Modelling and H2 control of a single-link flexible manipulator

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Modelling and H2 control of a single-link flexible manipulator R P Sutton1, G D Halikias2, A R Plummer1* and D A Wilson2 1School of Mechanical Engineering, The University of Leeds, UK 2School 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 of a single-link flexible manipulator. The manipulator is modelled including gravity terms, and it is verified experimentally that the horizontal and the vertical motions are decoupled for a cylindrically symmetrical link and payload. The mathematical model is used to design a mixed-sensitivity H2 controller. The sensitivity weighting function is used to obtain the required disturbance rejection properties, including zero sensitivity to a force disturbance in the steady state (i.e. integral action control ). The control sensitivity weighting function is chosen to guarantee stability despite variation in the payload mass. The controller is compared experimentally with a proportional-plus-integral controller with velocity feedback and shown to give improved vibration control performance in the vertical plane. Keywords:

flexible manipulator, robust control, H2 control, vibration control

NOTATION A ,A y z

B ,B y z C ,C y z C yi E F (x, t), F (x, t) y z g g n G(s) G (s) i

h h coefficient matrices of state space plant model (the horizontal and the vertical planes respectively) driving matrices of state space plant model (the horizontal and the vertical planes respectively) output matrices of state space plant model (the horizontal and the vertical planes respectively) damping coefficient for mode i in the horizontal plane Young’s modulus of elasticity for the link material forces acting on the link in the horizontal and the vertical planes respectively acceleration due to gravity component of g normal to the longitudinal axis of the link plant model transfer function plant model transfer function (the vertical plane, payload i )

The MS was received on 20 May 1998 and was accepted after revision for publication on 24 December 1998. * Corresponding author: School of Mechanical Engineering, The University of Leeds, Leeds LS2 9JT, UK. I02498 © IMechE 1999

min

I j

b

J ,J hy hz J p J Ty K(s) L L g m M b M p M (x, t), M (x, t) y z Q (t), Q (t) yi zi

H2 norm of the cost function minimum H2 norm of the cost function second moment of area for the link cross-section polar moment of inertia of the link per unit length about the longitudinal axis hub moments of inertia in the horizontal and the vertical planes respectively payload moment of inertia total inertia of hub, link and payload in the horizontal plane, referred to the hub controller transfer function length of the link position of the hub centre of gravity, as a distance from the hub axis mass per unit length of the link link mass payload mass bending moments in the horizontal and the vertical planes respectively temporal descriptions of motion for mode i in the horizontal and the vertical planes respectively Proc Instn Mech Engrs Vol 213 Part I

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R R(s) S(s) S (x), S (x) yi zi t T(s) T g T (t), T (t) y z u ,u y z V (x, t), V (x, t) y z W (s) 1 W (s) 2 W (s) 3 x x y y y Y(x, t) Z(x, t) a D i h flex h g h ,h ,h hub yhub zhub

h ,h ,h tot ytot ztot

h yr w i t v ,v yi zi

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

distance between the hub centre and the root of the flexible link control sensitivity function sensitivity function mode shape i in the horizontal and the vertical planes respectively time complementary sensitivity function motor torque to support the link weight in the vertical plane torque developed by motors for the horizontal and the vertical motion respectively inputs to the state-space plant model in the horizontal and the vertical planes respectively shear forces in the horizontal and the vertical planes respectively sensitivity weighting function control sensitivity weighting function complementary sensitivity weighting function position along the link state vector (horizontal plane) output vector in the state space model (the horizontal plane) flexible displacement in the horizontal plane flexible displacement in the vertical plane link twist additive modelling error angular displacement of the link tip due to structural deformation position of the hub centre of gravity, as an angle above h hub angular displacements of the hub (any plane, the horizontal plane and the vertical plane respectively) total angular displacements of the link tip (any plane, the horizontal plane and the vertical plane respectively) rigid mode angular displacement in the horizontal plane general mode shape torsional moment acting on the link element natural frequency of the ith mode in the horizontal and the vertical planes respectively

Proc Instn Mech Engrs Vol 213 Part I

1 INTRODUCTION Industrial robot manipulators are currently designed with links which are sufficiently stiff for structural deflection to be negligible during normal operation. If this design constraint were not used, so that link flexing was allowed to occur, the following benefits may accrue: 1. Increased payload capacity. The ratio of allowable payload weight to robot weight is greater. 2. Reduced energy consumption. Lighter links require less powerful actuators. 3. Cheaper construction. Lighter robots will require fewer materials and smaller actuators. 4. Faster movements. Lighter links and the acceptance of appreciable link deformation will allow higher accelerations. 5. Longer reach. A more slender construction allows longer reach where there are access and space restrictions. 6. Safer operation. The compliance and low inertia of a flexible manipulator decrease the likelihood that damage will result from physical interaction between the arm and the working environment. However, link flexibility causes significant technical problems. Most importantly, control algorithms are required which compensate for both the vibration and the static deflection which this flexibility will cause. This is the main research issue and is addressed in this paper. Initial research on the modelling of flexible links was undertaken by Book [1]. The first significant experimental control results were obtained by Cannon and Schmitz [2]. Through the 1980s, there was a general increase in the scope and quantity of research conducted on both the modelling and the control of flexible manipulator systems. This was partly motivated by the space programme, with projects such as the development of the lightweight Canada arm for the Space Shuttle [3]. This was accompanied by the general desire to control a wider set of flexible structures, particularly lightweight space structures. A variety of modelling approaches have been applied to flexible manipulators. In a survey of modelling techniques for single-link manipulators the following approaches were identified [4, 5]: (a) the Lagrange equations and modal expansion (assumed modes method ) [6 ], (b) the Lagrange equation and finite element method [7], (c) the Newton–Euler and modal expansion [8], (d ) the Hamilton principle and modal expansion [2] and (e) the frequency domain and singular perturbation techniques [9]. There are additional hardware requirements for flexible manipulators, as measurement of joint angles alone is no longer sufficient to determine end-effector position I02498 © IMechE 1999

MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

and orientation. It is usual that measurements of the structural deformation of the link, in terms of the link tip displacement, are also required. However, a number of solutions which do not use the tip position feedback have been tried and these include strain gauges [10] and accelerometers at the link tip [11]. A wide range of control strategies have been proposed for flexible manipulators. Proportional-plus-integralplus-derivative (PID) controllers have been used by some investigators. However, even though de Luca and Siciliano [12] showed that a simple proportional–derivative controller could asymptotically stabilize robots with flexible links, in most cases the performance is likely to be inadequate. A linear quadratic Gaussian (LQG) approach was used by Cannon and Schmitz [2], and variants have been implemented by a number of researchers. To address the robustness problems associated with the LQG technique, H2 controller designs have been proposed. Lenz et al. [13] developed a mixed-sensitivity H2 controller that showed considerable promise. The majority of the work that has been conducted has concentrated on single payload loading conditions. However, Cashmore [14] used an uncertainty model to take into account some variation in payload conditions, thereby utilizing the robustness of the H2 controller. Recent work on the use of H2 controllers with flexible manipulators has been published by Matsuno and Tanaka [15], Estiko et al. [16 ] and Landau et al. [17]. All these workers highlighted the difficulties associated with choosing the weighting functions for the controller design and this was emphasized by Jovik and Lennartson [18] where the choice of weighting functions was considered in detail. Linked to the choice of weighting function is the level of performance achieved from the controller. Estiko et al. [16 ] proposed the use of a feedforward controller in conjunction with the H2 controller to counter the response overshoot experienced in previous studies [19].

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In the present study, the aim is to investigate simultaneous motion in the horizontal and the vertical planes, including experimental verification of the results. Very little research has included gravitational effects; thus modelling and control in the vertical plane is of particular interest. In addition, the design of controllers which are robust to payload variations is also addressed. The experimental results are used to verify a mathematical model developed using the Newton–Euler method, and to compare an H2 controller with a classical controller, concentrating on the vertical plane. 2 EXPERIMENTAL FLEXIBLE MANIPULATOR 2.1 Overview An experimental single-link two-degree-of-freedom flexible manipulator has been constructed ( Fig. 1). The two degrees of freedom allow the end effector to be moved in the horizontal and vertical planes. Figures 2 and 3 show the main components schematically. The experimental manipulator can be divided into four main subsystems: the flexible link and payload, the motors and amplifiers, the sensor arrangement, and the computer and interfacing hardware. Table 1 contains the component specifications. 2.2 Flexible link and payload The flexible link is a homogeneous cylindrical aluminium rod with constant properties along its length. The motion of the link is not artificially constrained in any plane. The link is symmetrical about its longitudinal axis, and this axis intersects both the horizontal and the vertical drive axes at the hub. The ratio of the diameter to the length of the flexible link is approximately 0.01; this is small enough for the link to be considered slender [20]. This allows for the

Fig. 1 A photograph of the experimental flexible manipulator I02498 © IMechE 1999

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a precision potentiometer (h ). This is combined with flex the second component, the motion of the hub relative to the base of the system, measured by an incremental optical encoder (h ). The potentiometer measuring hub h is part of a mechanical linkage which adds inertia flex and weight to the manipulator and would not be suitable for implementation as part of an industrial manipulator. However, this arrangement gives accurate and reliable results for the experimental set-up.

2.5 Computer and interfacing

Fig. 2 The main components of the experimental flexible manipulator

effects of ‘wrinkling’ of the link to be ignored for practical purposes in the modelling process. Each of the interchangeable payloads is symmetrical about the longitudinal axis of the link, with the centre of gravity positioned coincident with the link tip.

2.3 Motors and amplifiers Two direct drive d.c. torque motors are used, so that backlash and friction are minimized in the drive system. Brushed d.c. motors are used for low-torque ripple. The two motors have identical specifications.

A PC is used as the computing platform for controller implementation, with A/D and encoder expansion cards for interfacing. Figure 5 shows the interfacing between the computer and the manipulator. Controllers are implemented using SimulinkA. A controller can be designed in the Matlab environment, specified in terms of a Simulink block diagram (Fig. 6), automatically converted to C code using the real-time code generator and then used to control the manipulator. This approach allows rapid controller prototyping and easy comparison of real results with those predicted using Simulink’s simulation capability. The continuous controllers designed in this study are converted to discrete time using an approximate numerical integration technique. This conversion is part of the automatic code generation.

3 MODELLING 2.4 Sensor arrangement

3.1 Link element model

The vertical and horizontal displacements of the tip of the link are measured. Each measurement is composed of two parts as shown in Fig. 4. Firstly, the position of the link tip in relation to the hub axis is measured using

The Newton–Euler method for modelling transverse vibrations in beams will be used. The approach broadly follows that of Meirovitch [21], with extensions to accommodate gravitational effects in the vertical plane. A three-

Fig. 3 The flexible manipulator motor, shaft and sensor layout Proc Instn Mech Engrs Vol 213 Part I

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

Interchangeable payloads

Hub Frameless DC torque motors ( Kollmorgen Inland QT-6205)

Incremental rotary encoders (Heidenhain ERO 1325) Potentiometers (Novotechnik P2701A502) AD interface (Advantech PCL-814B/816-DA-1) Encoder interface card (Advantech PCL-833) Personal computer (PC ) computer platform

Length L Mass M b Offset of root from motor axes R Flexural rigidity EI Mass M and moment of inertia about hub p Payload 1 Payload 2 Payload 3 Horizontal inertia J hy Vertical inertia J hz Maximum torque Rotor inertia Electrical time constant Torque constant Ratio of motor torque to amplifier volts Line count Resolution (×4 counting) Linearity Linearity related to link tip position Voltage range A/D (0–10 V differential inputs) D/A (±10 V outputs) Digital outputs Quadrature input channels

1.00 m 0.34 kg 0.06 m 72.2 N m2 0.075 kg, 0.084 kg m2 0.414 kg, 0.465 kg m2 0.753 kg, 0.846 kg m2 0.468 kg m2 0.249 kg m2 33 N m 0.013 kg m2 2.41 ms 1.2 N m/A 3.18 N m/V 3600 4.4×10−4 rad ±0.2% ±3×10−3 rad 0–10 V 16×14 bit 2×16 bit 16 channels 3×24 bit

IntelA Pentium processor

90 MHz

Similarly, in the vertical plane, the link element motion is described by F (x, t)−EI z

q2Z(x, t) q4Z(x, t) =m qx4 qt2

(3)

In this case the weight of the link acts as a distributed force; therefore equation (3) becomes −mg −EI n Fig. 4 The components of the link tip position

dimensional free-body diagram of a small element, of length Dx, of the flexible link is shown in Fig. 7. In Fig. 7, the shear forces are denoted by V and V , y z the bending moments by M and M and the torsional y z moment by t. Neglecting angular inertia and shear deformation of the element, and neglecting second-order effects, the following link element equation for horizontal motion can be derived: F (x, t)−EI y

q4Y(x, t) q2Y(x, t) =m qx4 qt2

(1)

where Y(x, t) is the horizontal displacement of the element in Earth axes. The angular motion of the link can be represented by linear motion of its elements if the angular deviation is small. There is no distributed force in the horizontal plane; therefore equation (1) becomes −EI

q4Y(x, t) q2Y(x, t) =m qx4 qt2

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(2)

q2Z(x, t) q4Z(x, t) =m qx4 qt2

(4)

where g is the gravitational component normal to the n axis of the element. Implicit in equations (2) and (4) is the assumption that the flexural rigidities EI are the same in the horizontal and the vertical planes. Furthermore, it is assumed that the link is cylindrically symmetrical, so that the flexural rigidity remains constant despite link twisting. Twisting can occur owing to the combined horizontal and vertical deflection. This can be appreciated by considering the link element in torsion: Dt(x, t)+V (x, t) Dx z

qY(x, t) qZ(x, t) −V (x, t) Dx y qx qx

= j Dx b K

q2a(x, t) qt2

(5)

q2a(x, t) qY(x, t) qZ(x, t) +V (x, t) −V (x, t) z y qx2 qx qx =j b

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

Fig. 5 Computer to manipulator interfacing

Fig. 6 Simulink block diagram for controller implementation

where a is the link twist, j is the polar moment of inertia b about the longitudinal axis per unit length and K is the torsional stiffness (the product of the modulus of rigidity and the polar moment of area). Thus a combination of Proc Instn Mech Engrs Vol 213 Part I

horizontal and vertical deflections of the link can generate element twisting. An example scenario is horizontal motion of the link which has a static vertical deflection due to gravity. I02498 © IMechE 1999

MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

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

payload gives

3.2 Boundary conditions Four boundary conditions can be derived for each direction of motion (horizontal and vertical ). Considering horizontal motion, there is a geometric boundary condition at the root of the link: =R

qY(x, t) qx

K

(6) x=0 Taking moments about the hub axis gives a second boundary condition at the root of the link: Y(x, t)|

x=0

T (t)−REI y

q3Y(x, t) qx3

K

+EI x=0

q2Y(x, t) qx2

K

x=0 q3Y(x, t) =J hy qt2 qx

K

(7)

x=0 is the hub inertia

where T is the motor torque and J y hy in the horizontal plane. Two boundary conditions are associated with the payload. The centre of gravity of the payload lies on the link axis at x=L. Resolving forces acting on the payload horizontally gives q3Y(x, t) qx3

K

=M p

q2Y(x, t) qt2

K

(8) x=L x=L where M is the mass of the payload. Finally, considering p the moments acting about the centre of gravity of the EI

I02498 © IMechE 1999

−EI

q2Y(x, t) qx2

K

q3Y(x, t) =J p qt2 qx

K

(9)

x=L x=L where J is the moment of inertia of the payload. Note p that the payload is assumed to be cylindrically symmetrical about the longitudinal axis of the link, so that the link twist will not alter J . Thus the payload moments p of inertia are the same for horizontal and vertical motions. The boundary conditions for vertical motion have the same form, except for the inclusion of payload weight, and a motor torque T required to support any out-ofg balance hub weight: Z(x, t)| =R x=0 T (t)−REI z

qZ(x, t) qx

q3Z(x, t) qx3

K

K

(10) x=0

x=0

+EI

q2Z(x, t) qx2

K

−T g

x=0 q3Z(x, t) =J (11) hz qt2 qx x=0 q3Z(x, t) q2Z(x, t) EI −M g =M (12) p n p qx3 qt2 x=L x=L q3Z(x, t) q2Z(x, t) =J (13) −EI p qx2 qt2 qx x=L x=L

K

K

K

K

K

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into equations (6) to (9):

3.3 Mode shapes A set of mode shapes will be determined by considering the free response of the system. These will then be used to develop the solution to the forced vibration problem. In the horizontal plane, the free response is defined by the element equation (2) and the boundary conditions (6) to (9), with the motor torque set to zero in equation (7). A solution to these equations is assumed to have the form

S (x)| =R y x=0

Y(x, t)=S (x)Q (t) (14) y y Substituting this solution into equation (2) allows the variables to be separated:

EI

EI 1 d4S (x) 1 d2Q (t) y =− y (15) Q (t) dt2 m S (x) dx4 y y For the two sides of the equation (15) to be equal for all values of x and t, they must be constant; let this constant be −v2, where v is real. Hence y y d2Q (t) y +v2 Q (t)=0 (16) y y dt2 d4S (x) v2 m y − y S (x)=0 dx4 EI y

(17)

Equation (16) represents a second-order system with zero damping and natural frequency v . Equation (17) y describes the amplitude variation along the link, i.e. the mode shape. The boundary conditions for this differential equation are derived by substituting equation (14)

−REI

dS (x) y dx

d3S (x) y dx3

d3S (x) y dx3 d2S (x) y dx2

K K

K

x=L

x=0

K

(18)

x=0 d2S (x) y +EI dx2

K

x=0 dS (x) =−J v2 y hy y dx

=−M v2 S (x)| p y y x=L =J v2 p y

dS (x) y dx

K

(19) x=0 (20)

K

(21) x=L x=L Solving equation (17) with these boundary conditions gives a set of mode shapes of arbitrary relative amplitude. For the experimental manipulator of Section 2, with payload 2 (M =0.414 kg, see Table 1), the mode p shapes are as shown in Fig. 8. The amplitudes of these 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 torque or gravitational forcing terms gives a completely analogous set of equations. The mode shapes are only different because of a difference between the vertical and the horizontal hub inertias. EI

3.4 Horizontal plane model The general solution of equation (2) with boundary conditions (6) to (9) is a sum of modal terms:

Fig. 8 The mode shapes (i=1, ..., 6) for the horizontal plane Proc Instn Mech Engrs Vol 213 Part I

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MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

2 (22) Y(x, t)= ∑ S (x)Q (t) yi yi i=0 Substituting equation (22) into equation (2) gives

C

D

d2Q (t) d4S (x) 2 yi +EI yi ∑ mS (x) Q (t) =0 yi yi dt2 dx4 i=0

hub angle respectively: y =C x y y y (32) y =[h h ]T y ytot yhub S (L) S (L) S (L) y1 yn y0 0 0, ··· , 0 L+R L+R L+R C = y S (0) S (0) S (0) y0 y1 yn 0 0, ··· , 0 R R R

C

(23)

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

C C

D DP

2 d2Qyi (t) +v2 Q (t) mS (x)=0 ∑ yi yi yi dt2 i=0 2 d2Qyi (t) ∑ +v2 Q (t) yi yi dt2 i=0

L

0

(24)

Substituting in the orthogonality condition (70) and equations (78) to (80) from Appendix 1 gives (26)

J = Ty

Thus the time responses for the individual modes are independent. It is expected that in general, the higher the frequency, the less significant is the mode’s contribution to the overall response. Hence a finite number of the lower frequency modes, i=0 to n, will provide a good approximation. So equation (26) can be used to construct a state space model of finite dimension: x˙ =A x +B u y y y y y where

N−v2 −C 0 0 N 0 y0 0 y0 0 1 N 0 0 −v2 −C A = y1 y1 y N e e e N e 0 0 0 N 0 v 0 0 0 0

C

S (0) y0 B = 0 y R

S (0) y1 0 ··· R

0

··· ···

0 0

0 0

···

0

0

e ···

e 0

e 1

D

(30)

u =T (t) (31) y y and where C are damping coefficients included to repyi resent the low levels of structural and mechanical damping which will be present. The following output equation defines two system outputs: the total angular deflection of the link tip and the I02498 © IMechE 1999

u N N N N N N w

··· −v2 −C yn yn (29)

S (0) T yn 0 R

P

L

0

m(x+R)2 dx+J +M (L+R)2+J hy p p (35)

1 S (L) S (0) y0 = y0 = L+R R √J

(28) 0

(33)

Thus J is the total inertia of the hub, link and payload Ty in the horizontal plane referred to the hub. Note that the mode 0 terms in the B and C matrices can be simpliy y fied, as from equation (34)

(27)

˙ ]T ˙ ... Q Q ˙ Q Q x =[Q Q yn yn y y0 y0 y1 y1 1 0 0 ··· t 0

D

Mode 0 can be interpreted as a rigid mode, i.e. the following solution to equations (17) to (21) obtained with v =0: y0 1 S (x)= (x+R) (34) y0 √J Ty The ‘amplitude’ of the mode is determined by normalizing using the orthogonality condition; thus substituting equation (34) into equation (70) and putting i= j give

mS (x)S (x) dx=0 yi yj (25)

d2Q (t) S (x)| yi +v2 Q (t)=T (t) yi x=0 yi yi y dt2 R

93

(36)

Ty Thus, from the state space model, the rigid mode contribution to the angular deflection, h , can be found to be yr d2h dh 1 yr +C yr = T (t) (37) y0 dt y dt2 J Ty Although the model has been developed using smallangle approximations, it can be seen that these rigid mode dynamics are valid for large movements. Hence the state space model is applicable throughout the manipulator work space.

3.5 Vertical plane model The general solution of equation (4) with boundary conditions (10) to (13) is a sum of modal terms: 2 Z(x, t)= ∑ S (x)Q (t) (38) zi zi i=0 Substituting equation (38) into equation (4) gives

C

D

d2Q (t) d4S (x) 2 zi +EI zi ∑ mS (x) Q (t) =−mg zi n zi dt2 dx4 i=0 (39) Proc Instn Mech Engrs Vol 213 Part I

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C C

D DP P

2 d2Qzi (t) ∑ +v2 Q (t) mS (x)=−mg zi n zi zi dt2 i=0 L

2 d2Qzi (t) ∑ +v2 Q (t) zi zi dt2 i=0

0

=

(40)

mS (x)S (x) dx zi zj

L

−mg S (x) dx n zj

(41)

0 Substituting in the orthogonality condition from Appendix 1 and equations (75) to (77) gives d2Q (t) zi +v2 Q (t) zi zi dt2 S (x)| x=0 =[T (t)−T ] zi z g R −M g S (x)| − p n zi x=L

P

Fig. 9 The plant model frequency response (payload 2, vertical plane)

L

mg S (x) dx (42) n zi 0 The influence of the weight of the link and the payload depends on the relative orientation of the gravity vector. The following approximation will be used:

The input vector is now

C

g =g cos(h ) (43) n ztot The angle h is the angular displacement of the tip, or ztot payload, above the horizontal. This is a good angle to use as in many configurations of the experimental manipulator it is the payload weight which is by far the most significant. Even if this is not the case, the approximation will be good if the flexible displacement is small compared with the total range of manipulator movement. If the centre of gravity of the hub is at an angle h g above the hub angle and displaced L from the hub axis, g then

4 MODEL VERIFICATION

T =M gL cos(h +h ) (44) g h g zhub g Equation (42) can be expressed in the state space form entirely analogously to the horizontal plane model. Only the input terms are different. The B matrix now has extra columns to account for the effect of gravity acting on the payload:

t 0 NS (0) N z0R N N 0 NSz1 (0) B = z N R N e N N 0 NSzn (0) v R

0 S (0) −M gL z0 h g R 0 S (0) −M gL z1 h g R

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Simulation and actual responses can be compared to verify that the models obtained in the previous section are accurate. As the open-loop plant is only marginally stable, a comparison of open-loop responses is im0

−S (L)M g− z0 p

0 −S (L)M g− z1 p

L

P e

0

0 −S (L)M g− zn p

u N mgS (x) dxN z0 0 N N L N mgS (x) dx z1 N 0 N N N L N mgS (x) dx w zn

P

e S (0) −M gL zn h g R

D

T (t) z u = cos(h (46) +h ) z zhub g cos(h ) ztot To illustrate the result of this modelling process, Appendix 2 contains models for the vertical plane motion for each of the three payloads. Note that three modes (n=0, 1, 2) are included in each case. The magnitude of the frequency response of the tip angle h to ztot the motor torque for payload 2 is shown in Fig. 9. The rapid roll-off with frequency indicates that the inclusion of two flexible modes should be sufficient. The other models exhibit similar trends in this respect.

(45)

P

0

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MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

practical. Hence closed-loop control is required, and a proportional-plus-integral-plus-velocity (PIV ) feedback controller is used for this purpose. A comparison between simulated and experimental link tip position responses in the horizontal and vertical planes is shown in Figs 10 to 13. In each case three modes are simulated and payload 2 is used.

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A good correlation between the simulated and experimental responses is seen. It is found that including more modes in the model gives no improvement in the simulated response [23], which further strengthens the argument that a three-mode model will be adequate for controller design. In the horizontal plane, it can be seen 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

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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 response. This is a result of Coulomb friction, which is not included in the model. Coulomb friction also affects vertical plane motion, but because of the more oscillatory nature of the transient response its influence is less pronounced. However, a friction effect can be seen in Fig. 13 between 15 and 20 s, where the simulated steady state control signal is lower than at (say) 5–10 s because the change in angle reduces the gravity related torque, but the experimental control signal has arrested the motion when at a higher value. It is an important observation that Coulomb friction Proc Instn Mech Engrs Vol 213 Part I

is significant despite using direct drive torque motors for actuation in order to minimize friction. The use of lowbacklash geared motors would be a more economical drive solution but would further increase any frictionrelated difficulties. 5 CONTROL 5.1 Gravity compensation In the vertical plane model, the gravity-related terms in equations (45) and (46) will be neglected for the purI02498 © IMechE 1999

MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

poses of linear controller design. Instead, an additional component will be added to the control signal to provide a torque which partially supports the static weight of the manipulator. This component is proportional to the vertical hub angle. The response of the PIV controller with and without this gravity compensation is seen in Fig. 14. The integral action in the PIV controller reduces the steady state error to zero much more rapidly when the weight compensation is present.

5.2 Mixed-sensitivity H2 controller design The PIV controller has been designed, by trial and error, to give the best achievable compromise between the speed of response, the settling time and the steady state error. However, it is apparent that the response, particularly in the vertical plane, is far from acceptable. Thus an H2 controller has also been designed for this manipulator, with the objective of improving the vibration control performance while in addition guaranteeing stability despite the payload variation. The results are presented for the vertical plane. A mixed-sensitivity H2 controller is used. The structure of the controller is shown in Fig. 15. The cost function consists of the weighted sensitivity function, the weighted control sensitivity function and the weighted complementary sensitivity function. These three sensitivity functions are defined as follows: Sensitivity function: S(s)=[1+K(s)G(s)]−1

(47)

97

Control sensitivity function: R(s)=K(s)[1+K(s)G(s)]−1

(48)

Complementary sensitivity function: T(s)=K(s)G(s)[1+K(s)G(s)]−1

(49)

Thus the mixed-sensitivity H2 controller is the controller K(s) which stabilizes the system and minimizes h, where h is the H2 norm of the cost function:

LC

DL

W ( jv)S( jv) 1 (50) W ( jv)R( jv) h= 2 W ( jv)T( jv) 3 2 and where W (s), W (s) and W (s) are weighting func1 2 3 tions. Suitable choices for the weighting functions in this case are 4s+1 W (s)= 1 s2(s+4) W (s)= 2

(51)

5×10−4(s+20) s+1

(52)

2s+1 W (s)= 3 s+700

(53)

The magnitudes of these weighting functions are plotted against frequency in Fig. 16. The sensitivity weighting function W (s) is chosen to be high at low frequencies 1 to give good low-frequency disturbance rejection. In fact by including two poles at s=0 this forces the sensitivity function S(s) to have two zeros at s=0; the plant has one pole at s=0 which appears as a zero in equa-

Fig. 14 A comparison of the tip position response with and without gravity compensation I02498 © IMechE 1999

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

Fig. 15

The controller structure

which leads to plant model G (s), then the additive mod3 elling errors when the other payloads are used are D ( jv)=G ( jv)−G ( jv) (56) 1 1 3 D ( jv)=G ( jv)−G ( jv) (57) 2 2 3 The magnitudes of these additive modelling errors are plotted in Fig. 17, and it is shown that the control sensitivity weighting function has been chosen to over-bound these errors. With the specified weighting functions, and the payload 3 model G (s) shown in Appendix 2, the minimum 3 of the H2 norm [equation (50)] can be calculated to be h =0.96. Hence the inequality (54) is satisfied, which min guarantees robustness to the payload variation for which the system has been designed. Fig. 16 The weighting functions

tion (47); therefore the other s=0 zero will result from the controller. In other words the controller is forced to have integral action. The factorization method of Zhou et al. [22] is used to solve the H2 minimization problem with weighting function poles on the imaginary axis. The complementary sensitivity weighting function W (s) is chosen to penalize the complementary sensi3 tivity more severely at high frequencies. As the complementary sensitivity can be interpreted as the response of the output position to measurement noise, this will tend to attenuate the effect of high-frequency noise. The control sensitivity weighting function W (s) is 2 used as a bound on the additive modelling error. If the minimum H2 norm, denoted h , obeys min h |W ( jv)| 2 |R( jv)|

for all v

Figure 18 presents the step response results with the actual plant being the same as the nominal. The results show that the system has a fast response and a small settling time. The performance of the controller is significantly better than that achieved using the PIV controller.

(55)

Hence the control sensitivity reciprocal will be larger than the additive modelling error at all frequencies; this is sufficient to guarantee stability [22]. For this application there will be a modelling error when the payload is different from that used to form the model. If the nominal payload is payload 3 (see Table 1), Proc Instn Mech Engrs Vol 213 Part I

5.3 Experimental H2 controller results for the vertical plane motion

Fig. 17 The additive modelling error I02498 © IMechE 1999

MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

99

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

The presence of the integral action can be seen in the torque trace shown in Fig. 19. The torque increases to the level necessary to support the arm with zero steady state error at the tip. The results presented in Figs 20 and 21 are the tip displacement and torque plots for the step response of the flexible arm across the range of payloads (1, 2 and 3). Although the H2 controller exhibits stability robustness

Fig. 19 I02498 © IMechE 1999

across this range as expected, there is distinct variation in the achieved level of performance.

5.4 Combined horizontal and vertical motion The motions of the arm in the horizontal and the vertical planes should not interact if the modelling assumptions

The torque for the experimental vertical plane response (H2 controller) Proc Instn Mech Engrs Vol 213 Part I

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

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 the same mixed-sensitivity H2 control approach with integral action that is described in Sections 5.2 and 5.3. Figures 22 and 23 show the step response of the arm in the horizontal and the vertical planes simultaneously. The reference signals for the two planes have been offset to ensure that any coupling effects are easily visible. An individual mixed-sensitivity H2 Proc Instn Mech Engrs Vol 213 Part I

controller with integral action is implemented for each plane. The coupling effect between the two planes of motion is very small, with only minor disturbances being visible in the vertical displacement trace caused by the step change in the horizontal plane. This is thought to be exaggerated at 25 s owing to stiction; the vertical displacement ‘jumps’ from one value to another due to a stick–slip phenomenon. I02498 © IMechE 1999

MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

101

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 As shown in this work, the degree of sophistication required in the controller for a flexible manipulator is dramatically greater than its rigid counterpart. In turn the modelling required to support the controller design is much more involved. Techniques which will successfully control general multiple-degree-of-freedom manipulators with flexible links are not yet available. As part of the development towards a control scheme I02498 © IMechE 1999

for general flexible manipulators, an experimental manipulator has been constructed, with a single link which can be driven in the horizontal and the vertical planes. The effect of gravity, almost always excluded from other studies, is included in the modelling of this system. A close correlation between the response of the model and that of the experimental system is found. Small discrepancies are attributable to friction, despite minimizing friction in the design of the manipulator by using direct drive torque motors for actuation. Proc Instn Mech Engrs Vol 213 Part I

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

A PIV controller, tuned experimentally, will give a stable closed-loop response. However, with this manipulator, the best balance of performance that can be achieved still gives a very oscillatory response in the vertical plane. Hence the motivation to design a modelbased controller. Not only does a mixed-sensitivity H2 controller give the freedom to manipulate the dynamic response for the nominal plant, but also robustness to parameter variation can be explicitly incorporated. In this work the variation in the payload within a known range of masses is considered. The experimental results show the improved performance with the H2 controller and verify that stability is maintained with payload mass variation. However, the performance robustness requires improvement. Integral action is incorporated in the controller; this is particularly important in the vertical plane to ensure that gravity is counteracted but has also been found to be useful for horizontal motion where otherwise friction can introduce some steady state error. Both theoretically and experimentally, it is shown that there is no interaction between horizontal and vertical motion. However, this only holds when both the link and the payload are cylindrically symmetrical. Thus adding an asymmetric payload or additional links to the manipulator may result in appreciable interaction. This is the subject of further work.

ACKNOWLEDGEMENTS The authors would like to thank both the Engineering and Physical Sciences Research Council and British Nuclear Fuels plc for their financial support of this project.

REFERENCES 1 Book, W. J. Modelling, design and control of flexible manipulator arms. PhD thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA, 1974. 2 Cannon Jr, R. H. and Schmitz, E. Initial experiments on the end-point control of a flexible one-link robot. Int. J. Robotics Res., 1984, 3(3), 62–74. 3 Singh, S. N. Control and stabilisation of a non-linear uncertain elastic robotic arm. IEEE Trans., 1988, AES-24, 148–155. 4 Kanoh, H., Tzafestas, S., Lee, H. G. and Kalat, J. Modelling and control of flexible robot arms. In Proceedings of the 25th Conference on Decision and Control, Athens, 1996, pp. 1866–1870 (IEEE, New York). 5 Tokhi, O. and Azad, A. K. M. Modelling of a single link flexible manipulator system: theoretical and practical investigations. Robotica, 1996, 14, 91–102. 6 Fraser, A. R. and Daniel, R. W. Perturbation techniques for flexible manipulators, 1991, ( Kluwer, Deventer). 7 Theodore, R. J. and Ghosal, A. Comparison of the assumed Proc Instn Mech Engrs Vol 213 Part I

8

9

10

11

12

13

14 15

16

17

18

19

20

21 22

23

modes and finite element models for flexible multi-link manipulators. Int. J. Robotics Res., 1995, 14(2), 91–111. Sakawa, Y., Matsuno, F. and Fukushima, S. Modelling and feedback control of a flexible arm. J. Robotic Systems, 1985, 2(4), 453–472. Truckenbrodt, A. Truncation problems in the dynamics and control of flexible mechanical systems. In Proceedings of the Eighth Triennial IFAC Congress, 1982, pp. 1909–1914. Devasia, S., Paden, B. and Bayo, E. Vibration control of an experimental two link manipulator. In Proceedings of the American Control Conference, 1993, pp. 2093–2098. Khorrami, F. and Jain, S. Non-linear control with endpoint acceleration feedback for a two-link flexible manipulator: experimental results. J. Robotic Systems, 1993, 10(4), 505–530. de Luca, A. and Siciliano, B. Regulation of flexible arms under gravity. IEEE Trans. Robotics Automn, 1993, 9(4), 463–467. Lenz, K., Ozbay, H., Tannebaum, A., Turi, J. and Morton, B. Frequency domain analysis and robust control design for an ideal flexible beam. Automatica, 1991, 27, 947–961. Cashmore, R. D. Adaptive and robust control of a flexible manipulator system. PhD thesis, University of Leeds, 1993. Matsuno, F. and Tanaka, M. Robust force and vibration control of a flexible arm without a force sensor. In Proceedings of the 34th IEEE Conference on Decision and Control, 1995, pp. 2835–2840 (IEEE, New York). Estiko, R., Tanaka, H., Moran, A. and Hayase, M. H2 control of flexible arm considering motor dynamics and optimum sensor location. In Proceedings of IPEC, Yokohama, 1995, pp. 1440–1445. Landau, I. D., Langer, J., Rey, D. and Barnier, J. Robust control of 360° flexible arm using the combined pole placement/sensitivity function shaping method. IEEE Trans. Control Systems Technology, 1996, 4(4), 369–383. Jovik, I. and Lennartson, B. On the choice of criteria and weighting functions of an H2/H2 design problem. In Proceedings of the 13th Triennial World Congress, 1996, pp. 373–378. Trautman, C. and Wang, D. Experimental H2 control of a single flexible link with a shoulder joint. In Proceedings of the IEEE International Conference on Robotics and Automation, 1995, pp. 1235–1241 (IEEE, New York). Timoshenko, S., Young, D. H. and Weaver Jr, W. Vibration Problems in Engineering, 4th edition, 1994 (John Wiley, New York). Meirovitch, L. Elements of Vibration Analysis, 1986 (McGraw-Hill, New York). Zhou, K., Doyle, J. C. and Glover, K. Robust and Optimal Control, 1996 (Prentice-Hall, Englewood Cliffs, New Jersey). Sutton, R. P. The control and modelling of a highly flexible single link robot arm. PhD thesis, The University of Leeds, 1997.

APPENDIX 1 Mode orthogonality The orthogonality properties of the free vibration modes are required. Let w (x), or simply w , represent a mode i i I02498 © IMechE 1999

MODELLING AND H2 CONTROL OF A SINGLE-LINK FLEXIBLE MANIPULATOR

shape, either S (x) (the horizontal plane) or S (x) (the yi zi vertical plane). If w and w are two modes with the i j associated frequencies v and v , then commensurate i j with equation (17) EIw+∞ =v2 mw i i i and

(58)

EIw+∞ =v2 mw (59) i j j Multiplying equation (58) by w and equation (59) by j w gives i v2 m (60) w w+∞ = i w w j i EI j i v2 m w w+∞ = j w w i j EI i j

(61)

Integrating equations (60) and (61) by parts gives the expressions v2 m i EI

P

L

0

C

w w dx= w w+ −w∞ w◊ + j i j i j i

P

L

0

D

w◊ w◊ dx j i

L

0 (62)

and v2 m j EI

P

L

0

C

w w dx= w w+ −w∞ w◊ + i j i j i j

P

L

0

D

w◊ w◊ dx i j

L

0 (63)

G

0, i≠ j d = ij 1, i= j

103

(71)

When i= j, the indeterminate condition in equation (69) allows the arbitrary choice in equation (70) of the lefthand side equalling unity. This choice normalizes, i.e. fixes the amplitude, of the modes. To prove separability of the mode time responses in equations (25) and (41), the orthogonality condition can be used, together with additional expressions for the mass and inertia terms. These expressions are derived below. Considering the vertical plane as the more general case, and substituting equation (38) into the boundary conditions (11) to (13), 2 T −T + ∑ [−REIw+ (0)+EIw◊ (0)]Q i i zi z g i=0 2 ¨ = ∑ J w∞ (0)Q (72) hz i zi i=0 2 2 ¨ ∑ EIw+ (L)Q −M g = ∑ M w (L)Q i zi p n p i zi i=0 i=0

(73)

2 2 ¨ ∑ −EIw◊ (L)Q = ∑ J w∞ (L)Q p i zi i zi i=0 i=0

(74)

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

Subtracting equation (62) from equation (63) gives m (v2 −v2 ) i j EI

P

L

w w dx j i

0 =−[w w+ −w∞ w◊ +w∞ w◊ −w w+ ]L (64) i j i j j i j i 0 Restating the boundary conditions (18) to (21), w (0)=Rw∞ (0) (65) i i −REIw+ (0)+EIw◊ (0)=−J v2 w∞ (0) (66) i i h i i EIw+ (L)=−M v2 w (L) (67) i p i i EIw◊ (L)=J v2 w∞ (L) (68) i p i i Substituting in equation (64) for w+ (0) and w∞ (0) using i i equations (66) and (65), and for w+ (L) and w◊ (L) using i i equations (67) and (68), and similarly for w , j L (v2 −v2 )m w w dx i j i j 0 =−(v2 −v2 )[J w∞ (0)w∞ (0)+M w (L)w (L) i j h i j p i j +J w∞ (L)w∞ (L)] (69) p i j from which the orthogonality condition follows:

P

m

P

L

0

w (x)w (x) dx+J w∞ (0)w∞ (0)+M w (L)w (L) i j h i j p i j

+J w∞ (L)w∞ (L)=d (70) p i j ij where d is the Kronecker delta given by ij I02498 © IMechE 1999

2 ¨ +v2 Q )=(T −T )w∞ (0) ∑ J w∞ (0)w∞ (0) (Q hz i j zi zi zi z g j i=0 (75) and substituting equation (67) into equation (73) gives 2 ¨ +v2 Q )=−M g w (L) ∑ M w (L)w (L) (Q p i j zi zi zi p n j i=0 (76) and finally substituting equation (68) into equation (74) gives 2 ¨ +v2 Q )=0 ∑ J w∞ (L)w∞ (L) (Q zi zi zi p i j i=0

(77)

The equations for the horizontal plane equivalent to equations (75) to (77) are of the same form but with the gravity-related terms set to zero: 2 ¨ +v2 Q )=T w∞ (0) ∑ J w∞ (0)w∞ (0) (Q yi yi yi y j hy i j i=0

(78)

2 ¨ +v2 Q )=0 ∑ M w (L)w (L) (Q yi yi yi p i j i=0

(79)

2 ¨ +v2 Q )=0 ∑ J w∞ (L)w∞ (L) (Q p i j yi yi yi i=0

(80)

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APPENDIX 2

Model with payload 3 (nominal plant)

Plant models for the vertical plane Model with payload 1

C C

0

1

0 −3.4 0 0

A = z1 0 0

0 0

0

0

B = z1

C

0

0

0 0

0 1

−2530 −0.9 0 0 0

0

C D C D

0

0

0

0 0

0 0

0 0

0 1

0

1.37 0

−5.47 0

−1.06 0

−1.33 0

5.30 0

−1.47 0

−0.320

1.27

1.11

1.37 0

1.886

0 −1.429 0

C C

0 A = z2 0

0 0

0 0

0 0

B = z2

C

0 0

0 1 −1440 −0.9 0 0

0 0

D

0 −4.27

0 −4.61

0 −1.64

0 6.55

0 −3.40

0 −0.357

0 1.42

0 1.73

D

0 0

0 0

0 0

0 0

0.790 0 −0.401 0 −1.657 0 −0.370 0

Proc Instn Mech Engrs Vol 213 Part I

0

0

0

0

0 0

0 1

0 0

0 0

0 0

0 1

−1240 −0.9 0 0 0

0

0

0

0.908 0

−3.62 0

−7.11 0

−1.75 0

6.97 0

−3.96 0

−0.364

1.45

1.82

0.506

D

−54 800 −0.9

0

D

0 −0.232 0

−1.790 0 −0.366 0

D

Note that the transfer function plant models relating motor torque to link tip position can be obtained thus:

0 1 −56 800 −0.9

0 1.07

1.07 0 C = z2 1.07 0

0

C

−1.305 0 −0.311 0

0 0

0

0.908 0 C = z3 0.908 0

Model with payload 2 0 1 0 −2.1

0 0

B = z3

0

C = z1 1.37 0

A = z3 0 0

−70 700 −0.9

0

1

0 −1.5 0 0

D

G (s)=C∞ (sI−A )−1B∞ 1 z1 z1 z1 G (s)=C∞ (sI−A )−1B∞ 2 z2 z2 z2 G (s)=C∞ (sI−A )−1B∞ 3 z3 z3 z3 where B∞ and C∞ are the first column and first row of zi zi B and C respectively. zi zi

D

I02498 © IMechE 1999