulation of dynamic equations for robot manipulators with elastic links”, Journal of robotzc systems, vol. 4, no. 3, pp. 435-456, Jan. 1987.

[6] S.N. Singh and A.A. Schy, “Decomposition and state variable feedback control of elastic robotic systems”, Proceedzngs of the Ameracan Control Reference, pp. 375-380, 1985.

[7] D.R. Meldrum J.T. Wen and M.J. Balas, “Discrete-time model reference adaptive control using the cgt concept”, Robotzcs: theory and applzcatzons, vol. 3, pp. 17-22, dec 7 1986.

[8] K.S. Yeung and Y. P. Chen, “A new controller design for manipulators using the theory of variable structrure systems”, IEEE Transactzons on automatzc control, vol. 33, no. 2, pp. 200-206, Feb. 1988.

[9] K. S. Yeung and Y.P. Chen, “Regulation of a one-link flexible robot arm using sliding mode technique”, Internatzonul Journal of Control, vol. 49, no. 6, pp. 1965-1978, Feb. 1989.

[lo] Jean Jacques E. SIotine and Weiping Li, Applzed nonlznear control, Prentice-Hall International Editions, 1991.

23

24 REFERENCES

THE EIGENVALUEPROBLEM

Let us consider the eigenvalueproblem related to the partial differential equation 2.2-2.3 which is given by,

@ A @ i j ( X i ) - @;;'(xi) = o (O 5 zi 2 Li ) (A.la) @ i j ( O ) = q j ( o ) = @ij(Li) = o (A.lb)

EiIi@;;(Li) - Mi@ij(Li)w:j = O (A.lc)

where ,í3$ = p i A i ~ : ~ / E i I i

Note that the boundary conditions A. lb also depends on the eigenvalue w ! ~ . The general solution of A.la is given by,

@i(z i ) = C ~ ~ C O S ( P ~ Z ~ ) + Ciacosh(P;zi) + Cissin(Pizi) + Ci4sinh(P;zi) ( A 4 where Cih = constant for h = 1. . .4

W ~ J = natural frequency of the j th mode for i = 1 , 2 and j = 1 . . . ri

By using the boundary conditions A. la we see that the boundary value problem has solutions if Pij

statisfies ,

(A.3) Mi Pij

pi Ai 1 + cosh(PijLi)cos(PijLi) + - COS(@^^ Li)sinh(&jLi) - sin(Pij L i ) ~ û ~ h ( @ i j L i ) ] = O

This problem can be solved iteratively. The fundamental eigenvalue can be written as a function of the payload M L . In table A. l we can find the values of the eigenvalue which are derived by software package Maple.

ML = 0.0 kg E Ij P 2 I 1.6799 1.4669 1.3437

Table A.l Eigenvalues of the two-links

The corresponding eigenfunctions can be calculated as,

25

26 APPENDIX A

Let us determine the constants Ci so that the set {(a(.)} of the eigen functions forms an orthonormal system in H (a Hilbert space) [4, Sak 851. Namely we have to calculate Ci such that ( ( a i j a i j ) = 1. After complicated calculations, we obtain,

For the two link flexible manipulator the Ml and M2 are given by,

where ML = load of the manipulator MG = mass of gripper Mti = mass of joint between link 1 and 2

B PARAMETERS OF THE TWO LINK FLEXIBLE

MANIPULATOR

B e l THE DYNAMICAL EQUATION

As discussed in chapter 2 the dynamical equation can be derived by using the Lagrange’s approach. The dynamical equation which represent a two link one mode flexible manipulator (Y = 1) can be written as follows,

Note that B. l .

where Mv

M

H

B.2 are called, respectively, the ’rigid’ part and the ’flexible’ part of the equation.

1 mrïi mri2

mr12 m r 2 2

m r f i i n Z r f 2 1 = I

= [ 0 1 l H,,]

- -

m f l 2 m f 2 2

- -

- -

27

APPENDIX B

THE DYNAMICAL EQUATION IN STATE SPACE NOTATION

Its now easy to write equation B.4 in state space notation as follows,

- 33 = f(i) + &)u + P(2)b

where g = is an n x 1 vector of generalized coordinates - U

- d f(g)

= = =

is an n x 1 vector of generalized forces applied at each joint is an B x 1 vector of generalized disturbances is n x 1 vector representing the Corolis, centripetal and gravitational forces

Parameters of the two link flexible manipulator

ML = 0.0 ML = 0.2 w1 4.4749 3.9581 ~2 7.2193 5.5047

Meaning

ML = 0.4 3.5868 4.6189

mass density modulus of elasticity

thickness height length

mass of gripper mass of the joint

pavload

29

Value 2712.6 kg.m-3 71.10’ N.m-2 0.009 m 0.03 m 1.0 rn 0.1 k g 0.2 k g 10.0 0.2 0.4 1 leg

Table B.l Numerical data for both links

B.3 PARAMETERS OF THE LINKS

The numerical data for the both links are represented in table B.3.

Because of the variational load the eigenfrequency wi of both links are a function of the payload. The values of wi are represented in table B.3 ( Note wi = ,@fi ).

Table B.2 Eigenfrequency of both link with variational load

30 APPENDIX B

CONTROL PARAMETERS

C.l EXPANSION OF T* The expansion of equation 3.10 is tedious and long. Defening z = [ B I 02 q 1 q 2 I t , T* can be expressed as follows,

where,

The coefficients on the right-hand of C.2 with a superscript * depend in general upon 01 &y1 and y2 but not upon their derivatives. Based on this fact and the assumption that the deflection @ i e q i is smaller than 0.1Li upper bounds for the coefficients exist and can be determined. Let these be.

31

32

Desired trajectories e d l

Bd2 Parameters in 71

C1

c"l

a11

a 2 1

a 2 1

k

ai1

Case 1 First-link regulation

0.5 0.5

20.73 36.95 -57.78 909.65 150.50

-4237.46 O

First-link ramp-tracking

0.5 -0.lt 0.5

20.73 36.95

909.65 150.50

-4237.46 O

-57.78

Case 2 First-link

parabola-tracking

0.08t2 - 0.4t + 0.5 0.5

32.33 78.40 -51.09 749.09 150.46

73.90 -3936.46

APPENDIX C

Case3 Firs t-link

regultion(g $: O)

-1.0 0.5

31.35 63.97 51.44

776.18 150.84

3945.46 77.62

Table C.l Parameters of the slide-mode control

CONTROL DESIGN FOR SEPARATE CASES

In the appendix different cases will be discussed to design the sliding mode controller. The controller is based upon a reguIation of the second Iink and a desired trajectory for the first-link.

D.l CASE 1: FIRST-LINK RAMP-TRACKING OF THE MANIPULATOR ON A HORIZONTAL PLANE

Here, the desired trajectory of the link can be expressed as B d l = ut + b. This means û d = O and fgj = O. The total system is given from 3.16 as,

where M,!f and M j are the linearized terms. Equation D.l is a linear time-invariant system with the input q . In order to stabilize this equation a PD compensation is chosen,

Using D.l and D.2 yields,

The characteristic equation of D.3 can be obtained as,

If we select stable poles for the characteristic equation, the overall system will stabilize. Hence that the degree of )r equals the number of unknown paramaters. One could specify the pole locations and determine these unknown parameters. It follows that e l and q both approach zero at t i co.

33

34 APPENDIX D

D.2 CASE 2: FIRST-LINK PARABOLA-TRACKING OF THE MANIPULATOR ON A HORIZONTAL PLANE

Here, the desired trajectory of the link can be expressed as B d l = ut2 + bt + e. This means id # O and fsf = O. In order to stabilize this equation D.2 a PID compensation is chosen,

71 = c l& + & e 1 + P I & + P1qi + P Z Q Z + P 2 q 2 + k (et1 + e t z ) (D.5) .I Once et2 has been driven to zero, the overall system can be expressed, using D.5 and D.2 yields,

The characteristic equation of D.6 can be obtained as,

(D.6a) (D.6b)

(D.6c)

As in case 1, if we select stable poles for the characteristic equation, the overall system will stabilize. Hence that the degree of X equals the number of unknown parameters. One could specify the pole locations and determine these unknown parameters. If the roots are selected to be stable, 213 = ûti - û d l -+ O as t i 00.

D.3 CASE 3: FIRST-LINK REGULATION OF THE MANIPULATOR IN A VERTICAL PLANE

Here, the desired trajectory of the link can be expressed as B d l = ut + b. This means Bd = O and fgf # O. The total system is given from 3.16 as,

where

(D.8a) (D.8b)

Due to the presence of the cosine functions the overall system is not linear. After linearization of the cosine function, the overall system becomes,

ë1 1 = M:fël + Mfoq + Hq + hel

- 71 -h*

(D. 1 Oa)

(D.lOb)

Control design for separate cases

P-

35

As described in case 2 a PID compensator is used to stabilize the overall system. By using D.8 and D.10 the corresponding characteristic equations can be obtained,

( D . l l )

Following a simular argument as described in case 2, we can place suitable stable roots in for stabilization the manipulator. An overall feedback structure for the above three cases is shown in figure D . l .

I - I I u equation 2.14

I' I I Scope I

-

Gewenst equation 3.2 Traject

Figure D.l Block diagram of the sliding-mode control

36 APPENDIX D

PROGRAMS

E.1 PROGRAM OF AN ONE LINK I-MODE FLEXIBLE MANIPULATOR

This program is adopted from article by Chen and Yeung. It is translated in a S-function, so it can be used in SIMULINK.

% The model is derived by the article of Yeung en Chen 1989 % VERSION : date 19 juli 1994 % one flexible link

function [sys,xO] = linkm(t,x,u,flag);

load para; % The dynamical equations derived as procedure chapter 2 if abs(f1ag) == I, % The system matrices

% load parameters needed for programm

alpha00 = Jls + Bls*x(3)*x(3); alpha01 = w i s ; alpha11 = B I S ; M = [alpha00 alpha01 ; alpha01 alpha111 ; Fr I Ffl = - HlI*x(3) + Bls*x(3)*~(2).̂ 2;

= -2*x( 2) *B~s*x( 3) *X (4) ;

% The state space notation Mthetp = inv(M)*[Frl;Ffl] + inv(M)*[u(l);Ol ;

sys(2) = Mthetp(1);

sys(4) = Mthetp(2);

sys(1) = x(2);

sys(3) = x(4);

sys(1) = x(1); sys(2) = x(2); sys(3) = x(3); sys(4) = x(4);

elseif flag == 3,

elseif flag == O,

37

38 APPENDIX E

sys = [4;0;4;1;0;01; xo = c0;0;0;01;

sys = c1; else

end

% Initiatial conditions

E.2 PROGRAM OF THE TWO LINK 1-MODE FLEXIBLE ARM

This programm is adopted from article written by Yeung and Chen. It is translated in a S-function, so it can be used by SIMULINK.

% Model of the two-flexible link by Yeung and Chen % version: 29 august 1994 % files: input.m, beam.m

function Csys,xOl = beamm(t,x,u,flag);

load paramet er ; load conpara;

% De systeemvergelijkingen opgebouwd als artikel if abs(f1ag) == i,

x(6) = u(3); x(7) = u(6); x(8) = u(4); Ui = Phii*x(6); u2 = Phi2*x(8); uie = Phii*x(6); u2e = Phi2*x(8); alpha = i + x(6).-2*(Phii/Li)-2; alphap = (2*(Phii/Li)"2)*~(6)*~(5) ;

x(5) = u(5);

% Bepaling van de systeem matrices mrll = Jis + Bis*x(6).-2 t alpha*(J2s+B2s*x(8).̂ 2 t 2*Li*P2s*cos(x(4)) . . .

mr12 = alpha*(J2s+x(8)*B2s*x(8) + cos(x(4))*P2s-sin(x(4))*Li*Q2s*x(8)); mr22 = alpha*(J2s+x(8)*B2s*x(8)); mriip = 2*~(5)*~(6)*Bis+alpha*(2*~(7)*~(8)*B2~ - 2*x(3)*sin(x(4))*Li*P2s . . .

- 2*x(3)*cos(x(4))*Li*Q2s*x(8)-2*sin(x(4))*Li*Q2~*~(7)) . . . + alphap*(J2s+B2s*x(8).~2t2*cos(x(4))*Li*P2~-2*sin(x(4))*Li*Q2~*~(8));

- x(3)*cos(x(4))*L2*Q2s*x(8)-sin(x(4))*12*Q2s*x(7)) . . . + alphap* (J2s+B2s*x( 8). -2tcos (x(4) )*Li*P2s-sin(x(4) )*Li*Q2s*x(8) ) ;

- 2*Q2s*Li*sin(x(4))*~(8)) ;

mri2p = alpha*(2*~(7)*B2~*~(8)-~(3)*sin(x(4))*Li*P2~ . . .

Programs

mr22p = alpha*(2*~(7)*~(8)*B2s)+alphap*(J2~+~(8).^2*B2~);

39

Mr = [mrii mri2 ; mri2 mr221; Mrp = [mrilp mrl2p;mri2p mr22pl ;

mrfil = wis + Phil/Li*(J2s+x(8)*B2s*x(8) + 2*cos(x(4))*Li*P2s . . .

mrf21 = alpha*(w2s +Li*cos(x(4))*Q2s); mrfi2 = Phii/Li*(J2s+x(8)*B2s*x(8) + cos(x(4))*LI*P2~ . . .

- 2*sin(x (4) ) *Q2s*EI*x (8) ) ;

- sin(x(4))*Q2s*Li*x(8)) - Phii/Li*uie*(sin(x(4))*P2s . . . + cos (x(4) )*Q2s*x(8) ) ;

mrf22 = alpha*w2s; mrfiip = Phii/L1*(2*x(7)*x(8)*B2s-2*x(3)*s~n(x(4))*L~*P2~ . . .

mrf 2ip = alpha* (-Li*x(S)*sin(x (4)) *Q2s) +alphap*( w2s + Li*cos (x (4) ) *Q2s) ; mrfi2p = Phii/Li*(S*x(7)*~(8)*B2s - x(3)*sin(x(4))*Li*P2s . . .

- 2*x( 3) *cos (x(4) ) *Li*QLs*x( 8) -2*sin( x (4) ) *Li*Q2s*x(7) ) ;

- x(3)*cos(x(4))*Ll*Q2s*x(8)-sin(x(4))*Li*Q2~*~(7)) . . . - Phii^2/Lí*x (6) * (x( 3) *cos (x(4)) *P2s-x (3) *sin(x(4) ) *Q2s*x (8) . . . + cos(x(4))*Q2s*x(7))-Phii~2/Li*x(5)*(sin(x(4) )*P2s+cos(x(4) )*QSs*x(8)) ;

mrf22p = alphap*w2s;

Mrf = [mrf li mrf 12 ; mrf2i mrf221; Mrfp = [mrfiip mrfi2p ; mrf2ip mrf22pl ;

mfii = BIS + (Phii/Li)-2*(J2s+x(8)*B2s*x(8) + Phil-2 . . .

mf12 = (ule*B2s*x(8)/(L1-2) - (sin(x(4))*uie*Q2s-w2s)/Li . . .

mf22 = alpha*B2s; mf lip

mf 12p = Phil*( (ule*B2s*x(8)+Phii*x(5)*B2s*x(8))/(Ll~2) . . .

+ 2*cos(x(4))*Li*P2~-2*sin(x(4) )*Q2s*Ll*x(8)) ;

+ cos(x(4)*Q2s) )*Phil;

= (Phii/Li)~2*(2*x(7)*x(8)*B2s-2*x(3)*sin(x(4))*Ll*P2~ . . . - 2*x (3)*cos (x(4) ) *Li*QSs*x( 8) -2*sin(x (4) ) *Li*Q2s*x ( 7 ) ) ;

- (x( 3) *cos (x(4) ) *uie*Q2s +sin(x(4) ) *Phii*x (5) *Q2s) /LI . . . - x (3)*sin(x(4)) *Q2s) ;

mf22p = alphap*B2s;

Mf = [mfii mfi2 ; mfí2 mf221; Mfp = [mfiip mfl2p;mfi2p mf22pl;

Mthet = Mr; %- Mrf’*inv(Mf)*Mrf; Mq = Mf - Mrf*inv(Mr)*Mrf’; phi1 = i ;phi2 = I; Fgri = mi*grav*xi*cos(x(2)+u~/xi)+m2*grav*(L~*cos(x(2)+u~e/L~) . . .

+ x2*cos (~(2)+~(4)+uie/Li+u2/~2) )+Mt i*grav*Li*cos (x(2) +uie/Li) . . . + Mt2*grav* (LI*cos (x( 2) +ule/Li) +L2*cos (x (2) +x(4) +uie/Li+u2e/L2) ) ;

+ Mt 2*grav*L2*cos (x (2 ) +x (4) +u1 e/L i+u2e/L2) ;

+ x2*Phii/Li*cos(x(2)+x(4)+uie/Li+u2/x2)) . . .

Fgr2 = m2*grav*(x2*cos(x(2)+x(4)+uie/Ll+u2/x2)) . . .

Fgf i = mi*grav*phil*cos (x( 2)+ui/xl)+m2*grav*(Phii*cos (x(2)+uie/Ll) . . .

40 APPENDIX E

+ Mti*Phii*grav*cos(x(2)+u~e/L~)+Mt2*grav*(Phii*cos(x(2)+u~e/Lí) . . . + L2/Li*Phii*cos(x(2)+~(4)+uie/Li+u2e/L2));

+ x(4)+uie/Li+u2e/L2) ; Fgf2 = m2*grav*phi2*cos(x(2)+x(4)+ule/Li+u2/x2)+~t2*grav*Phi2*~0~(~(2) . . .

Fgr = [Fgrl;Fgr2] ; Fgf = [Fgfi ;Fgf 21 ; Psi = [x(i);x(3);~(5);~(7>1;

% Mtheta voor de Fr mt21 = alpha*(-2*sin(x(4))*Li*P2~-2*cos(x(4 mt22 = alpha*(-sin(x(4))*Li*P2s-cos(x(4))*L

)*Li*Q2s*x *Q2s*x(8))

8));

mt23 = Phii/L1*(-2*sin(x(4))*Li*P2~-2*cos(x(4))*Li*Q2~*~(8)); mt24 = alpha*(-Li*sin(x(4))*Q2s); mt25 = Phil/Li*(-sin(x(4))*L~*P2s-cos(x(4))*Li*Q2s*x(8)) . . .

- Phil*ule/L1*(cos(x(4))*P2s-sin(x(4))*Q2s*x(8)); mt26 = (Phii/Li)̂ 2* (-2*sin(x(4) )*Li*P2~-2*cos (x(4) )*Li*Q2s*x(8) ) ; mt27 = Phil*(-cos(x(4) )*uie*Q2s/Ll-sin(x(4) )*Q2s) ; Mth = Cmt2i mt22 mt23 mt24;

mt22 0.00 mt25 0.00; mt23 mt25 mt26 mt27; mt24 0.00 mt27 0.001 ;

Mtha = Psi’*Mth*Psi;

% Mqi voor Fgl mqli = 2*Bis*x(6) + 2*x(6)*(Phii/L1)^2*(J2s+B2s*x(8).~2+2*cos(x(4~)~Li*P2~ . . .

mqi2 = 2*x(6)*(Phil/L1)^2*(J2s+B2s*x(8).^2+cos(x~4~~*Li*P2~ . . .

mqi3 = 2*x(6)*(Phil/Li)^2*(w2s+Ll*cos(x(4))*Q2s~; mqi4 = 2*~(6)*(Phii/Li)^2*(J2s+B2~*~(8).-2); mqí5 = -Phii~2/Li*(sin(x(4))*B2s+cos(x(4))*Q2s*x(8)); mqi6 = 2*x(6)*(w2s); mqi7 = (Phii*B2s*x(8)/LiA2 - sin(x(4))*Q2s*Phii/Ll)*Phil; mq18 = 2*~(6)*(Phil/Ll)-2*(B2s);

- 2*sin(x(4))*Li*Q2s*x(8));

- sin(x(4))*Ll*Q2s*x(8));

Mqh = Cmqii mqi2 0.00 mqi3; mqi2 mqi4 mq15 mqi6; 0.00 mq15 0.00 mq17; mqi3 mqi6 mqi7 mqi81;

Mql = Psi’*Mqh*Psi;

% Mq2 voor de Fg2 mq2i = alpha*(2*B2s*x(8)-2*sin(x(4))*Li*Q2s); mq22 = alpha*(2*B2s*x(8)-sin(x(4))*Li*Q2s); mq23 = Phil/Ll*(2*B2~*~(8)-2*sin(x(4))*Li*Q2~); mq24 = alpha*(2*B2s*x(8)); mq25 = Phii/L1*~2*B2s*x(8)-sin(x(4))*L~*Q2s)-Phii*ule/Ll*~cos(x(4))*Q2s); mq26 = (Phil/Lí)-2*(2*B2~*~(8)-2*sin(x(4))*Li*Q2~);

Programs

mq27 = ule*B2s*Phil/Li^2;

Mqi = imq21 mq22 mq23 O. O0 ; mq22 mq24 mq25 0.00; mq23 mq25 mq26 mq27;

0.00 0.00 mq27 0.001; Mq2 = Psi’*Mqi*Psi;

Fr = Mrp*[x(l);x(3)] t Mrfp’*[x(5);x(7)] - CO;O.ó*Mthal; Ff = Mrfp*[x(l);x(3)] + Mfp’*[x(5);x(7)] - [0.5*Mq1;0.5*Mq21;

% De state space notatie van de vergelijking

M = [Mr Mrf ’ ; Mrf Mf] ;

Mthetp = inv(Mthet)*(- Fr - Fgr t Mrf’*inv(Mf)*( Ff t ~Hii*x(6);H21*x(8)] ... + Fgf) + Cu(l) ;u(2)1) ;

sys(1) = Mthetp(1);

sys(3) = Mthetp(2); sys(2) = x(i);

sys(4) = x(3);

elseif flag == 3, sys(1) = x(i); sys(2) = x(2); sys(3) = x(3); sys(4) = x(4);

41

elseif flag == O, sys = [4;0;4;6;0;0]; xo = c0;0;0;01;

sys = [I; else

end