A NOVEL DYNAMIC CONTROL DESIGNSCHEME FOR FLEXIBLE-LINK

MANIPULATORSV. Gavriloiu, V. Yurkevich (*), and K. Khorasani

Department of Electrical and Computer EngineeringConcordia University

Montreal, Quebec H3G 1M8 CANADAContact E-mail: kash@ece.concordia.ca

(*) Visiting Professor at Concordia University from Novosibirsk State Technical UniversityNovosibirsk, Siberia RUSSIA

Abstract— The problem of designing a robust dynamical con-troller for solving the tracking control problem for a flexible-link manipulator is studied. The design methodology is basedon construction of a two-time scale dynamical motion of theclosed-loop system. It is shown that for a sufficiently smallperturbation parameter associated with the dynamical controllerconsisting of high order derivatives of the output signal, a twotime-scale separation of the fast and slow modes are induced inthe closed-loop system. Stability conditions imposed on the fastand slow subsystems can then ensure that the full-order closed-loop system achieves the desired properties, thereby obtainingoutput performance that is insensitive to parameter variationsand external disturbances. Numerical simulations for both a two-link rigid manipulator and a flexible link manipulator are shownto demonstrate the advantages of the proposed design strategy.

I. INTRODUCTION

A large number of methods have been developed andused to solve the problem of controller design for roboticmanipulators in the past three decades. For example, VariableStructure Systems (VSS) theory [6], [10], [3], adaptive controlapproaches [1], learning controllers [2], control laws basedon Nonlinear Inverse Dynamics (NID) method [5], to namea few are among the techniques reported in the literature.The control law based on NID method may generally beused provided that the dynamics of the system is exactlyknown. However, under dynamic and parametric uncertaintiesand incomplete information about external disturbances, robustcontrol approaches are needed. An approach to the solution ofthe NID problem was suggested by [7] through the applicationof higher order derivatives of the output along with a highgain control law. Singular perturbation method was used toanalyze the closed loop system properties in this type ofcontrol systems in [8], and [9].

In the present paper the tracking control problem of aflexible-link manipulator based on the methodology developedin [11] to design a control system under uncertainty is in-vestigated. This methodology is a generalization and furtherdevelopment of the results reported in [7], [8], and [9].

II. PROBLEM STATEMENT

The tip position tracking control problem of a structurallyflexible robot manipulator is challenging due to the nonlinearand internally unstable characteristics of the system. Even fora single-link flexible arm, it is well known that the transferfunction from the torque input to the tip position output isnon–minimum phase [12]. Most of the early experimentalwork in this area have addressed the end-point regulationproblem [28], [29]. For a causal controller, the non-minimumphase property hinders perfect asymptotic tracking of a desired

tip trajectory with a bounded control input. Thus, to achieveperfect tracking using a causal controller, the flexible systemshould be minimum phase. The minimum phase property maybe achieved by output redefinition, as performed in [12], [13],[14] and [25], or by a redefinition of the output into slow andfast outputs in the context of integral manifold theory in [17],[18], and [27]. Inversion techniques using redefined outputshave also been discussed in [21] and [13].

In this paper, we develop a control strategy based on input–output linearization of the flexible–link system. The output re–definition concept is employed to determine points near the tipoutputs such that stable zero–dynamics may be achieved. It isassumed that the vibrations are mainly lateral vibrations aboutthe axis of each link. In other words, for each link it is assumedthat a considerable amount of potential energy is stored in thedirection of bending corresponding to the axis of rotation ofthat link, and that the potential energies due to deflectionsin other directions are negligible. This may be achieved byproper mechanical structure design. A planar manipulator withrectangular cross sections in which the height to thickness ratioof each cross section is large is an example of such a system.

A closed–loop stability analysis is performed and conditionsfor achieving stable closed–loop behavior are stated. Thetheoretical developments are further enhanced by a number ofsimulation studies for a two–link rigid manipulator as well asa single-link flexible manipulator to demonstrate the potentialand capabilities of the proposed robust dynamic controller.In particular, stable closed–loop performance with small tipposition tracking errors is achieved with relatively large controlgains, resulting in reduced closed–loop system sensitivity thatwould otherwise not be achieved by conventional methods.

A. Model of Flexible-Link ManipulatorThe input–state representation of a flexible–link manipulator

is not feedback linearizable [15], although the system islocally input–output linearizable. In order to apply input-output linearization technique to flexible–link manipulators,consider the dynamics of a multi–link flexible manipulator[16], [27], and [26] i.e.,

M(q, δ)[

qδ

]+

[f1(q, q) + g1(q, q, δ, δ)

f2(q, q) + g2(q, q, δ, δ) + Kδ

]=

[u0

]

(1)

where q is the n × 1 vector of joint variables, δ is them × 1 vector of deflection variables, f1, f2, g1, and g2 arethe terms due to gravity, Coriolis, and centripetal forces, M isthe positive–definite inertia matrix, K is the positive–definite

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

TA1.3

0-7803-9354-6/05/$20.00 ©2005 IEEE 595

d

yq

l ie

ii

i

Fig. 1. The output of link i (yi).

stiffness matrix, and u is the n × 1 vector of input torques(clamped mode shapes have been assumed). Let us define

H(q, δ) = M−1(q, δ) =[

H11 H12H21 H22

]

Consequently, (1) can be written in the state–space form

x = f(x) + g(x)u (2)

where xT =[

qT δT qT δT]

and

f(x) =

⎡⎢⎣

qδ

−H11(f1 + g1) − H12(f2 + g2 + Kδ)−H21(f1 + g1) − H22(f2 + g2 + Kδ)

⎤⎥⎦ ,

g(x) =

[O(m+n)×n

H11(q, δ)H21(q, δ)

].

Referring to Figure 1, since the beam deflection is usuallysmall with respect to the link length, we have

yi = qi + αidie/li, i = 1, 2, · · · , n (3)

In (3), αi is a variable which takes values between −1 and+1, with αi = 1, 0,−1 corresponding to the tip, joint angle,and reflected tip output positions, respectively. Defining theoutput vector as

y = q + Ψn×m(α)δ (4)

where Ψ(α) is the matrix depending on modal shape functionsand the vector αT = [α1 · · ·αn] defining the physical outputlocations on the links for ensuring stable zero–dynamics [25].

B. Input-Output Model of the Flexible ManipulatorThe input–output description of (1) with the output de-

scribed by (4) is then obtained by differentiating the vectory with respect to time until the input vector appears, which isgiven by

y = a(α, x) + B(α, q, δ)u (5)

where y = [y1, y2, . . . , yn]T is the output available for mea-surement, u = [u1, u2, . . . , un]T is the control vector (vectorof joint torques in (5)), and

B(α, q, δ) = H11 + Ψn×mH21 (6)

and

a(α, x) = −(H11 + ΨH21)(f1 + g1) (7)− (H12 + ΨH22)(f2 + g2 + Kδ)

Let us specify a finite region around the desired joint anglereference trajectories qr, qr given by

Ωr = x : | q − qr |< κ1, | q − qr |< κ2, (8)

| δ |< κ3, | δ |< κ4

where κi (i = 1, · · · , 4) are some positive constants.Assumption 2.1: Let detB(α, q, δ) = 0 ∀x ∈ Ωr, where

x ∈ Ωr ⊂ R2(n+m) and Ωr is a bounded set of R2(n+m).Assumption 2.2: Let the vector α, the matrices H(0, 0),K,

and E2 be such that the matrix

A(α) =[

O I−P0K −P0E2

](9)

with P0 given by

P0 = [H22 − H21(H11 + ΨH21)−1

(H12 + ΨH22)]|(0,0) (10)

is a Hurwitz matrix.Consequently, the origin of the manipulator is locally asymp-totically stable, and the original nonlinear model (2), (4) islocally minimum phase.

Assumption 2.1 is basically a controllability–like assump-tion [22] and is guaranteed to hold when for instance α = 0. Inthis case, B = H11 is positive-definite and therefore invertible.Thus, in a neighborhood of α = 0, B is guaranteed to beinvertible since it is continuously dependent on α.

C. Output Tracking Control ProblemLet us denote

e(t) = r(t) − y(t) (11)

as the tracking error where y(t) is the manipulator end-effectoroutput and r(t) is the reference input signal. The dynamiccontroller designed is to ensure and guarantee the followingcondition

limt→∞ e(t) = 0 (12)

Moreover, the output transients for y(t) should have a desiredbehavior which does not depend either on the external distur-bances or on the possibly varying parameters of the flexible-link manipulator model (2) and (4).

III. CONTROL PROBLEM REFORMULATION

A. Desired Dynamic EquationsLet us form a reference model for the output transients of

y(t) according to the following vector differential equation

y = F (y, y, r, r) (13)

For example, (13) may have the form of a linear vectorequation

y = −Ad1y − Ad

0y + Bd1 r + Bd

0r (14)

By selecting Adj and Bd

j as diagonal matrices, we may requirethe decoupling of the control channels.

B. Insensitivity ConditionLet us denote

eF = F (y, y, r, r) − y (15)

where eF is the realization error characteristics of the desireddynamics which is assigned by F (y, y, r, r). Accordingly, ifthe condition

eF = 0 (16)

is satisfied, the desired behavior of y(t) with prescribeddynamics of (13) is achieved. The expression (16) correspondsto the insensitivity condition of the output transient perfor-mance with respect to the external disturbances and varyingparameters of the flexible-link manipulator model (2) and(4). In other words, the control design problem (12) may bereformulated as the requirement (16).

596

IV. STRUCTURE OF THE CONTROL LAW

To satisfy the requirement of (16), let us construct thecontrol law according to the following differential equation

µ2v + µD1v + D0v = K1eF , v(0) = v0 (17)

whereu = K0v (18)

and v = [v, v]T . It is ssumed that D1, D0 are diagonal ma-trices, and where µ is a sufficiently small positive parameter,K0 is a nonsingular matrix and K1 = diagk1, . . . , kn.

Consequently, by combining equations (15) and (17), thedynamic control law (17) may be re-written in the form

µ2v + µD1v + D0v

= K1−y − Ad1y − Ad

0y + Bd1 r + Bd

0r (19)

V. MAIN RESULTS

The analysis below for the slow and fast motion dynamicsproperties assumes that the states of the manipulator, namely,q and δ are bounded within an open set. This is also consistentwith the assumption of internally stable dynamics of theflexible manipulator system.

A. Fast-Motion Subsystem

Theorem 5.1: Associated with the closed-loop system (5),(17) as µ → 0, the fast-motion subsystem (FMS) is governedby the following equation

µ2v + µD1v + D0 + K1B(α, q, δ)K0v= K1F (y, y, r, r) − a(α, x) (20)

where it is assumed that y ≈ 0, r ≈ 0 and y ≈ const, r ≈const during the fast-time scale transients of the system.

Proof. The closed-loop input-output governing system equa-tions have the following form

y = a(α, x) + B(α, q, δ)K0v (21)µ2v + µD1v + D0v = K1e

F (22)

From (13), (15) it follows that the closed loop system equa-tions may be rewritten in the form

y = a(α, x) + B(α, q, δ)K0v (23)µ2v + µD1v + D0 + K1B(α, q, δ)K0v

= K1F (y, y, r, r) − a(α, x) (24)

The equations (23), (22) may be re-written in the followingstate space representation form

d

dtη1 = η2,

d

dtη2 = a(α, x) + B(α, q, δ)K0v1, (25)

µd

dtv1 = v2,

µd

dtv2 = −D0 + K1B(α, q, δ)K0v1 − D1v2

+K1F (η2, η1, r, r) − a(α, x) (26)

where η1 = y, η2 = y, v1 = v, and v2 = v. Following thestandard singular perturbation procedure, let us introduce anew fast-time scale t0 = t/µ, where we now have

d

dt0η1 = µη2,

d

dt0η2 = µa(α, x) + B(α, q, δ)K0v1, (27)

d

dt0v1 = v2,

d

dt0v2 = −D0 + K1B(α, q, δ)K0v1 − D1v2 +

+K1F (η2, η1, r, r) − a(α, x) (28)

By setting µ = 0 in the above equations we get

d

dt0η1 = 0,

d

dt0η2 = 0, (29)

d

dt0v1 = v2,

d

dt0v2 = −D0 + K1B(α, q, δ)K0gv1 − D1v2

+K1F (η2, η1, r, r) − a(α, x) (30)

In the original time scale t = µt0, we obtain the followingfast-motion subsystem

ηi = const, i = 1, 2

µd

dtv1 = v2,

µd

dtv2 = −D0 + K1B(α, q, δ)K0v1 − D1v2

+K1F (η2, η1, r, r) − a(α, x) (31)

The above fast-motion subsystem equations may be alterna-tively re-written in the form (20).

Remark 5.1: Asymptotic stability, desired transients behav-ior and desired settling time for v(t) can be achieved by aproper choice of the control law parameters.

Remark 5.2: In order to provide stability of the fast-motionsubsystem (20) the matrix K0 should be nonsingular such thatK0B is positive definite (or e.g. if K0 ≈ B−1.)

B. Slow-Motion SubsystemTheorem 5.2: If µ → 0 and the FMS of (20) is asymptoti-

cally stable then

y = F (y, y, r, r)+K−1

1 D0K−11 D0 + BK0−1a(α, x) − F (32)

represents the slow-motion subsystem.Proof. If the fast-motion subsystem (20) is stable, by takingthe limit µ → 0 in (23) and (24) the slow-motion subsystem(32) may be obtained.

Remark 5.3: If D0 = 0 and ki 1 ∀ i = 1, n then theslow-motion subsystem (32) approaches the form of (13). IfD0 = 0 and ki > 0 ∀ i = 1, n then the slow-motion subsystem(32) is the same as (13).

Remark 5.4: If µ → 0 then from (23), (24) it follows thatthe behavior of y(t) tends to the solution of the reference

597

model, and accordingly the controlled output transients in theclosed loop system have desired performance specificationsafter the fast FMS transients.The overall closed-loop system stability as well as the upperbound on the perturbation parameter of the controller havebeen developed but are not included due to space limitations.

VI. SIMULATION RESULTS

In this section we will provide simulation results on atwo-link rigid manipulator as well as a single-link flexiblemanipulator to demonstrate and illustrate the performanceproperties and capabilities of our proposed dynamical controlstrategy.

A. Rigid Link ManipulatorThe system studied here is a two-link rigid manipulator.

The simulation is conducted under the assumption that theinitial conditions of the manipulator and the controller arematched. For this simulation the following parameters areused: the mass of the links are m1 = m2 = 1, and thelengths of the links are: l1 = l2 = 1. The initial conditionsare q1i = −π/2, q2i = 0 (the manipulator is at rest). Themanipulator is commanded with a step signal to qd1 = −π andqd2 = π/3. The simulation results are presented in Figures2-3. For the sake of comparison, let us consider the samerigid manipulator, but instead the standard computed torquecontrol is now used. The gain parameters are KP = 100 andKD = 14. The simulations results for this case are presentedin Figures 4-5.

B. Flexible Link ManipulatorNext, we present the simulation results using our proposed

control method for a single-link flexible manipulator. In thesesimulations, we consider the following parameters: the massof the link is set to m = 1.356 and the length of the linkis set to l = 1.2. The initial condition is y = 0, and thedesired final position is set to y = 1. In the first set ofsimulations, as shown in Figures 6-7, we use the controllerthat is designed for the flexible manipulator. In the second setof simulations, as shown in Figures 8-9, we have the sameflexible manipulator, but instead the controller is based onthe rigid model and parameters. The above experiment is alsorepeated for the same control strategy but now for the trackingproblem. The desired reference trajectory is considered as asinusoidal input given by y = Asinωt, where A = 1 andω = 0.1. For the scenario when a controller based on theflexible case conditions is used, the corresponding simulationresults are shown in Figure 10.

VII. CONCLUSIONS

The main result of this paper is development of a dy-namic control procedure to analyze both the fast and slowmotions of a controlled system that is parameterized by asufficiently small parameter that is designed for a flexible-link manipulator. It is shown that under certain conditionsthe slow motion subsystem achieves a stable and, accordingly,follows the desired output transients that are specified in theclosed loop system under conditions of incomplete informationabout external disturbances and varying parameters of thesystem. The resulting output feedback controller for the flex-ible manipulator has a simple form consisting of a low ordernonlinear dynamical filter. It can be shown that if a sufficienttimescale separation between fast and slow modes in thecorresponding closed-loop system is kept and stability of theFMS are guaranteed, then SMS equation has the desired form,and consequently after damping out of the fast transients the

end-effector performance specifications become insensitive toparameter variations of the system and external disturbances.Moreover, when µ → 0, the control signal converges to thesolution of the Nonlinear Inverse Dynamics (NID) problem.

Analyzing the simulation results for the rigid link manip-ulator, it can be stated that the proposed method performsbetter that the conventional compute torque. The regulationerrors between the desired position and the achieved positionfor each link is increased from e1 = 0.008 for the link 1 andfrom e2 = 0.02 for the second link for the case of our proposedmethod to e1 = 0.18 for the first link and to e2 = 0.39 forthe second link for the case of the computed torque method.For the one-link flexible manipulator we are able to achieve anerror bound of e = 0.001 using the dynamical control method.Under the assumption of using a controller that is based onrigid model assumptions to the actual flexible system, theclosed loop system becomes unstable. The controlled outputin the latter case is the joint angle q while in the former casethe controlled output is the re-defined output y = q + Ψδ.

REFERENCES

[1] Abdallah, C., et al Survey of robust control for rigid robots. IEEE ControlSystem Magazine, 11, 24-30, 1994.

[2] Park, P.H., T.-Y. Kuc and J.S.Lee (1996) Adaptive learning control ofuncertain robotic systems. Int. J.Control, 65, no.5, 725–744.

[3] Slotine, J.-J. and W. Li (1991). Applied non-linear control. Prentice Hall.[4] Spong, M. and M.Vidyasagar (1989) Robot Dynamics and Control. John

Wiley and Sons.[5] Timofeev, A.V. (1980) Adaptive control system design with prescribed

motions. Leningrad, Energia.[6] Utkin, V.I. (1978). Sliding modes and their application in variable

structure systems (In Russian). Mir Publishers, Moskow.[7] Vostrikov, A.S. (1977). On the synthesis of control units of dynamic

systems. Systems Science, Techn. Univ., Wrocław, 3(2), 195–205.[8] Vostrikov, A.S. and V.D. Yurkevich (1991). Decoupling of multi-channel

non-linear time-varying systems by derivative feedback. J. on SystemsScience, Wroclaw, 17(4), 21–33.

[9] Vostrikov, A. S. and V.D. Yurkevich. (1993). Design of control systems bymeans of localisation method. Preprints of 12-th IFAC World Congress,8, 47-50.

[10] Young, K.-K.D. (1978). Controller design for a manipulator using theoryof variable structure systems. IEEE Trans., vol.SMC-8, 101–109.

[11] Yurkevich, V.D. (1995). Decoupling of uncertain continuous systems:Dynamic Contraction Method. Proc. of the 34-th IEEE Conference onDecision and Control. Evanston, 1, 196-201.

[12] Wang, D. and M. Vidyasagar (1989). Transfer Functions for a SingleFlexible Link, Proc. IEEE International Conference on Robotics andAutomation, Vol. 2, pp. 1042–1047.

[13] De Luca, A. and L. Lanari (1991). Achieving Minimum Phase Behaviorin a One–Link Flexible Arm, Proc. International Symposium on Intelli-gent Robotics, Bangalore, India.

[14] Madhavan, S. K. and S. N. Singh (1991), Inverse Trajectory Controland Zero–Dynamics Sensitivity of an Elastic Manipulator, Int. J. Rob.and Autom., Vol. 6, No. 4.

[15] Wang, D. and M. Vidyasagar (1989), Feedback Linearizability of Multi–Link Manipulators with One Flexible Link, Proc. 28th IEEE Conferenceon Decision and Control, Tampa, Florida.

[16] Book, W. J. (1984), Recursive Lagrangian Dynamics of Flexible Ma-nipulator Arms, Int. J. Rob. Res., Vol. 3, No. 3, pp. 87–101.

[17] Hashtrudi–Zaad, S. and K. Khorasani (1995), Control of NonminimumPhase Singularly Perturbed Systems with Applications to Flexible LinkManipulators , Proceedings of Workshop on Advances in Control andits Applications, eds. H.K. Khalil, J.H. Chow and P.A. Ioannou, LectureNotes in Control and Information Sciences, Springer–Verlag.

[18] Hashtrudi–Zaad, K. and K. Khorasani (1996), An Integral ManifoldApproach to Tracking Control for a Class of Non–minimum Phase LinearSystems by Using Output Feedback, Automatica, Vol. 32, No. 11, pp.1533–1552.

[19] Hirschorn, R. M. (1979), Invertibility of Nonlinear Control Systems,SIAM J. Contr. and Opt., Vol. 17, No. 2, pp. 282–297.

[20] Byrnes, C. I. and A. Isidori (1985), Global Feedback Stabilizationof Nonlinear Systems, Proc. 24’th IEEE Conference on Decision andControl, pp. 1031–1037.

[21] De Luca, A., P. Lucibello and G. Ulivi (1989), Inversion Techniques forTrajectory Control of Flexible Robot Arms, J. Rob. Sys., Vol. 6, No. 4,pp. 325–344.

[22] Slotine, J-J.E. and W. Li (1991), Applied Nonlinear Control, Prentice–Hall, Englewood Cliffs, NJ.

[23] Khalil, H. K. (1992), Nonlinear Systems, Macmillan, New York.

598

0 1 2 3 4 5 6 7

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Angles joints 1&2

t [sec]

q [r

ad]

Joint 1Joint 2

Fig. 2. Dynamic control scheme applied to the two-link rigid manipulator -joint angles

0 1 2 3 4 5 6 7−20

−15

−10

−5

0

5Torques joints 1&2

t [sec]

torq

ue [N

m]

Torque 1Torque 2

Fig. 3. Dynamic control scheme applied to the two-link rigid manipulator -joint torques command

[24] Descusse, J. and C. Moog (1985), Decoupling with Dynamic Compen-sation for Strong Invertibel Affine Nonlinear Systems, Int. J. Contr., Vol.42, No. 6, pp. 1387–1398.

[25] Moallem, M., R. V. Patel and K. Khorasani (1996), ”An InverseDynamics Control Strategy for Tip Position Tracking of Flexible Multi–Link Manipulators,” International Federation of Automatic Control 13thWorld Congress, San Francisco, CA, Vol. A, pp. 85–90.

[26] Khorasani, K. (1992) “Adaptive Control of Flexible Joint Robots”, IEEETransactions on Robotics and Automation, Vol. 9, No. 2, pp. 250-267.

[27] Moallem, M., K. Khorasani and R. V. Patel (1997), ”An IntegralManifold Approach for Tip Position Tracking of Flexible Multi–LinkManipulators,” IEEE Tran. Robotics and Automation, Vol. 13, No. 6, pp.823-837.

[28] Cannon, R. H. and E. Schmitz (1984), Initial Experiments on the End–Point control of a Flexible One–Link Robot, Int. J. Rob. Res., Vol. 3, No.3, pp. 62–75.

[29] Hastings, G. and W. J. Book (1985), Experiments in the Optimal Controlof a Flexible Link Manipulator, American Control Conference, pp. 728–729, Boston.

[30] Isidori, A. (1995), Nonlinear Control Systems, Springer–Verlag, NewYork.

0 1 2 3 4 5 6 7

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Angles joints 1&2

t [sec]

q [r

ad]

Joint 1Joint 2

Fig. 4. Computed torque control scheme applied to the two-link rigidmanipulator - joint angles

0 1 2 3 4 5 6 7−20

−15

−10

−5

0

5Torques 1&2

t [sec]

torq

ue [N

m]

Torque 1Torque 2

Fig. 5. Computed torque control scheme applied to the two-link rigidmanipulator - joint torques command

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5Redefined output

t [sec]

y [r

ad]

Redefined outputReal output

Fig. 6. Dynamic control scheme applied to a flexible manipulator - redefinedoutput

599

0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Torque

t [sec]

torq

ue [N

m]

Torque

Fig. 7. Dynamic control scheme applied to a flexible manipulator - torquecommand

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

9 Redefined output

t [sec]

y [r

ad]

Redefined outputReal outputDesired output

Fig. 8. Dynamic control scheme applied to a flexible manipulator designedbased on the rigid model - redefined output

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−10

−8

−6

−4

−2

0

2

4x 10

30 Torque

t [sec]

torq

ue [N

m]

Torque

Fig. 9. Dynamic control scheme applied to a flexible manipulator designedbased on the rigid model - torque command

0 10 20 30 40 50 60 70 80 90 100−1.5

−1

−0.5

0

0.5

1

1.5Redefined output

t [sec]

y [r

ad]

Redefined outputReal outputDesired output

Fig. 10. Dynamic control scheme applied to a flexible manipulator - redefinedoutput tracking

600