multi-input multi-output direct adaptive control for a distributed parameter flexible...

Scientia Iranica, Vol. 15, No. 1, pp 65{74c Sharif University of Technology, February 2008

Multi-Input Multi-Output DirectAdaptive Control for a DistributedParameter Flexible Rotating Arm

B. Gharesi Fard1, M. Azadi2 and M. Eghtesad�

In this paper, a Multi-Input Multi-Output (MIMO) Model Reference Adaptive Control (MRAC)scheme for a exible rotating arm is developed. In order to construct a reference model to befollowed by this distributed parameter system, a �nite element method is used to approximatethe behavior of the arm. An input error direct adaptive control algorithm is utilized as thecontrol approach to account for parameter uncertainty. Assuming the same approximation andstructure as the model for the actual system, the stability analysis of the proposed controller willbe straightforward. Simulation results are provided to illustrate the performance of the proposedalgorithm in the presence of disturbance and uncertainties. Also, the proposed algorithm resultsare compared with those of a conventional PD controller.

INTRODUCTION

In many robotics applications where high motionspeed, low energy consumption, and a wide operationrange are required, long lightweight robot manipulatorsare commonly used. The combined e�ect of extendedlength and lightweight increases the structural exibil-ity of the manipulator; therefore, for accurate positiontracking of a robot manipulator, the elasticity e�ect inthe exible structure cannot be neglected.

There is rich literature on the control of ex-ible robots and a number of di�erent control ap-proaches have been suggested, as for example, in-verse dynamics, feedback linearization and inversionbased techniques [1-6], sliding mode, H1 and robustcontrollers [7-13], predictive control [14,15], hybridforce/position control [16,17], singular perturbationand two-time scale control [18-20], neural networkand fuzzy controllers [21-26], active vibration con-trol [26,27], optimal control [28,29], passivity-based

1. Department of Applied Mathematics, Queens University,Kingston, Ontario, Canada.

2. Department of Mechanical Engineering, Shiraz Univer-sity, Shiraz, I.R. Iran.

*. Corresponding Author, Department of Mechanical Engi-neering, Shiraz University, Shiraz, I.R. Iran.

control [30,31], command or input shaping meth-ods [32] and many more contributions, which are notquoted here to save space.

When the parameters of the system are unknown,adaptive controllers can be used. One attractivefeature of the adaptive controllers is that the controlimplementation does not require a priori knowledgeof unknown system parameters. In a robotic sys-tem, some of the parameters, such as payload mass,link exibility or friction coe�cients, are di�cult tocompute or measure; therefore, adaptive controllersrepresent an important step in high-speed/precisionrobotic applications.

Most of the adaptive control schemes proposedfor exible robots are model-independent methods, inwhich fuzzy modeling, neural network approximationand energy-based approaches are used, in combinationwith adaptation mechanisms, to adjust neural networkweights [33], to tune fuzzy controller gains [34], togenerate reference models [35], to tune the internalmodel for unknown disturbances [36], or to suppressthe vibrations [37]. A few direct adaptive controlschemes have been proposed in which the modal fre-quencies [32,38,39] or controller gains [40] are updatedon-line rather than adapting the model parameters.

To study alternative modeling and/or controlschemes, theoretically or experimentally, for robot

66 B. Gharesi Fard, M. Azadi and M. Eghtesad

manipulators with exible behavior, usually, �rst, asimple one-link exible robot arm is considered andthen the results are extended to more complex (multi-link) robots. Also, mostly, this one-link robot arm isconsidered with a rotational joint, a payload and a hub(e.g., see [1,3,8,23]).

The exible link robot is governed by a set ofpartial di�erential equations and the dynamics of thisrobot can be modeled by assuming the exible objectto be a distributed parameter system or by assumingthat it consists of lumped masses and springs (in�niteor �nite dimensional modeling). Usually, �nite elementapproximation and mode summation procedures areused for �nding the �nite number of the mode shapesand natural frequencies of the robot.

The nontrivial extension of Single-Input Single-Output (SISO) adaptive control algorithms leadsto MIMO adaptive control schemes, which maybe achieved by using Matrix-Fraction Descriptions(MFDs). The use of MFDs allows one to develop astate space realization of the system [41]. The multi-port control of a distributed parameter system allowsa more appropriate scheme, which is suitable for theirdistributed nature. Little work has been done in MIMOcontrol of distributed parameter systems, although anindirect MIMO scheme with Auto Regressive (AR)model representation of the distributed parametersystems was developed to dampen the vibration of acantilever beam and a Recursive Least Square (RLS)model was used for the estimation algorithm [42].

In this paper, an input error direct model ref-erence adaptive control scheme with unknown modelparameters is developed for a exible rotating arm.The exible arm is assumed to be �xed in a rigid hubat one end and carrying a payload in its tip on theother end. The reference model structure is obtainedby FEM approximation and by dividing the actualsystem into N elements, where each element has twonodes and each node has two degrees of freedom, i.e., itstransverse displacement and slope. The control schemeis developed from the idea of the MIMO adaptivecontrol scheme, described in [43,44]. The controllerapplies input signals (forces and torques) on nodes.Assuming the same structure for the actual systemand the reference model, the stability analysis of theproposed controller will be the same as that for linearMIMO systems. Simulation results are provided todemonstrate the performance of the proposed algo-rithm in the presence of disturbance and uncertainties.

FLEXIBLE ROTATING ARM (PLANT)MODELING

Consider a exible beam, one end of which is clampedto a control motor shaft and rotated by its rotor. It isassumed that the beam satis�es the Euler-Bernoulli hy-

Figure 1. Flexible rotating arm with payload, mo, hub'smoment, JM of inertia, Jm and length l.

pothesis [45] and the shear deformations are negligible.Also, the tip body of the exible beam is consideredas a concentrated mass, as shown in Figure 1. So, thedynamic equations of motion for the system, assumingno damping e�ects, can be modeled as follows [46]:

@2y(x; t)@t2

+EI�@4y(x; t)@x4 = �x��(t);

0 < x < l; t > 0;

y(0; t) =@y@x

(0; t) = 0;

EI@2y@2x

(l; t) = 0;

m�y(l; t)� EI @3y@x3 (l; t) = �ml��(t);

y(x; 0) = y0(x);

_y(x; 0) = y1(x); (1)

where l, EI, � and m are the length, uniform exuralrigidity, uniform mass density and the mass attached tothe tip, respectively. Also, y = y(x; t) is the transversedisplacement of the beam, with respect to the oatingframe, XY and �(t), shown in Figure 1, is the oatingframe's rotation angle, with respect to the inertialframe, X0Y0 [46].

Finally, the equation of motion of the controlmotor can be written as:as:

Jm��(t) = �(t) + EI@2y@x2 (0; t); (2)

MIMO MRAC Scheme for a Flexible Rotating Arm 67

where Jm and �(t) are, respectively, the rotationalmass moment of inertia and the input torque of themotor.

For simulation purposes, one may use �nite dif-ference or �nite element or any other approximationmethods to obtain the time description of the aboveequations. The �nite element method has been usedfor this, since using FEM approximation is practicallyacceptable and e�cient. One of the most importantproblems with approximation of the partial di�erentialequations of motion of a exible link robot is ignoringhigher frequencies, which can be avoided by dividingthe link into more elements.

CONTROL SCHEME

Model Reference Adaptive Control

A model reference adaptive control can be regardedas an adaptive servo system, in which the desiredperformance is expressed in terms of a referencemodel. It is assumed that the plant parameters areunknown and the controller parameters are updatedrecursively using an identi�er. In a direct adaptivecontrol scheme, which will be used, an identi�cationscheme is designed that directly identi�es the controllerparameters. Internal signals in the system, calledauxiliary signals, construct the system output. Outputerror and input error approaches can be used for errorestimation. In MIMO systems, the same approachis used, although the scheme is much more compli-cated [43,44].

Reference Model Dynamics

A reference model is used to specify the ideal be-havior of the adaptive control system to the externalcommands. To construct the appropriate referencemodel, a �nite element approximation of the governingequations of the system is utilized.

Finite Element Method (FEM) ApproximationFor FEM approximation of the above system, it isassumed that the beam is divided into N elements, eachof which has two nodes at its ends. The weak formu-lation for this equation is obtained using variationalmethods and, then, by inserting boundary conditions,the domain may be discretized. Since two degrees offreedom are assumed in each node, due to transversedisplacement and slope, the Hermite family of inter-polation functions may be used for approximation asfollows:

ye(x) =X

uej :�ej(x); (3)

�e1(x) = 1� 3�xhe

�2

+ 2�xhe

�3

;

�e2(x) = �x:�

1� xhe

�2

;

�e3(x) = 3�xhe

�2

� 2�xhe

�3

;

�e4(x) = �x: �

xhe

�2

� xhe

!; (4)

where he is the length of each element [47,48].Therefore, the equation of motion of each element

can be obtained as follows, by substituting Equation 3in Equation 1, multiplying by a test function and, then,integrating in the domain [47]:

[M ]ef�yge + [K]efyge = fFge; (5)

where [M ]e and [K]e are the mass and sti�ness matricesof each element, respectively as follows:

Me = che420

2664 156 �22h 54 13h�22h 4h2 �13h �3h2

54 �13h 156 22h13h �3h2 22h 4h2

3775 ; (6)

Ke =(EI)eh3e

2664 12 6 �12 66 4 �6 2�12 �6 12 �6

6 2 �6 4

3775 ; (7)

where c and (EI)e are the longitudinal density andsti�ness of each element.

The equations of motion of each element may beassembled into one matrix equation, of the followingform:

[Mm]f��mg+ [Km]f�mg = fFmg; (8)

where:

�m=�y1m �1m y2m �2m � � � yN+1m �N+1m

T2N+2 ;

(9)

in which, yim and �im are the displacement and slopefor each node of the reference model, respectively.

The Newmark method has been used for timesolution approximation [47,48]. The input to thereference model is r(t) as shown in Figure 2.

Model Reference MIMO Adaptive Control fora Flexible Arm

The objective of the present work is to train a controlscheme for the trajectory tracking of a exible rotating


Figure 2. MIMO controller structure-adaptive form.

arm. With respect to the dynamic equations of themotion of the arm and by using FEM approximation,the following vector shows the degrees of freedom ofthe system:

�p =�y1p �1p y2p �2p � � � yN+1p �N+1p

T2N+2 ; (10)

where yip and �ip , respectively, are the displacementand slope for each node of the actual system on whichthe measurement instruments should be mounted(compare �p with the de�nition of Equation 9).

To construct the model reference MIMO adaptivecontrol, the reference model transfer matrix and theHermite normal form should be obtained [41].

The reference model transfer matrix may befound using a modal matrix, f�mg [43], (which isthe eigenvectors square matrix of the arm motion) forEquation 8:

�Tm[Mm]�mf�xmg+ �Tm[Km]�mfxmg = �TmfFmg;�m = �m:x: (11)

The products �TmMm�m and �TmKm�m are diagonalmatrices due to orthogonality and are called general-ized mass and sti�ness matrices, respectively, so themodel transfer matrix in this new coordinate systemcan be presented, using generalized mass and sti�nessmatrices [45].

The Hermite normal form, H, for this model isselected to have the following form [43]:

H =

26641

(s+a)2 0 � � �0

. . ....

... � � � 1(s+a)2

3775 : (12)

Therefore, the reference model can be described by:

M = HM0 2 R(2N+2)�(2N+2)(2N+2);0 (s); (13)

where M0 is a proper stable transfer matrix shown inFigure 2.

Then, the model state space representation willbe:

M =�TmfFmg(s)fxmg(s)

=

266641

mgm11s2+Kgm110 � � �

0. . .

...... � � � 1

mgm2N+2s2+Kgm2N+2

37775 ;(14)

and:

M = ��Tm M 0��1m ; (15)

where mgmii and Kgmii are elements of generalizeddiagonal mass and sti�ness matrices, respectively [43].So, r = M0(r) will be the input to the plant (Figure 2).

Identi�er Structure

Figure 2 shows the block diagram for the input errormodel reference MIMO adaptive control algorithm.W (1)i and W (2)

i are the auxiliary signals generatedby �ltering the plant input and �p (de�ned in Equa-tion 10), respectively. C0, Ci, D0 and Di are (2N+2)�(2N +2) matrices, which include controller parametersand are de�ned in order to make the plant behave


similar to the model; so the matrix of the controllerparameters can be described as follows:

� =

8>>>>>>>>>>>><>>>>>>>>>>>>:

[C0](2N+2)�(2N+2)[C1](2N+2)�(2N+2)

...[Cv�1](2N+2)�(2N+2)[D0](2N+2)�(2N+2)[D1](2N+2)�(2N+2)

...[Dv�1](2N+2)�(2N+2)

9>>>>>>>>>>>>=>>>>>>>>>>>>;(4v(N+1))�(2N+2)

(16)

By de�ning �(s) as a monic, Hurwitz polynomial ofdegree v� 1 (where v is the observability index, whichis the maximum of the row degree indices in the leftfraction of P [41] set equal to 2 for mechanical systemsgoverned by Newton's second law) and de�ning �(s) 2R2N+2

2N+2(s), such that:

�(s) = diag(�(s)); (17)

the auxiliary �ltered signals will be:

W (1)i =

si�1

�(u);

W (2)i =

si�1

�(�p); i = 1 � � � v � 1: (18)

So, the regressor vector is de�ned as follows, andconsists of internal signals:

W =nr W (1)

1 � � � W (1)v�1 �p W (2)

1 � � � W (2)v�1

oT=�

r(2N+2)�1W 2(2v�1)(N+1)�1

�(4v(N+1))�1

:(19)

The input vector to the controller would be [43]:

u = �TW: (20)

Now, L(s) can be de�ned, as follows:

L(s) = diag[l(s)] 2 R2N+22N+2(s); (21)

where l(s) is a monic, Hurwitz polynomial, such thatits rank is equal to the rank of H�1 [43]. Thistransfer function makes the control procedure capableof handling plants with relative degrees higher thanone. So, by de�ning:

V =�

(HL)�1�pL�1(W )

�; V = L�1(W ); (22)

the input error will be:

e2 = �TV � L�1(u): (23)

Letting the controller parameter, �, be updated withtime, the scheme will be made adaptive. Extendingthe update laws from a SISO to a MIMO case, thenormalized gradient algorithm may be used [43] asfollows:

_� = �K1V :eT

1 +K2V TV; K1;K2 > 0: (24)

Stability

To perform a stability analysis for the described sys-tem, �rst it is assumed that the actual system isapproximated with the same FEM procedure, whichis practically acceptable and e�cient. In this case, thereference model and the system have the same structurebut di�erent parameters; then, the stability proof isstraightforward and is similar to the SISO cases (seeLemma 3.6.2, Proposition 6.3.3 and Section 6.3 in [43])in which:

- All states of the adaptive system are bounded func-tions of time;

- The output error 2 L2 and tends to zero as t!1;- -The regressor error 2 L2 and tends to zero as t !1.

SIMULATION RESULTS

Simulation results will be presented for a exible rotat-ing arm with a payload attached to its end tip, to showthe performance of applying an input error MRAC in�ve di�erent cases. The system parameters are givenfor di�erent cases in Table 1. With respect to theseproperties, the exible link has a small bending sti�-ness, (EI = 20). To achieve control purposes, whichare de�ned as following, a rigid link behavior and lessde ection in the arm's tip, a exible model with highbending sti�ness (EI = 200) and di�erent parameters

Table 1. Plant's material properties, length and inertiaparameters.

Plant Parameters

Case 1 2 3 4 5

EI (N/m2) 20 20 20 20 20

� (kg/m3) 2000 2000 2000 2000 2000

l (m) 0.5 0.5 0.5 0.5 0.5

mo (gr) 10 10 10 10 10

Jm (kg.m2) 0.5 0.5 0.5 0.5 0.5EI = sti�ness, � = density, l = length, mo = payload massand Jm = hub's moment of inertia.


Table 2. Reference model's material properties, lengthand inertia parameters.

Reference Model Parameters

Case 1 2 3 4 5

EI (N/m2) 200 200 200 200 200

� (kg/m3) 3000 3000 3000 3000 3000

l (m) 0.5 0.5 0.5 0.5 0.5

mo (gr) 10 30 10 10 10

Jm (kg.m2) 0.5 0.5 0.5 0.5 0.5EI = sti�ness, � = density, l = length, mo = payload massand Jm = hub's moment of inertia.

are chosen (see Table 2). So, de�ning this model shouldlead to less deformation in the tip. The plant andreference model are simulated by an FEM model, inwhich the rotating link is divided into N(10) elementswith the same length and sti�ness. The Newmarkfamily of time integration scheme (Gallerkin method)is used for structural dynamics approximation [47,48].Then, the controller and identi�er structure, which arerepresented in state space form, together with FEMsubfunctions, will be solved in the time domain.

The manner used in Figure 3 to show di�erentnode displacements and slopes is not repeated andassumed known for the other �gures.

The system's response has been simulated for�ve di�erent cases. In all, the number of elements isassumed to be 10 and a constant input, ri(t) = 0:02,i = 1; � � � ; 2N + 2, is applied to all degrees of freedom,except Case 2. The responses of the system to di�erentdisturbances and uncertainties are presented in the �rstfour cases and, in the last one, a comparison betweenthe controller and a simple PD controller is discussed.

1. Figure 4 illustrates a numerical comparison betweenthe reference model and the plant de ections in

Figure 3. The outpu �p = [y; tet]T where yi and tetistand for transverse displacement and slope in node i,respectively.

Figure 4. Reference model and plant's outputs (�m and�p (meter)) are plotted, respectively in 1st and 2nd rows.The left hand �gures show the reference model and plantconvergence signi�cantly after about 30 seconds (Case 1).

local coordinates for constant inputs. These resultsshow that the plant has a signi�cant trackingperformance of the reference model after 30 seconds(less than � rad. rotation). It should be noted thatthe model has a high exural sti�ness, which willlead to less de ections in all its nodes, including itstip. Convergence performance of input and outputerrors is shown in Figures 5 and 6, respectively;

2. In this case, an uncertainty of 20 gr. in thepayload mass is considered. The results are shownin Figure 7. It can be seen that the control systemis able to handle this parameter uncertainty (twiceas much as the plant payload mass) and the plantstill follows the reference model output;

Figure 5. Input error e2 = �TV � L�1(u) vs. time(second) in Case 1.


Figure 6. Output error convergence, �m � �p (meter), vs.time (second) in Case 1.

Figure 7. The uncertainties e�ect on parameterestimation (mo = 10 gr and 30 gr in plant and referencemodel, respectively). �m and �p (meter) vs. time (second).

3. The e�ect of a more complex input, ri(t) = 0:02 +0:02 sin

� 10�:tT

�; i = 1; � � � ; 2N + 2, to the reference

model is shown in Figures 8 and 9, where T isthe period of the motion. The results again showa considerable similarity between the plant andreference model outputs;

4. In this case, a sinusoidal disturbance is consideredin motor torque with an amplitude of about 10percent of the actual input torque. Figures 10and 11 show the e�ect of this disturbance in thesystem. The controller structure demonstrates agreat robustness;

5. In the �nal case, the results of the presented controlscheme have been compared with a simple PD

Figure 8. E�ect of complex input presented in Case 3.1st and 2nd row's �gures show reference model and plantoutputs, respectively. �m and �p (meter) vs. time(second).

Figure 9. Input error e2 = �TV � L�1(u) vs. time(second) for Case 3.

controller [49,50]. The results (Figure 12) showthat the proposed MIMO adaptive controller has amuch better performance. Also, the PD controlleris more sensitive to disturbance and controller gainsand cannot overcome the plant uncertainties e�ects.

The consequence of dividing a exible link intomore elements and having uncertainty in the motorhub inertia are also simulated, but not presented in thispaper. The results show that using more elements willlead to better convergence, although it needs smallertime steps and more computational loads. So, the


Figure 10. The disturbance e�ect on plant's inputs,where 1st and 2nd row's �gures show reference model andplant outputs, respectively. �m and �p (meter) vs. time(second) for Case 4.

Figure 11. Input error e2 = �TV � L�1(u) vs. time(second) for Case 4.

convergence will be more dependent on controller gains,�(s); L(s) and the time step.

CONCLUSION

A multi-input multi-output input error direct adaptivecontrol scheme is presented to make a distributedparameter plant's output track the desired trajectories(reference model's output). The reference model ismade by an FEM approximation of the distributedparameter system. It is desired that the plant behavesas a link with high sti�ness (almost rigid). It appearsthat the proposed controller can handle uncertaintiesin estimated parameters EI, � and mo in the referencemodel. The e�ects of a more complex input to thesystem and a disturbance in the motor torque are

Figure 12. The comparison between MIMO adaptive(up) and PD (bottom) controllers. �p (meter) vs. time(second).

considered, too. The controller shows a signi�cantrobustness in these cases. Dividing a exible link intomore elements leads to more appropriate results, butit will be more sensitive to gains and time steps. Thecomparison between the MIMO adaptive and popularPD controller is also discussed and it can be stated thatthe proposed controller performs very well, especiallyin cases of parameter uncertainty and disturbance.

NOMENCLATURE

l length of the beamEI uniform exural rigidity of the beam� uniform mass density of the beamm the mass attached to the tip of the

beamy = y(x; t) beam transverse displacement with

respect to the oating frame, XY�(t) oating frame's rotation angle with

respect to the inertial frame, X0Y0

Jm rotational mass moment of inertia ofthe motor

�(t) input torque of the motorhe length of each elementf�mg the modal matrix[M ]e mass matrix of each element[K]e sti�ness matrix of each elementc longitudinal density of each element(EI)e sti�ness of each element


yi displacement of the ith node�i slope of the ith noder(t) input to reference modelmgmii an element of generalized diagonal

mass matrixKgmii n element of generalized diagonal

sti�ness matrixW (1)i ;W (2)

i auxiliary signals generated by �lteringplant input and �p

�p vector of measured degrees of freedomof the system

l(s) a monic, Hurwitz polynomial such thatits rank is equal to the rank of H�1

e2 input errorC0; Ci; D0; Di the (2N + 2) � (2N + 2) matrices,

which include controller parameters� vector of controller parameters

P the plant

M reference model transfer matrixv observability index

�(s) a monic, Hurwitz polynomial of degreev � 1

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