modelling and h2control of a single-link ﬂexible manipulatorgeorge1/[a.14]...

85

Modelling and H2 control of a single-link flexiblemanipulator

R P Sutton1, G D Halikias2, A R Plummer1* and D A Wilson21School of Mechanical Engineering, The University of Leeds, UK2School of Electronic and Electrical Engineering, The University of Leeds, UK

Abstract: The aim of this study is to investigate motion in the horizontal and the vertical planes ofa single-link flexible manipulator. The manipulator is modelled including gravity terms, and it isverified experimentally that the horizontal and the vertical motions are decoupled for a cylindricallysymmetrical link and payload. The mathematical model is used to design a mixed-sensitivity H2controller. The sensitivity weighting function is used to obtain the required disturbance rejectionproperties, including zero sensitivity to a force disturbance in the steady state (i.e. integral actioncontrol ). The control sensitivity weighting function is chosen to guarantee stability despite variationin the payload mass. The controller is compared experimentally with a proportional-plus-integralcontroller with velocity feedback and shown to give improved vibration control performance in thevertical plane.

Keywords: flexible manipulator, robust control, H2 control, vibration control

h H2 norm of the cost functionNOTATIONhmin minimum H2 norm of the cost

Ay, A

zcoefficient matrices of state functionspace plant model (the I second moment of area for thehorizontal and the vertical link cross-sectionplanes respectively) j

bpolar moment of inertia of the

By, B

zdriving matrices of state space link per unit length about theplant model (the horizontal and longitudinal axisthe vertical planes respectively) Jhy, Jhz hub moments of inertia in the

Cy, C

zoutput matrices of state space horizontal and the verticalplant model (the horizontal and planes respectivelythe vertical planes respectively) Jp payload moment of inertia

Cyi

damping coefficient for mode i JTy total inertia of hub, link andin the horizontal plane payload in the horizontal plane,

E Young’s modulus of elasticity referred to the hubfor the link material K(s) controller transfer function

Fy(x, t), F

z(x, t) forces acting on the link in the L length of the link

horizontal and the vertical Lg position of the hub centre ofplanes respectively gravity, as a distance from the

g acceleration due to gravity hub axisgn component of g normal to the m mass per unit length of the link

longitudinal axis of the link Mb link massG(s) plant model transfer function Mp payload massGi(s) plant model transfer function M

y(x, t), M

z(x, t) bending moments in the

(the vertical plane, payload i ) horizontal and the verticalplanes respectively

The MS was received on 20 May 1998 and was accepted after revision Qyi

(t), Qzi

(t) temporal descriptions of motionfor publication on 24 December 1998.

for mode i in the horizontal and* Corresponding author: School of Mechanical Engineering, TheUniversity of Leeds, Leeds LS2 9JT, UK. the vertical planes respectively

I02498 © IMechE 1999 Proc Instn Mech Engrs Vol 213 Part I

86 R P SUTTON, G D HALIKIAS, A R PLUMMER AND D A WILSON

1 INTRODUCTIONR distance between the hub centreand the root of the flexible link

R(s) control sensitivity function Industrial robot manipulators are currently designedS(s) sensitivity function with links which are sufficiently stiff for structuralSyi

(x), Szi

(x) mode shape i in the horizontal deflection to be negligible during normal operation. Ifand the vertical planes respectively this design constraint were not used, so that link flexing

t time was allowed to occur, the following benefits may accrue:T(s) complementary sensitivity

1. Increased payload capacity. The ratio of allowablefunctionpayload weight to robot weight is greater.Tg motor torque to support the link

2. Reduced energy consumption. Lighter links require lessweight in the vertical planepowerful actuators.T

y(t), T

z(t) torque developed by motors for

3. Cheaper construction. Lighter robots will requirethe horizontal and the verticalfewer materials and smaller actuators.motion respectively

4. Faster movements. Lighter links and the acceptanceuy, u

zinputs to the state-space plant

of appreciable link deformation will allow highermodel in the horizontal and theaccelerations.vertical planes respectively

5. Longer reach. A more slender construction allowsVy(x, t), V

z(x, t) shear forces in the horizontal

longer reach where there are access and spaceand the vertical planesrestrictions.respectively

6. Safer operation. The compliance and low inertia of aW1(s) sensitivity weighting functionflexible manipulator decrease the likelihood thatW2(s) control sensitivity weightingdamage will result from physical interaction betweenfunctionthe arm and the working environment.

W3(s) complementary sensitivityHowever, link flexibility causes significant technicalweighting function

problems. Most importantly, control algorithms arex position along the linkrequired which compensate for both the vibration andx

ystate vector (horizontal plane)

the static deflection which this flexibility will cause. Thisyy

output vector in the state spaceis the main research issue and is addressed in this paper.model (the horizontal plane)

Initial research on the modelling of flexible links wasY(x, t) flexible displacement in theundertaken by Book [1]. The first significant experimen-horizontal planetal control results were obtained by Cannon and SchmitzZ(x, t) flexible displacement in the[2]. Through the 1980s, there was a general increase invertical planethe scope and quantity of research conducted on both

a link twist the modelling and the control of flexible manipulatorDi

additive modelling error systems. This was partly motivated by the space pro-hflex angular displacement of the link gramme, with projects such as the development of the

tip due to structural lightweight Canada arm for the Space Shuttle [3]. Thisdeformation was accompanied by the general desire to control a wider

hg position of the hub centre of set of flexible structures, particularly lightweight spacegravity, as an angle above hhub structures.

hhub, hyhub , h

zhub angular displacements of the hub A variety of modelling approaches have been applied(any plane, the horizontal plane to flexible manipulators. In a survey of modelling tech-and the vertical plane niques for single-link manipulators the followingrespectively) approaches were identified [4, 5]:

htot, hytot , hztot total angular displacements of(a) the Lagrange equations and modal expansionthe link tip (any plane, the

(assumed modes method) [6 ],horizontal plane and the vertical(b) the Lagrange equation and finite element methodplane respectively)

[7],hyr rigid mode angular displacement

(c) the Newton–Euler and modal expansion [8],in the horizontal plane(d) the Hamilton principle and modal expansion [2] and

wi

general mode shape(e) the frequency domain and singular perturbation

t torsional moment acting on thetechniques [9 ].link element

vyi

, vzi

natural frequency of the ith There are additional hardware requirements for flex-ible manipulators, as measurement of joint angles alonemode in the horizontal and the

vertical planes respectively is no longer sufficient to determine end-effector position

I02498 © IMechE 1999Proc Instn Mech Engrs Vol 213 Part I

87MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

and orientation. It is usual that measurements of the In the present study, the aim is to investigate simul-taneous motion in the horizontal and the vertical planes,structural deformation of the link, in terms of the link

tip displacement, are also required. However, a number including experimental verification of the results. Verylittle research has included gravitational effects; thusof solutions which do not use the tip position feedback

have been tried and these include strain gauges [10 ] and modelling and control in the vertical plane is of particu-lar interest. In addition, the design of controllers whichaccelerometers at the link tip [11].

A wide range of control strategies have been proposed are robust to payload variations is also addressed. Theexperimental results are used to verify a mathematicalfor flexible manipulators. Proportional-plus-integral-

plus-derivative (PID) controllers have been used by model developed using the Newton–Euler method, andto compare an H2 controller with a classical controller,some investigators. However, even though de Luca and

Siciliano [12 ] showed that a simple proportional–deriva- concentrating on the vertical plane.tive controller could asymptotically stabilize robots withflexible links, in most cases the performance is likely to

2 EXPERIMENTAL FLEXIBLE MANIPULATORbe inadequate.A linear quadratic Gaussian (LQG) approach was

2.1 Overviewused by Cannon and Schmitz [2 ], and variants have beenimplemented by a number of researchers. To address the

An experimental single-link two-degree-of-freedom flex-robustness problems associated with the LQG technique,ible manipulator has been constructed (Fig. 1). The twoH2 controller designs have been proposed. Lenz et al.degrees of freedom allow the end effector to be moved[13] developed a mixed-sensitivity H2 controller thatin the horizontal and vertical planes. Figures 2 and 3showed considerable promise. The majority of the workshow the main components schematically.that has been conducted has concentrated on single pay-

The experimental manipulator can be divided intoload loading conditions. However, Cashmore [14 ] usedfour main subsystems: the flexible link and payload, thean uncertainty model to take into account some vari-motors and amplifiers, the sensor arrangement, and theation in payload conditions, thereby utilizing the robust-computer and interfacing hardware. Table 1 contains theness of the H2 controller.component specifications.Recent work on the use of H2 controllers with flexible

manipulators has been published by Matsuno andTanaka [15 ], Estiko et al. [16 ] and Landau et al. [17]. 2.2 Flexible link and payloadAll these workers highlighted the difficulties associatedwith choosing the weighting functions for the controller The flexible link is a homogeneous cylindrical aluminium

rod with constant properties along its length. The motiondesign and this was emphasized by Jovik and Lennartson[18] where the choice of weighting functions was con- of the link is not artificially constrained in any plane.

The link is symmetrical about its longitudinal axis, andsidered in detail. Linked to the choice of weighting func-tion is the level of performance achieved from the this axis intersects both the horizontal and the vertical

drive axes at the hub.controller. Estiko et al. [16 ] proposed the use of a feed-forward controller in conjunction with the H2 controller The ratio of the diameter to the length of the flexible

link is approximately 0.01; this is small enough for theto counter the response overshoot experienced in pre-vious studies [19 ]. link to be considered slender [20 ]. This allows for the

Fig. 1 A photograph of the experimental flexible manipulator



a precision potentiometer (hflex). This is combined withthe second component, the motion of the hub relativeto the base of the system, measured by an incrementaloptical encoder (hhub). The potentiometer measuringhflex is part of a mechanical linkage which adds inertiaand weight to the manipulator and would not be suitablefor implementation as part of an industrial manipulator.However, this arrangement gives accurate and reliableresults for the experimental set-up.

2.5 Computer and interfacing

A PC is used as the computing platform for controllerimplementation, with A/D and encoder expansion cardsfor interfacing. Figure 5 shows the interfacing betweenFig. 2 The main components of the experimental flexible

manipulator the computer and the manipulator.Controllers are implemented using SimulinkA. A con-

effects of ‘wrinkling’ of the link to be ignored for practi- troller can be designed in the Matlab environment, speci-cal purposes in the modelling process. Each of the inter- fied in terms of a Simulink block diagram (Fig. 6),changeable payloads is symmetrical about the automatically converted to C code using the real-time codelongitudinal axis of the link, with the centre of gravity generator and then used to control the manipulator. Thispositioned coincident with the link tip. approach allows rapid controller prototyping and easy

comparison of real results with those predicted usingSimulink’s simulation capability. The continuous control-

2.3 Motors and amplifiers lers designed in this study are converted to discrete timeusing an approximate numerical integration technique.

Two direct drive d.c. torque motors are used, so that This conversion is part of the automatic code generation.backlash and friction are minimized in the drive system.Brushed d.c. motors are used for low-torque ripple. Thetwo motors have identical specifications.

3 MODELLING

3.1 Link element model2.4 Sensor arrangement

The vertical and horizontal displacements of the tip of The Newton–Euler method for modelling transversevibrations in beams will be used. The approach broadlythe link are measured. Each measurement is composed

of two parts as shown in Fig. 4. Firstly, the position of follows that of Meirovitch [21], with extensions to accom-modate gravitational effects in the vertical plane. A three-the link tip in relation to the hub axis is measured using

Fig. 3 The flexible manipulator motor, shaft and sensor layout



Table 1 Experimental manipulator specifications (A/D, analogue to digital; D/A, digital to analogue)

Flexible link Length L 1.00 mMass Mb 0.34 kgOffset of root from motor axes R 0.06 mFlexural rigidity EI 72.2 N m2

Interchangeable payloads Mass Mp and moment of inertia about hubPayload 1 0.075 kg, 0.084 kg m2Payload 2 0.414 kg, 0.465 kg m2Payload 3 0.753 kg, 0.846 kg m2

Hub Horizontal inertia Jhy 0.468 kg m2Vertical inertia Jhz 0.249 kg m2

Frameless DC torque motors Maximum torque 33 N m( Kollmorgen Inland QT-6205) Rotor inertia 0.013 kg m2

Electrical time constant 2.41 msTorque constant 1.2 N m/ARatio of motor torque to amplifier volts 3.18 N m/V

Incremental rotary encoders Line count 3600(Heidenhain ERO 1325) Resolution (×4 counting) 4.4×10−4 rad

Potentiometers Linearity ±0.2%(Novotechnik P2701A502) Linearity related to link tip position ±3×10−3 rad

Voltage range 0–10 VAD interface A/D (0–10 V differential inputs) 16×14 bit

(Advantech PCL-814B/816-DA-1) D/A (±10 V outputs) 2×16 bitDigital outputs 16 channels

Encoder interface card Quadrature input channels 3×24 bit(Advantech PCL-833)

Personal computer (PC) IntelA Pentium processor 90 MHzcomputer platform

Similarly, in the vertical plane, the link element motionis described by

Fz(x, t)−EI

q4Z(x, t)qx4

=mq2Z(x, t)

qt2(3)

In this case the weight of the link acts as a distributedforce; therefore equation (3) becomes

−mgn−EIq4Z(x, t)

qx4=m

q2Z(x, t)qt2

(4)

Fig. 4 The components of the link tip position where gn is the gravitational component normal to theaxis of the element.

Implicit in equations (2) and (4) is the assumptiondimensional free-body diagram of a small element, oflength Dx, of the flexible link is shown in Fig. 7. that the flexural rigidities EI are the same in the hori-

zontal and the vertical planes. Furthermore, it isIn Fig. 7, the shear forces are denoted by Vy

and Vz,

the bending moments by My

and Mz

and the torsional assumed that the link is cylindrically symmetrical, sothat the flexural rigidity remains constant despite linkmoment by t. Neglecting angular inertia and shear defor-

mation of the element, and neglecting second-order twisting. Twisting can occur owing to the combined hori-zontal and vertical deflection. This can be appreciatedeffects, the following link element equation for hori-

zontal motion can be derived: by considering the link element in torsion:

Fy(x, t)−EI

q4Y(x, t)qx4

=mq2Y(x, t)

qt2(1) Dt(x, t)+V

z(x, t) Dx

qY(x, t)qx

−Vy(x, t) Dx

qZ(x, t)qx

where Y(x, t) is the horizontal displacement of theelement in Earth axes. The angular motion of the link = jb Dx

q2a(x, t)qt2 (5)can be represented by linear motion of its elements if the

angular deviation is small.There is no distributed force in the horizontal plane; K

q2a(x, t)qx2

+Vz(x, t)

qY(x, t)qx

−Vy(x, t)

qZ(x, t)qxtherefore equation (1) becomes

−EIq4Y(x, t)

qx4=m

q2Y(x, t)qt2

(2) = jbq2a(x, t)

qt2



Fig. 5 Computer to manipulator interfacing

Fig. 6 Simulink block diagram for controller implementation

where a is the link twist, jb is the polar moment of inertia horizontal and vertical deflections of the link can gener-ate element twisting. An example scenario is horizontalabout the longitudinal axis per unit length and K is the

torsional stiffness (the product of the modulus of rigidity motion of the link which has a static vertical deflectiondue to gravity.and the polar moment of area). Thus a combination of



Fig. 7 The three-dimensional free-body diagram for a link element

3.2 Boundary conditions payload gives

Four boundary conditions can be derived for each direc-−EI

q2Y(x, t)

qx2 Kx=L

=Jpq3Y(x, t)

qt2 qx Kx=L

(9)tion of motion (horizontal and vertical ). Consideringhorizontal motion, there is a geometric boundary con-

where Jp is the moment of inertia of the payload. Notedition at the root of the link:that the payload is assumed to be cylindrically symmetri-cal about the longitudinal axis of the link, so that theY(x, t) |

x=0=RqY(x, t)

qx Kx=0

(6)link twist will not alter Jp . Thus the payload momentsof inertia are the same for horizontal and vertical

Taking moments about the hub axis gives a second motions.boundary condition at the root of the link: The boundary conditions for vertical motion have the

same form, except for the inclusion of payload weight,Ty(t)−REI

q3Y(x, t)

qx3 Kx=0

+EIq2Y(x, t)

qx2 Kx=0

and a motor torque Tg required to support any out-of-balance hub weight:

=Jhyq3Y(x, t)

qt2 qx Kx=0

(7) Z(x, t) |x=0=R

qZ(x, t)

qx Kx=0

(10)

where Ty

is the motor torque and Jhy is the hub inertiain the horizontal plane. T

z(t)−REI

q3Z(x, t)

qx3 Kx=0

+EIq2Z(x, t)

qx2 Kx=0

−TgTwo boundary conditions are associated with the pay-load. The centre of gravity of the payload lies on the

=Jhzq3Z(x, t)qt2 qx K

x=0(11)link axis at x=L. Resolving forces acting on the payload

horizontally gives

EIq3Z(x, t)

qx3 Kx=L

−Mpgn=Mpq2Z(x, t)

qt2 Kx=L

(12)EIq3Y(x, t)

qx3 Kx=L

=Mpq2Y(x, t)

qt2 Kx=L

(8)

where Mp is the mass of the payload. Finally, considering −EIq2Z(x, t)

qx2 Kx=L

=Jpq3Z(x, t)qt2 qx K

x=L(13)

the moments acting about the centre of gravity of the



3.3 Mode shapes into equations (6) to (9):

A set of mode shapes will be determined by considering Sy(x) |

x=0=RdS

y(x)

dx Kx=0

(18)the free response of the system. These will then be usedto develop the solution to the forced vibration problem.In the horizontal plane, the free response is defined by −REI

d3Sy(x)

dx3 Kx=0

+EId2S

y(x)

dx2 Kx=0the element equation (2) and the boundary conditions

(6) to (9), with the motor torque set to zero in equa-=−Jhyv2y

dSy(x)

dx Kx=0

(19)tion (7). A solution to these equations is assumed tohave the form

EId3S

y(x)

dx3 Kx=L

=−Mpv2ySy(x) |

x=L (20)Y(x, t)=Sy(x)Q

y(t) (14)

Substituting this solution into equation (2) allows theEI

d2Sy(x)

dx2 Kx=L

=Jpv2ydS

y(x)

dx Kx=L

(21)variables to be separated:

Solving equation (17) with these boundary conditions1

Qy(t)

d2Qy(t)

dt2=−

EI

m

1

Sy(x)

d4Sy(x)

dx4(15)

gives a set of mode shapes of arbitrary relative ampli-tude. For the experimental manipulator of Section 2,

For the two sides of the equation (15) to be equal for with payload 2 (Mp=0.414 kg, see Table 1), the modeall values of x and t, they must be constant; let this shapes are as shown in Fig. 8. The amplitudes of theseconstant be −v2

y, where v

yis real. Hence modes have been set by normalization in line with the

orthogonality condition derived in Appendix 1.In the vertical plane, the free response without motor

d2Qy(t)

dt2+v2

yQy(t)=0 (16)

torque or gravitational forcing terms gives a completelyanalogous set of equations. The mode shapes are onlyd4S

y(x)

dx4−

v2ym

EISy(x)=0 (17) different because of a difference between the vertical and

the horizontal hub inertias.

Equation (16) represents a second-order system withzero damping and natural frequency v

y. Equation (17) 3.4 Horizontal plane model

describes the amplitude variation along the link, i.e. themode shape. The boundary conditions for this differen- The general solution of equation (2) with boundary con-

ditions (6) to (9) is a sum of modal terms:tial equation are derived by substituting equation (14)

Fig. 8 The mode shapes (i=1, . .., 6) for the horizontal plane



hub angle respectively:Y(x, t)= ∑

2

i=0Syi

(x)Qyi

(t) (22)yy=C

yxy

yy= [h

ytot hyhub ]T(32)Substituting equation (22) into equation (2) gives

∑2

i=0 CmSyi

(x)d2Q

yi(t)

dt2+EI

d4Syi

(x)dx4

Qyi

(t)D=0

Cy=CS

y0(L)

L+R0

Sy1

(L)

L+R0, ·· · ,

Syn

(L)

L+R0

Sy0

(0)

R0

Sy1

(0)

R0, ·· · ,

Syn

(0)

R0D(23)

and substituting equation (17) into equation (23) gives

(33)∑2

i=0 Cd2Qyi

(t)

dt2+v2

yiQyi

(t)D mSyi

(x)=0 (24)Mode 0 can be interpreted as a rigid mode, i.e. the fol-lowing solution to equations (17) to (21) obtained withvy0=0:∑

2

i=0 Cd2Qyi

(t)

dt2+v2

yiQyi

(t)D P L0

mSyi

(x)Syj

(x) dx=0

(25) Sy0

(x)=1

√JTy(x+R) (34)

Substituting in the orthogonality condition (70) andThe ‘amplitude’ of the mode is determined by normaliz-

equations (78) to (80) from Appendix 1 givesing using the orthogonality condition; thus substitutingequation (34) into equation (70) and putting i= j gived2Q

yi(t)

dt2+v2

yiQyi

(t)=Ty(t)

Syi

(x) |x=0

R(26)

JTy= P L0

m(x+R)2 dx+Jhy+Mp(L+R)2+JpThus the time responses for the individual modes are(35)independent. It is expected that in general, the higher

the frequency, the less significant is the mode’s contri- Thus JTy is the total inertia of the hub, link and payloadbution to the overall response. Hence a finite number of in the horizontal plane referred to the hub. Note thatthe lower frequency modes, i=0 to n, will provide a the mode 0 terms in the B

yand C

ymatrices can be simpli-

good approximation. So equation (26) can be used to fied, as from equation (34)construct a state space model of finite dimension:

Sy0

(L)

L+R=

Sy0

(0)

R=

1

√JTy(36)x

y=A

yxy+B

yuy

(27)

where Thus, from the state space model, the rigid mode contri-bution to the angular deflection, h

yr , can be found to bexy= [Q

y0Qy0

Qy1

Qy1

.. . Qyn

Qyn

]T (28)d2h

yrdt2

+Cy0

dhyr

dt=

1JTy

Ty(t) (37)

Although the model has been developed using small-angle approximations, it can be seen that these rigidA

y=

tNNNNNNv

0 1 0 0 · · · 0 0

−v2y0

−Cy0

0 0 ·· · 0 0

0 0 0 1 · · · 0 0

0 0 −v2y1

−Cy1

·· · 0 0

e e e e e e e0 0 0 0 · · · 0 1

0 0 0 0 · · · −v2yn

−Cyn

uNNNNNNw

mode dynamics are valid for large movements. Hencethe state space model is applicable throughout themanipulator work space.

(29)3.5 Vertical plane model

By=C0

Sy0

(0)R

0Sy1

(0)R

··· 0Syn

(0)R DT The general solution of equation (4) with boundary con-

ditions (10) to (13) is a sum of modal terms:(30)

Z(x, t)= ∑2

i=0Szi

(x)Qzi

(t) (38)uy=T

y(t) (31)

and where Cyi

are damping coefficients included to rep- Substituting equation (38) into equation (4) givesresent the low levels of structural and mechanical damp-ing which will be present. ∑

2

i=0 CmSzi

(x)d2Q

zi(t)

dt2+EI

d4Szi

(x)dx4

Qzi

(t)D=−mgnThe following output equation defines two system out-puts: the total angular deflection of the link tip and the (39)



∑2

i=0 Cd2Qzi

(t)

dt2+v2

ziQzi

(t)D mSzi

(x)=−mgn (40)

∑2

i=0 Cd2Qzi

(t)

dt2+v2

ziQzi

(t)D P L0

mSzi

(x)Szj

(x) dx

= P L0

−mgnSzj

(x) dx (41)

Substituting in the orthogonality condition fromAppendix 1 and equations (75) to (77) gives

d2Qzi

(t)dt2

+v2zi

Qzi

(t)

= [Tz(t)−Tg]

Szi

(x) |x=0

RFig. 9 The plant model frequency response (payload 2,

vertical plane)−MpgnSzi

(x) |x=L− P L

0mgnS

zi(x) dx (42)

The input vector is nowThe influence of the weight of the link and the payloaddepends on the relative orientation of the gravity vector.The following approximation will be used:

uz=C T

z(t)

cos(hzhub+hg)

cos(hztot) D (46)

gn=g cos(hztot) (43)

The angle hztot is the angular displacement of the tip, or To illustrate the result of this modelling process,

payload, above the horizontal. This is a good angle to Appendix 2 contains models for the vertical planeuse as in many configurations of the experimental motion for each of the three payloads. Note that threemanipulator it is the payload weight which is by far the modes (n=0, 1, 2) are included in each case. The magni-most significant. Even if this is not the case, the approxi- tude of the frequency response of the tip angle h

ztot tomation will be good if the flexible displacement is small the motor torque for payload 2 is shown in Fig. 9. Thecompared with the total range of manipulator rapid roll-off with frequency indicates that the inclusionmovement. of two flexible modes should be sufficient. The other

If the centre of gravity of the hub is at an angle hg models exhibit similar trends in this respect.above the hub angle and displaced Lg from the hub axis,then

4 MODEL VERIFICATIONTg=MhgLg cos(h

zhub+hg) (44)

Simulation and actual responses can be compared toEquation (42) can be expressed in the state space formverify that the models obtained in the previous sectionentirely analogously to the horizontal plane model. Onlyare accurate. As the open-loop plant is only marginallythe input terms are different. The B matrix now has extrastable, a comparison of open-loop responses is im-columns to account for the effect of gravity acting on

the payload:

Bz=

tNNNNNNNNNNv

0 0 0

Sz0

(0)

R−MhgLg

Sz0

(0)

R−S

z0(L)Mpg− P L

0mgS

z0(x) dx

0 0 0

Sz1

(0)

R−MhgLg

Sz1

(0)

R−S

z1(L)Mpg− P L

0mgS

z1(x) dx

e e e0 0 0

Szn

(0)R

−MhgLgSzn

(0)R

−Szn

(L)Mpg− P L0

mgSzn

(x) dx

uNNNNNNNNNNw

(45)



practical. Hence closed-loop control is required, and a A good correlation between the simulated and exper-imental responses is seen. It is found that including moreproportional-plus-integral-plus-velocity (PIV ) feedback

controller is used for this purpose. A comparison between modes in the model gives no improvement in the simu-lated response [23], which further strengthens the argu-simulated and experimental link tip position responses in

the horizontal and vertical planes is shown in Figs 10 ment that a three-mode model will be adequate forcontroller design. In the horizontal plane, it can be seento 13. In each case three modes are simulated and pay-

load 2 is used. that the actual link tip comes to rest after the first over-

Fig. 10 A comparison of the real and simulated step responses in the horizontal plane

Fig. 11 A comparison of the real and simulated step response control signals in the horizontal plane



Fig. 12 A comparison of the real and simulated step responses in the vertical plane

Fig. 13 A comparison of the real and simulated step response control signals in the vertical plane

shoot, whereas oscillation continues in the simulated is significant despite using direct drive torque motors foractuation in order to minimize friction. The use of low-response. This is a result of Coulomb friction, which isbacklash geared motors would be a more economicalnot included in the model. Coulomb friction also affectsdrive solution but would further increase any friction-vertical plane motion, but because of the more oscillatoryrelated difficulties.nature of the transient response its influence is less pro-

nounced. However, a friction effect can be seen in Fig. 13between 15 and 20 s, where the simulated steady state 5 CONTROLcontrol signal is lower than at (say) 5–10 s because thechange in angle reduces the gravity related torque, but 5.1 Gravity compensationthe experimental control signal has arrested the motionwhen at a higher value. In the vertical plane model, the gravity-related terms in

equations (45) and (46) will be neglected for the pur-It is an important observation that Coulomb friction



poses of linear controller design. Instead, an additional Control sensitivity function:component will be added to the control signal to provide R(s)=K(s)[1+K(s)G(s)]−1 (48)a torque which partially supports the static weight of the

Complementary sensitivity function:manipulator. This component is proportional to the ver-tical hub angle. The response of the PIV controller with T(s)=K(s)G(s)[1+K(s)G(s)]−1 (49)and without this gravity compensation is seen in Fig. 14.

Thus the mixed-sensitivity H2 controller is the control-The integral action in the PIV controller reduces theler K(s) which stabilizes the system and minimizes h,steady state error to zero much more rapidly when thewhere h is the H2 norm of the cost function:weight compensation is present.

h=LCW1( jv)S( jv)

W2( jv)R( jv)

W3( jv)T( jv)DL2

(50)5.2 Mixed-sensitivity H2 controller design

The PIV controller has been designed, by trial and error,and where W1(s), W2(s) and W3(s) are weighting func-to give the best achievable compromise between thetions. Suitable choices for the weighting functions in thisspeed of response, the settling time and the steady statecase areerror. However, it is apparent that the response, particu-

larly in the vertical plane, is far from acceptable. ThusW1(s)=

4s+1s2(s+4)

(51)an H2 controller has also been designed for this manipu-lator, with the objective of improving the vibration con-trol performance while in addition guaranteeing stability W2(s)=

5×10−4(s+20)s+1

(52)despite the payload variation. The results are presentedfor the vertical plane.

W3(s)=2s+1

s+700(53)A mixed-sensitivity H2 controller is used. The struc-

ture of the controller is shown in Fig. 15. The cost func-tion consists of the weighted sensitivity function, the The magnitudes of these weighting functions are plotted

against frequency in Fig. 16. The sensitivity weightingweighted control sensitivity function and the weightedcomplementary sensitivity function. These three sensi- function W1(s) is chosen to be high at low frequencies

to give good low-frequency disturbance rejection. In facttivity functions are defined as follows:by including two poles at s=0 this forces the sensitivity

Sensitivity function:function S(s) to have two zeros at s=0; the plant hasone pole at s=0 which appears as a zero in equa-S(s)= [1+K(s)G(s)]−1 (47)

Fig. 14 A comparison of the tip position response with and without gravity compensation



Fig. 15 The controller structure

which leads to plant model G3(s), then the additive mod-elling errors when the other payloads are used are

D1( jv)=G1( jv)−G3( jv) (56)

D2( jv)=G2( jv)−G3( jv) (57)

The magnitudes of these additive modelling errors areplotted in Fig. 17, and it is shown that the control sensi-tivity weighting function has been chosen to over-boundthese errors.

With the specified weighting functions, and the pay-load 3 model G3(s) shown in Appendix 2, the minimumof the H2 norm [equation (50)] can be calculated to behmin=0.96. Hence the inequality (54) is satisfied, whichguarantees robustness to the payload variation for whichthe system has been designed.

Fig. 16 The weighting functions5.3 Experimental H2 controller results for the vertical

plane motiontion (47); therefore the other s=0 zero will result fromthe controller. In other words the controller is forced to Figure 18 presents the step response results with thehave integral action. The factorization method of Zhou actual plant being the same as the nominal. The resultset al. [22] is used to solve the H2 minimization problem show that the system has a fast response and a smallwith weighting function poles on the imaginary axis. settling time. The performance of the controller is sig-

The complementary sensitivity weighting function nificantly better than that achieved using the PIVW3(s) is chosen to penalize the complementary sensi- controller.tivity more severely at high frequencies. As the comp-lementary sensitivity can be interpreted as the responseof the output position to measurement noise, this willtend to attenuate the effect of high-frequency noise.

The control sensitivity weighting function W2(s) isused as a bound on the additive modelling error. If theminimum H2 norm, denoted hmin , obeys

hmin<1 (54)

then it follows that

1|R( jv) |

>|W2( jv) | for all v (55)

Hence the control sensitivity reciprocal will be largerthan the additive modelling error at all frequencies; thisis sufficient to guarantee stability [22].

For this application there will be a modelling errorwhen the payload is different from that used to form the

Fig. 17 The additive modelling errormodel. If the nominal payload is payload 3 (see Table 1),



Fig. 18 The experimental vertical plane response (H2 controller)

The presence of the integral action can be seen in the across this range as expected, there is distinct variationin the achieved level of performance.torque trace shown in Fig. 19. The torque increases to

the level necessary to support the arm with zero steadystate error at the tip.

The results presented in Figs 20 and 21 are the tip 5.4 Combined horizontal and vertical motiondisplacement and torque plots for the step response ofthe flexible arm across the range of payloads (1, 2 and 3). The motions of the arm in the horizontal and the vertical

planes should not interact if the modelling assumptionsAlthough the H2 controller exhibits stability robustness

Fig. 19 The torque for the experimental vertical plane response (H2 controller)



Fig. 20 The experimental vertical plane response for payloads 1, 2 and 3 (H2 controller)

Fig. 21 The torque for the vertical plane response for payloads 1, 2 and 3 (H2 controller)

are accurate. The results presented in this section use controller with integral action is implemented for eachplane. The coupling effect between the two planes ofthe same mixed-sensitivity H2 control approach with

integral action that is described in Sections 5.2 and 5.3. motion is very small, with only minor disturbancesbeing visible in the vertical displacement trace causedFigures 22 and 23 show the step response of the

arm in the horizontal and the vertical planes simul- by the step change in the horizontal plane. This isthought to be exaggerated at 25 s owing to stiction;taneously. The reference signals for the two planes

have been offset to ensure that any coupling effects the vertical displacement ‘jumps’ from one value toanother due to a stick–slip phenomenon.are easily visible. An individual mixed-sensitivity H2



Fig. 22 The combined motion in the horizontal and the vertical planes (payload 2)

Fig. 23 The torque for the combined motion in the horizontal and the vertical planes (payload 2)

6 CONCLUSIONS for general flexible manipulators, an experimentalmanipulator has been constructed, with a single linkwhich can be driven in the horizontal and the verticalAs shown in this work, the degree of sophisticationplanes. The effect of gravity, almost always excludedrequired in the controller for a flexible manipulator isfrom other studies, is included in the modelling of thisdramatically greater than its rigid counterpart. In turnsystem. A close correlation between the response of thethe modelling required to support the controller designmodel and that of the experimental system is found.is much more involved. Techniques which will suc-Small discrepancies are attributable to friction, despitecessfully control general multiple-degree-of-freedomminimizing friction in the design of the manipulator bymanipulators with flexible links are not yet available.

As part of the development towards a control scheme using direct drive torque motors for actuation.



modes and finite element models for flexible multi-linkA PIV controller, tuned experimentally, will give amanipulators. Int. J. Robotics Res., 1995, 14(2), 91–111.stable closed-loop response. However, with this manipu-

8 Sakawa, Y., Matsuno, F. and Fukushima, S. Modelling andlator, the best balance of performance that can befeedback control of a flexible arm. J. Robotic Systems, 1985,achieved still gives a very oscillatory response in the ver-2(4), 453–472.tical plane. Hence the motivation to design a model- 9 Truckenbrodt, A. Truncation problems in the dynamics and

based controller. Not only does a mixed-sensitivity H2 control of flexible mechanical systems. In Proceedings ofcontroller give the freedom to manipulate the dynamic the Eighth Triennial IFAC Congress, 1982, pp. 1909–1914.response for the nominal plant, but also robustness to 10 Devasia, S., Paden, B. and Bayo, E. Vibration control ofparameter variation can be explicitly incorporated. In an experimental two link manipulator. In Proceedings of

the American Control Conference, 1993, pp. 2093–2098.this work the variation in the payload within a known11 Khorrami, F. and Jain, S. Non-linear control with end-range of masses is considered. The experimental results

point acceleration feedback for a two-link flexible manipu-show the improved performance with the H2 controllerlator: experimental results. J. Robotic Systems, 1993,and verify that stability is maintained with payload mass10(4), 505–530.variation. However, the performance robustness requires

12 de Luca, A. and Siciliano, B. Regulation of flexible armsimprovement. Integral action is incorporated in the con- under gravity. IEEE Trans. Robotics Automn, 1993, 9(4),troller; this is particularly important in the vertical plane 463–467.to ensure that gravity is counteracted but has also been 13 Lenz, K., Ozbay, H., Tannebaum, A., Turi, J. andfound to be useful for horizontal motion where otherwise Morton, B. Frequency domain analysis and robust control

design for an ideal flexible beam. Automatica, 1991, 27,friction can introduce some steady state error.947–961.Both theoretically and experimentally, it is shown that

14 Cashmore, R. D. Adaptive and robust control of a flexiblethere is no interaction between horizontal and verticalmanipulator system. PhD thesis, University of Leeds, 1993.motion. However, this only holds when both the link

15 Matsuno, F. and Tanaka, M. Robust force and vibrationand the payload are cylindrically symmetrical. Thuscontrol of a flexible arm without a force sensor. Inadding an asymmetric payload or additional links to the Proceedings of the 34th IEEE Conference on Decision and

manipulator may result in appreciable interaction. This Control, 1995, pp. 2835–2840 (IEEE, New York).is the subject of further work. 16 Estiko, R., Tanaka, H., Moran, A. and Hayase, M. H2

control of flexible arm considering motor dynamics andoptimum sensor location. In Proceedings of IPEC,Yokohama, 1995, pp. 1440–1445.ACKNOWLEDGEMENTS

17 Landau, I. D., Langer, J., Rey, D. and Barnier, J. Robustcontrol of 360° flexible arm using the combined pole

The authors would like to thank both the Engineering placement/sensitivity function shaping method. IEEEand Physical Sciences Research Council and British Trans. Control Systems Technology, 1996, 4(4), 369–383.Nuclear Fuels plc for their financial support of this 18 Jovik, I. and Lennartson, B. On the choice of criteria and

weighting functions of an H2/H2 design problem. Inproject.Proceedings of the 13th Triennial World Congress, 1996,pp. 373–378.

19 Trautman, C. and Wang, D. Experimental H2 control of aREFERENCES single flexible link with a shoulder joint. In Proceedingsof the IEEE International Conference on Robotics and

1 Book, W. J. Modelling, design and control of flexible Automation, 1995, pp. 1235–1241 (IEEE, New York).manipulator arms. PhD thesis, Department of Mechanical 20 Timoshenko, S., Young, D. H. and Weaver Jr, W. VibrationEngineering, Massachusetts Institute of Technology, Problems in Engineering, 4th edition, 1994 (John Wiley,Cambridge, Massachusetts, USA, 1974. New York).

2 Cannon Jr, R. H. and Schmitz, E. Initial experiments on 21 Meirovitch, L. Elements of Vibration Analysis, 1986the end-point control of a flexible one-link robot. Int. (McGraw-Hill, New York).J. Robotics Res., 1984, 3(3), 62–74. 22 Zhou, K., Doyle, J. C. and Glover, K. Robust and Optimal

3 Singh, S. N. Control and stabilisation of a non-linear uncer- Control, 1996 (Prentice-Hall, Englewood Cliffs, Newtain elastic robotic arm. IEEE Trans., 1988, AES-24, Jersey).

23 Sutton, R. P. The control and modelling of a highly flexible148–155.single link robot arm. PhD thesis, The University of4 Kanoh, H., Tzafestas, S., Lee, H. G. and Kalat, J.Leeds, 1997.Modelling and control of flexible robot arms. In

Proceedings of the 25th Conference on Decision andControl, Athens, 1996, pp. 1866–1870 (IEEE, New York).

APPENDIX 15 Tokhi, O. and Azad, A. K. M. Modelling of a single linkflexible manipulator system: theoretical and practicalinvestigations. Robotica, 1996, 14, 91–102. Mode orthogonality

6 Fraser, A. R. and Daniel, R. W. Perturbation techniquesThe orthogonality properties of the free vibration modesfor flexible manipulators, 1991, ( Kluwer, Deventer).

7 Theodore, R. J. and Ghosal, A. Comparison of the assumed are required. Let wi(x), or simply w

i, represent a mode



shape, either Syi

(x) (the horizontal plane) or Szi

(x) (thedij=G0, i≠ j

1, i= j(71)vertical plane). If w

iand w

jare two modes with the

associated frequencies vi

and vj, then commensurate

with equation (17) When i= j, the indeterminate condition in equation (69)EIw+∞

i=v2

imw

i(58) allows the arbitrary choice in equation (70) of the left-

hand side equalling unity. This choice normalizes, i.e.andfixes the amplitude, of the modes.

EIw+∞i=v2

jmw

j(59) To prove separability of the mode time responses in

equations (25) and (41), the orthogonality condition canMultiplying equation (58) by wj

and equation (59) bybe used, together with additional expressions for thew

igives

mass and inertia terms. These expressions are derivedbelow. Considering the vertical plane as the more general

wjw+∞i=

v2im

EIwjwi

(60)case, and substituting equation (38) into the boundaryconditions (11) to (13),

wiw+∞j=

v2jm

EIwiwj

(61)

Tz−Tg+ ∑

2

i=0[−REIw+

i(0)+EIw◊

i(0)]Q

ziIntegrating equations (60) and (61) by parts gives theexpressions

= ∑2

i=0Jhzw∞i(0)Q

zi(72)v2

im

EI P L0

wjwidx=Cwjw+i −w∞

jw◊i+ P L

0w◊jw◊i

dxDL0

(62) ∑2

i=0EIw+

i(L)Q

zi−Mpgn= ∑

2

i=0Mpwi(L)Q

zi(73)

and

∑2

i=0−EIw◊

i(L)Q

zi= ∑

2

i=0Jpw∞i(L)Q

zi(74)v2

jm

EI P L0

wiwj

dx=Cwiw+j −w∞iw◊j+ P L

0w◊iw◊j

dxDL0

(63)Then substituting equation (66) into equation (72) gives

Subtracting equation (62) from equation (63) gives

∑2

i=0Jhzw∞i(0)w∞

j(0)(Q

zi+v2

ziQzi

)=(Tz−Tg)w∞

j(0)(v2

i−v2

j)

m

EI P L0

wjwidx

(75)=− [w

iw+j−w∞

iw◊j+w∞

jw◊i−w

jw+i

]L0

(64)and substituting equation (67) into equation (73) gives

Restating the boundary conditions (18) to (21),

wi(0)=Rw∞

i(0) (65) ∑

2

i=0Mpwi(L)w

j(L) (Q

zi+v2

ziQzi

)=−Mpgnwj(L)−REIw+

i(0)+EIw◊

i(0)=−Jhv2i w∞i(0) (66) (76)

EIw+i

(L)=−Mpv2i wi(L) (67)and finally substituting equation (68) into equation (74)

EIw◊i(L)=Jpv2i w∞i(L) (68)

givesSubstituting in equation (64) for w+

i(0) and w∞

i(0) using

equations (66) and (65), and for w+i

(L) and w◊i(L) using

∑2

i=0Jpw∞i(L)w∞

j(L) (Q

zi+v2

ziQzi

)=0 (77)equations (67) and (68), and similarly for wj,

The equations for the horizontal plane equivalent to(v2i−v2

j)m P L

0wiwj

dxequations (75) to (77) are of the same form but withthe gravity-related terms set to zero:=−(v2

i−v2

j)[Jhw∞i(0)w∞

j(0)+Mpwi(L)w

j(L)

+Jpw∞i(L)w∞j(L)] (69)

∑2

i=0Jhyw∞i(0)w∞

j(0)(Q

yi+v2

yiQyi

)=Tyw∞j(0) (78)

from which the orthogonality condition follows:

m P L0

wi(x)w

j(x) dx+Jhw∞i(0)w∞

j(0)+Mpwi(L)w

j(L) ∑

2

i=0Mpwi(L)w

j(L) (Q

yi+v2

yiQyi

)=0 (79)

+Jpw∞i(L)w∞j(L)=d

ij(70)

∑2

i=0Jpw∞i(L)w∞

j(L) (Q

yi+v2

yiQyi

)=0 (80)where d

ijis the Kronecker delta given by



APPENDIX 2 Model with payload 3 (nominal plant)

Plant models for the vertical plane

Model with payload 1

Az3=C0 1 0 0 0 0

0 −1.5 0 0 0 0

0 0 0 1 0 0

0 0 −1240 −0.9 0 0

0 0 0 0 0 1

0 0 0 0 −54 800 −0.9D

Az1=C0 1 0 0 0 0

0 −3.4 0 0 0 0

0 0 0 1 0 0

0 0 −2530 −0.9 0 0

0 0 0 0 0 1

0 0 0 0 −70 700 −0.9D

Bz3=C 0 0 0

0.908 −3.62 −7.11

0 0 0

−1.75 6.97 −3.96

0 0 0

−0.364 1.45 1.82D

Bz1=C 0 0 0

1.37 −5.47 −1.06

0 0 0

−1.33 5.30 −1.47

0 0 0

−0.320 1.27 1.11D C

z3=C0.908 0 0.506 0 −0.232 0

0.908 0 −1.790 0 −0.366 0DNote that the transfer function plant models relatingmotor torque to link tip position can be obtained thus:

Cz1=C1.37 0 1.886 0 −1.429 0

1.37 0 −1.305 0 −0.311 0D G1(s)=C∞z1

(sI−Az1

)−1B∞z1

G2(s)=C∞z2

(sI−Az2

)−1B∞z2

G3(s)=C∞z3

(sI−Az3

)−1B∞z3

Model with payload 2

where B∞zi

and C∞zi

are the first column and first row ofBzi

and Czi

respectively.

Az2=C0 1 0 0 0 0

0 −2.1 0 0 0 0

0 0 0 1 0 0

0 0 −1440 −0.9 0 0

0 0 0 0 0 1

0 0 0 0 −56 800 −0.9D

Bz2=C 0 0 0

1.07 −4.27 −4.61

0 0 0

−1.64 6.55 −3.40

0 0 0

−0.357 1.42 1.73D

Cz2=C1.07 0 0.790 0 −0.401 0

1.07 0 −1.657 0 −0.370 0D


modelling and h2control of a single-link ﬂexible manipulatorgeorge1/[a.14]...

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