F:/Publications/IAENG/IAENG Flexible Joint · WCECS 2011. III. MTHEMATICALA MODEL We consider a...

Sliding Mode Control for Flexible Joint usingUncertainty and Disturbance EstimationPramod D. Shendge*,Member, IAENG,Prasheel V. Suryawanshi**,Member, IAENG

Index Terms—sliding mode control, uncertainty and distur-bance estimation, flexible joint.

Abstract—This paper proposes sliding mode control basedon uncertainty and disturbance estimator (UDE), for trajectorytracking control of flexible joint robotic system. UDE is usedto estimate plant uncertainty and disturbances. The controllerdoes not requires knowledge of plant uncertainty and externaldisturbance. Reaching phase is eliminated for robustification.The perturbation is efficiently compensated by feedback of theestimated value. The proposed reference model is to track theplant states according to this model. The closed loop stability forthis model with uncertainty and disturbance is also proposed.

Index Terms—sliding mode control, uncertainty and distur-bance estimation, flexible joint.

I. I NTRODUCTION

The problem of joint flexibility has received considerableattention as the major source of compliance in most presentday manipulator designs. This joint flexibility typically arisesdue to gear elasticity, shaft windup, etc., and is important inthe derivation of control law. Perhaps it is more critical to ac-count for the joint flexibility when dealing with force controlproblems, than it is for pure position control. Joint flexibilitymust be taken into account in both modeling and control inorder to achieve better tracking performance, for practicalapplications. Unwanted oscillations due to joint flexibility,imposes bandwidth limitations on all algorithm designs;based on rigid robots and may create stability problemsfor feedback controls that neglect joint flexibility. Controlof flexible joint has been an important research topic andreceived considerable attention after 1990. The importanceof joint flexibility in the modeling, control, and performanceevaluation of robot manipulators has been established byseveral researchers. Spong used a singular perturbation modelof the elastic joint manipulator dynamics and showed forcecontrol techniques developed for rigid manipulators can beextended to the flexible joint case [1]. A completely linearalgorithm is proposed for composite robust control of flexiblejoint robots. Moreover, the robust stability of the closed loopsystem in presence of structured and unstructured uncertain-ties is analyzed. To introduce the idea, flexible joint robotwith structured and unstructured uncertainties is modeled andconverted into singular perturbation form [2].

In literature, a number of feedback control schemes havebeen proposed to address the issue of joint flexibility. Asliding mode control based strategy [3] is proposed thatneeds knowledge of the bounds of uncertainty and also thecomplete state vector for its implementation. A dynamic

Manuscript received July 4, 2011; revised August 12, 2011* Pramod D. Shendge is with the Department of Instrumentation &

Control, College of Engineering Pune, India. e-mail: [email protected]** Prasheel V. Suryawanshi is with Maharashtra Academy of Engineering,

Alandi (D), Pune, India. e-mail: [email protected]

feedback controller for trajectory tracking control problemof robotic manipulators with flexible joints is proposed in[4]. The design requires position measurements on the linkas well as the motor side and the velocities required in thecontroller are estimated through a reduced order observer.Further, robustness of the closed loop system is establishedby assuming that the uncertainties satisfy certain conditions.A singular perturbation approach is employed for the sametask [5], wherein the controller needs measurements ofposition and elastic force. A nonlinear sliding mode stateobserver is used for estimating the link velocities and elasticforce time derivatives. A Feedback Linearization (FL) basedcontrol law made implementable using extended state ob-server (ESO) is proposed for the trajectory tracking control ofa flexible joint robotic system in [6]. Controller design basedon the integral manifold formulation [7], adaptive control[8], adaptive sliding mode [9] and back-stepping approach[10] are some other approaches reported in the literature.Most of the schemes that appeared in literature have certainissues that require attention. Firstly many of them requiremeasurements of all state variables or at least the positionvariables on link and motor side. Next robustness whereverguaranteed, is often highly model dependent. Also some needknowledge of certain characteristics of the uncertainties, suchas its bounds. A variable structure observer that requires onlymeasurement of link positions to estimate the full state of aflexible joint manipulator is proposed in [11]. Additionallya reduced adaptive observer that requires the measurementof link and motor positions is reported in [12] and a MIMOdesign for the strongly coupled joints in [13].

The design of robust, model following, sliding mode, loadfrequency controller for single area power system based onuncertainty and disturbance estimator (UDE) is discussedin [14]. The literature on UDE also mentions control ofuncertain LTI systems [15], model following sliding modecontrol [16], Ackermann’s formula for reaching phase elimi-nation [17], robust model following based on UDE [18]. Thecontrol proposed does not require the knowledge of boundsof uncertainty and disturbance and is continuous.

In this paper, SMC is proposed to control flexible jointmanipulator with uncertainty and disturbance. A nonlineardisturbance is considerd here and reaching phase is elimi-nated for robustification. The plant model is controlled tofollow the desired states and the uncertainty and disturbanceis estimated with UDE.

The paper is organized as follows: Section II describes theproblem statement. The mathematical model is explained inSection III and Section IV explains the stability analysis.A numerical example is explored in Section V. Simulationresults and discussions are presented in Section VI and thepaper concludes in Section VII.

Proceedings of the World Congress on Engineering and Computer Science 2011 Vol I WCECS 2011, October 19-21, 2011, San Francisco, USA

ISBN: 978-988-18210-9-6 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

WCECS 2011

II. PROBLEM FORMULATION

Reviewing the continuous plant defined as,

x(t) = Ax(t) + Bu(t) + d(t) (1)

y = Cx(t)

A = Anc + ∆A B = Bnc + ∆B

Here ‘nc’ denotes the normal part of uncertain continuoustime system.x(t) is n-dimensional plant state vector,u(t)is control input vector,d(t) is external disturbance.∆A and∆B are the uncertainties in the system matrix.

Assumption 1:The uncertainties∆A,∆B and distur-banced(x, t) satisfy matching conditions given by,

∆A = BD ∆B = BE d(x, t) = Bv(x, t) (2)

The system (2) can now be written as,

x(t) = Ax(t) + Bu(t) + Be(t) (3)

y = Cx(t)

where,e(x, t) = Dx + Eu + v(x, t).Reference model that generates desired trajectory as a LTI

system can be defined as ,

xm(t) = Amxm(t) + Bmum(t) (4)

ym(t) = Cmxm(t)

Assumption 2:The choice of model is such that,

A − Am = BL (5)

Bm = BM (6)

Control objective is to find a control input ‘u’ that makesthe states of the plant; asymptotically track the response ofa reference model (4).

Assumption 3:The lumped uncertaintye(x, t) is suchthat,

e 6= 0, for i = 1, 2, . . . , (r − 1) (7)

e = 0, for i = r (8)

where,r is any positive integer.

A. Design of Control

The main objective of this controller is to eliminateuncertainty and disturbance in the system and command adesired tip angle position.

In this section, a model following control is designedwith help of method suggested in [16].

Define a sliding surface [17]

σ = bT x + z (9)

where,

z = −bT Amx − bT bmum z(0) = −bT x(0) (10)

Equation (10) for the auxiliary variablez defined here isdifferent from that given in [17]. By virtue of the choice ofthe initial condition onz, σ = 0 at t = 0. If a controlu canbe designed ensuring sliding, thenσ = 0 implies;

x = Amx + bmum (11)

and hence fulfills the objective of the model following.Using (4), (9) and (10) gives,

σ = bT Ax + bT bu + bT be(x, t) − bT Amx − bT bmum (12)

= bT bLx − bT bMum + bT bu + bT be(x, t)

Let the required control be expressed as,

u = un + ueq (13)

Selecting,

ueq = −Lx + Mum − (bT b)−1kσ (14)

where,k is a positive constant.From (12) and (14) we get,

σ = bT bun + bT be(x, t) − kσ (15)

The lumped uncertaintye(x, t) can be estimated; as givenin [15]. Rewriting this equation,

e(x, t) = (bT b)−1(σ + kσ) − un (16)

It can be seen that lumped uncertaintye(x, t) can becomputed from (16), which cannot be done directly.

Let the estimate of the uncertainty be defined as,

e(x, t) = e(x, t) Gf (s) (17)

Using (16) and (17)

e(x, t) = [(bT b)−1(σ + kσ) − un]Gf (s) (18)

where,Gf (s) is strictly proper order, low pass filter, withunity gain and enough bandwidth. With such a filter,

e(x, t) ∼= e(x, t) (19)

Error in the estimation is,

e(x, t) = e(x, t) − e(x, t) (20)

B. UDE with first order filter

If Gf (s) is proper first order, low pass filter, with unitygain defined as,

Gf (s) =1

τs + 1(21)

where,τ is small positive constant.

With the aboveGf (s) and in view of (16), (18) and (20),

e(x, t) = (1 − Gf (s))[(bT b)−1(σ + kσ) − un] (22)

= τ e(x, t)Gf (s)

The error in estimation varies withτ , enabling design ofun as,

un = −e(x, t) (23)

Combining (23) and (18)

un = −(bT b)−1(σ + k)Gf (s) + Gf (s)un (24)

Solving for un gives,

un =(bT b)−1

τ(σ +

kσ

s) (25)



WCECS 2011

III. M ATHEMATICAL MODEL

We consider a single link manipulator, with revolute jointactuated by DC motor and model the elasticity of the jointas a linear torsional spring with stiffnessK. The equationsof motion for this system as taken from [19] are,

Iq1 + MgL sin(q1) + K(q1 − q2) = 0

Jq2 − K(q1 − q2) = u (26)

where,q1 andq2 are the link and motor angles respectively,I is the link inertia,J being the inertia of motor,K is thespring stiffness,u is the input torque, andM andL are themass and length of link respectively. The tracking problemfor the system of (26) is to find a control which ensuresq⋆1(t)−q1(t) = 0 for given initial states whereq⋆

1(t) is desiredtrajectory forq1(t).

The equations of motion for the Quanser’s Flexible Jointmodule as given in [20] are,

θ + F1θ −Kstiff

Jeqα = F2Vm

θ − F1θ +Kstiff (Jeq + Jarm)

JeqJarmα = −F2Vm (27)

where,

F1

∆=

ηmηgKtKmK2g + BeqRm

JeqRm

and

F2

∆=

ηmηgKtKm

JeqRm

The parameters are :θ is motor load angle,α is linkjoint deflection,ηm is the motor efficiency,ηg is the gearboxefficiency,Kt is the motor torque constant,Km is the backEMF constant,Kg is the gearbox ratio,Beq is the viscousdamping coefficient,Rm is the armature resistance,Jeq isthe gear inertia,Kstiff is the spring stiffness,Jarm is the linkinertia, andVm is the motor control voltage.

Considering the output of the system asy = θ + α, thedynamics (27) in terms ofy andθ is re-written as,

y =Kstiff

JeqF3y −

Kstiff

JeqF3θ (28)

θ =Kstiff

Jeqy −

Kstiff

Jeqθ − F1θ + F2Vm (29)

whereF3

∆=

(1 −

Jeq + Jarm

Jarm

).

Defining the state variables as,x1 = y, x2 = y = x1,x3 = θ, x4 = θ = x3, the dynamics (28)–(29) become,

x1 = x2

x2 =Kstiff

JeqF3(x1 − x3)

x3 = x4

x4 =Kstiff

Jeq(x1 − x3) − F1x4 + F2Vm (30)

The state space form for (30) can be written as,

x = A x + B Vm (31)

where,x = [x1 x2 x3 x4]T

A =

0 1 0 0Kstiff

JeqF3 0 −

Kstiff

JeqF3 0

0 0 0 1Kstiff

JeqF3 0 −

Kstiff

JeqF3 −F1

B =

000F2

For the desired output, the relative outputy can bedifferentiated in proper manner. In order to satisfy the modelfollowing conditions, the above system (31) is converted tophase variable form by using the transformation,

Z = Tx

Then the Eq. (31) can be written as [6],

z = A z + B Vm (32)

where,z = [z1 z2 z3 z4]T

A =

0 1 0 00 0 1 00 0 0 1

0 −KstiffF1

Jarm−

Kstiff (Jeq + Jarm)

JeqJarm−F1

B =

000

Kstiff F2

Jarm

IV. STABILITY ANALYSIS

Using Eq. (17)

e(x, t) = e(x, t) Gf (s) (33)

ChoosingGf (s) from Eq. (21)

e(x, t) = e(x, t)1

τs + 1e(x, t)(τs + 1) = e(x, t)

e(x, t)(τs) + e(x, t) = e(x, t) (34)

Simplifying Eq. (34) and using Eq. (20)

τ ˙e + e = e

Adding and subtractingτ e on LHS and simplifying,

τ ˙e + τ e − τ e = e − e

−τ ˙e = e − τ e

˙e = −1

τe + e (35)

Using Eq. (15), (20) and (23)

σ = −bT b e + bT b e − kσ

= bT b e − kσ (36)

The dynamics of the flexible joint can be represented instate space form using equations (35) and (36) as,

[σ˙e

]=

[−k (bT b)0 − 1

τ

] [σe

]+

[01

]e (37)

This satisfies the separation principle. The eigen values aredecided by constantk and filter constantτ , thus ensuringstability. The appropriate choice ofk and τ ensures slidingvariableσ and lumped uncertaintye tend to zero.



WCECS 2011

V. EXAMPLE

In order to illustrate the proposed control algorithm, asingle flexible link with flexible joints is considered. Sim-ulation is performed to demonstrate the effectiveness, of theproposed control algorithm. Numerical dynamic model ofthis system from [20] is as follows;

x1

x2

x3

x4

=

0 1 0 00 0 1 00 0 0 10 −10007 −837 −28

x1

x2

x3

x4

+

000

10007

Vm (38)

The initial conditions arex(0) = [0 0 0 0]. Thisnominal values of the various flexible joint parameters arefrom [20]: Kstiff =1.248 N − m/rad, ηm=0.69, ηg=0.9,Kt=0.00767 N − m, Kg = 70, Jeq=0.00258 kg − m2,Jarm=0.00352kg − m2, Rm=2.6 Ω.

The model to be followed is assumed as;

xm1

xm2

xm3

xm4

=

0 1 0 00 0 1 00 0 0 10 −560 −320 −85

xm1

xm2

xm3

xm4

+

000

160

Vm (39)

with initial conditions arexm(0) = [0 0 0 0]. Thedisturbanced(t) = 2 sin(t), is sinusoidal with amplitude 2and frequency 1rad/sec and uncertainty in the plant is 40%.

VI. SIMULATION

The simulation studies reveal the results as shown in Fig.1 – Fig. 4. Fig. 1 and Fig. 2 show the results fork = 1and k = 5 respectively, when the value ofτ is 10 ms.Fig. 1(a)–1(d) are the plant states i.e. displacement, velocity,acceleration and jerk, whenk = 1 andτ = 10 ms. The plantand model states are plotted in this window. Fig. 1(e) showsthe control torque required and the Fig. 4(f) shows slidingvariable (σ). The uncertainty in the plant is considered 40%(in both i.e. state matrixA and input matrixB). The trackingperformance is improved as the gaink is increased to 5. Thisis shown in Fig. 2(a)–2(f).

Fig. 3 and Fig. 4 shows the results fork = 1 andk = 5respectively, when the value ofτ is 1 ms. The figure revealsthe ability of the controller, to drive the system to followthe reference model. It is easily observed that system isrobust even in presence of parameter variations and externaldisturbance. Controller is able to force the plant to followthe given model inspite of parameter variations.

VII. C ONCLUSION

In this paper, a trajectory tracking controller for flexiblejoint system, based on uncertainty and disturbance estimation(UDE) is proposed. The uncertainties and disturbance isestimated and compensated in the system performance. This

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Fig. 1: State tracking, Control and Sliding variable forτ = 10ms andk = 1



WCECS 2011

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WCECS 2011

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is demonstrated through simulation. The control strategyincludes Ackerman’s method, which eliminates reachingphase to robustify the system. UDE is used to estimatethe uncertainties and disturbance. The model is decided andcontrol is designed to force the plant trajectories to track themodel states. The plant follows the model states, even in thepresence of uncertainties and disturbance. The results provethat the system performance is robust to parameter varia-tions and external disturbances. The tracking performance isimproved as the filter time constantτ becomes small. Theperformance also shows marked improvement as the valueof k is increased. The performance can be further improvedby using a higher order filter.

REFERENCES

[1] M. W. Spong, “On the force control problem for flexible joint manipu-lators,” IEEE Transactions on Automatic Control, vol. 34, pp. 107-111,1989.

[2] H. Taghirad and M. Khosravi, “A robust linear controller for flexiblejoint manipulators,”Proc. of 2004 IEEE/RSJ Int. Conf. on IntelligentRobots and Systems, Sendai, Japan, vol. 3, pp. 2936-2941, 2004.

[3] S. K. Spurgeon , L. Yao and X.Y. Lu, “Robust tracking via sliding modecontrol for elastic joint manipulators,”Proc. IMechE, Part I: J. Systemsand Control Engineering,vol. 215, pp. 405-417, 2001.

[4] Y. C. Chang, B. S. Chen and T. C. Lee, “Tracking control of flexiblejoint manipulators using only position measurements,”InternationalJournal of Control, vol. 64, no. 4, pp. 567-593, 1996.

[5] J. Hernandez and J. Barbot, “Sliding observer-based feedback controlfor flexible joints manipulator,”Automatica, vol. 32, no. 9, pp. 1243-1254, 1996.

[6] S. E. Talole, J. P. Kolhe and S. B. Phadke, “Extended state observerbased control of flexible joint system with experimental validation,”IEEE Tran. on Industrial Electronics, vol. 57, no. 4, pp. 1411-1419,April 2010.

[7] M. W. Spong, “Modeling and control of elastic joint robots,”ASMEJournal of Dynamic Systems, Measurement, and Control, vol. 109,pp. 310-319, 1985.

[8] F. Ghorbel, J. Y. Hung and M. W. Spong, “Adaptive control of flexiblejoint manipulators,”IEEE Control Systems Magazine, pp. 9-13, 1989.

[9] M. Farooq, Dao Bo Wang and N. U. Dar, “Adaptive sliding modehybrid/force position controller for flexible joint robot,”Proc. of IEEEInt. Conf. on Mechatronix and Automation, 2008, pp. 724-731.

[10] J. H. Oh and J. S. Lee, “Control of flexible joint robot system byback-stepping design,”Proc. of the IEEE Int. Conf. on Robotics andAutomation, Albuquerque, New Mexico,pp. 3435-3440, 1997.

[11] P. Sicard and N. Lechevin, “Variable structure observer design forflexible joint manipulator,”Proc. of the American Control Conference,Albuquerque, New Mexico,pp. 1864-1866, 1997.

[12] N. Lechevin and P. Sicard, “Observer design for flexible joint manip-ulators with parameter uncertainties,”Proc. of the IEEE InternationalConference on Robotics and Automation, Albuquerque, New Mexico,pp. 2747-2552, 1997.

[13] T. L. Le, S. A. Albu and H. Gerd, “MIMO state feedback controllerfor a flexible joint robot with strong joint coupling,”IEEE InternationalConference on Robotics and Automation, Roma, Italy,pp. 3824-3830,2007.

[14] P. D. Shendge, B. M. Patre and S. B. Phadke, “Robust Load FrequencySliding Mode Control based on Uncertainty and Disturbance,”Advancesin Industrial Engineering and Operations Research,Springer, pp. 361-374, 2008.

[15] Q. C. Zhong and D. Rees, “Control of uncertain LTI systems based onan uncertainty and disturbance estimator,”ASME Journal of DynamicSystems, Measurement and Control,vol. 126, pp. 905-910, 2004.

[16] S. E. Talole and S. B. Phadke, “Model following sliding mode controlbased on uncertainty and disturbance estimator,”ASME Journal ofDynamic Systems, Measurement, and Control,vol. 130, pp. 1-5, 2008.

[17] J. Ackermann and V. Utkin, “Sliding mode control design based onAckermann’s formula,”IEEE Transactions on Automatic Control,vol.43, no. 2, pp. 234-237, 1998.

[18] P. D. Shendge and B. M. Patre, “Robust model following loadfrequency sliding mode controller based on UDE and error improvementwith higher order filter,” IAENG International Journal of AppliedMathematics,vol. 37, pp. 27-32, 2007.

[19] M. W. Spong and M. Vidyasagar,Robot Dynamics and Control, Wiley,New York, USA, 1989.

[20] Quanser Inc., Canada,Rotary Flexible Joint User Manual, 2008.



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