An Exponentially Convergent Adaptive Sliding Mode Control of Robot Manipulators

7/28/2019 An Exponentially Convergent Adaptive Sliding Mode Control of Robot Manipulators

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Adaptive Control for Robot Manipulators with Sliding Mode ErrorCoordinate System: Free and Constrained Motions

Vicente Parra-Vega * and Suguru ArimotoMathematical Engineering and Information Physics Department

Faculty of Engineering, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan.

AbstractBased on a novel error coordinate system which in-duces a sliding mode, two passivity-based adaptive con-trollers are proposed. The control strategies, similar t o[2 ] for free motion and [3]-[4/ or constrained motion,give rise t o the exponential convergence of tracking

errors. Computer simulat ion d a t a for f re e and con-strained motions show a high performance.KE Y W O R D S : ADAPTIVEL I D I N G MODE C O N -

T R O L , POSITION-FORCERACKING,XPONENTIALC O N V E R G E N C E ,O B U S T N E S S .

1 IntroductionAn effective combintion of adaptive control and

sliding mode control is pursued in this papers. We fo -cus on the energy-motivated passivity-based approach[l].The contribution of this paper is a novel designof an error coordinate representation which inducesan sliding mode and gives rise t o a robust exponentialconvergence. In the constrained motion case, orthogo-nalized sliding modes arises to ensure the exponentialconvergence of position and and force tracking errors.The control structure can be seen as composed by twocontrol loops: an outer adaptive control loop compen-sates for parametric uncertainty while an inner slidingmode control loop gives the missing energy to yield theexponential convergence robust performance typical ofVSS systems. Since the inner loop does not deals di-rectly with th e parametric uncertainty, it requires onlya small magnitude of the chattering. This struc turecontrasts t o previous "adaptive sliding mode" algo-rithms in which the control law had to induce a slid-ing mode on the s tat e variable of the closed loop errorequation, a task which is more demanding. Furthre-more, it is presented b oth free and constrained motion

'Email: vegaQarimoto1ab.t.u-tokyo.ac.jp'one forpositionerrors and other one for force tracking errors

cases together with a simulation study. The layout ofthis paper is as follows. In Section 2 deals with freemotion case and Section 3 deals with constrained mo-tion case. Section 4 hows computer simulations forboth cases and in Section 5 some conclusions are pre-sented.

2 Free Motion Case2.1 Robot Dynamics

The dynamic model of a rigid serial n-link with allrevolute joints described in joint coordinates is givenas follows:

H i + B o + T H + S q + G = U (1){ 1 . 1where H =H ( q ) denotes the n x n symmetric positivedefinite inertial matr ix, Bo stands for an n xn positivedefinite matrix of damping coefficients, = s(q, i )is a skew symmetric matrix, G =G(q)models thegravity forces and U the torque input . Left hand sideof (1) can be parametrized linearly in terms of unknowparameters and a nominal reference ir . In this waythe representation of (1 ) in error coordinates arises asfollows [2]:

HSr+ B ~ + ~ H + sr = U - Y r O (2){ 1 . -1where S, = q - r is termed as the nominal slidingsurface and Y,O =Hi,. +Bo+$ H +S)q, .+G wherethe regressor Y, E RnXp s composed of known non-linear functions and 0 E RP is assumed to representunknown but constant parameters.

The tipical approach to design either adaptive orsliding mode or a combination of both has been tochoose qr and U in such a way that Sr has exponen-tial stable dynamics driven by an L:!nput ( i . e . [2]).To combine adaptive and sliding mode strategies we

I E E E International Conferenceon Robotics and Automation -591-0-7803-1965-6/95 $4.0001995 I EEE
http://vegaqarimoto1ab.t.u-tokyo.ac.jp/http://vegaqarimoto1ab.t.u-tokyo.ac.jp/


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2.4 Exponential Convergence with En-hanced Parameter StabilityDuring ideal sliding mode, Sq=0 and Sq=0. On

the other hand (Bo+!jk+S)Sr can be parametrizedlinearly in terms of Yc(q,4. ,ST)0 here YC E R n x p .Since the sliding mode exists a t t = o then at t = o +Sfor S >0 the following quantity is available from (2)and (5)

thus we have

where Yw =Yr +Y c .Theorem 2 Consider the robot dynamics (1 ) inclosed loop with the controller given by:U = Yr6-I


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3.3 Holonomic Robot DynamicsDuring constraint motion a holonomic constraint is

imposed by the forward kinematic mapping. In jointspace it can be expressed in terms of q

v ( q >=0 (19)where p ( q ) :Rn -+ R". Differentiating (19) yields

&J,(q)d. =0 , Jvp(q)=- -Jx (20)aq axwhere Jx denotes the standard jacobian matrix of therobot manipulator. Physical interpre tation of (20)suggests th at , in joint space , the velocity vector arisesonto the space tangent a t the contact point t o the sur-face p ( q ) , and hence J 9 ( q ) denotes the map that spansthe subspace normal at the contact point.

At the contact point two components arise: a work-less force term in the norma1 direction and a slidingfriction force term in the direction of the velocity vec-tor onto the tangent plane. In th e Lagragian dynamicsthese terms appear as follows

1 .H ? j + ( B o + 5 H + S ) Q + G = U + J T sX - T.f944 ) = 0 (21)

where J:+. = J T + ( q ) = J: (q) (JV(q)J: (q) ) - ' is thepseudoinverse of J:(q) and X plays a role of the Lagra-gian multiplier in t he calculus of variations and physi-cally stands for the force applied at the contact point.For simplicity, we assume a sliding viscous frictionTJ =Cjd. where C j =C ( l l X l l ) J T J z with C(llXll)2 .3.4 Joint Space Decomposition

Two transformations that span exclusively all theDOF on the joint space and allow to state the globalstability results are derived now. To this end, con-sider a partition of th e joint space coordinate q asfollows q = [q T q r ] where q1 E 2" and 4 2 E Rn-".Due to the kinematic constrain p ( q ) = 0, there arem dependent coordinates which we represent them asq1 . Now consider ( 2 0 ) with its corresponding parti-where J9pl(q) E Rnxm and J 9 z ( q ) E E"'("-").Solving this expression for d.1 yields q 1 =Rq2 where=- [J9pl(q)] -1JVa(q) . Thu s the velocity of the gen-eralized coordinates can be written q = Q(i2 whereQ E R n X ( n - m )s given by

tion given by J , ( Q ! d . =[ J ,T1 (Q) J5.(4)IT[41 42 1 =0

RQ = [ In-m ]

Remark 7. - C O O R D I N A T ERANSFORMATION Q ISWELL-POSED.See [4] and [14].

3.5 An Orthogonalized Sliding Mode Er-ror Coordinate SystemConsider the nominal reference q,. as follows:qr =& { h i - f A Q 2- d p - 1 S g n ( S q p ( T ) ) d T }

+pJ: (q){SqF + 21" s ! ? n ( S p F( r ) ) d T } .Jo'Provided that d. = Q & , the nominal sliding surfaceS,. =4- r arises as follows

S r =& S u p - l J ; ( q ) s v F (23)where Su pand S u ~re defined by:

S U P = S q p + Y l J' f d S g p ( T ) ) d T (24)t oS,F = S q ~~2 ll S g n ( S q F ( T ) ) d T (25)

withs q p = s p - d p SqF =S F - S d FSp = A & $ a A q z S F = A F

s d p = sp(tO)xp-P(l-to)S d F = ~ F ( t 0 )xp-o(t-lo)

where A q 2 =qz - 2d , F =st:X(T)dT, A F =F - dand p, a , 7, p , y 1 , and 7 2 are positive constants.3.6 Control Design

The regressor for constrained motion contains ad-ditionally the friction at the contact point 7-f =( j q .Adding this regressor to (21) and using (23) gives riseto,H S r + Bo +2 H +S +C j S r =U + ;+X - rO

(26)1.{

Consider t he following the control law U :U = Yr6 -


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Proof.- An analysis of passivity suggest the followingLyapunov function [4]

1V = ,{$ITS,. + ?STFSu~ AO*I-lAO)Straighforward derivations yields the derivative of Vseminegative definite in its a rgume nts

Applying Barbalats Theorem and assuming tha t q d EC3 and A d E C1, t turns out

Since S, and S,,F converge and Q is full rank th en Supalso converges. Arguing similar analisis as in eqs.(8)-(9) t can be proved the existence of sliding modes onS,, =0 and S,F =0 and hence we can concluded theexponential convergence of position, velocity and inte-gral of force tracking errors. Th e proof of convergenceof force tracking error itself follows closely [15]and istherefore ommited.

Remark 8.- E N H A N C E DARAMETERTABI LI TY.Remarks stated in the section 2 are valid fo r this sec-tion. Furthermore a controller with enhanced param-eter stability , similar t o subsection (2.4), can be de-signed. Details are omitted due to space limitations.

4 Computer simulations4.1 Free Motion

A simple two degrees of freedom manipulator inthe vertical plane X -Z with viscous friction at eachjoint wa s simulated. Masses and friction coeficientswere estimates with zero initial conditions. Th e tra-jectories to follow are qld = cos(t) and q 2 d = sin(t)and wrong initial conditions were chosen. ml =4Iigand m2 = 2119. In order to smooth out the chat-tering a saturation function s o t ( s ) =s as usedwith 6 =0.01 instead of the sgn(s) in q,.. In this case,GUUB arises instead of GES. The following parame-ter were used in all the figures: a =4, y =10, /? =2,I-l = 6 , 6 = 0.05 and I(d = 10. Figures 1 and 2show a performance predicted in both theorems 1 and2 with smooth control input.

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4.2 Constrained MotionThe end-effector travels with a speed of 0.25mtls

in the Z-axis direction and th e desired co ntact force is

A d =20+5sin(t)N. The initial configuration in AZ =-0.2 mt while keeping contact with th e wall. Massesand inertials parameters of both links were considereduncertain parameters with 25 % error. The controlgains were chosen as follows: -yi =10, Kdj =10, ? =5,I=12, a p =7=2 , and y =5 for i =1 , . . 4 an d j =1,2. Figure 3 shows the convergence of the positiontracking error of the end-effector in 2 - x i s and thecontact force tracking error with smooth control input.

5 Discussion and conclusionsAn adaptive controller is designed over a novel error

coordinate system to attain exponential convergenceand enhanced parameter stability without any persis-tent excitation condition on the regressor. Computersimulation dat a validate the predicted performance forboth free and constrained motion tasks. Experimentalverification is under way and experimental data willsoon be reported elsewhere. Thi s result represents asystematic combination of model-based adaptive con-trol a nd sliding mode control for free and constrainedmotion tasks.A cknowledgernen s

Th e work of the first autho r was supported in partby The CIEA-IPN of Mexico and Th e Monbusho Fel-lowship of Japan. Th e authors are grateful to acknowl-edge discussions with Dr. Yunhui Liu from ETL andAssistant Professor Tomohide Naniwa.

References[l] Ortega, R, Spong, M., Adaptive Control of

Robot Manipulators: A Tutor ial, Automatica,25, 877-888 (1989).

[2] Slotine, J. J. and Li, W. , O n the Adaptive Con-trol of Manipulators, Int. Journa l o f Robotics Re -search, 6, 49-59 (1987).

[3] Arimoto, S., Liu, Y.H. and Naniwa, T., Model-Based Adaptive Hybrid Control for Geometri-cally Constrained Robots, Proc. of the 1993IEEE Int. Conf. of Robotacs an d Automation,618-623 (1993) .

[4] Parra-Vega, V., Arimoto, S . , Liu, Y.N. andNaniwa, T ., Model-Based AdapLive Hybrid Con-trol of Robot Manipulators under HolonomicConstraints, SYROCO94, It al j, 275-480 (1994).

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[5] Zanassi R., Sliding Mode Using DiscontinuousControl Algorithms of Integral Type, Int. J. ofControl, 57, 1079-1099 (1993).

[6] Slo tine, J.J.E . Hedrick,J.H. and Miyasawa, E.A. ,O n Sliding Observer for Nonlinear Systems,Journal of Dynamic Systems, Measurements andControl, 109, 245-252 (1987).

[7] Y. Stepanenko and Ch.Y. S u , Variable StructureControl of Robot Manipulators with NonlinearSliding Manifolds, Int . J . Control, 58 (a), 285-300 (1993).

[8] Raibert, M.H., and Craig, J.J., HybridPosition-Force Control of Mnaipulators, IEEETrans.Syst. Man. and Cyber., 12 , 266-275 (1982).

[9] Duffy, J., Fallacy of Modern Hybrid ControlTheory that is Based on Orthogonal Compli-ments of Twist and Wrench Spaces, J. RoboticSystems, 7, 39-144 (1990).

[lo] Khatib, O., A Unified Approach for Motion andForce Control of Robot Manipulators: The Op-erational Space, I E E E Trans. of Robotics andAutomation, 3, 43-53 (1987).

[l l] McClamroch, N. and Wang, D ., Feedback Sta-bilization and Tracking of Constrained Motion,IE E E Transaction of Automatic Control, 33,419-42 6 (1988).

[12] Grabbe, M.T., Robust Control of Constrainedand Unconstrained Robot Manipulators with Ex-ponential Position Tracking Error Convergence:IEEE Proc. Int. Conf. on Robotics and Automa-tion, Nice, France, 2146-2151.

[13] Yao, B. Tomizuka, M., Adaptive Control ofRobot Manipulators in Constrained Motion,American Control Conference (1993)

[14] Blajer, W., A Projective Criterion to the C o-ordinate Partitioning Method for Multibody Dy-namics, Archive of Applied Mechanics, 64 , 147-153 (1994).

[15] Naniwa T . V., Arimoto, S . , Model-Based Adap-tive Control for Geometrically Constrained RobotManipulators, The Proc. of The Japan- U SASymposium on Flexible Automation, Kobe, 23-30(1994).

-10 2 4 6 8 10 0 2 4 6 8 10

Nm Control input joint 1

0 2 4 6 8 1 0 0 2 4 6 8 10Kg Estimate mass, link2

0 2 4 6 8 10 0 2 4 6 8 10Figure 1: Tracking errors, controls and estimates ofmasses using the controller (5).

deg Trackina error, ioint24 /

eg Trackina error. ioint 14 f

Nm Control input, joint2p Control input, oint 1

-2fvlJ w: jvv-v2 4 6 8 10 0 2 4 6 8 10

K!& Mass estimate link 1

0 2 4 6 8 10, Y i0 2 4 6 8 10

Figure 2: Tracking errors, controls and estimates ofmasses using the controller (14)-(15).

Forceexerted bv theend-effectoron the enviroment20

actual

i0 2 4 6 s[16] Utkin,V. , Variable Structure Systens: Control Figure 3: End-effector position and force trajectory

tracking in Cartesian space.nd Optimization, Kluwer.

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An Exponentially Convergent Adaptive Sliding Mode Control of Robot Manipulators